gauge mechanics

Gauge Mechanics

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Gauge Mechanics

L. Mangiarotti University of Camerino, Italy

G. Sardanashvily Moscow State University, Russia

World Scientific Singapore *New Jersey London 'HongKong

Published by

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GAUGE MECHANICS

Copyright © 1998 by World Scientific Publishing Co Pte. Ltd All rights reserved. This book, or parts thereof, may not he reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-3603-4

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Printed in Singapore by Uto-Print

Preface This book presents in a unified way modern geometric methods in analytical

mechanics, based on the application of jet manifolds and connections. As is well known, the technique of Poisson and symplectic spaces provide the adequate Hamil-tonian formulation of conservative mechanics. This formulation, however, cannot be extended to time-dependent mechanics subject to time-dependent transformations. We will formulate non-relativistic time-dependent mechanics as a particular field theory on fibre bundles over a time axis.

The geometric approach to field theory is based on the identification of classical fields with sections of fibred manifolds. Jet manifolds provide the adequate mathematical language for Lagrangian field theory, while the Hamiltonian one is phrased in terms of a polysymplectic structure. The 1-dimensional reduction of Lagrangian field theory leads us in a straightforward manner to Lagrangian time-dependent mechanics. At the same time, the canonical polysymplectic form on a momentum phase space of time-dependent mechanics reduces to the canonical exterior 3-form which plays the role similar to a symplectic form in conservative mechanics. With this canonical 3-form, we introduce the canonical Poisson structure and formulate Hamiltonian time-dependent mechanics in terms of Hamiltonian connections and Hamiltonian forms.

Note that the theory of non-linear differential operators and the calculus of variations are conventionally phrased in terms of jet manifolds. On the other hand, jet formalism provides the contemporary language of differential geometry to deal with non-linear connections, represented by sections of jet bundles. Only jet spaces enable us to treat connections, Lagrangian and Hamiltonian dynamics simultaneously.

In fact, the concept of connection is the main link throughout the book. Connections on a configuration space of time-dependent mechanics are reference frames. Holonomic connections on a velocity phase space define non-relativistic dynamic equations which are also related to other types of connections, and can be written as non-relativistic geodesic equations. Hamiltonian time-dependent mechanics deals with Hamiltonian connections whose geodesies are solutions of the Hamilton

v

PREFACE vi

equations. The presence of a reference frame, expressed in terms of connections, is the main

peculiarity of time-dependent mechanics. In particular, each reference frame defines an energy function, and quantizations with respect to different reference frames are not equivalent.

Another important peculiarity is that a Hamiltonian fails to be a scalar function under time-dependent transformations. As a consequence, many well-known constructions of conservative mechanics fail to be valid for time-dependent mechanics, and one should follow methods of field theory.

At the same time, there is the essential difference between field theory and time-dependent mechanics. In contrast with gauge potentials in field theory, connections on a configuration space of time-dependent mechanics fail to be dynamic variables since their curvature vanishes identically. Following geometric methods of field theory, we obtain the frame-covariant formulation of time-dependent mechanics. By analogy with gauge field theory, one may speak about gauge time-dependent mechanics.

In comparison with non-relativistic time-dependent mechanics, a configuration space of relativistic mechanics does not imply any preferable fibration over a time. To construct the velocity phase space of relativistic mechanics, we therefore use formalism of jets of submanifolds. At the same time, Hamiltonian relativistic mechanics is seen as an autonomous Hamiltonian system on the constraint space of relativistic hyperboloids.

With respect to mathematical prerequisites, the reader is expected to be familiar with the basics of differential geometry of fibre bundles. For the convenience of the reader, several mathematical facts and notions are included as an Interlude, thus making our exposition self-contained.

Contents Preface v

Introduction 1

1 Interlude: bundles , J e t s , Connect ions 9 1.1 Fibre bundles 9 1.2 Multivector fields and differential forms 20 1.3 Jet manifolds 35 1.4 Connections 42 1.5 Bundles with symmetries 46 1.6 Composite fibre bundles 53

2 Geometry of Poisson Manifolds 57 2.1 Jacobi structure 57 2.2 Contact structure 61 2.3 Poisson structure 66 2.4 Symplectic structure 73 2.5 Presymplectic structure 81 2.6 Reduction of symplectic and Poisson structures 84 2.7 Appendix. Poisson homology and cohomology 89 2.8 Appendix. More brackets 96 2.9 Appendix. Multisymplectic structures 100

3 Hami l ton ian Sys tems 105 3.1 Dynamic equations 106 3.2 Poisson Hamiltonian systems I l l 3.3 Symplectic Hamiltonian systems 114 3.4 Presymplectic Hamiltonian systems 118 3.5 Dirac Hamiltonian systems 123 3.6 Dirac constraint systems 129

vii

viii CONTENTS

3.7 Hamiltonian systems with symmetries 133 3.8 Appendix. Hamiltonian field theory 139

4 Lagrangian time-dependent mechanics 151 4.1 Fibre bundles over K 152 4.2 Dynamic equations 159 4.3 Dynamic connections 162 4.4 Non-relativistic geodesic equations 170 4.5 Reference frames 175 4.6 Free motion equations 179 4.7 Relative acceleration 182 4.8 Lagrangian systems 186 4.9 Newtonian systems 195 4.10 Holonomic constraints 206 4.11 Non-holonomic constraints 211 4.12 Lagrangian conservation laws 218

5 Hamiltonian time-dependent mechanics 227 5.1 Canonical Poisson structure 228 5.2 Hamiltonian connections and Hamiltonian forms 231 5.3 Canonical transformations 242 5.4 The evolution equation 247 5.5 Degenerate systems 248 5.6 Quadratic degenerate systems 262 5.7 Hamiltonian conservation laws 269 5.8 Time-dependent systems with symmetries 271 5.9 Systems with time-dependent parameters 274 5.10 Unified Lagrangian and Hamiltonian formalism 282 5.11 Vertical extension of Hamiltonian formalism 285 5.12 Appendix. Time-reparametrized mechanics 296

6 Relativistic mechanics 299 6.1 Jets of submanifolds 299 6.2 Relativistic velocity and momentum phase spaces 303 6.3 Relativistic dynamics 307 6.4 Relativistic geodesic equations 311

CONTENTS ix

Appendix A. Geometry of BRST mechanics 317

Appendix B. On quantum time-dependent mechanics 327

Bibliography 332

Index 347

Introduction

The present book deals with first order mechanical systems, governed by the second order differential equations in coordinates or the first order ones in coordinates and momenta. Our goal is the description of non-conservative mechanical systems subject to time-dependent transformations, including inertial and non-inertial frame transformations and phase transformations.

Symplectic technique is well known to provide the adequate Hamiltonian formulation of conservative (i.e., time-independent) mechanics where Hamiltonians are independent of time [2, 6, 72, 116, 126]. The familiar example is a mechanical system whose momentum phase space is the cotangent bundle T'M of a configuration space M. This fibre bundle is provided with the canonical symplectic form

Q = dpi A dq', (1)

written with respect to the holonomic coordinates (g',p, = <?,) on T'M. A Hamiltonian H of a conservative mechanical system is defined as a real function on the momentum phase space T'M. Then a motion of this system is an integral curve of the Hamiltonian vector field

■d = ^d' + d%

on T'M which fulfills the Hamilton equations

ti\Q = -dH,

& = sht, ■di = -diH.

Lagrangian conservative mechanics is usually seen as a particular Hamiltonian mechanics on the tangent bundle TM of a configuration space M, which is endowed with the presymplectic form defined by a Lagrangian.

The Hamiltonian formulation of conservative mechanics cannot be extended in a straightforward manner to time-dependent mechanics because the symplectic form

1

2 INTRODUCTION

(1) is not invariant under time-dependent transformations, including the inertial frame transformations.

The existent formulation of time-dependent mechanics implies a preliminary splitting of a configuration space

Q = 1 x M, (2)

where M is a manifold, while 1 is a time axis (see [29, 31, 50, 110, 136, 146, 166] and references therein). FYom the physical viewpoint, it means that a certain reference frame is chosen. Then we have the corresponding splitting of the velocity phase space

R x T M (3)

and that of the momentum phase space

R x T'M. (4)

The momentum phase space (4) is provided with the presymplectic form

prjQ = dp, A dqx (5)

which is the pull-back of the canonical symplectic form Q. (1) on the cotangent bundle T'M [27]. By a time-dependent Hamiltonian H is meant a real function on the momentum phase space R x T'M, while trajectories of motion are integral curves of the time-dependent vector field

■d : R x T'M -> TT'M

which satisfies the Hamilton equations

0" = d'H, A = -diH.

The problem is that the splittings (2) - (4) are broken by any time-dependent transformation, and so is the presymplectic form (5). Therefore the familiar methods of conservative mechanics and their extensions to the product spaces (2) - (4) fail to be valid for mechanical systems subject to time-dependent transformations.

We will formulate non-relativistic time-dependent mechanics as a particular field theory whose configuration space is a fibred manifold over a time axis R [14, 55, 57, 106, 114, 132, 159, 161].

INTRODUCTION 3

Geometric formalism of field theory is based on the identification of classical fields with sections of a fibred manifold Y —> X. The corresponding velocity phase space is the first order jet manifold J 1 Y of sections of Y —* X, while the momentum one is the Legendre bundle

U = VY®CAT*X), n = dim X, (6)

over Y [28, 56, 57, 73, 96, 158, 159]. In the case of X = R of time-dependent mechanics, its configuration space is a

fibred manifold

7T : Q — R,

equipped with fibred coordinates (t,ql). The base R is parameterized by the Cartesian coordinates t with the transition functions t' = £+const. Relative to these coordinates, the time axis R is provided with the standard vector field dt and the standard 1-form dt which is also the volume element on R. Of course, this is not the case of relativistic mechanics (see Chapter 6) nor of the models with a time reparametrization (see Section 5.12).

The velocity phase space of non-relativistic time-dependent mechanics is the first order jet manifold JlQ of sections of the fibred manifold Q —» R. It is equipped with the adapted coordinates (t,q%,q\). There is the canonical imbedding

X:JlQ^TQ, (7)

A = dt + q\du

of the velocity phase space JXQ into the tangent bundle TQ of the configuration space Q. From now on we will identify J1Q with its image in TQ given by the coordinate conditions

t = l, <?• = < ? ; • (8)

This is an affine subbundle of TQ —* Q modelled over the vertical tangent bundle VQ -> Q of the fibred manifold Q — R.

The morphism (7) plays a prominent role in our formulation of time-dependent mechanics. It enables us to treat the jet manifold JlQ as a velocity phase space

4 INTRODUCTION

of a mechanical system. Due to this morphism, every connection T on the fibred manifold Q -> R can be identified with the nowhere vanishing vector field

r : Q -» JlQ C TQ, (9)

r = at + r%,

on Q which is the horizontal lift of the standard vector field dt on R by means of this connection T. We will continue to call (9) a connection in order to refer to the standard properties of connections without additional explanation. From the physical viewpoint, a connection (9) sets a tangent vector at each point of the configuration space Q, which characterizes the velocity of an "observer" at this point. It follows that a connection T on the fibred manifold Q —» R defines a reference frame [57, 132, 161]. In particular, one can think of the difference q\ - P as being the relative velocity with respect to the reference frame T, whereas the notion of a relative acceleration is more intricate (see Section 4.7).

The momentum phase space of non-relativistic time-dependent mechanics is the Legendre bundle (6) where X = R. This phase space is isomorphic to the vertical cotangent bundle II = V'Q of the fibred manifold Q —► R, and is equipped with the holonomic coordinates (t,ql,p, = <ft). It should be emphasized that this is not the most general case of a momentum phase space of time-dependent mechanics, which is defined as a fibred manifold II —» R provided with a Poisson structure such that the corresponding symplectic foliation belongs to the fibration II —► R [74]. In fact, putting n = V'Q, we restrict our consideration to Hamiltonian systems which have the Lagrangian counterparts.

Note that Lagrangian and Hamiltonian formalisms are equivalent only if a Lagrangian is hyperregular, i.e., the Legendre map from the velocity phase space to the momentum one is a diffeomorphism. In general, a degenerate Lagrangian involves a set of associated Hamiltonians in order to exhaust solutions of the Lagrange equations (see Section 5.5). Nevertheless, there are physically interesting systems whose phase spaces fail to be the cotangent bundles of configuration spaces, and they do not admit any Lagrangian description [168]. The unified Lagrangian-Hamiltonian formalism of the joint velocity-momentum phase space

n = V V ' Q S JlV"Q

enables us to relate a Lagrangian system to any Hamiltonian one (see Section 5.10).

INTRODUCTION 5

Let us turn to the momentum phase space V'Q of time-dependent mechanics. It is endowed with the canonical exterior 3-form

il = dpi A dqx A dt (10)

which is the particular case of the canonical polysymplectic form on the Legendre bundle (6), when X = R [57, 161]. The exterior form (10) is invariant under all holonomic transformations of the momentum phase space V'Q.

In time-dependent mechanics, the canonical 3-form $7 (10) plays a role similar to the canonical symplectic form (1) in conservative symplectic mechanics. The form (10) yields the canonical Poisson structure on the momentum phase space V'Q, and provides the Hamiltonian formulation of time-dependent mechanics in terms of Hamiltonian connections and Hamiltonian forms. This formulation is compatible with the Lagrangian formulation of time-dependent mechanics on the velocity phase space JlQ, and is equivalent to the Lagrangian one in the case of hyperregular Lagrangians.

The following peculiarities of Hamiltonian time-dependent mechanics should be emphasized.

• The canonical Poisson structure defined by the 3-form fl (10) on the momentum phase space V'Q of time-dependent mechanics is degenerate.

• A Hamiltonian on a momentum phase space of time-dependent mechanics fails to be a scalar function, but reads

H = p , P + HT, (11)

where T is a connection on the fibred manifold Q —»IR, while Hr is a Hamiltonian function which is also an energy density with respect to the reference frame T.

• A Poisson bracket of a Hamiltonian (11) with functions on a momentum phase space is defined only locally. Being equal to zero with respect to some coordinates, it does not necessarily vanish with respect to other ones.

• As a consequence, the evolution equation in time-dependent mechanics is not reduced to a Poisson bracket, and integrals of motion are not functions in involution with a Hamiltonian. For the same reason, the familiar Dirac-Bergmann

6 INTRODUCTION

algorithm for describing constraint systems is not extended to time-dependent mechanics.

• Quantizations with respect to different reference frames are not equivalent.

For the sake of simplicity, throughout the book, a configuration space of time-dependent mechanics Q —> R is assumed to be a fibre bundle with a typical fibre M. We will call it a configuration bundle. It is always trivial. Note that, although such a configuration space Q is diffeomorphic to a direct product R x M, in general, it cannot be canonically identified to R x M. Its different trivializations

ip: Q ^ R x M (12)

differ from each other in fibrations Q —> M, while the fibration Q —> R is once for all. Given a trivialization (12) of a configuration space, there are the corresponding splittings of velocity and momentum phase spaces

JlQ ^ R x TM, V'Q = R x T'M.

We show that every trivialization (12) of the fibre bundle Q —» R defines a (complete) connection (9) on this fibre bundle, and vice versa. Consequently, every such a trivialization corresponds to a (complete) reference frame.

Note that a reference frame is one of the main ingredients in our picture of time-dependent mechanics. For instance, each reference frame defines an energy function. Following geometric methods of field theory, we obtain the formulation of time-dependent mechanics which involves necessarily connections T on a configuration bundle Q —► R, and thus is covariant under reference frame transformations. By analogy with the gauge field theory, one may speak about gauge time-dependent mechanics, though the term "gauge mechanics" also stands for mechanics of de-formable bodies [120] (see Section 5.11).

At the same time, there is an essential difference between field theory and time-dependent mechanics. Since a configuration space of time-dependent mechanics is a fibre bundle Q —» R over a 1-dimensional base, the curvature of any connection on a configuration bundle vanishes identically. In contrast with gauge potentials of gauge theories, these connections fail to be dynamic quantities because they can always be brought into the standard vector field r = dt by time-dependent transformations.

INTRODUCTION 7

Therefore, Lagrangians in time-dependent mechanics are covariant, but not invariant under reference frame transformations.

Connections play a prominent role in our formulation of time-dependent mechanics. As was mentioned above, connections on a configuration bundle Q —» R describe non-relativistic reference frames. Holonomic connections on the jet bundle JlQ —» K define non-relativistic dynamic equation which, in turn, are associated with connections on the affine jet bundle JlQ —> Q and the tangent bundle TQ —> Q. As a result, every non-relativistic dynamic equation can be seen as a geodesic equation on the tangent bundle TQ —» Q that furnishes the relationship between non-relativistic and relativistic dynamics (see Section 6.4). Hamiltonian time-dependent mechanics deals with Hamiltonian connections whose geodesies are solutions of the Hamilton equations.

In comparison with non-relativistic mechanics, if a configuration space of a mechanical system has no preferable fibration Q —> R, we obtain the general formulation of relativistic mechanics, including Special Relativity on the Minkowski space Q = R4. The velocity phase space of relativistic mechanics is the first order jet manifold J\Q of 1-dimensional submanifolds of the configuration space Q [57, 161]. This notion of jets generalizes that of jets of sections of fibre bundles which we have utilized in field theory and non-relativistic mechanics. The jet bundle J\Q —» Q is projective, and one can think of its fibres as being spaces of the 3-velocities of a relativistic system. The 4-velocities of a relativistic system are represented by elements of the tangent bundle TQ of the configuration space Q, while the cotangent bundle T'Q, endowed with the canonical symplectic form, plays the role of the momentum phase space of relativistic theory. As a result, Hamiltonian relativistic mechanics can be seen as a constraint Dirac system on the hyperboloids of relativistic momenta in the momentum phase space T'Q.

Formalism of jets of submanifolds provides the common description of non-relativistic mechanics and relativistic theory. In particular, the tangent bundle TQ of a configuration space Q plays the role of the space of the 4-velocities both in non-relativistic and relativistic mechanics. The difference is only that, given a fibration Q —> R, the 4-velocities of a non-relativistic system live in the subbundle (8) of TQ, whereas the 4-velocities of a relativistic theory belong to the hyperboloids

g^ifq" = 1, (13)

where g is an admissible pseudo-Riemannian metric in TQ. Moreover, as was mentioned above, both relativistic and non-relativistic equations of motion can be seen

8 INTRODUCTION

as geodesic equations on the tangent bundle TQ, but their solutions live in its different subbundles (13) and (8).

Unless otherwise stated, we believe that all quantities are physically dimension-less. Following field theory, we will sometimes refer to the universal unit system where the velocity of light c and the Planck constant h are equal to 1, while the length unit is the Planck one

(Gftc-3)1/2 = G1/2 = 1.616-10-Mcm,

where G is the Newtonian gravitational constant. Relative to the universal unit system, the physical dimension of the spatial and temporal Cartesian coordinates is the [length], the physical dimension of a mass is the [length]-1, while an action functional and a metric tensor are physically dimensionless.

Chapter 1

Interlude: bundles, jets, connections

This Chapter does not claim to be a survey on modern differential geometry. The relevant material is presented in a fairly informal way. For details, we refer the reader to [57, 100, 157, 164, 170, 185].

Throughout the book, all maps are smooth, i.e., of class C°°, while manifolds are real, finite-dimensional, second-countable and, hence, paracompact. Unless otherwise stated, we assume that manifolds are connected.

We use the standard symbols ®, V, and A for the tensor, symmetric, and exterior products, respectively. The interior product (contraction) of vectors and forms is denoted by J. By dg are meant the partial derivatives with respect to the coordinates with indices g. The symbol o stands for a composition of maps.

1.1 Fibre bundles

Subsections: Fibre bundles, 9; Vector bundles, 12; A/fine bundies, 14; Tangent and cotangent bundles, 15; Tangent and cotangent bundles of fibre bundles, 16; Sheaves, 18.

Fibre bundles

By a fibre bundle is meant a locally trivial fibred manifold

■K:Y ->X (1.1.1)

9

10 CHAPTER 1. INTERLUDE: BUNDLES, JETS, CONNECTIONS

where a fibration (or a projection) n is a surjective submersion from a manifold Y, called a totai space, onto a base X. Unless otherwise stated, we put d imX = n. By definition, a base X of a fibre bundle (1.1.1) admits an open covering {£/J so that Y is locally isomorphic to the splittings

^ : T T 1 ^ ) -» U( x V,

called local bundJe trivializations, together with the transition functions

P K ■ {ui n trc) x v -»(t/j n pc) x v, V»e(«) = (p«°^c)(y). y e *-'((/< n£/c),

where V is the typicaj fibre of the fibre bundle (1.1.1). The bundle trivializations (U(,ilf() constitute an atlas

* = {(t/t,V«).P«} of a fibre bundle. Given an atlas ty, a fibre bundle Y is provided with the associated atlas of fibred coordinates (xx,yl), where

xx{y) = (xxon){y), yeY,

are coordinates on the base X, and

y'(y) = (y*oP^2°^)(v)

are coordinates on the typical fibre V. A fibre bundle Y —► X is called trivial if V is diffeomorphic to the product X x V.

Different trivializations of a fibre bundle differ from each other in projections Y —► V of the total space Y onto the typical fibre V.

THEOREM 1.1.1. [170]. Each fibre bundle over a contractible base is trivial. □

By a section (or a global section) of a fibre bundle (111) is meant a manifold morphism s : X —» Y such that -n o 5 = IdX. A section s is an imbedding, i.e., s(X) C Y is a submanifold of a total space Y which is also a topological subspace of Y. Similarly, a section s of a fibre bundle Y —> X over a submanifold N C X is a morphism s : N —» Y, such that

n o s = iN : N ■—> Y.

1.1. FIBRE BUNDLES 11

A section of a fibre bundle over an open subset of its base will be called simply a (local) section. A fibre bundle, by definition, admits a local section over an open neighbourhood of each point of its base.

T H E O R E M 1.1.2. [170]. A fibre bundle Y -* X whose typical fibre is diffeomorphic to Rm has a global section. A (smooth) section over a closed subset of X can always be extended to a global section. D

A Bbred morphism of two bundles n : Y —► X and 7r' : Y' —> X' is a pair of maps $ : Y -> Y' and / : X -* X' such that the diagram

Y -*->r i I X —>X'

/

(1.1.2)

is commutative, i.e., $ sends fibres to fibres. In brief, we will say that (1.1.2) is a fibred morphism

$ : Y —► Y' ! over / . If f = ldX, then

$ : Y —>Y' x

is called a fibred morphism over X.

R e m a r k 1.1.1. Unless otherwise stated, by the rank of a fibred morphism $ (1.1.2) over a diffeomorphism / is meant its rank minus dimX. •

A fibred morphism $ (1.1.2) over X (or its image $(Y)) is said to be a subbundle of the fibre bundle Y' -> X if $(Y) is a submanifold of Y'.

We deduce from the implicit function theorem the following useful criteria for an image and a pre-image of a fibred morphism to be a subbundle [149, 185].

T H E O R E M 1.1.3. Let # : Y —► Y' be a fibred morphism over X. Given a global section s' of the fibre bundle V" —> X such that s'(X) C I m $ , by the Jcernel of the fibred morphism $ with respect to the section s' is meant the pre-image

Kers.$ = $-1(s'(^))


of s(X) by $. If $ : Y —» V" is a fibred morphism of constant rank over X, then I m $ and Ker s /$ are subbundles of Y' and Y, respectively. □

An isomorphism of fibre bundles is a fibred morphism (1.1.2) such that $ is a diffeomorphism. A fibred morphism [isomorphism] of a fibre bundle V - > X t o itself is called an endomorphism (automorphism). An automorphism over I d X is said to be a vertical automorphism. Following physical terminology, automorphisms of fibre bundles will also be called gauge transformations.

Given a fibre bundle 7r : Y —> X and a manifold map / : X' —» X, the pull-back f'Y of Y by / is the fibre bundle over X' whose total space is

f'Y^{(x\y)eX'xY : n(y) = /(*')}

together with the natural projection (x1, y) i-» x'. Roughly speaking, the fibre of the pull-back f'Y over a point x' £ X' is that of Y over the point f(x') e X. If X' C X is a submanifold of X and ix> is the corresponding natural injection, then the pull-back

i'x,Y = Y \x,

is called the restriction of a fibre bundle Y to the submanifold X' C X. Let 7T : Y —> X and ir' : V" —> X be fibre bundles over the same base X. Then-

fibred product

YxY'

over X is defined as the pull-back

YxY' = n'Y' or V x Y' = Tr'V A- X

together with the natural projection onto X.

Vector bundles

A vector bundle is a fibre bundle Y —* X such that:

• its typical fibre V and all the fibres Yx = ir '(x), x € X, are real finite-dimensional vector spaces;

LI. FIBRE BUNDLES 13

• there is a bundle atlas <P = {(Ua,ipa)} of Y —> X whose trivialization mor-phisms 4>a restrict to linear isomorphisms

xPa(x) :YX^V, Vz 6 Ua.

Dealing with a vector bundle Y, we always use linear bundle coordinates ( I A , J / ' ) associated with the above-mentioned bundle atlas 9. We have

(pr2 o ipa)(y) = y'a,

y = y'e,(x) = yxipa(x) l(e<),

where {e,} is a fixed basis for the typical fibre V of Y, while {e,(z)} is the fibre basis (or the frame) for the fibre Yx of Y, which is associated with the bundle atlas 9.

By virtue of Theorem 1.1.2, vector bundles have global sections, e.g., the global zero section 0(X). If there is no risk of confusion, we write 0, instead of 0(X).

A morphism of vector bundles # : Y —► Y' is defined as a fibred morphism over / : X —► X' whose restriction $ x : Yx —» Y^^ to each fibre of Y is a linear map. It is called a linear bundle morphism over / .

The following assertion is a corollary of Theorem 1.1.3.

PROPOSITION 1.1.4. If Y —► X and Y' -» X are vector bundles and $ : Y -> V" is a linear bundle morphism of constant rank over X, then the image of $ and the kernel Kerjj$ of $ with respect to the zero section 0 of Y' —> X are vector subbundles of Y' —* X and Y —> X, respectively. Note that a vector subbundle of a vector bundle is a closed imbedded submanifold. □

Unless otherwise stated, by Ker $ of a linear bundle morphism $ is meant its kernel with respect to the zero section 0.

There are the following standard constructions of new vector bundles from old.

• Let Y —> X be a vector bundle with a typical fibre V. By Y" —► X is meant the dual vector bundle with the typical fibre V, dual of V. The interior product of Y and Y' is defined as a fibred morphism

\:Y®Y" — > X x R . J x


• Let Y —> X and Y' —» X be vector bundles with typical fibres V and V, respectively. Their Whitney sum Y © Y' is a vector bundle over X with the typical fibre V ®V.

• Let V —► X and Y' —» X be vector bundles with typical fibres V and V, respectively. Their tensor product Y ® Y' is a vector bundle over X with the typical fibre V ® V". Similarly, the exterior product Y AY is denned.

In particular, let $ : Y\ —* Y2 be an injection of a vector bundle iT\ : Y\ —» Xi to a vector bundle 7r2 : Y2 —► X2 over a diffeomorphism f : X\ —* X2. Then, there is the dual surjection

$• : y2* — y;, <*•(«),«> =<«,*(«)>, v^eTrrHr'o^^cy,, u e y2-, (1.1.3)

over the diffeomorphism / ' .

AfRne bundles

Let W : y —* X be a vector bundle with a typical fibre V. An afSne bundle modelled over the vector bundle Y —» X is a fibre bundle n : Y -+ X whose typical fibre V is an affine space modelled over V, while the following conditions hold.

• All the fibres Yx of Y are affine spaces over the corresponding fibres Yx of the vector bundle Y.

• There is a bundle atlas * = {(Ua,ipa)} of Y —► X whose local trivializations restrict to affine isomorphisms

i M i ) : Y* -» V, Vx e [4.

In particular, every vector bundle has a natural structure of an affine bundle. Dealing with an affine bundle, we use only affine bundle coordinates {xx,y')

associated with the above-mentioned bundle atlas * . We have the fibred morphisms

YxY —>Y, x x (y\V')^y' + y',

YxY —>Y, x x ( y i , j / ' < ) - 2 / ' - y ' i ,

where (y1) are linear coordinates on the vector bundle Y.


By virtue of Theorem 1.1.2, affine bundles have global sections. A morphism of affine bundles $ ; Y —» Y' is a fibred morphism over / whose

restriction $z : Yx —► Vi. , to each fibre of Y is an affine map. It is called an affine bundle morphism over / .

Every affine bundle morphism $ : Y —► Y' from an affine bundle Y modelled over a vector bundle Y to an affine bundle Y' modelled over a vector bundle Y determines uniquely the linear bundle morphism

<l>: Y - Y * , — 3$'

called the linear derivative of $>. Let Y x Y' be the fibred product of two affine bundles Y —» X and Y' —► X

x which are modelled over the vector bundles Y —» X and Y —» X, respectively. This product, called the Whitney sum, is also an affine bundle modelled over Y(BY.

Furthermore, let Y' —> X be an affine bundle modelled over a vector bundle Y —► X. Let Y C Y' be an affine subbundle modelled over a vector bundle Y —> X. Assume that Y is the Whitney sum of Y and a complementary vector bundle Z —<■ X. Then one can easily verify that the affine bundle Y' —» X decomposes in the Whitney sum

Y' = Y®Z. x

Tangent and cotangent bundles

The fibres of the tangent bundle

■Kz : TZ —» Z

of a manifold Z are tangent spaces to Z. Given a coordinate atlas

* z = {(%*e)}

of a manifold Z, the tangent bundle is provided with the (holonomic) atlas

* = { ( 7 r ; I [ / ? ) , ^ = T ^ ) } ,


where by Tfy is meant the tangent map to <j>^. The associated linear bundle coordinates (zx) on TZ are called the induced (or holonomic) coordinates with respect to the frames {d\} in tangent spaces TZZ. Their transition functions read

dz» Every manifold morphism / : Z —* Z' generates the fibred morphism of the tangent bundles

Tf :TZ —>TZ', I

i*oT/ = §£*. called the tangent map to / .

The cotangent bundle of a manifold Z is the dual

■K.Z :T'Z — Z

of the tangent bundle TZ —> Z. It is equipped with the (holonomic) coordinates (zx, ix) with respect to the coframes {dzx} dual of {d\}. Their transition functions read

., _ dz» . Zx ~ dz^2"-

Tangent and cotangent bundles of fibre bundles

Let TTY : TY —» Y be the tangent bundle of a fibre bundle ir : Y —> X. Given fibred coordinates (xA,y') on Y, the tangent bundle TY is equipped with the coordinates (xx,y',xx,y').

The tangent bundle TY is a fibre bundle

ix o Try : T Y -» X

over X, while the tangent map TIT to 7r defines the fibration

Tn:TY — T X

There is the commutative diagram

TY ^TX

I I y —► x

Tf


The tangent bundle TY -> Y of a fibre bundle Y -> X has the vertical tangent subbundle

VYd=KerTn

of TV, given by the coordinate relation i A = 0. This subbundle consists of the vectors tangent to fibres of Y. The vertical tangent bundle VY is provided with the coordinates {xx,yx,y') with respect to the frames {d,}.

Let T $ be the tangent map to a fibred morphism $ : Y —► Y'. Its restriction

V<f> = T$\VY : VY -» VY',

i / " oV$ = ^ $ ' = ^ $ ' , (1.1.4)

to VY is a linear bundle morphism of the vertical tangent bundle VY to the vertical tangent bundle VY', called the vertical tangent map to $ .

Every vector bundle Y —> X admits the canonical vertical splitting of the vertical tangent bundle

VY = Y x Y x (1.1.5)

because the coordinates y' on VY have the same transformation law as the linear coordinates y' on Y.

An affine bundle Y —> X modelled over a vector bundle Y —* X also admits the canonical vertical splitting of the vertical tangent bundle

VY^YxY x

(1.1.6)

because the coordinates y' on VY have the same transformation law as the linear coordinates y1 on the vector bundle Y.

The cotangent bundle T'Y of a fibre bundle Y —» X is equipped with the coordinates (xx,y1,i\,yi). There is its natural fibration T'Y —» X over X, but not over T'X.

The vertical cotangent bundle VY —» V of a fibre bundle Y —» X is defined as the vector bundle dual of the vertical tangent bundle VY —> Y. It should be emphasized that there is no canonical injection of VY into the cotangent bundle T'Y of Y, but we have the canonical projection

C : T'Y - VY, * Y

(1.1.7)

C : xxdxx + i/idy* i-> jfidj/*,


where (dy'} are the bases for fibres of VY, which are dual of the frames {9,} in the vertical tangent bundle VY.

With VY and V'Y, we have the following two exact sequences of vector bundles over Y:

0 — VY — TY -> Y x TX — 0, (1.1.8a)

0 -» Y x T'JT — T * r -» v * y -» 0. (1.1.8b)

For the sake of simplicity, we will denote the pull-backs

YxTX, x YxT'X x simply by TX and T 'X .

Example 1.1.2. Let us consider the tangent bundle TT'X of T'X and the cotangent bundle T'TX of TX. Relative to coordinates (xx, p\ = ±\) on T'X and (xx, xx) on TX, these fibre bundles are provided with the coordinates (xx,p\, xx,p\) and (xx,xx,±\,xx), respectively. By inspection of the coordinate transformation laws, one can show that there is the isomorphism

a : TT'X s T'TX, P\ <—► Xx, Px *—> i\ (1.1.9)

of these bundles over TX [43, 96]. Given a fibre bundle Y —► X, there is the similar isomorphism

av : VV'Y S W r , ft <—> j/i, p, <—► in (1.1.10)

over VY, where (xx,yt,pl,yi,pl) and (xx,y,,y\yi,y\) are coordinates on V V V and V'VY, respectively. •

Sheaves

There are several equivalent definitions of sheaves [16, 80). We will start from the following. A sheaf on a topological space X is a topological fibre bundle 5 —> X whose fibres, called the stalks, are Abelian groups Sx provided with discrete topology.

A presheaf on a topological space X is defined if an Abelian group Sy corresponds to every open subset U C X (Sj = 0) and, for any pair of open subsets V C U, there is the homomorphism

rv : Su —> Sy


such that

rv = Id Su, ru _ rv V Tw — ~w~v> W C V C U.

Example 1.1.3. Let X be & topological space, Su the additive Abelian group of all continuous functions on U C X, while the homomorphism

rv : Su —* Sy

is the restriction of these functions to V C X. Then {Sv, ry} is a presheaf. •

Every presheaf {Su, ry} on a topological space X yields a sheaf on X whose stalk Sx at a point x € X is the direct limit of the Abelian groups Su, x e U, with respect to the homomorphisms ry. It means that, for each open neighbourhood U of a point x, every element s e Su determines an element sx € Sx, called the germ of s at x. Two elements s £ Su and s' € S'v define the same germ at x if and only if there is an open neighbourhood W 3 x such that

ru . _ rv I rws — rws .

For instance, two real functions s and s' on X define the same germ sx if they coincide on an open neighbourhood of x. The sheaf generated by the presheaf in Example 1.1.3 is called the sheaf of continuous functions. The sheaf of smooth functions on a manifold X is defined in a similar way.

Two different presheaves may generate the same sheaf. Conversely, a sheaf defines a presheaf of Abelian groups T(U, S) of local sections of the sheaf S. This presheaf {T(U,S),ry} is called the canonical presheaf of the sheaf S. It is easily seen that the sheaf generated by the canonical presheaf {r(U, S),ry} of the sheaf S coincides with S. Therefore, we will further identify sheaves and canonical presheaves.

Example 1.1.4. Let Y —* X be a vector bundle. The germs of its sections make up the sheaf S(Y) of sections of Y —» X. The stalk SX(Y) of this sheaf at a point x e X consists of the germs of sections of Y —> X in a neighbourhood of x € X. The stalk SX(Y) is a module over the ring C™(X) of the germs at x G X of smooth functions on X. If we deal with a tangent bundle TX —» X, the stalk SX(TX) is a Lie algebra with respect to the Lie bracket of vector fields. •


1.2 Multivector fields and differential forms

Subsections. Vector fields, 20; Vector fields on fibre bundles, 21; Multivector fields, 22; The Schouten-Nijenhuis bracket, 24; Exterior forms, 25; Exterior forms on fibre bundles, 26; Interior products, 28; Bivector fields and 2-forms, 29; The Lie derivative, 31; Tangent-valued forms, 31; Distributions, 32; Foliations, 34.

Vector fields

A vector field on a manifold Z is defined as a global section of the tangent bundle TZ -» Z. The set 7~(Z) of vector fields on Z is both a module over the ring C°°{Z) of smooth functions on Z and a real Lie algebra with respect to the Lie bracket

[v, u] = (vxdxu" - v^dxv^d^, u = uxd\, v = vxd\.

A curve c : () —► Z, () C R, in Z is said to be an integraJ curve of a vector field u on Z if

c = u o c, cA(t) = uA(c(t)), t e ( ) .

Recall that, for every point z € Z, there exists a unique integral curve

c : ( -e ,e ) -Z , ( >0,

of a vector field u through z = c(0). A vector field u on an imbedded submanifold N c Z is said to be a section of

the tangent bundle TZ —► Z over N. It should be emphasized that this is not a vector field on a manifold N since u(N) does not belong to TN C TX in general. A vector field on a submanifold N C Z is called tangent to the submanifold iV if u(N) C TAT.

Let U C 2 be an open subset and t > 0. By a JocaJ 1-parameter group of local diffeomorphisms of Z defined on (—c, e) x (/ is meant a mapping

G :(-e,() xU3{t,z)^G,{z)eZ

which possesses the following properties:

• for each t e (—e, e), the mapping G( is a diffeomorphism of £/ onto the open subset Gt(l7) C Z;

1.2. MULTIVECTOR FIELDS AND DIFFERENTIAL FORMS 21

• Gt+t,(z) = (Gt o G,)(z) if t + H e (-«, e).

If such a mapping G is defined on 1 x Z, it is called a 1-parameter group of diffeo-morphisms of Z.

T H E O R E M 1.2.1. [100]. Each local 1-parameter group of local diffeomorphisms G on U C Z defines a local vector field u on U by setting u{z) to be the tangent vector to the curve s(t) = Gt{z) at ( = 0. Conversely, let u be a vector field on a manifold Z. For each z G Z, there exist a number e > 0, a neighbourhood (/ of 2 and a unique local 1-parameter group of local diffeomorphisms on (—e, e) x U, which determines u. D

In brief, every vector field u on a manifold Z is the generator of a local 1-parameter group of local diffeomorphisms. In particular, every exterior form (p on a manifold Z is invariant under a local 1-parameter group of local diffeomorphisms Gu with the generator u, i.e.,

g'4> = 0, V5 e G„,

if and only if its Lie derivative Lu4> along u vanishes. If a vector field u on a manifold Z is induced by a 1-parameter group of diffeo

morphisms of Z, then u is called a complete vector Held.

Vector fields on fibre bundles

A vector field u on a fibre bundle Y —> X is said to be projectable if it projects over a vector field ux on X, i.e., if the following diagram

Y JL^TY

X —*TX

is commutative. A projectable vector field has the coordinate expression

U = T l V ) & + « W ) 3 i . it* = uxdx-

A vector field r = TA&A on a base X of a fibre bundle V —» X can give rise to a vector field on Y, projectable over r, by means of some connection on this fibre bundle (see (1.4.7) below). Nevertheless, a tensor bundle

T=(®TX)®(®TmX),


admits the canonical lift

r = T*0„ + [duT^x^;0a- + ... °/3iT xv0i-0k • ' J o - a , am

(1.2.1)

of any vector field r o n l . In particular, there exist the canonical lift

f = T<^ + 3 „ T a ± " A (1.2.2)

of r onto the tangent bundle TX, and its canonical lift ri

f = r^dy, - d0Tvxu-— oxff

(1.2.3)

onto the cotangent bundle T'X. Hereafter, we will use the compact notation

dx = — dxx (1.2.4)

A projectable vector field u = u'<9, on a fibre bundle Y —* X is said to be vertical if it projects over the zero vector field ux = 0 on X.

Let Y —» X be a vector bundle. Using the canonical vertical splitting (1.1.5), we obtain the canonical vertical vector field

uY = y'd, (1.2.5)

on Y, called the Liouville vector Held. For instance, the Liouville vector field on the tangent bundle TX reads

UTX = ixd\. (1.2.6)

Accordingly, any vector field r = rxd\ on a manifold X has the canonical vertical lift

TV = TXdX (1.2.7)

onto the tangent bundle TX.

Multivector fields

A multivector field d of degree | t? |= r (or simply an r-vector field) on a manifold Z is a section

4= V 1 XrdXlA---AdXr r!


of the exterior product l\TZ —> Z. Let us denote by %{Z) the vector space of r-vector fields on Z. In particular, T\{Z) is the space of vector fields on Z (denoted by T(Z) for the sake of simplicity), while TQ(Z) is the vector space Cca{Z) of smooth functions on Z. All multivector fields on a manifold Z make up the real Z-graded vector space T,{Z) which is also a Z-graded exterior algebra with respect to the exterior product of multivector fields.

Given a manifold Z, the tangent lift tf onto TZ of an r-vector field d on Z is denned bv the relation

d{F,...,<j')=d{a\...^) (1.2.8)

where: (i) ak = akdxx are arbitrary 1-forms on the manifold Z, (ii) by

ak = x»dvO$dxx + akxdxx

are meant their tangent lifts (1.2.24) onto the tangent bundle TZ of Z, and (iii) the right-hand side of the equality (1.2.8) is the tangent lift (1.2.22) onto TZ of the function -d(or,..., a1) on Z [67]. We then have the coordinate expression

t? = - r t f A l V d A l A - - A d A r , r:

7"! (1.2.9)

tfAl"*'£&, A • • • A S A , A •■•A& r].

In particular, if r is a vector field on a manifold Z, its tangent lift (1.2.9) coincides with the canonical lift (1.2.2). If an r-vector field # is simple, i.e.,

tf = T ' A - - - A T ' ,

its tangent lift (1.2.9) reads T

■d= ^ T y A - - - A f i - - - A r ( ; , i = l

where T£ is the vertical lift (1.2.7) onto TZ of the vector field r*.

Example 1.2.1. The tangent lift of a bivector field

w^^w^d^hdv


is

w = - ( i A a A w ^ 4 A 3„ + I O " " ^ A dv + ufdn A dv)

•

Schouten-Nijenhuis bracket

The exterior algebra of multivector fields on a manifold Z is provided with the Schouten-Nijenhuis bracket which generalizes the Lie bracket of vector fields as Mows [13, 181]:

[., .]SN : %{M) x TS(M) -» T r+S_,(M), (1.2.10)

r! SI

[tf,u]sK = * * t » + ( - l ) " t ; * t f ,

* * " = ^ j ( ^ 2 A r ^ a ' a * ^ 2 A • • • A 3Ar A dol A • • • A da.).

There following relations hold:

[ M a , = (-1)WMM]SN. (1.2.11) [«/, 0 A V]SH = [u, tf]SN A v + (-1){M-DI«I^ A [v, u]SN, ( - i jMCM-D^ (tfi U]SN]SN + (_ i )W0" l -«^ [„, „]SN]SN +

(-l)M(l«l-i)[ t , j[V| t f] s s]SN = 0 .

(1.2.12)

(1.2.13)

Example 1.2.2. Let

w = -uiMI/dM A dv

be a bivector field. Its Schouten-Nijenhuis bracket reads

[U>,W]SN = w^d^w^d^ A dx, A dX3.

The Schouten-Nijenhuis bracket commutes with the tangent lift (1.2.9) of mul-tivectors [67], i.e.,

[tf,v]SN = [IMSN- (1.2.14)


Remark 1.2.3. Let us point out another sign convention used in the definition of the Schouten-Nijenhuis bracket [125]. This bracket, denoted by [., .]SN-, is

[IMSN' = - ( -1 ) ' " [1MSN. (1.2.15)

The relation (1.2.11) for this bracket reads

[MSN< = ( - i r ' - 'X '^MlsN' . (1.2.16)

The relation (1.2.12) keeps its form, i.e.,

[v, t? A v]SN, = [u, tf]SN< At; + (_l)CM-DWtf A [„, v]m,t (1.2.17)

while the relation (1.2.13) is replaced by

(_1 )(M-.)(M-i, [ l / ] [dMsN,]sN, + ( ^ M - i X W - D p , [Vf v ] sN , ] sN , +

( _ 1 ) (M- i ) (W- i ) [ w [ j / ^ s N , ] s N , = 0

(1.2.18)

The equalities (1.2.16) and (1.2.18) show that, with the modified Schouten-Nijenhuis bracket (1.2.15), the Z-graded vector space T.{Z) of multivector fields on a manifold Z is a graded Lie algebra, where the Lie degree of a multivector field $ is | d \ — 1. In particular,

a&d{v)^[d,v\sw (1.2.19)

is a graded endomorphism of degree | ■& | — 1 of the graded Lie algebra T,(Z). If ■& is a vector field, the endomorphism (1.2.19) is the Lie derivative

adtf(u) = L#v (1.2.20)

of the multivector field v along ti. •

Exterior forms

An exterior r-form on a manifold Z is a section

cj>^-d>x1...xrdzXlA---AdzXr

r!

of the exterior product A T'Z —» Z. We denote by Dr(Z) the vector space of exterior r-forms on a manifold Z. This is also a module over the ring D°(Z) = C°°{Z).


From now on we will use the notation C°°{Z) for the ring of smooth functions on a manifold Z, while D°(Z) stands for the vector space of these functions as a rule.

All exterior forms on Z constitute the exterior Z-graded algebra 0"{Z) with respect to the exterior product. The exterior differential is the first order differential operator

d : D r ( Z ) - » D r + 1 ( Z ) ,

d<t> = -S„0 A l Ardz" A dzx> A • • • dzx', r!

on D'{Z). It obeys the relations

dod = Q,

d{4> A a) = d{4>) A a + (-l)w<£ A d(a),

where | </> | is the degree of cj>. Given a manifold map / : Z —* Z*, by f'(j> is meant the pull-back on Z of an

r-form <j> on Z' by / , which is defined by the condition

r4>{v\ ■ ■ .y){z) = (PiTfiv1),..., Tf(vr))(f(z)), Vv\ ■•■■«/ eTzZ.

We have the relations

/ • ( ^ A a ) = / > A / V ,

df'4> = r{d4>).

For instance, if in ■ N —» Z is a submanifold, the pull-back i j ^ onto TV is called the restriction of an exterior form <j> to N.

Exterior forms on fibre bundles

Let n : Y —► X be a fibre bundle with fibred coordinates ( iA ,y ' ) . The pull-back on Y of exterior forms on X by n provides the inclusion

it* : 0*(X) -» D*(y).

Exterior forms

<&: y - A r*x,

<A = -7^A,...Ar<ii:Al A---AdxX r , r!


on Y such that d\<j> = 0 for arbitrary vertical vector field d on. Y are said to be horizontal forms. A horizontal n-form is called a horizontal density. We will use the notation

w = dx1 A •••Adz", ^A = d\\w. (1.2.21)

In the case of the tangent bundle TX —► X, there is a different way, besides the pull-back, to lift onto TX the exterior forms on X [67, 110, 189]. Let / be a function on X. Its tangent lift onto TX is defined as the function

/ = ±XdxJ- (1.2.22)

Let a be an r-form on X. Its tangent lift onto TX is said to be the r-form a given by the relation

^ ( n , . . . , f r ) = Cr(Ti,.. . ,Tr), (1.2.23)

where r4 are arbitrary vector fields on TX, and ?< are their canonical lifts (1.2.2 onto TX. We have the coordinate expression

a = -aXv.\rdxXi A - - - A d x v , r!

a = —[x}idliaxl...xrdxXi A • • • A dxAr + r! r

5^CV- ArdzAl A • • • A dxA' A • • • A dxAr]. t = l

(1.2.24)

The following equality holds:

da = da.

Example 1.2.4. Given a 2-form

fi = -Sl^dx* A dx"

on a manifold X, its tangent lift (1.2.24) onto TX reads

0 = hx^xSl^dx11 A dx" + il^dx" A dx" + fi^dx" A dx"). (1.2.25)

•


Interior products

The interior product (or the contraction) of a vector field u = u^d^ and an exterior r-form <j> is given by the coordinate expression

u\<t> = £ {~\-uA*<K.-K...A,«fa*' A • • • A d / ' A • • • A dzA' = *=l r-

7 7TT«M0^,2...ardza* A • • • A dzQ '. ( r - 1)!

(1.2.26)

It satisfies the relations

0 (u i , . . . ,U r ) = Ur\ ■■■Ui\4>, (1.2.27)

u\{<j>Aa) = u\4> A a + {-\)W(j> Au\a, (1.2.28)

[U,U' ]J0 = u\d{u'\4>) - u'\d{u\4>) - u'\u\d<j>, 0 6 O ' ( Z ) . (1.2.29)

The generalization of the interior product (1.2.26) for multivector fields is the left interior product

0\4> = W), l*!<IH ^eD'(Z), tf e r.(z),

of multivector fields and exterior forms, which is derived from the equality

(j>(u{ A--- A Mr) = ^ ( U i , . . . , ^ ) , * 6 0 ' ( Z ) , u, e T(Z) ,

for simple multivector fields. We have the relation

■d\v\<j> = (t)A #)J0 = ( - l ) | u | | %Jt f |0 , *eom i9,w<=T.(Z).

Example 1.2.5. The formula (1.2.29) can be generalized for multivector fields as follows [13]:

[ I M S N J 0 = ( - l ) M ( " | - I , 0 J d M 0 ) + (-l)l*l«jd(tfj«) - ujdjd*,

where | 0 | = | i? | + | u | - 1 . •

The right interior product

*!.* = *(*). Ul<l^l, tfeO'(Z), i? 6 T.(Z),


of exterior forms and multivector fields is given by the equalities

0 ( & , . . . , 0 r ) = 0 | V L * i . AeO'CZ), 0 e rr(Z),

^ = ( T Z ^ ^ - ^ - ' ^ A , A • • • A d0r_„ 0 6 D ' ( Z ) .

It satisfies the relations

(tf A u)|> = tf A (v|>) + (-1)M(T?L<W A U, ^ef l ' iZ) , tf(<M<r) = tfk[<£, 4>,aeO'(Z).

In particular, if | ■d |= | 0 |, we have the natural pairing

(,):T r(Z)xD r(Z)-C°°(Z), (0, 0) = tf|0 = T?[0 = #(0) = « W (1.2.30)

Bivector fields and 2-forms

Each bivector field

w = -w^df,. A d„

on a manifold Z defines the linear fibred morphism

Wt-.T'Z -*TZ, z w,(a) = —w(z)[a, a e r;z, (1.2.31)

IW'(Q) = w)w{z)aiLdl,,

which fulfills the relation

to(z)(a,/J) = w'(a)J/8 = w(z)[/?|a, zeZ, a,(3eT'zZ.

One says that a bivector field w is of rani r at a point z € Z if the morphism (1.2.31) has rank r at 2. If this morphism is an isomorphism at all the points z € Z, the bivector field w is said to be non-degenerate. Such a bivector field can exist only on an even-dimensional manifold.

The morphism (1.2.31) can be generalized to the homomorphism of graded algebras £>m(Z) —► T,(Z) in accordance with the relation

W*(^)(<Tl, - • • , <7r) = ( - l ) r ( ^ ( t U J ( f f 1 ) , . . . , iu'((7r)). (1.2.32)

0 e o m a i C O 1 ^ ) .

30 CHAPTER 1, INTERLUDE: BUNDLES, JETS, CONNECTIONS

This is clearly an isomorphism if the bivector field w is non-degenerate. Each 2-form

ft = ^zQ.^dz'1 A dzv

on a manifold Z defines the linear fibred morphism

ftb : TZ — T'Z,

n\v) = -n^iz^dz". (1.2.33)

One says that a 2-form ft is of rank r at a point z 6 Z if the morphism (1.2.33) has rank r at z. This is the maximal number Ik such that A Q(z) ^ 0.

The kernel of a 2-form ft is defined as the kernel

Kerftd= \J{v e TZZ : ujujft = 0, Vu 6 TZZ) zez

(1.2.34)

of the morphism (1.2.33). Its fibre Ker2ft at a point z e Z is a vector subspace of the tangent space TZZ whose codimension equals the rank of ft at z. If a 2-form ft is of constant rank, its kernel (1.2.34) is a subbundle of the tangent bundle TZ in accordance with Proposition 1.1.4.

A 2-form ft is called non-degenerate if its rank is equal to dim Z at all points z G Z. A non-degenerate 2-form ft can exist only on a 2m-dimensional manifold. Then A ft is nowhere vanishing, and can play the role of a volume element on Z.

On a 2m-dimensional manifold Z, there is one-to-one correspondence between the non-degenerate 2-forms ftm and the non-degenerate bivector fields WQ in accordance with the equalities

wn(<t>,°) = n«,(wJi(«?l)),wJi((7)), (1.2.35a)

nw(-d,v) = wn(ni(ti),{t(v)), (1.2.35b) 0,aeD'(2), tf,i/eT(Z),

where the morphisms WQ (1.2.31) and ftj„ (1.2.33) obey the relations

4 = (Ar\ ftiua/jU'n = °3<

i.e.,

wtittm = 0, <£(«&(*)) = <t>-


The Lie derivative

The Lie derivative of an exterior form 4> along a vector field u is given by the equality

Lu<£ = u\d(j> + d{u\4>).

In particular, if / is a function, then

Uf = u{}) = u\d}.

The relation

Lu(<£ A a) = Lucj> A a + <j> A Lucr

is fulfilled. Given the tangent lift <j> (1.2.24) of an exterior form 4>, we have

Lu(0) = u'4>

[67, 147]. The Lie derivative (1.2.20) of a multivector field v along a vector field u is

LUV = [U,V]SN' = [u,f]sN,

and it obeys the equality

L„(i9 Au) = Lutf A« + t)A Luw

in accordance with the relation (1.2.17).

Tangent -va lued forms

Elements of the tensor product OT(Z) ® T(Z) are called the tangent-valued r-forms

<b:Z-^AT,Z®TZ,

<j, = - ^ XrdzXl A • • • A dzXr ® d„.

There is one-to-one correspondence between the tangent-valued 1-forms <j> on a manifold Z and the linear bundle endomorphisms over Z:

4>:TZ -> TZ, 0 : T 2 Z 3 « ~ ? ; J t f > ( z ) e T 2 Z , (1.2.36)


and

$' : T'Z — T'Z, $' : T'ZZ 3 B ' H <t>{z)\v" e T ; Z . (1.2.37)

In particular, the canonical tangent-valued J-form

9Z = dzx <g> dx

on Z corresponds to the identity morphisms (1.2.36) and (1.2.37). Let Z = TX. There is the fibred endomorphism J of the tangent bundle TTX

of TX such that, for every vector field r on X, we have

J o f = TV , J o TV = 0,

where r is the canonical lift (1.2.2) and Tv is the vertical lift (1.2.7) onto TTX of a vector field r on TX. This endomorphism reads

JW0 = a*, J0X) = 0. (1.2.38)

It is readily observed that J o J = 0, and the rank of J equals n. The endomorphism J (1.2.38), called an almost tangent structure [110, 189], corresponds to the tangent-valued form

<j>j = dxx ® dx (1.2.39)

on the tangent bundle TX.

Distributions

An n-dimensional smooth distribution on a /c-dimensional manifold Z is an n-dimensional subbundle T of the tangent bundle TZ. We will say that a vector field v on Z is subordinate to a distribution T if it is a section of T —► Z. An integral curve of a vector field, subordinate to a distribution T, is called admissible with respect to T.

A distribution T is said to be involutive if the Lie bracket [u, v!\ is a section of T —> Z, whenever u and u' are sections of the distribution T —» Z.

A connected submanifold TV of a manifold Z is called an integral manifold of a distribution T on Z if the tangent spaces to N belong to the fibres of this distribution at each point of N. Unless otherwise stated, by an integral manifold we mean an


integral manifold of maximal dimension, equal to dimension of the distribution T. An integral manifold N is called maxima] if there is no other integral manifold which contains N.

T H E O R E M 1.2.2. [185]. Let T be a smooth involutive distribution on a manifold Z. For any point z £ Z, there exists a unique maximal integral manifold of T passing through z. □

In view of this fact, involutive distributions are also called completely integrable distributions.

If a distribution T is not involutive, there are no integral submanifolds of dimension equal to the dimension of a distribution. However, integral submanifolds always exist, e.g., the integral curves of vector fields, subordinate to T.

We refer the reader to [68] for a detailed exposition of differential and Pfaffian systems.

A differential system S on a manifold Z is said to be a subbundle of the sheaf S(TZ) of vector fields on Z whose fibre Sz at each point z € Z is a submodule of the Cf> (Z)-module SZ{TZ) (see Example (1.1.4)). The germs of sections of a distribution T obviously make up a differential system S(T).

The Bag of a differential system S is the sequence of differential systems

S , = S , S2 = [S,S], ••• S, = [Si-i,S].

Here [S,S']2 is the Cf (Z)-module generated by [v,u], v € S2, u e S^. Let S(T) be a differential system associated with a distribution T, and let

S(T) = S , c S j C -

be its flag. In general, S, is not associated with a distribution. If this is the case for all i, we may define the Sag of a distribution

T = Ti C T 2 C • • •. (1.2.40)

A distribution is called regular if its flag (1.2.40) is well defined. The sequence (1.2.40) stabilizes, i.e., there exists an integer r such that T r_i ^ T r = T r + i [184]; moreover T r is involutive. In particular, if r = 1, we are dealing with the integrable case. If T r = TZ, the distribution T is called totally non-holonomic.

A codistribution T* on a manifold Z is a subbundle of the cotangent bundle. For instance, the anniMator Ann T of an n-dimensional distribution T is a (k — n)-dimensional codistribution.


THEOREM 1.2.3. [185]. Let T be a distribution and AnnT its annihilator. Let A Ann T be the ideal of the exterior algebra O'(Z) which is generated by elements of Ann T. A distribution T is involutive if and only if the ideal AAnn T is a differential ideal, i.e., d(AAnnT) C AAnnT. D

COROLLARY 1.2.4. Let T be a smooth involutive r-dimensional distribution on a fc-dimensional manifold Z. Every point z £ Z has an open neighbourhood U B z which is a domain of a coordinate chart ( z \ . . . , zk) such that the restrictions of the distribution T and its annihilator Ann T to U are generated by the r vector fields

dz1''''' dzr

and the (k - r) 1-forms dzk r + 1 , . . . , dzk, respectively. It follows that integral manifolds of an involutive distribution make up a foliation. □

Example 1.2.6. Every 1-dimensional distribution on a manifold Z is integrable. Its section is a nowhere vanishing vector field u on Z, while its integral manifolds are the integral curves of u. By virtue of Corollary 1.2.4, there exist local coordinates (z ,..., zk) around each point z £ Z such that u is given by

u = d/dz1.

•

A Pfaffian system S' is asubmodule of the C°°(Z)-module D1(Z). In particular, sections of a codistribution constitute a Pfaffian system. Any Pfaffian system S" defines the ideal AS" of the exterior algebra D'(Z) which is generated by elements of 5*. Given a flag (1.2.40) of a regular distribution, one can introduce the coBag of the codistribution

Ann (T) D Ann (T2) D ■■■. (1.2.41)

The coflag (1.2.41) stabilizes. In particular, a distribution T is totally non-holonomic if and only if its coflag (1.2.41) shrinks to zero.

Foliations

An r-dimensional (regular) foliation on a k-dimensional manifold Z is said to be a partition of Z into connected leaves FL with the following property. Every point

2.3. JET MANIFOLDS 35

of Z has an open neighbourhood U which is a domain of a coordinate chart (za) such that, for every leaf Ft, the connected components Ft D U are described by the equations

z r + 1 = const., zk = const.

[90, 150]. Note that leaves of a foliation fail to be imbedded submanifolds, i.e., topological subspaces in general.

Example 1.2.7. Submersions n : Y —» X and, in particular, fibre bundles are foliations with the leaves 7r"'(x), x G 7r(V) C X. A foliation is called simple if it is a fibre bundle. Any foliation is locally simple. •

Example 1.2.8. Every real function / on a manifold Z with nowhere vanishing differential df is a submersion Z —» R. It defines a 1-codimensional foliation whose leaves are given by the equations

f(z) = c, c e f{Z) c R.

This is the foliation of level surfaces of the function / , called a generating function. Every 1-codimensional foliation is locally a foliation of level surfaces of some function on Z. •

The level surfaces of arbitrary function / ^ const, on a manifold Z define a singular foliation F on Z [90]. Its leaves are not submanifolds in general. Nevertheless if df(z) ^ 0, the restriction of F to some open neighbourhood U of z is a foliation with the generating function f\y-

1.3 J e t manifolds

Subsections: Jet manifolds, 35; Canonical horizontal splittings, 38; Second order jet manifolds, 38; The total derivative, 40; Higher order jet manifolds, 40; Differential operators and differential equations, 41.

Jet manifolds

Given a fibre bundle Y —» X with bundle coordinates (xx,yl), let us consider the equivalence classes j^s, x € X, of its sections s, which are identified by their


values s'(z) and the values of their first derivatives 3Ms'(i) at points i e X. The equivalence class j^s is called the first order jet of sections s at the point i £ l The set JXY of first order jets is provided with a manifold structure with respect to the adapted coordinate atlas

(z\<A</i),

{x\y\tfx){3ls) = {x\s>{x),dxs\x))}

^ = S(a"+^)2/"-(1.3.1)

It is called the jet manifold of sections of the fibre bundle Y —► X (or simply the jet manifold of the fibre bundle Y —> X).

The jet manifold admits the natural fibrations

TT1 : JlY 3 jls ■-» x 6 X, (1.3.2)

*\;JlY 3jls~s(x)eY, (1.3.3)

where (1.3.3) is an affine bundle modelled over the vector bundle

T'X®VY -» K Y

For the sake of convenience, the fibration JlY —> X is further called a jet bundie, while the fibration JlY —» Y is an affine jet bundle.

There are the following two canonical monomorphisms of the jet manifold JlY over y :

A: J 1 Y'- ->T*X®7T, (1.3.4)

A = dxx ® dA = dxx ® (dA + j/^a,),

where d\ is called the total derivative, and

0! : J'Y ^T'Y®VY, Y

(1.3.5)

Sl=p®d, = (dyl - y\dxx) ® d„

where

9' = dy' - y\dxx (1.3.6)


is called the contact form. In accordance with these monomorphisms, every element of the jet manifold JXY can be represented by the tangent-valued forms

dxx ® (dx + y\di) and (dy' - y\dxx) ® dx.

Each fibred morphism $ : F —» Y' over a diffeomorphism / is extended to the fibred morphism of the corresponding jet manifolds

J^:JXY — J 1 ?" , *

8(f~xY y'\ o J 1 * = (d,<DVM + 9 M * ' ) £ ^ J _ ,

called the jet prolongation of the morphism $. Each section s of a fibre bundle Y —» X has the jet prolongation to the section

(2 / , ,2 / l ) ° - / 1 s=(s ' (x ) ,^5 , (x) ) ,

of the jet bundle JlY —* X. A section s of the jet bundle JlY —> X is said to be hoionomic if this is the jet prolongation of some section of the fibre bundle Y —* X.

Any projectable vector field

u = ux(x")dx+u'(x>',y])d,

on a fibre bundle Y —» X admits the jet prolongation to the vector field

u = n o jlu ■. j 'y -»j'ry ->rj'y, G = uAaA + u% + (dxu> - fidxurffi, (1.3.7)

on the jet manifold JlY. One can show that the jet prolongation of vector fields u i—► u is the morphism of Lie algebras, i.e.,

[u,u'] = [u,lf}.

In order to obtain (1.3.7), we have used the canonical fibred morphism

r, : JlTY - TJlY,

yi°n = {y')x - yi±x-

* J^:jis^j}{x){^oSof-%

y'\ o J 1 * = (d^X + 9 M * ' ) £ ^ T - .

(^«)(x)^iia,

[u,u'J = [u.JZ'].

vi ° J-I = (y'h - y^v

(1.3.7)

(y\y\)oJls = (s>(x),dxs>(x)),

38 CHAPTER 1, INTERLUDE: BUNDLES, JETS, CONNECTIONS

In particular, there is the canonical isomorphism

VJlY = JXVY, (1.3.8)

y'x = (y')x-

Canonical horizontal splittings

The canonical morphisms (1.3.4) and (1.3.5) can be viewed as the morphisms

A : JlY x TX B dx >-» dx = dx\X € JlY x TY X Y

(1.3.9)

and

0i : JlY x V'Y 9 dy* H-. 0> = 9x\dy' € JlY x T'Y, (1.3.10)

where {dy'} are the bases for the fibres of the vertical cotangent bundle V'Y. These morphisms determine the canonical horizontal splittings of the pull-backs

J'Y xTY = X(TX) 0 VY, Y j i y

(1.3.11)

xx8x + y% = xx(dx + y\dx) + (y* - xxy\)d„

and

JlY x T'Y = T'X 0 0i(V*y), Y j i y

(1.3.12)

xxdxx + y,dy{ = ( iA + y,yx)dxx + y^dtf - y\dxx).

Second order jet manifolds

Taking the first order jet manifold of the jet bundle J ' V —* X, we come to the repeated jet manifold JlJlY, provided with the adapted coordinates

(* ,y\yx,ylM>2/J.A).

dxa

y'\x = d^(d°+ yfa)di + yiadj)y'y

There exist two different affine fibrations of J 1 J*Y over JlY:


• the familiar affine jet bundle (1.3.3)

*ii : J'J'Y - J ' y , y\°Tn = y\, (1.3.13)

modelled over the vector bundle

T'X ® VJlY — JlY, (1.3.14)

• and the affine bundle

J1^ : JxJlY -> / F , y* ° J^l = vU> (1.3.15)

whose underlying vector bundle

Jl(T'X®VY)^J1Y (1.3.16)

differs from (1.3.14).

In general, there is no canonical identification of these fibrations, but it can be made by means of a symmetric linear connection on X [57].

The points q € J1J1Y, where nn(q) = . / ' ^ ( g ) , make up the affine subbundle J*Y —> J1Y of J^J^Y, called the sesquiholonomic jet manifold. This is given by the coordinate conditions

V\x) = y\<

and is coordinated by (xx, y\j&,jfj,^). The second order jet manifold J2Y of a fibre bundle Y —> X is the affine sub-

bundle 7!-? : J2Y -» JlY of the fibre bundle pY — J 1 ^ , given by the coordinate conditions

vU = yU and coordinated by (xx,y',y\,y\ll = y'^)- It i s modelled over the vector bundle

VT'X ®

The second order jet manifold PY can also be seen as the set of the equivalence classes j2s of sections s of the fibre bundle Y —> X, which are identified by their

yy -» J1Y.


values and the values of their first and second order partial derivatives at points x€X:

y\(fxs) = dxs\x), VUJIS) = d^s'ix).

Let s be a section of a fibre bundle Y —♦ X and J*s its jet prolongation to a section of the jet bundle JlY —» X. The latter gives rise to the section JlJls of the repeated jet bundle Jl JlY —> X. This section takes its values into the second order jet manifold f*Y. It is called the second order jet prolongation of the section s, and is denoted by J2s.

PROPOSITION 1.3.1. Let s be a section of the jet bundle JlY —> X and Jls its jet prolongation to the section of the repeated jet bundle J1JiY —> X. The following three facts are equivalent:

• s = Jls where s is a section of the fibre bundle Y —> X;

• . / 's takes its values into J^Y;

• J^s takes its values into J2Y.

D

The total derivative

We will use the total derivative operator

d^d^ + yidi + yifl.

It satisfies the equalities

d\{4> A a) = d\(4>) A a + <t> A dx(a),

dx{d<t>) = d(dx(4>)).

Higher order jet manifolds

The k-orderjet manifold JkY of a fibre bundle Y ~* X comprises the equivalence classes j * s , x 6 X, of sections s of Y identified by the k + 1 terms of their Tailor


series at the points x e X. The jet manifold JkY is provided with the adapted coordinates

(xx,y\yl...,y\k...Xi), y\l.xlU*s) = dxr--dxlsi(x), 0<l<k.

Every section s of a fibre bundle Y —> X gives rise to the section Jks of the fibre bundle JkY -> X such that

j / A , . A t ° ^ = aA , . . .aA l Si , 0 < I < k.

Differential operators and differential equations

Let JkY be the fc-order jet manifold of a fibre bundle V —► X and E —* X a vector bundle over X.

DEFINITION 1.3.2. A fibred morphism

£:JkY^E x (1.3.17)

is called a fc-order differential operator on the fibre bundle Y —► X. It sends each section s(x) of Y —> X onto the section (£ o Jks)(x) of the vector bundle £ - » ! □

The kernel of a differential operator is the subset

Ker£ = £-1(0(X))cJkY, (1.3.18)

where 0 is the zero section of the vector bundle E —» X, and we assume that Q(X) C £{JkY).

DEFINITION 1.3.3. A system of Ar-order partial differential equations (or simply a differential equation) on a fibre bundle V —► X is defined as a closed subbundle € of the jet bundle JkY -» X [20, 57, 104]. D

Its (classical) solution is a (local) section s of the fibre bundle Y —> X such that its A;-order jet prolongation Jks lives in £.

For instance, if the kernel (1.3.18) of a differential operator £ is a closed sub-bundle of the fibre bundle JkY -> X, it defines a differential equation

£oJks = 0.


The following condition is sufficient for a kernel of a differential operator to be a differential equation.

PROPOSITION 1.3.4. Let the morphism (1.3.17) be of constant rank. By virtue of Theorem 1.1.3, its kernel (1.3.18) is a closed subbundle of the fibre bundle JkY —> X and, consequently, is a fc-order differential equation. □

1.4 Connections

Subsections: Connections, 42; The curvature of connections, 44; Linear connections, 44; Affine connections, 44; Flat connections, 45.

Connect ions

A connection on a fibre bundle Y —» X is defined as a global section

r : Y -* JlY, T = dxx®{dx + T\{x»,y*)di),

of the affine jet bundle JlY —» Y. Combining a connection T and the morphisms (1.3.9) and (1.3.10) gives the splittings

X o T : TX -» TY, 0i o r : V'Y - T'Y

of the exact sequences (1.1.8a) and (1.1.8b), respectively. Accordingly, substitution of the section y\ — r\ into the expressions (1.3.11) and (1.3.12) leads to the familiar splittings of the tangent bundle

TY = r(TX) © VY, Y

±xd, + y% = ±\dx + r\d,) + (y< - ixr\)dt, (1.4.1)

and the cotangent bundle

T'Y = T'X(BT(V'Y), Y

xxdxx + iudtf = (&x + T\ili)dxx + yi{dy' - r\dxx),

(1.4.2)

1.4. CONNECTIONS 43

of a fibre bundle Y —» X with respect to the connection T. In an equivalent way, the connection V defines the corresponding projection

T:TY3xxdx + y% -> (y* - ixr\)di G VY (1.4.3)

and the corresponding section

T = (dyl - T\dxx) <S> di (1.4.4)

of the fibre bundle T'Y ® VY -> Y. Y

Connections on a fibre bundle Y —• X constitute an affine space modelled over the linear space of soldering forms

a:Y -> T'X ® VY, Y

(T = a\dxx (8) di-

Any connection T on a fibre bundle Y —> X defines the first order differential operator on Y

Dr:JlY3z^[z- Tiirliz))] G T'X ® VY,

Dr = (y\-r\)dxx®di,

(1.4.5)

called the covarianfc differential. Its action on sections s of the fibre bundle Y reads

V r s = Dr o Jls = [dxs* - ( r o s)\]dxx ® a,. (1.4.6)

For instance, a section s is said to be an integral section for a connection T, if V r s = 0, i.e., T o s = J J s . For any section s of a fibre bundle Y —> X, there exists a connection r on Y —> X such that s is its integral section. This connection is an extension of the section s(x) i-» Jls(x) of the affine jet bundle J1Y —> Y over the closed submanifold s(X) C V in accordance with Theorem 1.1.2.

A connection T on a fibre bundle Y —» X defines the horizontal lift

rr = r\dx + r\d{) (1.4.7)

onto Y of each vector field T = rxd\ on X.


The curvature of connections

The curvature of a connection T on a fibre bundle Y —» X is said to be the 2-form on Y

R:Y ->AT'X®VY, Y

R= -R\tidxx Adx»®di,

R\„ = ^ r ; - d,r\ + r ^ r ; - r ^ r j , (1.4.8)

Linear connections

Let Y —» X be a vector bundle. A linear connection on Y —> X reads

r = dxA®[aA + rAiJ(xya,].

It defines the duai linear connection

r = dxx ® [5A - r^A(x)%a'] on the dual vector bundle Y* —» X. For instance, a linear connection .ft" on the tangent bundle TX, and the dual linear connection K* on the cotangent bundle T'X are given by the expressions

K = dxx <8> (3A + ^ a „ ( i ) i " a Q ) , (1.4.9)

A" = dxA ® (5A - Kf¥(x)*Jr)- (1.4.10)

Affine connections

Let Y —> X be an affine bundle modelled over a vector bundle Y —» X. An affine connection on Y —» X reads

r = dxA ® [dA + (r*'j(*y + r\(x))di}. It defines the linear connection

r = dxx®[dx + rxlJ(x)yi-?-}

on the vector bundle Y —» X.

1.4. CONNECTIONS 45

Flat connections

Each connection T on a fibre bundle Y —> X, by definition, yields the horizontal distribution T(TX) C TY on Y, generated by the horizontal vector fields (1.4.7). The following assertions are equivalent.

• The horizontal distribution is involutive.

• The connection T is flat (curvature-free), i.e., its curvature is equal to zero everywhere.

• There is an integral section for the connection T through any point y e Y.

Hence, a fiat connection r on Y —* X yields the integrable horizontal distribution, i.e., the horizontal foliation on Y, transversal to the fibration Y —> X. Its leaf through a point y 6 Y is defined locally by an integral section sy for the connection T through y. Conversely, let a fibre bundle Y —> X admit a horizontal foliation such that, for each point y e Y, the leaf of this foliation through y is locally defined by a section sy of Y —► X through y. Then the map

r : Y -» JlY,

r(?/) = jlxsv, n(y) = x,

introduces a flat connection on Y —> X. Thus, there is one-to-one correspondence between the flat connections and the horizontal foliations on a fibre bundle Y —» X.

Given a horizontal foliation on a fibre bundle Y —> X, there exists the associated atlas of bundle coordinates ( x \ y') of Y such that every leaf of this foliation is locally generated by the equations yl = const., and the transition functions yl —> 2/"(j/J) are independent of the base coordinates xA [23, 57]. This is called the atJas of constant local trivializations. Two such atlases are said to be equivalent if their union is also an atlas of constant local trivializations. They are associated with the same horizontal foliation. Thus, we come to the following assertion.

PROPOSITION 1.4.1. There is one-to-one correspondence between the flat connections r on a fibre bundle Y —> X and the equivalence classes of atlases of constant local trivializations of Y such that T\ = 0 relative to these atlases. □


1.5 Bundles with symmetries

Subsections: Tangent and cotangent bundles of Lie groups, 46; Principal bundles, 48; The linear frame bundle, 52.

Tangent and cotangent bundles of Lie groups

Let G be a real Lie group with dim G > 0 and flj [flr] its left [right] Lie algebra of ieft-invariant vector fields £i(g) = TLg(£i(e)) [right-invariant vector Selds £r(<?) = TRg(£r{e))} on the group G. Here, e is the unit element of G, while Lg and Rg

denote the action of G on itself on the left and on the right, respectively. Every left-invariant vector field £i(g) [right-invariant vector field £,■(<?)] corresponds to the element v = £i(e) [v = fr(e)] of the tangent space TeG provided with both left and right Lie algebra structures. For instance, given v £ TeG, let Vi(g) and vr(g) be the corresponding left-invariant and right-invariant vector fields. There is the relation

vl(g) = TLgoTR-g\vT{g)).

Let {em = em(e)} [{^m = £m(e)}] denote the basis for the left [right] Lie algebra, and let c^nn be the right structure constants:

[£m,en] = c^nek.

The mapping J H J ' yields the isomorphism

P ■ fll 3 (m >-» £m = -«m 6 0r (1.5.1)

of left and right Lie algebras. For instance, we have

[tm,en} = -c^nek.

The tangent bundle TTQ : TG —» G of the Lie group G is trivial. There are the isomorphisms

ft : TG 3 q - (g = nG(q),TL-1(q)) 6 G x ft>

P r : T G 3 ? « ( j = nG(q),TR-i(q)) 6 G x flr.

The left action L9 of a Lie group G on itself defines its adjoint representation g t-* Adg in the right Lie algebra gT and its identity representation in the left Lie algebra gj. Correspondingly, there is the adjoint representation

e' : e i—► ade'(£) = [e',e], adem(en) = c^nek,

1.5. BUNDLES WITH SYMMETRIES 47

of the right Lie algebra gr in itself. An action

G x Z B (g,z)>-> gz e Z

of a Lie group G on a manifold Z on the left yields the homomorphism

8, 3 e -» & 6 T(Z)

of the right Lie algebra gr of G into the Lie algebra of vector fields on Z such that

Udg(s) =Tgo£eog l (1.5.2)

[100]. Vector fields ££m are said to be the generators of a representation of the Lie group G in Z.

Let g" = T*G be the vector space dual of the tangent space TeG. It is called the dual Lie algebra (or the Lie coalgebra), and is provided with the basis {em} dual of the basis {em} for TeG. The group G and the right Lie algebra gr act on g* by the coadjoint representation

(Ad'g(e'),e)^{e\Adg-1(e)), £* G 9*, e e 0 r , (1.5.3) (adV(e*),e) = -(£*,[£',£]), e' e ft., ad*£m(e") = -cn

mkek.

R e m a r k 1.5.1. In the literature (see, e.g., [2]), one can meet another definition of the coadjoint representation in accordance with the relation

{Ad'g(e'), £) = <£*, Ad g(e)).

An exterior form <f> on the group G is said to be left-invariant [right-invariant] if 4>{e) = L'((j>{g)) \<j){e) = Rm((p(g))]. The exterior differential of a left-invariant [right-invariant] form is left-invariant [right-invariant]. In particular, the left-invariant 1-forms satisfy the Maurer-Cartan equations

<^(£,6') = - ^ ( M ) , e,e'eg,.

There is the canonical Qi-valued left-invariant 1-form

er-TcGBe^eegi


on a Lie group G. The components #p of its decomposition 0; = 0,mem with respect to the basis for the left Lie algebra g( make up the basis for the space of left-invariant exterior 1-forms on G:

tmW = C

The Maurer-Cartan equation, written with respect to this basis, reads

<*r = ^MnA0f.

Accordingly, the canonical gr-valued right-invariant 1-form

9T:TcGBe^e&QT

on the group G is defined. There are the relations

el{vg) = el{TL-g\vg)) = TL-g\vg), Vg G TgG, 8T(v9) = 8T(TR;\v9)) = TR;>(vg), p{0,(v9)) = -TLgoTR9

l9r(v9) = -Adg(er(v9)),

where p is the isomorphism (1.5.1).

Principal bundles

We refer the reader to [100, 170, 192] for the general theory of principal bundles. Let iTP : P —» X be a principal bundle with a real structure Lie group G. There

is the canonical free transitive action

Rc-.PxG-^P, x (1.5.4)

Rg P-> pg, peP, geG,

of G on P on the right. A principal bundle P is equipped with a bundle atlas * p = {(Ua,^)} whose

trivialization morphisms

Va ■ T P W -> Ua x G

obey the condition

pi2oip[oRg = go pr2 o t/£, VpeC.


Due to this property, every trivialization morphism tp£ uniquely determines a local section za:Ua-*P such that

pr2 o v £ o za = e.

The transformation rules for za read

z0(x) = za{x)pa0(x), x e Ua n Up, (1.5.5)

where pa0 are the transition functions of the atlas tfp. Conversely, the family {(Ua,za)} of local sections of P, which obey (1.5.5), uniquely determines a bundle atlas * P of P.

A principal bundle P —» X admits the canonical trivial vertical splitting

a:VP^ PxQt

such that a~l(em) are fundamental vector fields on P corresponding to the basis elements em for the left Lie algebra gj.

Taking the quotient of the tangent bundle TP —* P and the vertical tangent bundle V P of P by the tangent map TRg, we obtain the vector bundles

TGP = TP/G and VGP = VP/G (1.5.6)

over X. Sections of TGP —* X are G-invariant vector fields on P, while sections of VQP —» X are G-invariant vertical vector fields on P. Hence, the typical fibre of VQP —• X is the right Lie algebra gr of the right-invariant vector fields on the group G. The group G acts on this typical fibre by the adjoint representation.

The Lie bracket of vector fields on P goes to the quotients (1.5.6) and defines the Lie bracket of sections of the vector bundles TQP —» X and VGP —* X. It follows that VGP —> X is a fibre bundle of Lie algebras (the gauge algebra bundle in the terminology of gauge theories) whose fibres are isomorphic to the right Lie algebra gr of the group G.

Example 1.5.2. When P = X x G is trivial, we have

VGP = X x TG/G = X x 9 r .

Example 1.5.3. Given a local bundle splitting of P, there are the corresponding local bundle splitting of TGP and VGP. Given the basis {ep} for the Lie algebra gr,


we obtain the local fibre bases {d\,ep} for the fibre bundle TGP —► X and {ep} for the fibre bundle VCP — X. If

t, i;: X -» TGP,

e=^+^%, 77 = r/'d,, + r/%,

are sections, the coordinate expression of their bracket is

&»?] = K"fl^ - V^A)aA + (^a^r - vxdxe + C;,£VK-

Let JlP be the first order jet manifold of a principal bundle P —► X with a structure Lie group G. Bearing in mind that the jet bundle JlP —» P is an affine bundle modelled over the vector bundle

T'X®VP^ P, p

let us consider the quotient of the jet bundle JXP —» P by the jet prolongation JlRg

of the canonical action (1.5.4). We obtain the affine bundle

C = J1P/G-^X (1.5.7)

modelled over the vector bundle

C = T'X ® VGP -> X.

Hence, there is the canonical vertical splitting

VC^CxC. x In the case of a principal bundle P —» X, the exact sequence (1.1.8a) reduces to

the exact sequence

0 — VGP -^TGP — TX -» 0. x (1.5.8)

A principal connection A on a principal bundle P —» X is defined as a section A : P —* JlP which is equivariant under the action (1.5.4) of the group G on P, i.e.,

J1RgoA = AoRg, VpeG. (1.5.9)


Turning now to the quotients (1.5.6), such a connection defines the splitting of the exact sequence (1.5.8). It is represented by the tangent-valued form

A = dxx ® (dx + Ale,), (1.5.10)

where Apx are local functions on X.

On the other hand, due to the property (1.5.9), there is obviously one-to-one correspondence between the principal connection on a principal bundle P —> X and the global sections of the fibre bundle C -* X (1.5.7), called the bundie of principal connections.

Let a principal connection on the principal bundle P —* X be represented by the vertical-valued form A (1.4.4). Then the form

A:P -^T'P®VPl'^$T*P®Ql p

is the familiar g rvalued connection form on the principal bundle P. Given a local bundle splitting (U^,z^) of P, this form reads

A = dp - ~A\dxx ® £„

where Op is the canonical g;-valued 1-form on P, {tp} is the basis of gj, and Apx are

local functions on P such that

AUpg)cq = Al(p)Adg-\eq).

The pull-back z^A of A over U^ is the well-known local connection 1-form

At = -Aqxdxx ® e„ (1.5.11)

where Aqx = A9

xo z^ are local functions on X. It is readily observed that the coefficients A\ of this form are precisely the co

efficients of the form (1.5.10). Moreover, given a bundle atlas of P, the bundle of principal connections C is equipped with the associated bundle coordinates (xx, ax) such that, for any section A of C —» X, the local functions

A\ = a\oA

are again the coefficients of the local connection 1-form (1.5.11). In gauge theory, these coefficients are treated as gauge potentials. We will use this term to refer to sections A of the fibre bundle C —* X.


Let now

Y = (P x V)/G (1.5.12)

be a fibre bundle associated with the principal bundle P —> X whose structure group G acts on the typical fibre V of Y on the left. Let us recall that the quotient in (1.5.12) is defined by identification of the elements (p, v) and (pg,g~lv) for all g 6 G. Briefly, we will say that (1.5.12) is a P-associated Rbre bundle.

As is well known, the principal connection A (1.5.10) induces the corresponding connection on the P-associated fibre bundle (1.5.12). If Y is a vector bundle, this connection takes the form

A = dxx®(dx + AvxI'pdt),

where Ip are generators of the representation of the right Lie algebra gr on V. This is called the associated principal connection or simply a principal connection on Y-+X.

The linear frame bundle

Let X be an n-dimensional connected oriented manifold. Let

nLX ■ LX —> X

be the principal bundle of oriented linear frames {sa} in the tangent spaces to X (or simply the frame bundle). Its structure group is GL+(n,R). The frame bundle is associated with the tangent bundle TX and with the cotangent bundle T'X of X. Given frames {<9M} in the tangent bundle TX, every element {sa} of the frame bundle LX takes the form

SQ — S a C/^i,

where s^a are matrix elements of the group GL+(n, R). These matrix elements constitute the bundle coordinates

\X , s aj, S ° dx* "' on the frame bundle LX. With respect to these coordinates, the canonical action of GL+(n,R) on LX on the right reads

Rg : s*0 H-> s"bgba, g£GL+{n,R).

1.6. COMPOSITE FIBRE BUNDLES 53

The frame bundle LX is equipped with the canonical Rn-valued 1-form

OLX = s^dx" ® <a, (1.5.13)

where {ta} is a fixed basis for Rn, while s"^ are elements of the inverse matrix of o a-

The frame bundle, like tensor bundles, admits the canonical lift of any diffeo-morphism / of its base X to the automorphism

/ : ( x A , 5 A0 ) ^ ( / A ( x ) , ^ / V a ) .

These automorphisms and the corresponding automorphisms of associated bundles are called generai covariant transformations or holonomic automorphisms. For instance, in the case of the tangent bundle TX, the holonomic automorphisms f = Tf are the tangent maps to the diffeomorphisms / . Generators of general covariant transformations of tensor bundles are the canonical lifts r (1.2.1) of vector fields r on X.

1.6 Composite fibre bundles

Let us consider the composition of fibre bundles

Y -> E -► X, (1.6.1)

where

HYT. - Y -» E (1.6.2)

and

7T*-> v * ? , * X (1.6.3)

are fibre bundles. This is called a composite fibre bundle. The composite fibre bundle (1.6.1) is provided with an atlas of fibred coordinates (xx,am, y'), where (xM,(rm) are fibred coordinates on the fibre bundle (1.6.3) and the transition functions am —► am(xx,ak) are independent of the coordinates yx.

The following two assertions on composite fibre bundles are useful in applications to field theory and mechanics [57, 159].


PROPOSITION 1.6.1. Given a section h of the fibre bundle E -» X and a section sE

of the fibre bundle Y —» E, their composition

s = S£ o /i (1.6.4)

is a section of the composite fibre bundle Y —* X (1.6.1). Conversely, every section s of the fibre bundle Y —» X is the composition (1.6.4) of the section h = nys o s of the fibre bundle E —► X and some section s^ of the fibre bundle Y —► E over the submanifold h(X) C E. □

PROPOSITION 1.6.2. Given a composite fibre bundle (1.6.1), let h be a global section of the fibre bundle E —» X. Then the restriction

Yh = h'Y (1.6.5)

of the fibre bundle Y —> E to h(X) C E is a subbundle

ih ■ YK ^ Y

of the fibre bundle Y -> X. a

Let us consider the jet manifolds J ' E , J^Y, and JXY of the fibre bundles E —» X, Y —> E and Y —» X, respectively. They are parameterized by the coordinates

( „A _.m _.fn\ (x\<7m,2A&,24), ( x ^ . a " 1 , ? / 1 , ^ , ^ ) ,

respectively. There is the canonical map

p : J ' E x j ' y —» j ' y , r E v (1.6.6)

2/w = 2/>r+£-In particular, let

^ = dxx ® {dx + A\di) + dam ® (9m + 4,3,) (1.6.7)

be a connection on the fibre bundle Y —* E, and

r = dxx ® {dx + r?dm)

a connection on the fibre bundle E —» X. Then the connection

B = p o {A o TT* x r o TT1) = dxx ® \dx + r7<9m + (>i^r^ + A'x)d,} (1.6.8)

1.6. COMPOSITE FIBRE BUNDLES 55

on the composite fibre bundle Y —► X is defined. This is called the composite connection. In brief, we will write

B = A o r .

For instance, let us consider a vector field r on the base X, its horizontal lift IV onto E by means of the connection V and, in turn, the horizontal lift A{Vr) of TT onto Y by means of the connection A. Then A(FT) coincides with the horizontal lift BT of r onto Y by means of the composite connection (1.6.8).

Given a composite fibre bundle Y (1.6.1), there are the following exact sequences of vector bundles over Y:

0 — v E y <-> VY -> Y x VE -» 0, E

(1.6.9a)

o -> y x V*E «-»v*y - > y ? y -»o, E *-

(1.6.9b)

where Vj;y and V£Y are vertical tangent and cotangent bundles of the fibre bundle y —» E, respectively. Every connection A (1.6.7) on the fibre bundle Y —» E determines the splittings

VY = VEy ®A(Y x K E ) , K E

y-a, + amdm = (y> - A^d™)* + am(dm + A^d,), VY = (Y x V"E) © A(V£Y),

yidtf + amdam = y,(dyl ~ A^da™) + [am + A^da™,

(1.6.10a)

(1.6.10b)

of exact sequences (1.6.9a) and (1.6.9b), respectively. Using the splitting (1.6.10a), one can construct the first order differential operator

D: JlY ^T'X®VEY, Y

D = dxx® (y\ - 4 - 4 ^ R (1.6.11)

called the verticai covariant differential, on the composite fibre bundle 7 - > X . This operator can also be seen as the composition

D = pr, o DB : JlY ~* T'X ® VY -> T'X ® VyE , r Y

where DB is the covariant differential (1.4.5) relative to some composite connection (1.6.8), but D does not depend on T.


The vertical covariant differential (1.6.11) possesses the following important property. Let ft be a section of the fibre bundle E —► X and V), the subbundle (1.6.5) of the composite fibre bundle Y —► X, which is the restriction of the fibre bundle Y —► £ to h(X). Then the restriction of the vertical covariant differential D (1.6.11) to JHh(JlYh) C JlY coincides with the familiar covariant differential on Yh relative to the connection

Ah = dxx ® [ft + (A\ndxhm + (A o h)x)di]

on Yh, which is the restriction of the connection A to h(X) in accordance with the commutative diagram

j i y , J2% j i y

Ak ! | A E

Yk — Y

Example 1.6.1. Let r : Y —» JlY be a connection on a fibre bundle Y —» X. In accordance with the canonical isomorphism V J ' Y = JlVY (1.3.8), the vertical tangent map VT : VY —» VJlY to V defines the connection

VT-.VY ^ JlVY, vr = dxx ® (dx + r\d, + djTitfdi), (1.6.12)

on the composite vertical tangent bundle VY —> X. The dual connection on the composite vertical cotangent bundle VY —> X reads

VT : VY - » j ' v y ,

v*r = dxx ® (^ + rift - a»rift#). (1.6.13)

If y —» X is an affine bundle, the connection VT (1.6.12) can be seen as the composite connection (1.6.8) generated by the connection T on Y —► X and the linear connection

r = dxx ® (ft + dX\y>dx) + dy* ® ft (1.6.14)

on the vertical tangent bundle VY —» F . •

Chapter 2

Geometry of Poisson manifolds

This Chapter is devoted to the basic geometric structures on a manifold, which we meet in mechanics. We start from a Jacobi structure whose particular case is a Poisson structure.

Throughout this Chapter, by Z, unless otherwise stated, is meant a Ar-dimensional manifold with coordinates (zx).

For the sake of convenience, the Schouten-Nijenhuis bracket [., .]SN is denoted simply by [.,.].

2.1 Jacobi structure

A Jacobi bracket (or a Jacobi structure) on a manifold Z is defined as a bilinear map

D°(Z) x D°(Z) 9 (f,g) - {/,<?} G O0(Z)

on the vector space D°(Z) of real functions on Z. This map, by definition, satisfies the following conditions:

(Al) {g,f} = ~{f,g} (skew-symmetry), (A2) {/, {g, ft}} + {(?, {ft, / } } + {ft, {/, <?}} = 0 (Jacobi identity), (A3) the support of {/, g} is contained in the intersection of the supports of /

and g. A manifold Z endowed with a Jacobi structure is called a Jacobi manifold. A Jacobi bracket provides the space O0(Z) with a structure of a Lie algebra

because it is expressed by a bidifferential operator of not more than first order in each of its arguments [71, 98].

57

58 CHAPTER 2. GEOMETRY OF POISSON MANIFOLDS

PROPOSITION 2.1.1. [98, 118, 123]. Every Jacobi bracket on a manifold Z is uniquely defined in accordance with the relation

{/. 9} = w(rf/> dg) + u\ {fdg - gdf) (2.1.1)

by a pair (tu, u) of a bivector field w and a vector field u on Z such that

[u, w] = 0, [w, w] — 2u A w. (2.1.2)

□

Example 2.1.1. Taking w = 0 in (2.1.1), we find that every vector field u on a manifold Z provides the Jacobi bracket

{f,g}=u\(fdg-gdf).

The relations (2.1.2) are obviously satisfied. •

Example 2.1.2. The Jacobi bracket (2.1.1) with u = 0 is said to be a Poisson bracket. According to Proposition 2.1.1, a bivector field w on a manifold Z yields a Poisson bracket if it meets the condition

[w, w] = w^d^^dx, A dX2 Adx,=0-

It is called a Poisson bivector Geld. •

Let us consider the following examples of Jacobi manifolds which are not Poisson ones.

Example 2.1.3. Odd-dimensional contact manifolds are Jacobi manifolds in accordance with Proposition 2.2.6. •

Example 2.1.4. Let fi be a non-degenerate 2-form and cj> a closed 1-form on an even-dimensional manifold Z such that

dfi = <j> A fi.

The triple (Z, fi, <j>) is called a locally conformally symplectic manifold. This is a Jacobi manifold characterized by the bivector field wn (1.2.35a) and the vector field u = {nW<j>) [98, 118, 123]. If <t> = 0, we have a symplectic manifold. •

2.1. JACOBI STRUCTURE 59

Let Z\ and Z2 be Jacobi manifolds. A manifold map Q : Z\ —> Z2 is said to be a Jacobi morphism if, for every pair (f,g) of functions on Z2, we have

{f°Q,g°e}i = {f,g}2°e-

In particular, a vector field v on a Jacobi manifold (Z;w,u) is the generator of a local 1-parameter group of Jacobi morphisms (or an infinitesimal Jacobi morphism) if and only if

L„w = 0, L„u = 0.

DEFINITION 2.1.2. Given a function / £ D°{Z) on a Jacobi manifold (Z;w,u), the vector field

d,^w\df) + }u

is called the Hamiltonian vector Held for / . D

We have the relation

[*/.».] = w (2.1.3)

It follows that the map / •-» 1?/ is the Lie algebra homomorphism. The values of all Hamiltonian vector fields at all points of Z constitute the char

acteristic distribution T on the Jacobi manifold (Z;w,u). A glance at the formula (2.1.3) shows that this distribution is involutive, but it has different dimensions at different points z € Z in general. Therefore, this distribution is not a subbundle of the tangent bundle TZ and, consequently, is not a distribution as that in Section 1.2. A Jacobi manifold Z is said to be transitive if its characteristic distribution coincides with the tangent bundle TZ. Transitive Jacobi manifolds are proved to be either locally conformally symplectic manifolds, if dim Z is even, or the contact ones, if dim Z is odd [98, 118].

T H E O R E M 2.1.3. [44, 98]. The characteristic distribution on a Jacobi manifold is completely integrable in the sense of Sussmann [173] and defines on Z a singular Stefan foliation [171]. Each leaf of this foliation is endowed with a unique Jacobi structure such that its canonical injection into Z is a Jacobi morphism. □

The following definition generalizes for Jacobi manifolds the notion of coisotropic and Lagrangian submanifolds of a Poisson manifold [83] (cf. Definition 2.3.5).


DEFINITION 2.1.4. Let (Z\w,u) be a Jacobi manifold, with the characteristic distribution T, and N a submanifold of Z. The submanifold N is said to be:

• coisotropic if

w«(Ann7yV)C T*N> z€N,

where AnnTzN C T'Z is the annihilator o(TzN, and

• Lagrangian if

wl(AnnTzN)=TINnTz, ze N.

D

Different Jacobi structures can lead to the same characteristic distribution as follows. Let (Z\ w, u) be a Jacobi manifold and a a nowhere vanishing function on Z. Let us consider the bivector field wa and the vector field ua on Z given by

wa = aw, ua = wi(da) + au. (2.1.4)

Then the pair (wa,ua) (2.1.4) is a Jacobi structure on Z. It is called conformally equivalent to the Jacobi structure {w,u) because of the relation

{/,0}a = ~{af,ag}. a

It is readily observed that the Hamiltonian vector field for a function / with respect to the Jacobi structure (u», u) is also the Hamiltonian vector field for the function i / with respect to the conformally equivalent Jacobi structure (2.1.4). It follows that conformally equivalent Jacobi structures on a manifold Z define the same characteristic distribution on Z.

A Jacobi morphism from a Jacobi manifold to a conformally equivalent Jacobi manifold is said to be a conformal Jacobi morphism. Such a morphism preserves the characteristic distribution on a Jacobi manifold.

In particular, a vector field v on a Jacobi manifold (Z; w, u) is an infinitesimal conformal Jacobi isomorphism if and only if there exists a function b on Z such that

L„w = bw, hvu = w'(6) + bu.

2.2. CONTACT STRUCTURE 61

2.2 Contact structure

We will consider contact structures on odd-dimensional manifolds. By a contact structure is meant a strictly contact structure in the sense of [116].

DEFINITION 2.2.1. An exterior 1-form 6 on a (2m + l)-dimensional manifold Z is said to be a contact form if

8 A (d6)m 5* 0 (2.2.1)

everywhere on Z. O

If 6 is a contact form, so is ad where a is a nowhere vanishing function on Z. The pair (Z, 8) (or equivalently the pair (Z,a8)) is termed a contact manifold (we refer the reader to [15, 116] for the details of terminology).

If a manifold admits a contact structure, this manifold is orientable, and has the volume element (2.2.1). It follows from (2.2.1) that the exterior differential d9 of a contact form 8 is a presymplectic form of rank 2m.

There is the following variant of well-known Darboux's theorem [116].

T H E O R E M 2.2.2. Every point z of a (2m + l)-dimensional contact manifold (Z, 8) has an open neighbourhood U which is the domain of a coordinate chart (z°,..., z2m) such that the contact form 8 on U has the local expression

m 8 = dz°- 5>m + ,dz\ (2.2.2)

These coordinates are called the Darboux coordinates. □

A contact form 8 on a manifold Z defines the isomorphism

\>(u)d=u\d8+(u\8)6, u € T(Z), (2.2.3)

of the C°°(Z)-module T(Z) of vector fields on a manifold Z to the C°°(Z)-module £>' (Z) of exterior 1-forms on Z [83]. This isomorphism can be extended to a mapping from the exterior algebra T,(Z) of multivector fields to the exterior algebra D'(Z) of exterior forms by putting

b(«i A ••• Au r ) d =b(ui ) A • • • Abfa), u{ 6 T(Z). (2.2.4)


PROPOSITION 2.2.3. Let $ be a contact form on a manifold Z. There exists the unique nowhere vanishing vector field

E = b~1{9) (2.2.5)

on Z, called the Reeb vector field, such that

E\6=l, E\de = Q

[116]. It is readily observed that, relative to the local Darboux coordinates, this vector field reads E = d0. O

Example 2.2.1. Let M be a manifold with coordinates (ql) and Z the odd-dimensional manifold R x T'M, equipped with the coordinates {t,q',Pi = ft). The manifold R x T'M is well known to admit the contact form

6 = p,dq{ - dt. (2.2.6)

For the sake of convenience, this contact form is chosen to differ in minus sign from the Darboux one (2.2.2). •

Example 2.2.2. To generalize Example 2.2.1, let Q —» R be a fibre bundle, with fibred coordinates (t, q'), and Z = V'Q the vertical tangent bundle of Q, equipped with the coordinates (t,q\pl = q,). As mentioned above, V'Q is a phase space of time-dependent mechanics. Given a connection

r = dt g> {dt + Pdi)

on the fibre bundle Q —► R, the manifold V'Q is provided with the contact form

6r = pi{dq{ - T'dt) - dt

(see Proposition 5.2.11 below). The corresponding Reeb vector field reads

E = -{dt + Fdi-pjdiFdx).

Since a connection T on a fibre bundle Q —> R is flat, it defines the corresponding atlas of local constant trivializations such that, with respect to this atlas, T = dt®dt

and the contact form 9r is brought into the form (2.2.6). •


Given a (2m + l)-dimensional contact manifold (Z,6), the tangent bundle TZ of Z admits the splitting

TZ = Kerd6®KeiO,

where KerdO is the 1-dimensional vector subbundle generated by the Reeb vector field E [116]. In particular, every vector field u on Z is decomposed in a unique fashion into

u = (u\e)E+(u-(u\6)E).

By duality, the cotangent bundle T'Z of the contact manifold Z is found to have the splitting

T'Z = 6 e Ker E,

where 0 is the 1-dimensional vector subbundle generated by the contact form 6. As a consequence, every 1-form <j) on the contact manifold Z is decomposed into

4>=(E\4>)9 + (4>-(E\4>)e)-

There is the corresponding splitting of the isomorphism (2.2.3). This sends Kerd6 onto © and Ker 6 onto KerE. The restriction of b to Ker 8 coincides with the morphism (—d@f (1.2.33) defined by the presymplectic 2-form d8.

The 2m-dimensional subbundle Ker0 of the tangent bundle TZ of a contact manifold Z is called the contact distribution on Z. Any contact form a9, where a is a nowhere vanishing function on Z, defines the same contact distribution on Z. As is well known, there exist integral submanifolds of the contact distribution Ker 6 of dimension m, but none of higher dimension [15]. It is readily observed that the contact forms 6 and a6 have the same contact distribution.

DEFINITION 2.2.4. A submanifold N oia. (2m + l)-dimensional contact manifold Z is said to be a Legendre submanifold if it is an m-dimensional integral submanifold of the contact distribution on Z. □

Let (Zu0i) and (Z2,02) b e contact manifolds. A manifold map Q : Zx -» Z2 is said to be a contact transformation if

e'02 = o0i


where a is a nowhere vanishing function on Z [116]. A contact transformation preserves the contact distribution, but not a contact form in general. We consider especially the contact automorphisms of a contact manifold (Z,9), which keeps the contact form 9.

By definition, the Reeb vector field E (2.2.5) is the generator of a local 1-parameter group of local contact automorphisms (or an infinitesimal contact automorphism) of the contact manifold {Z,9), i.e.,

LE9 = 0.

It is readily observed that, if a vector field u on Z is an infinitesimal contact automorphism of the contact manifold (Z, 6), then u is a Hamiltonian vector field with respect to the presymplectic form d6 on Z, i.e., the exterior form u\d0 is exact (see Definition 2.5.2 below). The converse assertion is the following.

PROPOSITION 2.2.5. Given a contact manifold (Z, 6), let ds be a Hamiltonian vector field on Z for a function / with respect to the presymplectic form d.9, i.e.,

d,\d9 = -df.

Then the vector field

#/ = [f-{*/m]E+*f (2.2.7)

is an infinitesimal contact automorphism of the contact manifold Z, i.e.,

Lhe = o.

□ Proof. It follows from direct computation. QED

Note that the vector field (2.2.7), like dj, is a Hamiltonian vector field for the function / with respect to the presymplectic form d9.

A contact manifold is a Jacobi one as follows.

PROPOSITION 2.2.6. [123]. Each contact form 9 on a (2m+ l)-dimensional manifold Z yields the associated Jacobi bracket (2.1.2) on Z which is defined by the Reeb vector field E (2.2.5) and the bivector field w such that

w>(4>)\9 = 0, wt(<t>)\d9=-(<f,-(E\(f>)9) (2.2.8)


for any 1-form <f> on Z. □

Relative to the local Darboux coordinates for the contact form 9, the above-mentioned Jacobi bracket (2.1.2) reads

»-B£+»-"!■> * &=«. — - » • m

{/.»} = E(9m+,ffa,/ - a^ifdtg) + ({9}d0f - [f}dog),

where m

i= l m

[g}=^zm^dm+ig~g. i= i

In particular, let (Z,9) be a contact manifold and {w,u) the associated Jacobi structure on Z. A submanifold N of Z is a Legendre submanifold of the contact manifold (Z, 9) if and only if it is the Lagrangian submanifold of the Jacobi manifold (Z;w,u) [83]. Note that, in this case, the characteristic distribution of the Jacobi structure coincides with TZ, and we have

w{(AnnTzN) = TzN.

Let (Z, 0) be a contact manifold, v a vector field on Z, and b a function on Z. One can show [83] that the pair (v, b) is an infinitesimal conformal Jacobi morphism if and only if the pair (t>, —6) is an infinitesimal contact transformation, i.e.,

Lv9 = -66>.

Example 2.2.3. Let M be a manifold with coordinates (q') and Z the odd-dimensional manifold K x T"M, equipped with the holonomic coordinates (£, ql,Pi = Qi). This manifold is provided with the contact form (2.2.6). Hence, it is a Jacobi manifold where

w = -{di + p,dt) Ad' , E = -dt. (2.2.9)

•


2.3 Poisson structure

In accordance with Example 2.1.2, a bivector field

w = -w^dp A dv

on a manifold Z defines a Poisson bracket (or a Poisson structure)

{f,g}=w(df,dg) = w^dllfdl/g (2.3.1)

on Z if and only if

[w, w] = 0, w"k,d^wX2X3 + w^dpW*3* + w^d^w^2 = 0.

Besides the conditions (Al - A3), the Poisson bracket (2.3.1) satisfies the Leibniz rule

{h,fg} = {h,f}g + f{h,g}. (2.3.2)

A manifold Z endowed with a Poisson bivector field w is termed a Poisson manifold, while a real vector space D°(Z) of functions on Z forms a Poisson algebra, i.e., O0(Z) is a real associative and commutative algebra with unit with respect to pointwise multiplication, a real Lie algebra with respect to the Poisson bracket, and these two operations intertwine via the Leibniz rule (2.3.2).

Example 2.3.1. Each manifold admits a zero Poisson structure characterized by the zero bivector field w = 0. •

Example 2.3.2. Let u and v be vector fields on a manifold Z such that [u,v\ = 0 everywhere on Z. It is readily observed from the relation (1.2.12) that w = u A v is a Poisson bivector field. •

The Poisson structure (2.3.1) defined by a Poisson bivector field w is said to be regular if the associated morphism w' : T'Z —* TZ (1.2.31) is of constant rank. Hereafter, by a Poisson structure we mean only a regular Poisson structure.

The regular Poisson structure (2.3.1) defined by a Poisson bivector field w is said to be non-degenerate if the associated morphism w' : T'Z —» TZ (1.2.31) is of maximal rank. A non-degenerate Poisson structure can exist only on an even-dimensional manifold.

2.3. POISSON STRUCTURE 67

Example 2.3.3. A non-degenerate Poisson manifold (Z, w) is a symplectic manifold with the symplectic form fiw (1.2.35b) (see Proposition 2.4.4 below). •

Two functions / and g on a Poisson manifold (Z, w) are said to be in involution with each other if their Poisson bracket {/, g} equals zero. A function C on a Poisson manifold (Z, w) is called a Casimir function if it is in involution with any function on Zy i.e.,

{<?,/} = 0, V / 6 0° (Z) .

Casimir functions make up the centre of the Poisson algebra (D°(Z),u/). For instance, if a Poisson structure is non-degenerate, all the Casimir functions (on a connected manifold Z) are constant.

Let {Z\,wi) and {Z<i,wi) be Poisson manifolds. A manifold map g : Z\—> Z2 is said to be a Poisson morphism if

{f°Q,g°Q}i = {f,g}2°Q, Vf,geQ°(Z2),

or

W2 = TQOWI,

where Tg is the tangent map to g. If g is a Poisson morphism, the rank of W\{z) is grater or equal to that of W2{g{z)). If a Poisson morphism g is an immersion, tui(z) and w2{g{z)) are of equal rank.

A vector field u on a Poisson manifold [Z, w) is the generator of a local 1-parameter group of local Poisson automorphisms (or an infinitesimal Poisson automorphism) of (Z, w) if and only if

\JUW = [u, w] = 0. (2.3.3)

Such a vector field is called canonical Note that there are no pull-back or push-forward operations of Poisson structures

by manifold maps in general. The following assertion deals with Poisson projections [181].

PROPOSITION 2.3.1. Let (Z, w) be a Poisson manifold and i : Z - » V a projection. The following properties are equivalent:

• for every pair of functions (/, g) on Y and for each point j i e V , the restriction of the function {/ o n,g o n} to the fibre TT~1(y) is constant;


• there exists a Poisson structure w' on Y for which 7r is a Poisson morphism, i.e.,

{fon,goTr} = {f,g}' OTT.

□ One says that the Poisson structure w' in Proposition 2.3.1 is coinduced by the

projection IT. The direct product Z x Z' of Poisson manifolds (Z, w) and {Z',w') has the

Poisson structure defined by the bivector field w + w' on Z x Z'. It is called the direct product of Poisson structures. Obviously, the projections p ^ and pr2 are Poisson morphisms.

One can speak of a Poisson submanifold (Z1, w') of a Poisson manifold (Z, w) if Z' is a submanifold of Z and the natural injection Z' —> Z is a Poisson morphism.

DEFINITION 2.3.2. Given a real function / on a Poisson manifold (Z, w), the image

0/ = «*'(#), 0, = wTdJdu,

(2.3.4)

of its exterior differential d/ by the morphism wl (1.2.31) is called the Hamiitom'arj vector Geld for / with respect to the Poisson structure w. O

Example 2.3.4. The Hamiltonian vector field for a Casimir function equals zero.

It is readily observed that each Hamiltonian vector field is also a canonical vector field.

The Hamiltonian vector field $/ for a function / , by definition, obeys the relations

#f\dg = {f,g},

[0/.*«] = W

V<? e D°(Z),

(2.3.5)

Then it follows from (2.3.5) that / i—»• i?y is the Lie algebra homomorphism. In particular, we have

tf/Jd/ = 0


and, consequently, dj is tangent to the leaves of the singular foliation of the level surfaces of the function / at its regular points, where df ^ 0.

Remark 2.3.5. It is clear that not every vector field on a Poisson manifold Z is a Hamiltonian vector field. Given a vector field u on a manifold Z, one can try to construct a Poisson bracket and to find a function on Z such that u would be a Hamiltonian vector field for this function. A closely related subject is the inverse problem in Hamiltonian mechanics, which consists in trying to represent a given system of first order dynamic equations

zx = ux{z) (2.3.6)

on a manifold Z as the Hamilton equation with respect to some Poisson structure, called the generating Poisson structure on Z. In Section 3.2, we will present a general technique of constructing a generating Poisson structure for any dynamic equation (2.3.6) at least locally [59, 82]. •

The values of all Hamiltonian vector fields at all points of a Poisson manifold Z constitute an even-dimensional characteristic distribution T on the Poisson manifold (Z,w). By virtue of the relation (2.3.5), this distribution is involutive and, consequently, completely integrable.

T H E O R E M 2.3.3. [181]. A Poisson structure induces a symplectic structure on leaves of the characteristic foliation on Z, called a symplectic foliation. □

In particular, if a Poisson structure is non-degenerate, its characteristic distribution T coincides with TZ, and Z is a symplectic manifold.

Remark 2.3.6. It should be recalled that we are considering a regular Poisson structure, and the corresponding symplectic foliation is non-singular. •

A 2m-dimensional symplectic foliation on a /c-dimensional Poisson manifold Z admits the adapted coordinates (. . . , z 2 m + 1 , . . . , zk) described in Corollary 1.2.4. Moreover, one can choose these coordinates in such a way to bring the Poisson structure into the following canonical form [181, 186].

PROPOSITION 2.3.4. For any point 2 of a Poisson manifold, there exists a coordinate system

( < ?i , . . . , < r , p 1 , . . . , P m , z 2 m + i , . . . , 2

f c ) (2.3.7)


in a neighbourhood of z such that

{Pi,9>}=Sh {q\<?} = K f t } = {q\zA} = {Pi,zA} = {Az8} = 0.(2.3.8)

D

The coordinates (2.3.7) are called canonical coordinates. Given canonical coordinates, one says that the coordinates q' and p< are canonicaiiy conjugate.

In canonical coordinates (2.3.7), the Poisson bracket (2.3.1) takes the form

w = d' A di,

{f,9}=d'fd,g-djd\

while the Casimir functions are arbitrary functions of the coordinates zA. Accord ingly, the Hamiltonian vector field for a function / reads

■df = <97d, - djd'.

Let (Z, w) be a Poisson manifold with the characteristic distribution T, and N a submanifold of Z. The familiar definitions of coisotropic and Lagrangian sub-manifolds of a symplectic manifold are generalized for a Poisson manifold as follows [181, 187].

DEFINITION 2.3.5. The submanifold N is said to be

• coisotropic if

w»(AnnTA0CTAf,

• Lagrangian if

™"(Ann7W) = TATnT.

a

The following theorems generalize for Poisson manifolds the corresponding results on symplectic isomorphisms (see [178]).

THEOREM 2.3.6. [187]. Let $ : Z\ —» Z2 be a manifold morphism between Poisson manifolds (Z\,w{) and (Z2,11)2). This is a Poisson morphism if and only if its graph

A* = { ( z , $ ( z ) ) z e Zi} C Z, x Z2


is a coisotropic submanifold of the Poisson manifold [Z\ x Z^, W\ — w-i). □

T H E O R E M 2.3.7. [67]. Let u be a vector field on a Poisson manifold (Z, w). (i) In accordance with the equality (1.2.14), the tangent lift w (1.2.9) of the

Poisson bivector field w is a Poisson bivector field on the tangent bundle TZ of Z. (ii) A vector field v. is an infinitesimal Poisson automorphism of the Poisson

manifold (Z, w) if and only if u(Z) is a Lagrangian submanifold of the Poisson manifold (TZ,w). □

A Poisson structure on a manifold Z can be defined entirely by its symplectic foliation instead of by a Poisson bivector field [181].

T H E O R E M 2.3.8. Let Z be a manifold and F a foliation on Z such that every leaf Ft of F is endowed with a symplectic structure. Given a function / 6 Q°(Z) on Z, let ■d'j be a Hamiltonian vector field on a leaf F t for the function / \pL with respect to the symplectic structure on this leaf. Put

0f(')=rf{z), ze FL.

If -df is a differentiable vector field on the manifold Z for arbitrary function / , thei Z has a unique Poisson structure given by the Poisson bracket

{/,</}(*) =*/J<fo

whose symplectic foliation is F. □

Example 2.3.7. Let us consider the product Z = R x T'M with the coordinates if-, 9%Pi)- If the cotangent bundle T'M is provided with the canonical symplectic structure (see Example 2.4.2 below), the fibres of the projection pr, : Z —» R constitute the symplectic foliation on Z which satisfies the conditions of Theorem 2.3.8. Then, the manifold K x T'M is provided with the Poisson structure given by the Poisson bivector field

w = 9' A di. (2.3.9)

Obviously, this is the product of the zero Poisson structure on K and that defined by the canonical symplectic structure on the cotangent bundle T'M. •


Example 2.3.8. To generalize Example 2.3.7, let us consider a fibre bundle IT : Z —► X, parameterized by fibred coordinates (xx;q',pl), whose typical fibre V is a symplectic manifold with the Poisson bracket

{f,g} = d'fdtg-dxfd\ (2.3.10)

Each trivialization chart

V{ : rr-^Uf) - U( x V

is endowed with the Poisson structure described in Example 2.3.7 with the Poisson bracket (2.3.10). To generalize construction of symplectic vector bundles [116], we will say that Z —» X is a symplectic bundle if it admits a bundle atlas $ whose transition functions

pK(x) : {x} x V - {x} x V, x e c/f n u(,

provide isomorphisms of the symplectic manifold V. Then the fibre bundle 2 - > X is equipped with the Poisson structure given by the Poisson bracket (2.3.10).

For instance, let Z = V'Q be the vertical cotangent bundle of a fibre bundle Q —► K. The typical fibre of V'Q is the cotangent bundle T'M of the typical fibre M of the fibre bundle Q —> R. The fibre bundle V'Q —> R is a symplectic bundle, provided with the Poisson bracket (2.3.10) written with respect to the coordinates (£, q%, p() on V'Q. Indeed, it is readily observed that this Poisson bracket is invariant under all holonomic coordinate transformations of the vertical cotangent bundle V'Q. This is the canonical Poisson structure on a phase space V'Q of time-dependent mechanics. In Section 5.1, we will obtain it in a different way.

The notion of a symplectic bundle is naturally extended to that of a Poisson bundle. One should distinguish this notion from that of Jacobi bundles in the sense of Kirillov [98, 123] where a Jacobi bracket is generalized for sections of a 1-dimensional linear bundle, called a Jacobi bundle. •

Example 2.3.9. The Lie coalgebra g* of a Lie group G is provided with the canonical Poisson structure, called the Lie-Poisson structure (see [2, 116, 181]). This Poisson structure is given by the bracket

{f,g}^(e'Adf(e'),dg(e')}), /.seOV). (2.3.11)

2.4. SYMPLECTIC STRUCTURE 73

on g*, where df(e'),dg(e') 6 flr since we can regard them as linear mappings from Te'2' = Q* to R. The Lie-Poisson bracket (2.3.11) is defined by the Poisson bivector field

wmn = CkmnZky

where c£,n are the structure constants of the Lie algebra 0 r and zk are coordinates on g* with respect to a basis {ek}. We have the coordinate expression

{f,9}=ckmnzkV»fdng.

There is the following well-known theorem.

T H E O R E M 2.3.9. [186]. The symplectic leaves of the Lie-Poisson structure on the coalgebra g* of a connected Lie group G are the orbits of the coadjoint representation (1.5.3) of G o n g * . D •

2.4 Symplectic structure

Non-degenerate Poisson manifolds are symplectic manifolds (see Proposition 2.4.4 below).

DEFINITION 2.4.1. A non-degenerate exterior 2-form Q on a manifold Z is said to be symplectic if it is closed, i.e., dQ. = 0. A manifold equipped with a symplectic form is called a symplectic manifold. □

Every symplectic manifold (Z, £l) is 2m-dimensional. It is orientable, and

1 m V = — r A f i

m! (2.4.1)

is the volume element on Z.

DEFINITION 2.4.2. A morphism (, : Z -> Z' of a. symplectic manifold (Z, Q) to a symplectic manifold (Z', fi') is called a symplectic morphism if Q = C*ft'. □

By very definition, a symplectic morphism is an immersion.


Remark 2.4.1. One should distinguish a symplectic morphism from a symplecto-morphism in the terminology of [116]. The latter is a diffeomorphism. •

A vector field u on a symplectic manifold (Z, Q) is said to be a generator of a local 1-parameter group of symplectic automorphisms (or an infinitesimal symplectic automorphism) of a symplectic manifold (Z, fi) if and only if

L u n = 0.

Such a vector field is called canonical

Example 2.4.2. Let M be a manifold with coordinates (ql) and TT.M : T'M —> M its cotangent bundle, equipped with the holonomic coordinates (q',Pi). The cotangent bundle T'M is endowed with the canonical Liouville form

9 = p,dq\

This form is defined by the condition

v\6{p) =Tn.M(v)\p, Vu € TPT'M, p € T'M.

Its exterior differential is the canonical symplectic form

n = dd = dpxA dq* (2.4.2)

on T'M. All holonomic coordinates on the cotangent bundle T'M are canonical for this form.

It should be emphasized that the canonical symplectic form (2.4.2) is not a unique symplectic form on the cotangent bundle T'M. For every closed 2-form </> on a manifold M, the form

n* = n + ir;M<j> (2.4.3)

is also a symplectic form on T'M [116]. •

The canonical symplectic form (2.4.2) plays the fundamental role in view of Darboux's theorem [116].

T H E O R E M 2.4.3. Let (Z, Q) be a symplectic manifold. Each point of Z has an open neighbourhood U which is the domain of a canonical coordinate chart

(q\...,qm,pl,...,pm)


such that the symplectic form fi on U is given by the coordinate expression (2.4.2). D

This theorem is an immediate consequence of Proposition 2.3.4 and Proposition 2.4.4 below.

We have the following relationship between the non-degenerate Poisson structures and the symplectic ones.

PROPOSITION 2.4.4. [117]. On a 2m-dimensional manifold Z, there is one-to-one correspondence between the symplectic forms SI and the non-degenerate Poisson bivector fields w given by the equalities (1.2.35a) - (1.2.35b). □

The canonical coordinates for a symplectic form Q are also canonical for the corresponding Poisson bivector field w, and we have

fi = dp, A dql, w^&Adi.

With respect to these coordinates, the bundle isomorphism fr (1.2.33) reads

0k : v'di + Vidx i—► -Vidq' + v'dplt

while the isomorphism wl = (£r) (1.2.31) is

wl : Vidq' + vldpi i—» vxdx — Vid\

In view of Proposition 2.4.4, a Poisson structure is termed sometimes a cosym-plectic structure (one should distinguish this notion from the cosymplectic structure in Remark 4.8.8). A local structure of an arbitrary Poisson manifold in relation to a symplectic one is described by the following two theorems [181, 186].

THEOREM 2.4.5. Any point of a Poisson manifold has an open neighbourhood which is Poisson equivalent to a product of a manifold with the zero Poisson structure and a symplectic manifold. □

Let (Z,w) be a Poisson manifold. Its symplectic realization is said to be a symplectic manifold (Z1, £3) together with a projection Z' —> Z such that the Poisson structure w on Z is coinduced from the Poisson structure wn on Z'.

T H E O R E M 2.4.6. Any point of a Poisson manifold has an open neighbourhood which is realizable by a symplectic manifold. □


In local Darboux coordinates, this symplectic realization is seen as follows. The Poisson structure given by the Poisson bracket (2.3.8) with respect to the canonical coordinates is coinduced from the symplectic structure given by the symplectic form

Q = dpi A dq' + dlfy A dzA

with respect to the coordinates

(q , - - , q ,P\,--,Pm,Z , • • • , - Z , Z . • • • > 2 )

by the projection Cn1 nm T) n 7 2 m + l k ? 2 m + l yk\ (q,...,q ,pi,,,. ,Pm,Z ,...,z,z , . . . ,z / t—»

l<? , • • • , ? i P l , - ■ -,Pm,Z ,■ ■ ■ ,Z )■

Example 2.4.3. Let V'Q be the vertical cotangent bundle of a fibre bundle Q —» R as in Example 2.3.8. It is provided with the Poisson structure given by the Poisson bracket

{f,9} = d'fd,g-d,fd'g.

Its symplectic realization is the cotangent bundle T'Q of Q endowed with the canonical symplectic form

dE = dp Adt + dp, A dq', (2.4.4)

written with respect to the holonomic coordinates (t, q',p,Pi) (see Section 5.1). •

Let (Zi,fti) and (Z2, £h) be symplectic manifolds equipped with the associated Poisson structures w-i and w2, respectively. The map g : Z\ —> Z2 is a Poisson isomorphism if it is a symplectic isomorphism. Accordingly, a vector field on a symplectic manifold is canonical if and only if it is canonical for the associated Poisson structure. However, if dim Z\ > dim Z2 and g is a Poisson morphism, then this is not a symplectic morphism, i.e.,

w7=Tgowi, Qi ^ e'n2.

The notion of a Hamiltonian vector field on a Poisson manifold is restated for a symplectic one as follows.

DEFINITION 2.4.7. A vector field d on a symplectic manifold (Z, Q) is said to be locally Hamiltonian [Hamiltonian] if the exterior form tfjft is closed [exact]. □

As an immediate consequence of Definition 2.4.7, we find the following.


• A vector field ■d on a symplectic manifold (Z, il) is locally Hamiltonian if and only if it is an infinitesimal symplectic automorphism, i.e.,

USl = d(ti\n) = 0.

• A vector field i ) o n a symplectic manifold (Z, fi) is Hamiltonian if and only if it is a Hamiltonian vector field in accordance with Definition 2.3.2 for some function / on Z, and then

d;\U= -df. (2.4.5)

• The Poisson bracket defined by a symplectic form $1 reads

{/.g} = < W j n -

In the literature (see, e.g., [2]), one may meet a definition of Hamiltonian vector fields, which differs in the minus sign from (2.4.5).

Example 2.4.4. Let us consider the cotangent bundle T'M equipped with the canonical symplectic form fl (2.4.2). Let u = u'dx be a vector field on M. Then its canonical lift

u = uldi — PjdtvSd'

(1.2.3) onto T'M is a Hamiltonian vector field. We have

u]Q = —d(u'pi).

Let us turn now to submanifolds of a symplectic manifold. Let (Z, Q) be a 2m-dimensional symplectic manifold and w the corresponding

Poisson bivector field on Z. Let ./V be an n-dimensional submanifold of Z. The set

O r t h n T A r = (J {v e TZZ : v\u\Q = 0, Vu € TZN}, (2.4.6)

is called the orthogonal of TN relative to the symplectic form Q. or simply Q-orthogonal space ofTN.


Recall the useful formulas

Orthn(Orthn7W) = TN, Ann TAT = ft^OrthnTAT), iu»(AnnTAr) = OrthnTAr, Ann(OrthnTAr) = n''(TAr).

The inclusion TNt C TN2 is equivalent to OrthnTATi D OithnTN2. We have also

Orthn(TAf! nTJV2) = OrthnTAr, + OrthnTN2

and, in particular,

TN D OrthnTAr = Orth n (TN + OrthnTAT).

Note that, in general,

TN n OrthnTAf ^ 0,

while TZ \N is not the sum TN + OrthnTN. Henceforth, when there is no risk of confusion, we will write OrthTN instead of

OrthnTA^. Let fijv = 'Jv^ D e 'he restriction of the symplectic form ft to the submanifold

AT. This is a presymplectic form on A . Its kernel (1.2.34) is

Ker QN = TNH Orth^TN.

The following special submanifolds of a symplectic manifold (Z, ft) are usually considered. The presymplectic form QN on any of such special manifolds is always of constant rank.

DEFINITION 2.4.8. A submanifold N C Z of a 2m-dimensional symplectic manifold (Z, ft) is said to be (cf. Definition 2.3.5):

• coisotropic if OrthnAf C TN, n > m;

• isotropic if TN C Orth^A^, n<m;

• Lagrangian if OrthnAT = TN, n = m;

• symplectic if ft# is a symplectic form on N.


□

As an immediate consequence of this definition, a symplectic form restricted to isotropic or Lagrangian submanifolds is equal to zero.

One can classify the germs of special submanifolds of a symplectic manifold [48]. Recall that a germ of a submanifold N at a point z G Z is the equivalence class (N; z) of submanifolds of the manifold Z which pass through z and coincide with N in an open neighbourhood of z.

With respect to local canonical coordinates (q\pt), a Lagrangian submanifold of a symplectic manifold is given by the equations

q" = dbS, Pa = -daS, (2-4.7)

where S(qa,p0) is a function, called the generating function, of the m variable: {<7°,P(,; a 6 A, b E B} for some partition [A, B) of the set ( 1 , . . . , m). Then the germ of a Lagrangian submanifold (N; z) is symplectically equivalent to the gern of the subspace

{(q,p)€R2m: p, = 0 , t = l , . . . , m } (2.4.8)

at the point 0 6 R2m of the symplectic space M2m. The expression (2.4.8) results from (2.4.7) by means of the symplectic automorphisms

qb >-* ~Pb, Pb<->qb

and

Pi^Pi + d,S(qa,q").

Example 2.4.5. Let the cotangent bundle T'M of a manifold M be equipped with the canonical symplectic form Ci. Then the image 0(M) of the zero section 6 of T'M —» M is a Lagrangian submanifold {pi = 0} of the symplectic manifold (T'M,Q). •

The germ of a (2m — r)-dimensional coisotropic submanifold (N; z) is symplectically isomorphic to the germ of the subspace

{ ( g , p ) € R 2 m : gi = 0 , t = l , . . . , r }

at the point 0 € R2m of the symplectic space R2m.


The germ of a 2(m - r)-dimensional symplectic submanifold (N; z) is symplec-tically isomorphic to the germ of the subspace

{ ( g , p ) e R 2 m : q<=Pl = 0,l=l,...,r}

at the point 0 e R2m of the symplectic space R2m. We will need the following two constructions in the sequel [178].

Example 2.4.6. Let {ZuQi) and (Z2 ,n2) be symplectic manifolds. The 2-form

fii 0 &2 = pr'fii — prjf22

is clearly a symplectic form on the product Z\ x Zi- Then the graph of a symplectic diffeomorphism of (ZuVl{) onto (Z2,^2) i s a Lagrangian submanifold of the symplectic manifold (Z\ x Z2, Jli 0 f22). •

Example 2.4.7. Let Q be the canonical symplectic form (2.4.2) on the cotangent bundle T'M of an m-dimensional manifold M. Then its tangent lift ft (1.2.23) is the symplectic form

fi = dpi A dxf + dp, A dqx (2.4.9)

on the tangent bundle TT'M of T'M, written with respect to the holonomic coordinates (<7',g',p„Pi) on TT'M. Due to the isomorphism a : TT'M = T'TM (1.1.9), the form (2.4.9) is also the pull-back O'^T-TM of the canonical symplectic form J I T T M o n t n e cotangent bundle T'TM of the manifold TM.

Let H b e a function on the cotangent bundle T'M. One can think of H as being a Hamiltonian of conservative mechanics. The Hamiltonian vector field

dn = VUdi - diHff

for H defines the closed 2m-dimensional submanifold

q' = d"H, P, = -diH (2.4.10)

of the 4m-dimensional symplectic manifold (TT'M, fi). The restriction of the symplectic form (2.4.9) to the submanifold (2.4.10) is equal to 0. Hence, this is a Lagrangian submanifold with the generating function H(q',Pi).

2.5. PRESYMPLECTIC STRUCTURE 81

Let £ be a function on the tangent bundle TM. On can think of £ as being a Lagrangian of conservative mechanics. The exterior differential rf£ of £ yields the morphism

a - 1 odC : TM -» T'TM -» TT*M.

Its image, given by the coordinate relations

Pi = di£, p, = 4 £ , (2.4.11)

is a Lagrangian submanifold of the symplectic manifold (7T*M,ft). Its generating function is £(<?',(?<).

Furthermore, let cj> bb ea nxterior r-form oo nhe eangent tundle TM, treated aa a generalized Lagrangian [10]. Then we have the morphism

a-'o4>:TM.T'TM—TT'M

and the relation

(c-lo4>yh = d4>.

It follows that (a-'o<j>)(TM) is a Lagrangian submanifold of the symplectic manifold (TT'M, ft) if and only if the 1-form <j> on TM is closed. •

(2.4.11)

2.5 Presymplectic structure

As in the case of Poisson structures, there are no pull-back or push-forward operations of symplectic structures by manifold maps in general. Pull-backs of symplectic forms are the presymplectic ones.

DEFINITION 2.5.1. An exterior 2-form ftona manifold Z is said to be presymplectic if it is closed, but not necessarily non-degenerate. A manifold equipped with a presymplectic form is called a presymplectic manifold. □

Using the formula (1.2.29), one can justify the fact that the kernel Ker ft (1.2.34) of a presymplectic form n of constant rank is an involutive distribution, called the characteristic distribution [116]. It defines the characteristic foliation on a presymplectic manifold Z. The restriction of the presymplectic form O to the leaves of this foliation is equal to zero.


Remark 2.5.1. In contrast with a symplectic structure, a presymplectic one is not associated with any Poisson structure in a canonical way. Nevertheless, there is a construction [49] which may make a Poisson bivector field w correspond to a presymplectic form Cl on a manifold Z in accordance with the relations

w o f t o w = ur,

K e r w ' n l m n ^ O .

•

The notion of a Hamiltonian vector field on a symplectic manifold is extended in a straightforward manner to a presymplectic manifold.

DEFINITION 2.5.2. A vector field ■d on a presymplectic manifold (Z, ft) is said to be locally Hamiltonian [Hamiitonian] if the exterior form tf J Cl is closed [exact]. □

As in a symplectic case, we find the following.

• A vector field ■d on a presymplectic manifold (Z, Q) is locally Hamiltonian if and only if it is an infinitesimal automorphism of this presymplectic manifold, i.e.,

L,,fi = d(ti\Q) = 0.

• A Hamiltonian vector field tij for a function / on a presymplectic manifold obeys the relation

tij\df = 0

and, consequently, is tangent to the leaves of the singular foliation of the level surfaces of the function / at regular points, where df ^ 0.

Note however that, in contrast with Poisson manifolds, not each function on a presymplectic manifold admits an associated Hamiltonian vector field.

The pull-back of a symplectic form by a manifold map is obviously a presymplectic form. The converse is the following.

PROPOSITION 2.5.3. Any presymplectic form 0 on a manifold M can be represented as the pull-back of some symplectic form. □

2.5. PRESYMPLECTIC STRUCTURE 83

Proof. Let <f> be a presymplectic form on a manifold M. Then it is the pull-back

0 = 6*(ft + 7r;M^)

of the symplectic form ft0 (2.4.3) on the cotangent bundle T'M of M by the global zero section 0 of T'M —> M. QED

Moreover, it is readily observed that

Orth^TO(M) = (T0)(Kertf>) C T0(M).

It follows that the imbedding 6 of the presymplectic manifold (M, <j>) into the symplectic manifold (T'M, Q^) is coisotropic.

If a presymplectic form Q on a manifold M is of constant rank, the stronger result holds [62, 72].

PROPOSITION 2.5.4. Given a presymplectic manifold (M, fi) where fl is of constant rank, there exists a symplectic form on a tubular neighbourhood of the zero section of the dual bundle to the characteristic distribution Ker 0 , where M can be coisotropically imbedded. □

There is another well-known pull-back construction, where a presymplectic form is seen as a pull-back of a symplectic form by a surjective submersion.

PROPOSITION 2.5.5. Let a presymplectic form Q. on manifold M be of constant rank and its characteristic foliation be simple, i.e., its leaves are fibres of a fibre bundle 7r : M —> P. Then the base P of this fibre bundle is equipped with a symplectic form fi/> such that fl is the pull-back of Qp by n. □

Proof. The symplectic form Up on P is defined by the relation

Qp(„y)d!= fn(i7,tf)

for any v G T-n~l(v), vf 6 T T T " 1 ^ ' ) . QED

This pull-back construction is the key point of what is called a reduction of a symplectic structure.


2.6 Reduction of symplectic and Poisson structures

A submanifold N of a symplectic manifold (Z, Q) fails to be a symplectic one in general. At the same time, if the presymplectic form QN = i%& o n Y ' s °f constant rank and the characteristic foliation of fi/v is simple, i.e., its leaves are fibres of a fibre bundle N —> P, the base P of this fibre bundle is a symplectic manifold in accordance with Proposition 2.5.5.

This construction is called a reduction of a symplectic structure. It is generalized for Poisson structures. Important reduction processes appear in connection, e.g., with constraint Hamiltonian systems and Lie group actions on symplectic and Poisson manifolds.

DEFINITION 2.6.1. [2, 116]. A symplectic reduction of a symplectic manifold (Z, fi) is said to be a surjective submersion 7r : N —> P of a submanifold N C Z onto a symplectic manifold (P,QP), which satisfies

w'Qp = i*NQ.

One says that (P, fi/>) is the reduced symplectic manifold of the symplectic manifold (Z, Q) via the submanifold N. □

The above speculations show that, given a submanifold N of a symplectic manifold (Z, Q), there exists a symplectic reduction 7r : N —» P with connected fibres if and only if

• the presymplectic form ils = ?)v^ o n N ' s of constant rank,

• the characteristic foliation of Qfj is simple.

PROPOSITION 2.6.2. [116]. Let (Z, f2) be a symplectic manifold and N a submanifold of Z.

(i) If n : N —> P is a symplectic reduction, then the rank of the pull-back presymplectic form £lN on N is constant and equal to d imP, Kerfi/y = VN, and the connected components of fibres of 7r are the leaves of the characteristic foliation for fi/v. Furthermore, if n has connected fibres, the characteristic foliation of Qjv is simple, and is exactly the fibration N —> P.

(ii) Conversely, if the rank of the presymplectic form fijv on a submanifold N of a symplectic manifold (Z, fl) is constant and the characteristic foliation for f2^ is

2.6. REDUCTION OF SYMPLECTIC AND POISSON STRUCTURES 85

simple, the base P of the corresponding fibration N —> P has a unique symplectic form such that this fibration is a symplectic reduction of (Z, fi) via N. This is a unique symplectic reduction via N with connected fibres. Furthermore, if N —» P' is another symplectic reduction, there exists a surjection P —> P' which is a local symplectic isomorphism. □

A coisotropic submanifold N provides the most interesting case when the presym-plectic form fijv is of constant rank which is neither dim N nor 0.

Example 2.6.1. Let [M,<j>) be a presymplectic manifold, where a presymplectic form 4> is of constant rank and its characteristic foliation is simple. Combining the pull-back constructions in Propositions 2.5.3 and 2.5.5 gives the reduction of the symplectic structure 0.$ (2.4.3) on the cotangent bundle T*M via the coisotropic submanifold 0(M). •

The reduction procedure is extended to Poisson manifolds as follows.

DEFINITION 2.6.3. [129]. By a reductive structure of a Poisson manifold (Z, w) is meant a triple (Z, N, E) of a submanifold N of M and a vector subbundle E C TZ \N together with a submersion 7r : N —* P if the following conditions are satisfied:

(i) E fl TN is tangent to the fibres of the submersion N —> P;

(ii) if df and dg, where / , g € D°(Z), belong to Ann E, so is d{f, g}w.

A reductive structure is said to be a Poisson reduction if: (i) P above is a Poisson manifold with a Poisson bivector field W and (ii) for any local functions / , g on P and for any local extensions f,g onto Z of the pull-backs / o ir,g o ix such that df,dg C AnaE, the relation

{7,9~}w°iN = {f,g}w°n holds good. One says that (F, W) is the reduced Poisson manifold of (Z, w) via (N,E). a

The following assertion furnishes the necessary and sufficient conditions for a reductive structure of a Poisson manifold to be a Poisson reduction.

PROPOSITION 2.6.4. [129, 181]. Let (Z, N, E) be a reductive structure of a Poisson manifold (Z, w). This is a Poisson reduction if and only if

io»(Ann£) CTN + E. (2.6.1)


a

The functorality property of the Poisson reduction is given by the following.

PROPOSITION 2.6.5. [129]. Let (Z,N,E) and (Z',N',E') be Poisson reductions, and let $ : Z — Z' be a Poisson map such that ${N) C N', T$(E) C E', and $ sends the leaves of JV to the leaves of N'. Then $ induces a unique Poisson map $ : P —» P', called the reduction of $, such that n' o $ = $ o n. □

Example 2.6.2. Let Z = T'G, where G is a connected Lie group, N = Z, and £ comprise the tangent space to the right G-orbits. Then (Z, E) is a Poisson reduction, where ix(Z) = T'G/G = g' is the Lie coalgebra of G provided with the Lie-Poisson structure (2.3.11) [129, 130]. •

Example 2.6.3. To compare a Poisson reduction with a symplectic one, let (Z, fl) be a symplectic manifold and w the corresponding Poisson bivector field on Z. Let N be a submanifold of Z such that the presymplectic form QN = i%^ is of constant rank. Put E = OithTN. Then E l~l TN = KerftN is tangent to the characteristic foliation of QN- If there exists a symplectic reduction ir : N —» P of (Z, Q) to (P, ftp), one can show that (Z, N, OrthTTV) is a reductive structure of the Poisson manifold (Z, w). The condition (i) of Definition 2.6.3 holds since the integral manifolds of the distribution Ker fijv are connected components of fibres of N —» P. Furthermore, since we are on a symplectic manifold, df G Ann (OrthTW) if and only if the Hamiltonian vector field ■&; for / belongs to

iu ,(Ann(OrthTAf))=TAf. (2.6.2)

Then, dhdge TN implies

{#f,tis}=#{j,9)eTN.

Hence, condition (ii) of Definition 2.6.3 also holds. Moreover, the inclusion (2.6.1) obviously takes place because of (2.6.2). It follows that (Z, N, OrthTN) is a Poisson reduction and the corresponding reduced structure W on P is associated with the symplectic form fip. On the other hand, if we take E = Kerfi/v, the inclusion (2.6.1) requires N to be coisotropic, since

tu^Ann E)=TN + OrthTN,

2.6. REDUCTION OF SYMPLECTIC AND POISSON STRUCTURES 87

and one may apply the previous result. •

One can extend this Example to the Poisson manifold case as follows.

LEMMA 2.6.6. [181]. Let N be a. submanifold of a Poisson manifold (Z, w) such that

C(N)d^wl(AnnTN)C\TN (2.6.3)

is a distribution, i.e., is of constant rank. Then C(N) is involutive. Furthermore, if N is transversal to the leaves of the symplectic foliation F for w on Z, the distribution C(N) defines a subfoliation of F D TN which is transversally symplectic along each leaf of the latter. D

If the foliation C(N) is simple and corresponds to a fibration n : N —> P, then F n TN projects to a symplectic foliation n(F) on P, and P has a well-defined Poisson structure given by this foliation. This is called the leafwise reduction of (Z,w) via N. It follows in the same way as for the symplectic case discussed earlier that, if wl(AnnTN) has a constant dimension, then (Z, N, wÂnnTN)) is a reductive structure. Because of

wÂnniwÂnnTN))) C TN

and the relation (2.6.1), the reductive triple (Z, N, wt(Ann TN)) is also a Poisson reduction, while the reduced Poisson manifold P is exactly the one defined by the leafwise reduction.

Given a Poisson manifold (Z,w), let (P,W) be a reduced Poisson manifold of (Z, w) via a submanifold N. Then one can say that the Poisson algebra Q°(P) of smooth functions on P is the reduction of the Poisson algebra of Q°(Z) of smooth functions on Z. If N is a closed imbedded submanifold of a Poisson manifold Z, this reduction is described in an intrinsic way, i.e., in terms of ideals of the Poisson algebra D°(Z) as follows [97].

Let N be a closed imbedded submanifold of a Poisson manifold (Z,w). Let us consider the set / of real functions f on Z which vanish on N, i.e., i'Nf = 0. This set is the kernel / = Ker i'N of the linear morphism

t'N : 0°(Z) - Q°(N),


and it is an ideal of the associative commutative algebra Q°(Z). Since N is a closed and imbedded submanifold of Z, we have the isomorphism

Q°{Z)/I =* £>°(N) (2.6.4)

of associative commutative algebras. Let us consider the space of all vector fields on Z which restrict to vector fields

on N. It is given by

TN d^{u g T{Z) : u\df 6 / , V/ € / } . (2.6.5)

It is clear that 7]v ^ T(Z) |w . Suppose that -d; is a Hamiltonian vector field which restricts to a vector field on N. By the very definition (2.6.5), this means that

4f\dg = {f,g)zl, \fg€l.

Hence, the functions whose Hamiltonian vector fields restrict to vector fields on N are those functions in the normalizer of / , denoted by

7(/v)dJ= f{/e0°(Z): {/,<?} e / , v 5 e / } . (2.6.6)

It follows from the Jacobi identity that the normalizer (2.6.6) is a Poisson subalgebra of Q°(Z). Put

I'(N)d=I{N)nl. (2.6.7)

It is naturally a Poisson subalgebra of I{N). This subalgebra is generally non-zero since I2 C I'(N) due to the Leibniz rule.

The following Theorem leads us back to the reduction procedure.

THEOREM 2.6.7. [97]. Let N be a closed imbedded submanifold of a Poisson manifold Z. Let us assume that w^AnnTN) and C (2.6.3) are distributions of constant rank, and the foliation C is simple. Then (Z, N, u)"(Ann TN)) is a reductive structure such that there is the associative commutative algebra isomorphism

D°(F) S I(N)/I'(N). (2.6.8)

Since the right-hand side is naturally a Poisson algebra, this isomorphism defines a Poisson structure on P. O

Remark 2.6.4. The result is based on the fact that all sections of wi(AnnTN) are restrictions to N of Hamiltonian vector fields for elements in / , while all sections of

2.7. APPENDIX. POISSON HOMOLOGY AND COHOMOLOGY 89

C are Hamiltonian vector fields for elements in I'(N) restricted to N. In particular, if N is coisotropic, / C I(N), i.e., / = I'{N) is a Poisson subalgebra of D°(Z). •

Theorem 2.6.7 shows that a Poisson reduction procedure can be seen as an algebraic one; that leads to the following purely algebraic definition [97].

DEFINITION 2.6.8. Let V be a Poisson algebra, J be an ideal of V, J{N) its normalizer (2.6.6), and J'(N) = J{N) D J. One says that the Poisson algebra J(N)/J'(N) is the reduction of the Poisson algebra V via the ideal J. □

In particular, an ideal J of a Poisson algebra V is said to be coisotropic if J is a Poisson subalgebra of V.

The above algebraic reduction procedure has been generalized for Jacobi manifolds [87].

We meet the Poisson reduction in the sense of Definition 2.6.8 in connection with the BRST scheme [97, 169].

2.7 Appendix. Poisson homology and cohomology

This Section is concerned with the Koszul-Brylinski-Poisson homology and the Lichnerowicz-Poisson cohomology of a Poisson manifold, which find their application in the quantization procedure.

Remark 2.7.1. Let us recall briefly the basic notions of homology and cohomology of complexes [17, 122].

A sequence

n J?° n 3i D * h_ n aZ±L U * £>o * "1 * ' ' ' * av ' ' '

(2.7.1)

of Abelian groups Bp and homomorphisms dp is said to be a chain complex if

dpodp+1=0, V p e N ,

i.e., Imdp+i C Ker3p . The quotient

Hp(B.)d^Keidp/lmdp+i

is called the pth homology group of the chain complex B, (2.7.1). The chain complex (2.7.1) is said to be exact at an element Bp if HP(B.) = 0. It is called an exact sequence if it is exact at each element.


A sequence

0 ^ B ° ^ B l . * - , . . . £ ^ B P . * ♦ . . . (2.7.2)

of Abelian groups Bv and homomorphisms 6P is said to be a cochain complex if

«S" o 5*-1 = 0, V p e N .

The pth cohomology group of the cochain complex S" (2.7.2) is the quotient

Hp(B')d=Ker&'/lm6p-1.

The De Rham compiex of exterior forms on a manifold Z

... J - » 0 ' - i ( z ) -*-.0»(Z) - ^ D r + 1 ( Z ) -*- . . . .

exemplifies a cochain complex. Its cohomology group HP(Z), called the De Rham pth cohomology group, is the quotient of the space of closed p-forms by the subspace of exact p-forms. •

Given a Poisson manifold (Z, w), let us consider the operator

6w:Qr(Z)^Dr-\Z),

6W = w\ o d — d o w\, (2.7.3)

on the exterior algebra D*(Z), called the Poisson codifferential [19, 102, 181]. It is given by the coordinate expression

SModfi A • • • A dfT) = i > l ) , + 1{/o, h}dh A • • • A df, A • • • A dfT + i=i

£ (-l)*+Vod{/i, h) A dfx A • • ■ A dfi A • • • A dfi A - • • A dfr. l<i<j<r

Henceforth, when there is no risk of confusion, we shall write 5 instead of 5W. The operator 6 (2.7.3) obeys the equalities

6 o 6 = 0, do6 + 6od=0.

(2.7.4) (2.7.5)

Due to the nilpotency property (2.7.4), we have the chain complex

. . . J-£)P-i(z) J-DP{Z) ^ - D P + 1 ( Z ) J- ,


called the canonical complex of the Poisson manifold (Z, w). The homology H™n(Z) of this complex is termed the canonical or the Koszul-Brylinski-Poisson homology.

Furthermore, due to the property (2.7.5), the periodic double complex

ET,t(Z) = Ol~T(Z), 6 : Er,t(Z) -» Er^(Z), d:Eri{Z)^ET^{Z),

can also be defined. Let (Z, fi) be a 2m-dimensional symplectic manifold, provided with the volume

form V (2.4.1), and IUQ the corresponding Poisson bivector field. Imitating the well-known star isomorphism for Riemannian manifolds, one introduces the star operation

* : QT(Z) - D2m-r(Z),

where wl is the homomorphism (1.2.32). The (2m — r)-form *0 is called the adjoint of the r-form </> relative to the symplectic form Q. The following equalities

*{**) = <£, 0 6 D'(Z),

are fulfilled [116].

LEMMA 2.7.1. [19]. In the case of a symplectic manifold, the codifferential S (2.7.3) on £>T{Z) is given by the formula

6 = {-iy+l*d*. (2.7.6)

□ The relation (2.7.6) establishes an isomorphism of the canonical homology group

H^{Z) with the De Rham cohomology group H2m~'(Z) of a symplectic manifold as follows.

T H E O R E M 2.7.2. [19, 181]. If (Z, 0) is a 2m-dimensional symplectic manifold, then

H f ( Z ) = H^-^Z). (2.7.7)

*(cj>Aa) = 4W)J<**) = (-l)m°K(°)\(*<t>)

»0=u>?,(fljy,


a

Given a Poisson manifold (Z, w), let us now introduce the operation

w : %{Z) -* Tr+1(Z), (2.7.8)

on the exterior algebra 7;(Z) of multivector fields on Z, where [.,.] is the Schouten-Nijenhuis bracket (1.2.10). This operation satisfies the rules

w o w = 0,

w{ti Av) = ui(i?) A D + ( - l ) w t f A w(v),

(2.7.9) (2.7.10)

and is sometimes called a contravariant exterior differential [181]. Its relation to the familiar exterior differential is

iB(w>(<j>)) = -wl(d<j>), <t>eO'(Z), (2.7.11)

while that to the codifferential 6W (2.7.3) is given by the formula

w{-d)\<t> = ti\6m{4>) + {-\)w6w{d\<j>), | i ? H 0 ] - l . (2.7.12)

Example 2.7.2. It is readily observed that

-w(f) = [w, / ] = * , , / e To(Z),

is the Hamiltonian vector field for a function / on Z. •

Due to the nilpotency rule (2.7.9), the operation w (2.7.8) provides the cochain complex

••• ^%.,{Z) ^%{Z) ^Tr+l(Z) - ^ • • ,

called the Lichnerowicz-Poisson cochain complex. Cohomology groups HlP(Z,w) of this complex are said to be Lichnerowicz-Poisson or LP-cohomology groups of the Poisson manifold {Z,w).

Example 2.7.3. The LP-homology group //£p(Z,w) is the centre of the Poisson algebra (D°(Z),w). It consists of functions / € T0(Z) modulo constant functions whose Hamiltonian vector fields dj = —w(f) € KeriD vanish. •

gWg-K4 tfer.(Z),


Example 2.7.4. The first LP-homology group HlP(Z,w) is the space of canonical vector fields u for the Poisson bivector field w (i.e., huw = -w(u) = 0) modulo Hamiltonian vector fields -w(f), f e TQ{Z). •

Example 2.7.5. The second LP-homology group HlP{Z,w) has the distinguished element {w} whose representative is the Poisson bivector field w itself. We have {w} = 0 if there is a vector field u on Z such that w = w(u) = —huw. •

It is readily observed that, due to the property (2.7.10), HlP(Z,w) makes up a graded commutative algebra with respect to the product

{ i> }A{ i ; } d %At ; } .

It is also provided with the bracket

[W,M]"{fM) in accordance with the relation (1.2.13).

THEOREM 2.7.3. [181]. The relation (2.7.11) induces a homomorphism of the graded algebras

H'(Z) - H'h¥{Z) (2.7.13)

of the De Rham cohomology and the LP-cohomology. This is an isomorphism if the Poisson bivector field w comes from a symplectic structure on Z. D

In case of a symplectic manifold, the isomorphism (2.7.13) and the isomorphism (2.7.7) lead to the isomorphism

HT(Z) = Hlr\Z).

In general, we have the following relationship between the LP-cohomology and the canonical homology of a Poisson manifold.

PROPOSITION 2.7.4. By virtue of the relation (2.7.12), the natural pairing (1.2.30) induces the corresponding pairing

(,) : HUZ) x Hr(Z) - H^{Z),

(m,{<t»})=mm,


of vector spaces of the LP-cohomology and the canonical homology of a Poisson manifold. □

The constructions of the LP-cohomology and the canonical homology for Poisson manifolds can be extended to the Jacobi ones [115].

Given a Jacobi manifold (Z;w,u), let us define the operator

vm:Tr(Z)->TT+1{Z),

wud=-[w,#]+ruAi), d€Tr(Z),

which generalizes the contravariant exterior differential (2.7.8). This operator satisfies the rules

Lu o wu = wu o Lu, 552(i)) = - M A i o , i ) e T , ( Z ) , wu(-d A v) = wu{ti) Av+ ( - l ) 1 *^ A wu(v).

(2.7.14)

Its connection with the exterior differential is given by the relations

Lu(ti/'(0)) = w>(Lu<t>), wu(w*(<p)) = -wi{d<j>) + u/^uj^) A w.

(2.7.15) (2.7.16)

A glance at the expression (2.7.14) shows that the operator wu becomes nilpotent on the subspaces of u-invariant multivectors

Tr{Z) ={tf € %{Z) : Lutf = [u,■&] = 0}.

This allows us to introduce the cochain complex

IUU 3 = / rj\ M1U 7r I ry\ WU ~rf- j ly\ VJU

The cohomology groups H[^{Z,w) of this complex are called Lichnerowicz-Jacobi or LJ-cohomology groups of the Jacobi manifold (Z; w, u). It is clear that, if u = 0 and (Z, w) is a Poisson manifold, the LJ-cohomology is precisely the LP-cohomology.

Example 2.7.6. The first LJ-cohomology group H[p(Z,w) has the distinguished element whose representative is the vector field u, while the second LJ-cohomology group H{jp(Z,w) has the distinguished element whose representative is w. •


In the same way that the LP-cohomology groups are connected with the De Rham cohomology groups, the LJ-cohomology groups are connected with the cohomology groups of the subcomplex of basic forms of the De Rham complex.

An r-form </> on a Jacobi manifold (Z;w,u) is said to be basic if

uJ0 = O, LU0 = O.

It is easily seen that, if <fi is a basic form, so is its exterior differential d<p. Hence, basic forms constitute a subcomplex

••• ^OrB-1(Z) -^OT{Z)B J ^ D r + 1 ( Z ) B - * - . . - (2.7.17)

of the De Rham complex. On subspaces £>T

B{Z) C DT(Z) of basic forms, the relation (2.7.16) takes the form similar to (2.7.11):

wu(ui'(0)) = —w>(d(j)), 4> e DB(Z). (2.7.18)

Moreover, we deduce from the relation (2.7.15) that, if </> is a basic r-form, then wl((j>) € Tr(Z). In the same way as in Theorem 2.7.3, the equality (2.7.18) yields a homomorphism

H'B(Z) - H'LJ(Z)

from the cohomology of the complex (2.7.17), called the basic De Rham cohomology, to the LJ-cohomology.

To introduce the canonical homology of a Jacobi manifold (Z; w, u), let us consider the codifferential 6W (2.7.3) defined by the bivector field w. It was proved in [38] that, if ^ is a basic form, so is 8W(4>), and 8W o 6W = 0. Then the chain complex

••• h-O'-^Z) £^QPB{Z) &-&$l{Z) ^

is defined. The homology of this complex is called the canonicai homology of a Jacobi manifold. If u = 0, this is exactly the canonical homology of a Poisson manifold.

Similarly to Proposition 2.7.4, we have the following relationship between the LJ-cohomology and the canonical homology of a Jacobi manifold.

T H E O R E M 2.7.5. [115]. There is the pairing

(,) : H'LP(Z) x HT(Z) - m™(Z),

<{t f} ,M>=W0>}.


of vector spaces of the LJ-cohomology and the canonical homology of a Jacobi manifold. □

The result follows from the fact that, if tf € T.(Z) and 0 is a basic form, then their pairing (d, <j>) (1.2.30) is a basic function, and from the relation

wu{-d)\4> = d\6ul{<j>) + {-\)r6w(d\4>), <t>eOTB(Z), tfeTr_,(Z).

Note that, bearing additionally in mind the map (2.2.4), one can apply the construction of the LJ-cohomology and the canonical homology of a Jacobi manifold to contact manifolds as the particular case of Jacobi manifolds. We refer to [38, 115] for more details.

2 . 8 Appendix. More brackets

We will give a brief survey of several extensions of the Poisson bracket to multivectors and differential forms and its generalization to the bracket of n > 2 functions.

In this Section, we will return to the notation [., . ]S N for the Schouten-Nijenhuis bracket.

Let (Z, w) be a Poisson manifold. With the nilpotent operation iD (2.7.8), one can introduce the bracket

{d,v}wd=-[w{d),v]sN

on an exterior algebra T,(Z) of multivector fields on Z. This bracket has the property

[ti,v]w = -(-l)WM[v,i9]w + w([#,v}sri),

and is a graded skew-commutative Lie bracket on the quotient T,(Z)/w(T.(Z)), where the Lie degree of a multivector field i5 is | ■& | — 1.

Remark 2.8.1. Recall that multivector fields on a manifold Z constitute a graded Lie algebra with respect to the Schouten-Nijenhuis bracket [.,.]SN' (1-2.15), where the Lie degree of a multivector field <d is ] "& \ — 1 (see Remark 1.2.3). •

The Poisson bracket can be extended to exterior forms as follows. If the Poisson bivector w on a manifold Z is non-degenerate, i.e., (Z, w) is a

symplectic manifold, there is the isomorphism ti)" (1.2.32) of graded algebras Q'(Z)

2.8. APPENDIX. MORE BRACKETS 97

and T,(Z). This isomorphism extends the Schouten-Nijenhuis bracket to the algebra of exterior forms [133]:

{.,.}. :£)T(Z)xOs(Z)^Dr+s-1(Z), to'Ctoff},,) =[u>'0Vff]sN, 4>,a eD'(Z).

(2.8.1) (2.8.2)

In the general case of a Poisson manifold, we have the homomorphism w" (1.2.32) which satisfies the property (2.7.11). Therefore, in order to construct the bracket (2.8.1), let us consider the codifferential operator 6W (2.7.3). Then the Schouten-Nijenhuis bracket of exterior forms (2.8.1) is defined as

{4>, °}w = ( M ) A o + ( - l ) W ^ A {6wo) - 6W{4> A o) (2.8.3)

[102, 181]. This bracket has the properties

(_l)W(W-»){0, {a,6}w}w + (-l)M(W-«{ffi {<?,0U* +

Example 2.8.2. In particular, the bracket (2.8.3) of 1-forms reads

{0, cr}«; = L,„i0a - LwP(70 - d(io(tf>, a)) = (2.8.4)

It provides D1(Z) with a Lie algebra structure such that

w* : Q\Z) -> T{Z)

is a Lie algebra homomorphism (2.8.2). The relationship between the bracket (2.8.4) and the Poisson bracket (2.3.1) is

{df,dg}w = d{f,g}.

•

Using the nilpotency properties (2.7.4) and (2.7.5) of the codifferential 6W, one can obtain the formula

d{<j>,a}w = -{d<t>,o}w - (-l)w{<j>,da}w.

wl<f>\do - w'a^dcj) + d(w((/>,a)).

{*,o-}» = (-l)l*l""{^*}»,

C-1)WI"M){«.{«.^}.}« = 0-


Then the bracket

{4>, o}d = -{dct>, a}w (2.8.5)

on an algebra of exterior forms D'{Z) can be introduced [21, 133]. This is a graded skew-commutative Lie bracket

{*,*}< = ( - i ) w " W } . i

on the quotient D*(Z)/dD*(Z), where the Lie degree of an exterior form <f> is | <j> | —1. Let us turn now to an n-ary generalization of Poisson and Jacobi brackets.

DEFINITION 2.8.1. [84, 85). A generalized almost Jacobi bracket of order n on a manifold Z is said to be an n-linear map

{...} : xD°{Z)^0°(Z) (2.8.6)

which is

• skew-symmetric

| . . . , ji,..., jj,... | — \..., Jj,..., Ji,... I

with respect to any pair of arguments,

• and is a first order linear differential operator on Z with respect to each argument, i.e.,

{Si/i, ••■} = 9i{fx, ■••} + /i{ffi, ■■•} - Si / i{l , - . .}.

□

If {...} is a generalized almost Jacobi bracket of order n, then there exist an n-vector field w and an (n — 1)-vector field e on a manifold Z such that

{/i, - , / «} = w(dfu...,dfn)+ E ( - l ) ' + 7 . e ( r f / i , - , d / „ . . . , d / n ) t=i

(2.8.7)

and

{ l , / l , - , / n - l } = e ( d / 1 , . . . , d / n - l ) .

2.8. APPENDIX. MORE BRACKETS 99

Conversely, any pair (w,e) of an n-vector field w and an (n - l)-vector field e defines the generalized almost Jacobi bracket (2.8.6) given by the relation (2.8.7). A manifold Z provided with the bracket (2.8.7) is called a generalized almost Jacobi manifold. If e = 0, we obtain a generalized almost Poisson manifold.

In order to reproduce a Jacobi structure in the case of n = 2, one should add the integrability conditions, generalizing the Jacobi identity for the generalized almost Jacobi structure. In fact, two different types of integrability conditions have been suggested. They are

• the fundamental identity

{/l. • • • i / n - l , { f f l , - . - , 0 n } } = {{fl, ■ ■ ■ , fn-l, 9l}, 92, ■ ■ ■ , 9n) +

{9\,{fl,--,fn-l,92},--,9n} +

■■■ + { S l , - - , P n - l , { / l , - - , / n - l , 5 n } }

• and the generalized Jacobi identity

{/l> • • • i /n-l i {In,- ■ ■ )/2n-l}} + {/2. • • • , /n, {fn+l, ■ ■ ■ , /bn- l i / l}} H = 0 .

(2.8.9)

(2.8.8)

The bracket (2.8.7) satisfying the fundamental identity (2.8.8) is called a Nambu-Jacobi bracket and (Z;w,e) is a Nambu-Jacobi manifold [85, 128]. If e = 0, the bracket (2.8.7) is said to be a Nambu-Poisson bracket, and (Z, w) is a Nambu-Poisson manifold [9, 84, 175].

The bracket (2.8.7) satisfying the generalized Jacobi identity (2.8.9) is called a generalized Jacobi bracket, and (Z;w,e) is a generalized Jacobi manifold [84, 145]. If e = 0, the bracket (2.8.7) is said to be a generalized Poisson bracket, and (Z, w) is a generalized Poisson manifold [8, 9, 84].

The relationship between Nambu-Jacobi manifolds and generalized Jacobi manifolds can be shown directly by noticing that the generalized Jacobi identity (2.8.9) is the full antisymmetrization of the fundamental identity (2.8.8). It follows that every Nambu-Jacobi manifold is a generalized Jacobi manifold [9, 85].

For n even, the integrability conditions of a generalized Jacobi structure (w, e) are given by the following assertion [145].

PROPOSITION 2.8.2. A generalized almost Jacobi structure (w,e) of even order n = 2p is a generalized Jacobi structure if and only if the relations

[w, e]SN = 0, [w, W]SN = 2(2p - l J m A e ,


generalizing the conditions (2.1.2), hold good. □

At the same time, the integrability conditions of a Nambu-Jacobi structure {w, e) in particular imply that

• the multivector field w is a Nambu-Poisson structure of order n,

• the multivector field e is a Nambu-Poisson structure of order n — 1 [128],

• and every Nambu-Poisson multivector w can be written as an exterior product of vector fields [79].

Example 2.8.3. Let us consider the product Z = R x T'M with the holonomic coordinates (*,<?', p,) as in Examples 2.2.1, 2.2.3 and 2.3.7. It is equipped with both the Jacobi structure given by the pair (wi,Ei) (2.2.9) and the Poisson structure w? (2.3.9). Then Z is also provided with the generalized Jacobi structure of order 4 given by the multivector fields

W = W\ A W2, e = E Aui2

(see [85] for details). •

2.9 Appendix. Multisymplectic structures

We will give a brief exposition of multisymplectic and vector-valued generalizations of a symplectic structure.

Similarly to that a Poisson bivector field is replaced by a multivector one for a generalized Poisson structure or a Nambu-Poisson structure, one can consider an n-ary generalization of a symplectic form. This is a multisymplectic form.

Given an m-dimensional manifold M with coordinates (zx), let us consider the fibre bundle

C.AT'M^M

whose sections are exterior /t-forms on M. This fibre bundle is equipped with the holonomic coordinates (ZA ,PA), where A = (Aj < . . . < \k) are multi-indices of the

2.9. APPENDIX. MULTISYMPLECTIC STRUCTURES 101

length | A |= k. The manifold A T ' M is provided with the canonical exterior fc-form 0 defined by the relation

u„J.. .u1je(p) = TC(ti„)J...TC(u1)Jp1 p € AT 'M, u, e T p ( A T ' M ) .

Its coordinate expression is

e = (2.9.1)

where the sum is over all multi-indices A of the length k. The exterior differential dQ of the form (2.9.1) is the (k + l)-symplectic form

QM = de = (2.9.2)

which belongs to the class of multisymplectic forms in the sense of Martin [131]. If k = 1, the form Q.M (2.9.2) is the familiar symplectic form on the cotangent bundle T'M.

Example 2 .9 .1 . For instance, let Y —» X be a fibre bundle over a (1 < n)-dimensional base X. Let us consider the canonical form 9 (2.9.1) on the exterior product A T'Y. The homogeneous Legendre bundle over Y is said to be the fibre bundle

ZY=T'Y ACAT'X), (2.9.3)

equipped with the holonomic coordinates (xx,yi,pxyp), where the coordinate p has

the transformation law

'-«(£) ('-$£4 The canonical bundle monomorphism over Y

iz:T'YA{n/\T'X)^AT'Y

yields the pull-back form

5 = i"z@ =pcj + pxdyx A uix (2.9.4)

on the fibre bundle ZY, where we use the notation (1.2.21). Its exterior differential

Qz = dp A u + dpx A dy' A u>x (2.9.5)

A A--- AdzXk,

J^dp\1 \kAdzx> A---Adzx* A


is also called the multisymplectic form [63]. In particular, if Y —» R, the multisym-plectic form (2.9.5) leads to the canonical symplectic form (2.4.4) on the cotangent bundle ZY = T'Y. •

Let us touch on another generalization of a symplectic structure when an Rn-valued 2-form is considered [53, 142]. Let X be an n-dimensional manifold and LX —» X the principal frame bundle coordinated by (xx,s'ia). The frame bundle LX is provided with the canonical Revalued 1-form 6LX (1.5.13) which reads

OLX = s%dz" <g> ta,

where {ta} is a basis for Rn. Let us take its exterior differential

d0LX = ds% A dx" <8> ta,

called the n-symplectic form. It is readily observed that this form is non-degenerate, i.e., given a vector field

u = u*^ + uld»

on LX, the equality u\d0Lx = 0 holds if and only if u = 0. By analogy with the symplectic case, one can write

Uf\dOLX = -df, (2.9.6)

where / is an Rn-valued function on the frame bundle LX. There is the class of functions (e.g., which are constant on the fibres of LX) such that vector fields satisfying the equation (2.9.6) exist. Then their Jacobi bracket

{f,g} = Uf\dg

is defined, and it also belongs to the above-mentioned class of functions. Let now Y —> X be an arbitrary fibre bundle. The Legendre bundle over Y is

said to be the fibre bundle

n = AT*X®TX® VY a VY A (Vrx ) , Y Y

(2.9.7)

equipped with the holonomic coordinates (xx,y',p*) possessing the transition functions

Pi = (2.9.8) <V dx'x

dy1* dx» 6t {£)*

2.9. APPENDIX. MULTISYMPLECTIC STRUCTURES 103

It is provided with the TX-valued form

A = dpx A dy{ A u <g> dx, (2.9.9)

called the polysymplectic form. The Legendre bundle (2.9.7) plays the role of a finite-dimensional phase space in the Hamiltonian formulation of classical field theory [28, 56, 57, 73, 96, 158, 159].

In the case of X = K, the polysymplectic Hamiltonian formalism leads us to the Hamiltonian formulation of time-dependent mechanics (see Chapter 5).

The main peculiarity of polysymplectic formalism is that Hamiltonian connections on the Legendre bundle U —* X play a role similar to Hamiltonian vector fields on a symplectic manifold. A connection

T = da* ®{dx+ -$& + '&%)

on the fibre bundle n —♦ X is said to be locally Hamiltonian if the exterior form 7JA is closed (see Section 3.8 for details).

Remark 2.9.2. Note that generalization of the Poisson bracket for polysymplectic manifolds meets the difficulty (see [91, 92] and references therein). Nevertheless, in the particular case of an affine bundle Y —> X whose base X is provided with a non-degenerate metric g, the bracket of horizontal 1-forms on n —> X,

u. i at) (94>a dap da0d4>a\

can be globally defined. •

>/l7I«fa*. <j> = <t>x{p)dxx, a = <rx{p)dxx,

dpf dy{ dp? dy' 9a0

Chapter 3

Hamiltonian systems

The theory of Hamiltonian systems is a vast subject which can be studied from many different viewpoints. This Chapter provides a brief exposition of different types of Hamiltonian systems on Poisson, symplectic and presymplectic manifolds in conservative mechanics. We leave aside many interesting constructions of a Hamiltonian conservative mechanics and its straightforward extension to the direct product

[time] x [autonomous phase space]

that cannot be applied to time-dependent mechanics because a Hamiltonian is not a function under time-dependent transformations. Nevertheless, any object denned on a phase space V of conservative mechanics or on the product R x Z has the counterpart on a phase space

ifr-.W = Rx Z (3.0.1)

of time-dependent mechanics, where different trivializations tp (3.0.1) correspond to different reference frames.

For instance, there is one-to-one correspondence between the connections on a fibre bundle W —* R (3.0.1) and the time-dependent vector fields

1 x Z -^TZ z because of the diffeomorphism

J V = J*W = J\W xZ)=RxTZ.

However, this correspondence is not canonical, but depends on a trivialization (3.0.1).

105

106 CHAPTER 3. HAMILTONIAN SYSTEMS

Note that Hamiltonian relativistic mechanics can be seen as an autonomous Hamiltonian system on the hyperboloid of relativistic momenta, i.e., it exemplifies a Dirac constraint system (see Section 6.3). In this Chapter, we pay special attention to constraint systems, including symplectic Hamiltonian systems, Dirac Hamiltonian systems, and Dirac constraint systems, seen as particular presymplectic Hamiltonian systems.

3.1 Dynamic equations

There are several types of equations in conservative mechanics. These are first and second order dynamic equations, Hamilton equations with respect to Poisson, symplectic or presymplectic structures, and Lagrange equations. The relationship between them is the following.

• The dynamic equations, by definition, are differential equations which can be algebraically solved for the highest order derivatives.

• The second order dynamic equations on a manifold M, by definition, are particular first order dynamic equations on the tangent bundle TM of M.

• The first order dynamic equations on a manifold Z can be represented as Lagrange equations for Lagrangians on the tangent bundle TZ of Z, but they are not Hamilton equations on Z in general.

• The Hamilton equations are not necessarily first order dynamic equations.

• The Lagrange equations on a manifold M are derived as particular Hamilton equations on the tangent bundle TM, but they are not necessarily second order dynamic equations and differential equations in any strict sense.

• The second order dynamic equations on a manifold M fail to be represented as Lagrange equations for Lagrangians on the tangent bundle TM or as Hamilton equations on TM in general.

This Section is devoted to the notions of first and second order dynamic equations on a manifold, called autonomous dynamic equations.

3.1. DYNAMIC EQUATIONS 107

DEFINITION 3.1.1. Let Z, dimZ > 1, be a manifold, coordinated by (zx), and u a vector field on Z. The closed subbundle u(Z) of the tangent bundle TZ, given by the coordinate relations

iA = ux(z), (3.1.1)

is said to be an (autonomous) .first order dynamic equation on a manifold Z. This is a system of first order differential equations on the fibre bundle K x Z —> R in accordance with Definition 1.3.3 (see Example 5.2.1). □

By solutions of the first order dynamic equation (3.1.1) are meant integral curves of the vector field u.

Remark 3.1.1. Generalizing Definition 3.1.1, we will say that an (autonomous) first order differentia] equation on a manifold Z is a closed submanifold <£ of the tangent bundle TZ of Z. A solution of this differential equation is an integral curve of a vector field on a submanifold J V c Z , which takes its values into TN n <£. •

In this Chapter, we will deal with the first order dynamic equations (3.1.1) when vector fields u are Hamiltonian vector fields for Poisson, symplectic or presymplectic structure on a manifold Z, treated as a momentum phase space of conservative mechanics. However, they are not equivalent to the Hamilton equations in general. The Hamilton equations with respect to a Poisson structure, by definition, are first order dynamic equations, and so are the Hamilton equations with respect to a symplectic structure. This is not the case of Hamilton equations with respect to a presymplectic structure.

Let us bear in mind the well-known inverse problem in conservative Hamiltonian mechanics, which consists in trying to represent a first order dynamic equation on a manifold Z as Hamilton equations with respect to some Poisson structure on Z [32] (see the next Section). The important motivation of the inverse problem is that, with a Poisson structure, one may find integrals of motion of a dynamic equation and also quantize it.

Remark 3.1.2. As in the particular case of the general result [76, 81, 156], let us point out that every first order dynamic equation on a manifold Z can be represented as Lagrange equations for a Lagrangian on the tangent bundle TZ, but not in an explicit form. •


Let M be a manifold, coordinated by (q'). If it is a configuration space of conservative mechanics, the corresponding velocity phase space is the tangent bundle TM of M, while the cotangent bundle T'M plays the role of the momentum phase space.

DEFINITION 3.1.2. An autonomous second order dynamic equation on a manifold M is denned as a first order dynamic equation (3.1.1) on the tangent bundle TM, which is associated with a hoionomic vector field

E = q% + Ei(q^^)dl (3.1.2)

on TM. This vector field, by definition, obeys the condition

•/(•=•) = UTM,

where J is the endomorphism (1.2.38) and UTM is the Liouville vector field (1.2.6) on TM (see the notation (1.2.4)). □

The vector field (3.1.2) is called a dynamic vector Geld. In the literature, it is often termed a second order dynamic equation.

Let the double tangent bundle TTM be provided with the coordinates

(ql,?,$,?)■ (3.1.3)

With respect to these coordinates, the second order dynamic equation defined by the hoionomic vector field 5 (3.1.2) reads

q' = g', ? = =?(?,?). (3.1.4)

By solutions of the second order dynamic equation (3.1.4) are meant the curves c : () —» M in a manifold M whose tangent prolongations c : () —► TM are integral curves of the hoionomic vector field H or, equivalently, whose second order tangent prolongations c live in the subbundle (3.1.4). They satisfy the second order differential equations

c'(t) = S(c'(t),c'(t)).

A particular case of second order dynamic equations on a manifold M is that of geodesic equations on the tangent bundle TM.


Given a connection

K = dq>® (dj + K)di)

on the tangent bundle TM —> M, let

K :TMxTM -» TTM M

(3.1.5)

be the corresponding linear bundle morphism over TM which splits the exact sequence

0 —► VMTM >-+ TTM —>TM xTM —* 0.

DEFINITION 3.1.3. A geodesic equation on T M with respect to the connection tf is defined as the image

q' = <f, ? = K)q> (3.1.6)

of the morphism (3.1.5) restricted to the diagonal TM C TM x TM. D

By a solution of a geodesic equation on TM is meant a geodesic curve c in M, whose tangent prolongation c is an integral section (a geodesic vector Held) over c C M for the connection K. The geodesic equation (3.1.6) can be written in the form

fdtf = Ktf, where q'{qi) is a geodesic vector field (which exists at least on a geodesic curve), while q'di is the formal operator of differentiation (along a curve).

It is readily observed that the morphism K \TM is a holonomic vector field on TM. It follows that any geodesic equation (3.1.5) on TM is a second order equation on M. The converse is not true in general. Nevertheless, we have the following theorem.

T H E O R E M 3.1.4. [136]. Every second order dynamic equation (3.1.4) on a manifold M defines a connection K-= on the tangent bundle TM —> M whose components are

K) = \60. (3.1.7)

D

no CHAPTER 3. HAMILTONIAN SYSTEMS

However, the second order dynamic equation (3.1.4) fails to be a geodesic equation with respect to the connection (3.1.7) in general. In particular, the geodesic equation (3.1.6) with respect to a connection K determines the connection (3.1.7) on TM —» M which does not necessarily coincide with K. A second order equation E on M is a geodesic equation for the connection (3.1.7) if and only if E is a spray, i.e.,

[uTM, E] = =.,

where UTM is the Liouville vector field (2.4.9) on TM, i.e.,

H* = ayrywy. In Section 4.3, we will improve Theorem 3.1.4 (see Proposition 4.3.3 below).

An extensive literature is devoted to the inverse problem for second order dynamic equations in Lagrangian mechanics. In Section 4.9, we will touch on the following two aspects of this problem in time-dependent mechanics:

• the condition for an Euler-Lagrange type operator to be an Euler-Lagrange one as the particular inverse problem in field theory,

• and the corresponding relation between Newtonian and Lagrangian systems.

We refer the reader to [136] for a survey on the inverse problem in conservative Lagrangian mechanics. Recall that, in accordance with Remark 3.1.2, a second order dynamic equation on a manifold M, being a first order dynamic equation on TM, can always be thought of as Lagrange equations for a Lagrangian on the double tangent bundle TTM of M.

In this Chapter, we will give two examples of the inverse problem for second order dynamic equations in conservative Hamiltonian mechanics.

• As is well known, the tangent bundle TM fails to possess any canonical Pois-son, symplectic, or presymplectic structures. Nevertheless, there are procedures to study a second order dynamic equation as the Hamilton ones with respect to some Poisson structure on TM (see Remark 3.2.3 below).

• Lagrange equations for a Lagrangian £ on the tangent bundle TM can be also seen as Hamilton equations with respect to the presymplectic structure (or the symplectic structure, if £ is regular), defined by this Lagrangian on TM (see Examples 3.3.2 and 3.4.2 below).

3.2. POISSON HAMILTONIAN SYSTEMS 111

3.2 Poisson Hamiltonian systems

Let (Z, w) be a A:-dimensional Poisson manifold and H a real function on Z.

DEFINITION 3.2.1. A Poisson Hamiltonian system (w,H) on a manifold Z for an (autonomous) Hamiltonian H with respect to the Poisson structure w is the set

SH= \J{V€TZZ:V- w\dH){z) = 0}. 26Z

(3.2.1)

a

A solution of the Hamiltonian system (3.2.1) is a vector field tfona submanifold N C Z, which takes its values into TN n Sa

lt is readily observed that the Poisson Hamiltonian system Sw (3.2.1) has a unique solution which is the Hamiltonian vector field

&H = wl{dH) (3.2.2)

for the Hamiltonian H, that passes through any point of Sn- Hence, 5« is a first order equation in accordance with Definition 3.1.1. It is called the Hamilton equation for the Hamiltonian H with respect to the Poisson structure w.

With respect to the local canonical coordinates {q',Pi,zA) (2.3.7) for the Poisson structure w, the Hamilton equation (3.2.1) reads

while the vector field $n (3.2.2) takes the form

■&-H = PHdi - diHdK

Its integral curves r : () —► Z satisfy the equations

r* = d*n o r, U = —diH o r, fA = 0. (3.2.3)

Recall that, given a dynamic equation (3.1.1) associated with a vector field u on a manifold Z, the Lie derivative Lu of a function f on Z along u can be treated as the evolution of / along solutions of this dynamic equation. If

Uf = 0, (3.2.4)

the function / is constant on any solution of the above-mentioned dynamic equation or equivalently on integral curves of u. A function / ^ const, is called a first integral

g' = dm, pj = -djH, zA = 0,


of motion if it obeys the equation (3.2.4). Prom the geometric viewpoint, this means that the vector field u is tangent to the level surfaces of the function / at its regular points, where df =£ 0.

If a dynamic equation is a Hamilton equation with respect to a Poisson structure, one can find its first integrals of motion as functions in involution with a Hamiltonian as follows.

Let H be a Hamiltonian of a Poisson Hamiltonian system (w,H). The Lie derivative of an arbitrary real function / on Z along the Hamiltonian vector field tin for H reads

UJ = $H\d}={HJ}. (3.2.5)

The equality (3.2.5) is called the evolution equation. Substituting solutions r of the Hamilton equations (3.2.3) in (3.2.5), we obtain the evolution

dt(for) = {H,f}or

of a function / along the integral curves of the Hamiltonian vector field ■d-^. In particular, if

{HJ} = 0,

the function / is a first integral of motion of the Hamiltonian system (w,H).

Example 3.2.1. A Hamiltonian "H itself is obviously a first integral of motion of the Hamiltonian system (tu,W). •

It is easily seen that, if / and / ' are first integrals of motion of a Poisson Hamiltonian system, so is their Poisson bracket { / , / '} . It follows that first integrals of motions constitute a Lie algebra.

A vector field v on Z is called an infinitesimal symmetry of a Hamiltonian H if

KH = 0.

In particular, if / is a first integral of motion of a Poisson Hamiltonian system (w,H), the Hamiltonian vector field $/ for / is an infinitesimal symmetry of the Hamiltonian H, i.e.,

U,n = 4f\dH = -{H,f}=0.

3.2. POISSON HAMILTONIAN SYSTEMS 113

Let us turn now to the inverse problem in conservative Hamiltonian mechanics, mentioned in Remark 2.3.5. It has the following solution.

PROPOSITION 3.2.2. For any first order dynamic equation (3.1.1) on a manifold Z, there exists locally a generating Poisson Hamiltonian system (w, H) on Z such that (3.1.1) is locally a Hamilton equation. □

Proof. If u = 0, the statement is obvious. Let u{z) / 0 at a point z € Z. There exists a coordinate system (q1,. .. , qk) on an open neighbourhood of z such that

U = W and [u'v] = 0' v = w Then u A v is a local Poisson bivector field of rank 2 (see Example 2.3.2). It is readily observed that u is locally a Hamiltonian vector field for the function q2{zx) with respect to the Poisson structure u A » . QED

Proposition 3.2.2 leads us to the conditions for the inverse problem of a first order dynamic equation (3.1.1) to have a global solution.

PROPOSITION 3.2.3. [59, 82]. Let u be a nowhere vanishing vector field on a manifold Z. If there exist

• a nowhere vanishing vector field v on Z such that [u, v] = 0 everywhere on Z,

• and a function f on Z such that L u / = 0 and L„ / = 1,

then u is the Hamiltonian vector field for the function / with respect to the Poisson structure u A » . □

Example 3.2.2. Every first order dynamic equation (3.1.1) on a manifold Z is a Hamilton equation on the product RxZ with coordinates (t, zx) for the Hamiltonian Ti = t and with respect to the Poisson structure w = u A dt. We refer the reader to [59] for the physically interesting examples. •

R e m a r k 3.2.3. The method based on Propositions 3.2.2 and 3.2.3 can be applied to study the existence of a generating Poisson structure for a second order dynamic equation (3.1.4) on a manifold M defined by a holonomic vector field E (3.1.2) on the tangent bundle TM. In particular, for any second order dynamic equation


(3.1.4) on M, there exists locally a generating Poisson Hamiltonian system {w,H) on TM such that the associated holonomic vector field S is the Hamiltonian vector field for the Hamiltonian H with respect to the Poisson structure w. The necessary condition for a Poisson structure w on the tangent bundle TM to be a generating Poisson structure for a holonomic vector field 5 on TM is

L=(w) = [E,tu] = 0.

We refer the reader to [182] for a detailed analysis of this condition. •

3.3 Symplect ic Hami l ton ian systems

Let (Z, fi) be a symplectic manifold. The notion of a symplectic Hamiltonian system is a repetition of Definition 3.2.1 (see [127]).

DEFINITION 3.3.1. A symplectic Hamiltonian system (ft.H) on a manifold Z for a Hamiltonian H with respect to the symplectic structure fi is the set

Sn = \J{veTzZ: v\n + dH(z) = 0}. 2€Z

(3.3.1)

□

As in the general case of Poisson systems, the symplectic Hamiltonian system (fi, H) has a unique solution which is the Hamiltonian vector field tfK such that

V | fi = -dH

for the Hamiltonian H, that passes through any point of Sn- Hence, S-H is a first order dynamic equation, called the Hamilton equation.

With respect to the local canonical coordinates {q\pi) for the symplectic structure fi, the Hamilton equation (3.3.1) reads

q* = dxH, pj = -djTi. (3.3.2)

In the symplectic case, there is one-to-one correspondence between the Hamiltonian systems and the Hamiltonian vector fields on a symplectic manifold (Z, f2).

The first integrals of motion of a symplectic Hamiltonian system and infinitesimal symmetries of a Hamiltonian are defined as a repetition of those for a Poisson Hamiltonian system.

3.3. SYMPLECTIC HAMILTONIAN SYSTEMS 115

Recall that a Hamiltonian system on a 2m-dimensional symplectic manifold is called completely integrable if there exist m first integrals of motion which are pairwise in involution and whose differentials are linearly independent on a dense open subset of that manifold.

Example 3.3.1. Let Z = T*M be a symplectic manifold provided with the canonical symplectic form fi (2.4.2). In accordance with Example 2.4.7, any Hamiltonian system (ft, H) on the symplectic manifold (T'M, fi) is the Lagrangian submanifold Sn = $K{T'M) (2.4.10) of the symplectic manifold TT'M equipped with the symplectic form Q (2.4.9). The generating function of this submanifold, given by the coordinate relations

q' = &n, pi = -aoi, (3.3.3)

is the Hamiltonian 7i(qx,pi). Any Hamiltonian H on the symplectic manifold T'M defines the fibred morphism over M

H d= TT.TM o a o $H : T'M -^TT'M -> T'TM -» TM,

qio;R=d,n, (3.3.4)

called the Hamiltonian map. •

Example 3.3.2. Any non-degenerate autonomous Lagrangian system can be seen as a symplectic Hamiltonian system as follows.

An autonomous Lagrangian is defined as a real function C on the tangent bundle TM of an event space M. Throughout this Chapter, we will use the compact notation

7Ti = d{C, ■Kji — djdiC.

Using the tangent-valued form 4>j (1.2.39) on the tangent bundle TM, let us introduce the Poj'ncare-Cartan 1-form

He =4>j\d£ = ntdq\

Its differential

tie = dnt A dqi = ■K^dc? A dq' + djitidq1 A dq' (3.3.5)

is a symplectic form on TM if and only if the Lagrangian C is regular, i.e., its Hessian det (TT^) is nowhere vanishing.


If a Lagrangian C is regular, one can consider the symplectic Hamiltonian systen (Qci Ec) with respect to the symplectic form Qc for the Hamiltonian

Ec = v-TMJdC — C = q'lTi — £, (3.3.6)

called the energy function, where we make use of the Liouville vector field UTM (1.2.6) on TM. Then we come to the Hamilton equation for the Hamiltonian vector field dc such that

ticltoL = ~dEc. (3.3.7)

With respect to the coordinates (3.1.3) on the double tangent bundle TTM, the equation (3.3.7), called the Cartan equation for the Lagrangian £, reads

T«(q* - ?') = o, (3.3.8a) (3.3.8b)

Since the Lagrangian C is regular, the equation (3.3.8a) reduces to the equality

q' = <?', (3.3.9)

while the equation (3.3.8b) comes to the Lagrange equations

d,C — (pifji — qidj-Ki = 0 (3.3.10)

for the Lagrangian £. The condition (3.3.9) means that, in the case of a regular Lagrangian £, the

Hamiltonian vector field dc is holonomic. It defines the second order dynamic equation on TM which is equivalent to the Lagrange equations (3.3.10).

Let us consider the fibre bundles TT'M and T'TM provided with the coordinates (q',Pi,q',Pi) and (q',q', (?,, <ji), respectively, and recall the isomorphism a (1.1.9):

a : TT'M S T'TM, R — . ft, ft ^ q,.

The symplectic form fi£ (3.3.5) yields the fibred morphism (1.2.33) over TM:

fij. : 7TM -> TTM, q\ o nb

£ = q%y ,

diC - q'lTji - qJ9j7r, + (qJ - q')dtTX] = 0.

<7i o n b£ = -q>TT:i + q J(3j7rj - ^ T T , ) ,

3.3. SYMPLECTIC HAMILTONIAN SYSTEMS 117

and the corresponding fibred morphism

Q"1 o ClL : TTM ~> T'TM — TT'M, P, O a_ 1f^. = -q>TCjx + q'idiTTj - djTTt), p , O Q _ 1 O £lc = q*7Ty.

(3.3.11)

We have the relation

cic = -n£nT-™ = -{a~lo n).yn,

where fi£ is the tangent lift (1.2.25) of Q,c onto T T M and by CIT-TM is meant the canonical symplectic form on the cotangent bundle T'TM of TM.

It is readily observed that the system of Lagrange equations (3.3.10) is a Lagran-gian submanifold of the symplectic manifold (TTM, &c)- Accordingly, the image of the Lagrange equations (3.3.10) into TT'M by the morphism a - 1 o £lc (3.3.11) is the Lagrangian submanifold

Pi = ^TTjn pi = -d,C + tfdi-Kj

of the symplectic manifold (TT'M, fi). The generating function of this Lagrangian submanifold is the energy function Ec (3.3.6).

Every Lagrangian £ on the configuration space TM yields the fibred morphism over M

£d=irT.M o Q"1 o dC : TM — T'TM -> TT'M -> T'M;

p, = diC,

(3.3.12)

(3.3.13)

called the Legendre map. A Lagrangian £ is regular if and only if the corresponding Legendre map £ is a local diffeomorphism. A Lagrangian £ is called hyperregular if the Legendre map £ is a diffeomorphism.

The tangent map TC to the Legendre map C is the fibred morphism

TC : TTM -> TT'M,

PioT£ = TTi, p , = ix%Jq' + djiTiq>,

over M. Then the image of the Lagrange equations (3.3.10) into TT'M by TC is another Lagrangian submanifold

Px = TTt, Pi = di (3.3.14)

fi£ = - [ d ( - f Try + q*(flbi - djTT,)) A <tf + rf(qJ'%) A <#)],


of the symplectic manifold (TT'M, ft). This is exactly the Lagrangian submanifold (2.4.11), defined by the differential dC (see Example 2.4.7). Its generating function is C.

One can find the conditions of a hypen-egular Lagrangian £ on the tangent bundle TM and a Hamiltonian H on the cotangent bundle T'M to define the same Lagrangian submanifolds (3.3.14) and (3.3.3) of the symplectic manifold (TT'M, Q). This takes place if and only if

uT-M\dH-H = CoH, pidlH-H = C(q\diH),

(3.3.15)

where UT-M is the Liouville vector field (1.2.5) on T'M —» M, or equivalently

uTM\dC - £ = H o £,

qidiC-C = H(qi,diC),

where UTM is the Liouville vector field (1.2.6) on TM —► M. Taking the differential of the equation (3.3.15), we find that

H = C~\ (diC) oH= -dtH.

Then we have the commutative diagram

TM - ^ TTM

C \ \ TC

T'M —>TT'M

•

The constructions in Examples 3.3.1, 3.3.2 lead to description of Hamiltonian and Lagrangian dynamics of autonomous systems in terms of Lagrangian submanifolds and to constructing unified Lagrangian-Hamiltonian formalism on the phase space TT'M [111, 176, 177] (see Example 3.4.2 below).

3.4 Presymplectic Hamiltonian systems

The notion of a Hamiltonian system is naturally extended to presymplectic manifolds (see [12, 60, 138, 172]).

3.4. PRESYMPLECTIC HAMILTONIAN SYSTEMS 119

DEFINITION 3.4.1. Let Z be a fc-dimensional manifold equipped with a presym-plectic form fi, and let H be a real function on Z. A presymplectic Ha.miltonia.ii system for a Hamiltonian H, like (3.3.1), is the set

Snd= \J{ve TZZ : vjfi + dH(z) = 0}. zez

(3.4.1)

□ A solution of the presymplectic Hamiltonian system (3.4.1) is a Hamiltonian

vector field d-u for H which lives in Sn- It satisfies the equality

tfwjfi = -<m. (3.4.2)

In comparison with the symplectic case, the form £1 is degenerate, the set (3.4.1) fails to be a submanifold in general, and a solution of the equation (3.4.2) does not necessarily exist everywhere on the manifold Z.

Note that, for any point z e Z, the fibre

{v€TzZ : v\n + dH(z) = 0 } (3.4.3)

of the set S-H over z is an affine space modelled over the vector space

Ker2Q = {v G TZZ : u\v\Q = 0, Vu 6 TZZ}

which is the fibre over z of Kerfl (1.2.34) of the presymplectic form fi. The fibre (3.4.3) may be empty.

PROPOSITION 3.4.2. The equation

v\il + dH{z) = 0, v G TZZ, (3.4.4)

has a solution only at the points of the subset

N2 = {z G Z : u\<m{z) = 0, Vu G Ker2Q} C Z, (3.4.5)

i.e., where Kerzft C Ker zdH [60, 138]. □

Proof. Let a vector v G TZZ, satisfying the equation (3.4.4), exist. Then, the contraction of the right-hand side of the equation (3.4.4) with an arbitrary element u G KerzQ leads to the equality u\dH(z) = 0. In order to prove the converse assertion, it suffices to show that

dH(z) G Imnb,

http://Ha.miltonia.ii


that results from the inclusions

dft(z) £ Ann(Ker dH{z)) C Ann(Ker2fi) = Imfi*.

QED

It follows that, if Z ^ N2a. presymplectic Hamiltonian system is not a differential equation on Z. If N2 = Z, it is a differential equation, but not a dynamic equation.

From now on, let us suppose that a presymplectic form f2 is of constant rank, and that N2 (3.4.5) is a submanifold of Z, but not necessarily connected. Then, by virtue of Proposition 1.1.4, the kernel Kerfi is a closed vector subbundle (moreover, an involutive distribution) of the tangent bundle TZ. Accordingly, Sn \N7 is an affine bundle over N2. It admits a global section, but this section does not live necessarily in TN2 C TZ, i.e., is not a vector field on the submanifold N2 in general.

To obtain a solution of the presymplectic Hamiltonian system (3.4.1), let us consider a submanifold N C N2 C Z such that

Sn\NnTxN^9, V z e N,

or equivalently

dH(z) 6 Q\TN), Vze N.

If the morphism 0* \TN is of constant rank, then S-H H TN —> N is an affine bundle modelled over the vector bundle Ker Q n TN —» TV which may be 0. Sections of the affine bundle S-H fl TN —» TV are Hamiltonian vector fields $x (3.4.2) for the Hamiltonian W o n a submanifold N. If such a submanifold exists, it may be found by the following well-known algorithm [12].

Let us take the intersection

SH\N2nTN2

and project it to Z. We obtain the subset

N3 = KZ{SH\N,OTN2)CZ.

If N3 is a submanifold, let us take the intersection

SH\N3nTN3.

Its projection to Z gives a subset N4 C Z, and so on. Since a manifold Z is finite-dimensional, the procedure is stopped after a finite number of steps by one of the following results.

3.4. PRESYMPLECTIC HAMILTONIAN SYSTEMS 121

• There is i > 2 such that a set TVj is empty. This means that a presymplectic Hamiltonian system has no solution.

• A set N,, i > 2, fails to be a submanifold. It follows that a solution may exist, but not everywhere on A/j.

• If Ni+i = Nt for some i > 2, this is the desired submanifold N. If the morphism \TN is of constant rank, a solution of the presymplectic Hamiltonian system

(3.4.1) exists everywhere on N.

Remark 3.4.1. Since N2 ^ Z in general, one can consider another variant of how to solve a presymplectic Hamiltonian system. Let N2 (3.4.5) be a submanifold of Z. It can be provided with the pull-back presymplectic form Q2 = i)v2^- Then the pull-back presymplectic Hamiltonian system

SH, = U iv e TZN2 :v\Q2 + dH2(z) = 0} zew2

for the pull-back Hamiltonian H2 = i'^Ti on ./V2 can be examined. As for the initial system S-H, we find that the equation

v\Q2 + dH2(z)=0, veTzN2,

has a solution on the subset

N*i = {zeNi: u\i'N2dn{z) = 0, Vu € Ker2ft2 C TN2} C N2,

and so on. We obtain the chain of submanifolds

N2 D N'3 D N'4 ■ ■ ■

which is finite, and is stopped by one of the following results.

• There is i > 2 such that a set N- is empty.

• A set AT/, i > 2, fails to be a submanifold.

• If N[+l = N? for some t > 2, this is a desired submanifold N' C Z. If the presymplectic form i'N,Q, on N' is of constant rank, then there exists a solution of the presymplectic Hamiltonian system (i*N,£l, i^H) everywhere on N'.


It is readily observed that any solution $n C TN of the equation (3.4.2) on a submanifold N C Z also obeys the equation

tiH\i'N{n + dH) = 0

and, consequently, is a solution of the pull-back presymplectic system on N. It follows that N C N'. •

Solutions of the presymplectic Hamiltonian system 5« (3.4.1) on a submanifold N constitute an affine space modelled over the linear space of sections of the vector bundle KerCl n TN —» N. If this vector bundle is not 0, different solutions of the Hamilton equation (3.4.1) correspond to different first order dynamic equations on ./V. Therefore, a Hamilton equation with respect to a presymplectic structure on a manifold Z is not equivalent to a first order dynamic equation on Z in general.

Sections of the vector bundle Kerfi —» Z are sometimes called gauge fields in order to emphasize that, being solutions of the presymplectic Hamiltonian system (fi,0) for the zero Hamiltonian, they do not contribute to a physical state, and are responsible for the gauge freedom [12, 172]. At the same time, there are physically interesting presymplectic Hamiltonian systems, e.g., in relativistic mechanics when a Hamiltonian is equal to zero (see also Remark 4.8.9 below). In this case, Ker dH = TZ and the Hamilton equation (3.4.4) has a solution everywhere on a manifold Z.

Another pull-back construction is connected with the above-mentioned gauge freedom. Let a presymplectic form fi on manifold Z be of constant rank and let its characteristic foliation be simple, i.e., its leaves are fibres of a fibre bundle n : Z —» P over a symplectic manifold (P, $7/>) in accordance with Proposition 2.6.2. Then ft is the pull-back ir'Vlp of a symplectic form ftp on the base P by this fibration. Let a Hamiltonian H be the pull-back n'Hp of a function Tip on P. Then we have

Kerfi = VN C Ker dH,

and the presymplectic Hamiltonian system (f2, Ti) has solutions everywhere on the manifold Z. Any such solution tin is projected onto a unique solution of the symplectic Hamiltonian system (dp, Tip) on the manifold P, while gauge fields are vertical vector fields on the fibre bundle Z —» P. This is the case of a gauge invariant Hamiltonian system, where P is called a physical phase space.

In the case of a presymplectic Hamiltonian system on a manifold Z, this manifold fails to be provided with a Poisson structure in general (see Remark 2.5.1), that is an essential problem for quantization. Nevertheless, by virtue of Propositions 2.5.3 and

3.5. DIRAC HAMILTONIAN SYSTEMS 123

2.5.4, every presymplectic form on a manifold Z can be represented as a pull-back of a symplectic form by the coisotropic imbedding. It follows that each presymplectic Hamiltonian system can be seen as a Dirac constraint system [26] that we will investigate in Section 3.6.

Example 3.4.2. Autonomous Lagrangian systems with constraints are often treated as presymplectic Hamiltonian systems (see [30, 61, 112, 113, 138] and references therein). Every Lagrangian £ on the tangent bundle TM of a configuration manifold M provides TM with the presymplectic form Qc (3.3.5). Let us consider the presymplectic Hamiltonian system (Qc, Ec) on TM for the Hamiltonian Ec (3.3.6). This is exactly the Cartan equation (3.3.8a) - (3.3.8b) for the Lagrangian £. Its restriction to the submanifold q' = q' (3.3.9) of the double tangent bundle TTM of M leads to the Lagrange equations (3.3.10). This means that solutions of the Lagrange equations are solutions of the Cartan equation which are dynamic vector fields on TTM.

Conversely, a symplectic Hamiltonian system (fi, H) on the cotangent bundle T'M can be seen as a degenerate Lagrangian system on the velocity phase space TT'M of the momentum phase space T'M as follows. Put the Lagrangian

£n = Piq' -U{q\pi) (3.4.6)

on this velocity phase space. It is readily observed that the pre-image in TTT'M of the Hamilton equation (3.3.2) on TT'M for the Hamiltonian H by the projection TTT'M -> TT'M is exactly the Lagrange equations (3.3.10) on TTT'M for the Lagrangian (3.4.6). It follows that they have the the same solutions r : () —► T'M.

It should be emphasized, that, in general, neither Cartan equations nor Lagrange equations are differential equations in a strict sense. They are differential equations, e.g., when the Hessian matrix 7iy, of £ is of constant rank. •

3.5 Dirac Hamiltonian systems

There is an extensive literature devoted to autonomous mechanical systems with constraints (see, e.g., [121, 127, 137, 166, 172] and references therein). In this and the next Sections, we will deal only with autonomous Hamiltonian systems with constraints. We leave the constraints in time-dependent Lagrangian and Hamiltonian mechanics for Chapters 4 and 5.


Let (Z, fi) be a 2m-dimensional symplectic manifold and H a Hamiltonian on Z. Let N be a (2m - n)-dimensional closed imbedded submanifold of Z, called the primary constraint space or simply a constraint space. We aim to investigate the following two problems:

(A) solutions of the Hamiltonian system (Q, H) on a manifold Z, which live in the constraint space N,

(B) and the restriction of the Hamiltonian system (fi, H) to the constraint space N.

Remark 3.5.1. The following local relations will be useful in the sequel. Let the constraint space N be given locally by the equations

fa{z) = 0, o = l , . . . , n , (3.5.1)

where / a(z) are local functions on Z, called the primary constraints. Let us consider the set

IN = Ken'NcQ°{Z) (3.5.2)

of real functions f on Z which vanish everywhere on N, i.e., i'Nf = 0. This is an ideal of the associative commutative algebra 0°(Z) such that

D°(Z)/IN = 0°(N)

(cf. (2.6.4)). Its elements / are written locally in the form

/ = (3.5.3)

where ga are functions on Z. This means that the ideal //v is locally generated by the constraints /„. We will call {/a} a local basis for the ideal 1^.

Let dIN be the submodule of the D°(Z)-module O'(Z) , which is generated locally by the exterior differentials df of functions f £ IN. Its elements are the finite sums

a = / , € IN, 9' € 0°(Z).

By virtue of (3.5.3), they are given by local expressions

a=J2(9adfa + fa4>a), a = l

(3.5.4)

T.9'df„ i

a = l


where g" are functions and 4>a are 1-forms on Z. It should be emphasized that the expressions (3.5.3) and (3.5.4) are fulfilled

locally on a domain where the constraint space N is given by the equations (3.5.1).

The solution of problem (A) is obvious. It exists if a Hamiltonian vector field ■d-H, restricted to the constraint space N, is tangent to N, i.e.,

SH\N = \J{veTzZ: v\Q + dH{z) = 0} C TN. (3.5.5)

Then integral curves of the Hamiltonian vector field tin do not leave N. The condition (3.5.5) is satisfied if and only if

{H, IN} C IN, (3.5.6)

i.e., if and only if the Hamiltonian H belongs to the normalizer I(N) (2.6.6) of the ideal IN- With respect to the local basis {fa}, the condition (3.5.6) reads

#n\dfa = (3.5.7)

where gc are functions on Z. If the relation (3.5.6) does not hold, let us introduce the secondary constraints

{HJa} = 0.

If the collection of primary and secondary constraints is not closed with respect to the relation (3.5.7), let us add the tertiary constraints

{H,{H,fa}} = 0,

and so on. If a solution exists anywhere on N, the procedure is stopped after a finite number of steps by constructing the complete system of constraints. The complete system of constraints defines the final constraint space, which do not include the points of Z where the Hamiltonian vector field i?M is transversal to the primary constraint space N.

From the algebraic viewpoint, we find an ideal 7nn of D°(Z) which is the minimal extension of the ideal IN such that

{W, /fin} C /fin-

{«,/«} n

•


In the framework of the algebraic theory of constraints, problem (A) can be reformulated as follows.

Let AT be a closed imbedded submanifold of a symplectic manifold (Z, ft) and IN

(3.5.2) the ideal of functions which vanish everywhere on N. All its elements are said to be constraints. We aim to find a Hamiltonian, called an admissible HamiltoniaD, on Z such that the symplectic Hamiltonian system (ft, H) has a solution everywhere on the constraint space N. One can treat N as a final constraint space. Moreover, we may take an ideal IN whether it is an ideal of functions vanishing on some submanifold N C Z or not.

In accordance with the condition (3.5.6), any element of the normalizer I(N) (2.6.6) is an admissible Hamiltonian.

Let us consider the intersection I'{N) = I(N) n IN (2.6.7). Its elements are called the first-class constraints, while the remaining elements of IN axe the second-class constraints. The first-class constraints make up a Poisson subalgebra of the normalizer I(N) which, in turn, is a Poisson subalgebra of the Poisson algebra D°{Z) on the symplectic manifold (Z, ft) (see Section 2.6). Recall that IN C I'(N), i.e., the products of second-class constraints are also first-class constraints.

Example 3.5.2. If N is a coisotropic submanifold of Z, then IN C I{N) and I'{N) = IN- It follows that we have only the first-class constraints. •

The admissible Hamiltonians which are not first-class constraints are the representatives of the non-zero elements of the quotient I(N)/I'(N), which is the reduction of the Poisson algebra D°(Z) via the ideal IN (see Definition 2.6.8). In particular, let the presymplectic form i'Nfi on N be of constant rank and its characteristic foliation be simple, i.e., it defines a fibration 7r : N —» P over a symplectic manifold P. In view of the isomorphism (2.6.8), one can think of elements of the quotient I(N)/I'(N) as being the Hamiltonians on the physical phase space P. It follows that the restriction of an admissible Hamiltonian H to the constraint space N coincides with the pull-back onto N of some function / on P , i.e.,

i-NH = *•/, f e o°(P).

Let us turn now to problem (B). Given a Hamiltonian Wonasymplectic manifold (Z, Q) and a constraint space N C Z, a Dirac Hamiltonian system (ft, H, N) on the constraint space N is defined as the subset

S.H = \J{ve TZZ : i'N(v\n + dri(z)) = 0} (3.5.8)


[127, 137]. Note that

SH\N C S.n.

Therefore, the set (3.5.8) is not empty. We will say that a Dirac Hamiltonian system on N has a solution if there is a

vector field on a neighbourhood of each point z 6 /V, which lives into the subset S.n

(3.5.8). For instance, every solution of problem (A) is also a solution of problem (B).

Example 3.5.3. Problem (B) has a complete solution in the germ terms in the following case. Let Z = T'M be a symplectic manifold equipped with the canonical symplectic form fi, and N its coisotropic submanifold. Then TN is a coisotropic submanifold of the symplectic manifold TT'M equipped with the symplectic form Q (2.4.9). Indeed, let utTTN and v€TTT'M\TTN. There exists a vector field TU on JV such that its canonical lift TU onto TN contains tt. There is also a vector field T„ on T'M such that its canonical lift TV onto TT'M contains v. Let u and v be projected over the same element q £ TN. By the virtue of the relation (1.2.23), one can then write

v\u\a = n&}Jf t t(«)Jfi = Kvz)K(z)J f t ) , z = xz{q).

In the case of an arbitrary u 6 TqTN, this expression differs from 0 because there exists a vector field T„ on a neighbourhood of z, which does not belong to OrthnTN since N is coisotropic. It follows that Orth-TTN does not include elements of TTT'M\TTN.

Let 7{ be a Hamiltonian on T'M. Then the symplectic Hamiltonian system Sn (3.3.1) is a Lagrangian submanifold of the symplectic manifold (TT'M,fy (see Example 2.4.7). A Dirac Hamiltonian system {Sl,H,N) on a (2m - n)-dimensional coisotropic submanifold TV has a solution if the coisotropic submanifold TN C TT'M and the Lagrangian submanifold Sw are not transversal. The germ of any such pair (TN,S\.) is symplectically equivalent to the pair

({,' = • . • = ,*» = 0}, {qa = 8PS{p)) a = 1 , . . . . 2m}} 0)) (q,p) eR4m,

where S is the germ of a function such that the rank of the tangent map to the morphism

P - » (d^S,.,.,d2nS)


at 0 is not maximal. We refer the reader to [48] for a detailed classification of the germs of such generating functions S. •

A Dirac Hamiltonian system (Q, H, N) is called compieteiy integrable if there exists its solution through each point of the set S.n- The obvious necessary condition for a Dirac Hamiltonian system to be completely integrable is the inclusion S.n C TN. In this case, a Dirac Hamiltonian system {Q, H, N) reduces to the symplectic Hamiltonian system (i'NQ, J/yW) on the constraint space N.

Let us inspect the above-mentioned necessary condition. Note that, for each point 2 G N, the fibre

{v e TZZ : i'N(v\Sl + dH{z)) = 0}

of the set S.n over z € N is a non-empty affine space modelled over the vector space

{v e TZZ : u\v\Q. = 0, Vu € TZN} (3.5.9)

which is the fibre of the Q-orthogonal space OrthnTW (2.4.6) of TN. It follows that OrthnT/V C TN. Therefore, a Dirac Hamiltonian system on the constraint space N is completely integrable only if N is a coisotropic submanifold of the symplectic manifold (Z, Q). Then we have

u\dH = 0, Vu e OrthnN.

In order to formulate a sufficient condition for a Dirac Hamiltonian system to be completely integrable, let us assume that the vector spaces (3.5.9) for all z G N are of the same dimension. Then the fi-orthogonal OrthnTTV of TN is a vector bundle over N, while the Dirac Hamiltonian system S.n —► ,/V is an affine bundle over ./V.

PROPOSITION 3.5.1. [127], Let a submanifold iVofa symplectic manifold (Z,f2) be coisotropic. Then OrthnTW is an involutive distribution on N. If the exterior differential dH is constant on the leaves of the corresponding foliation, then the Dirac Hamiltonian system (3.5.8) is completely integrable. □

Proof. Since the set S.n is not empty, it suffices to show that this set belongs to TN. Let v £ S.n and z = TTZ(V). Whenever u € OrthnTAf C TN over z, the equality

v\u\Sl = u\dH = 0

3.6. DIRAC CONSTRAINT SYSTEMS 129

holds. It follows that v € TZN. Since S.« C TN, there is a vector field ■d on N through any point of the Dirac Hamiltonian system S.«. Such a solution is not unique. If ■& is the above-mentioned vector field, then any vector field ■d + v where v lives in OxianTN C TN is also a solution of the Dirac Hamiltonian system. QED

Thus, one should consider completely integrable Dirac Hamiltonian systems on coisotropic submanifolds N. In this case, a Dirac Hamiltonian system (fl, H, N) reduces to the presymplectic Hamiltonian system on the constraint space N for the pull-back Hamiltonian i'NH with respect to the pull-back presymplectic form i'NQ. This is the case of a Dirac constraint system.

3.6 Dirac constraint systems

Let (Z, fi) be a 2m-dimensional symplectic manifold, and let Nbea closed imbedded submanifold of Z. Let H be a Hamiltonian on Z.

DEFINITION 3.6.1. A Dirac constraint system on the constraint space N is the set

SN-H™ \J{v e TZN : v\i%(fl + dH{z)) = 0}. (3.6.1)

D

In particular, we are dealing with Dirac constraint systems when a Hamiltonian description of degenerate autonomous Lagrangian systems is investigated (see, e.g., [58, 172] and references therein). They are called generalized Hamiltonian systems. In this case, a momentum phase space Z is the cotangent bundle T'M of a configuration space M. This phase space is provided with the canonical symplectic form. A primary constraint space iV C Z is an image of the Legendre map L (3.3.12) defined by a degenerate Lagrangian £ on the velocity phase space TM. In fact, one introduces some initial Hamiltonian H on T'M such that the energy function Ec (3.3.6) for C is the pull-back C"H of this Hamiltonian H by the Legendre map C. Thus, we come to a Dirac constraint system on the Lagrangian constraint space N. The goal consists in constructing a Hamiltonian H' on the momentum phase space T'M (but not uniquely in general) such that the solution dw of the symplectic Hamiltonian system (fi, H') on T'M provides a solution of the above-mentioned Dirac constraint system on a final constraint space.


In fact a Dirac constraint system SN-n is the pull-back presymplectic Hamil-tonian system (i'NQ, i'NH) on a submanifold N C Z for the Hamiltonian i'NH and with respect to the presymplectic form i'N£l on N. For instance, Proposition 3.4.2 shows that the equation

i'N(v\Q. + dH{z)) = 0, v g TZN,

has a solution only at the points of the subset

N2 = {z e N : u\i'NdH(z) = 0, Vu £ KerzfiN}

which is assumed to be a manifold. Such a solution, however, fails to be tangent to N2. Then one should repeat the algorithm for presymplectic Hamiltonian systems in Remark 3.4.1. Nevertheless, one can say more since the presymplectic system SN-H (3.6.1) on TV is the pull-back of the symplectic system (Q, H) on Z.

Let a (2m — n)-dimensional closed imbedded submanifold TV of Z be a finai constraint space of the Dirac constraint system (3.6.1). This means that the equation

v\QN + dHN(z) = 0,

has a solution at each point z £ N. By virtue of Proposition 3.4.2, this is equivalent to the condition

u\dHN(z) = 0, Vu G Ker2Qw , VzG N,

or

Ker nN = TNC\ OrthnTN C Ker H. (3.6.2)

We will reformulate these condition in algebraic terms. Let Is be the ideal of functions on Z which vanish everywhere on the final constraint space N. Let I(N) be the normalizer (2.6.6) of 1^ and I'{N) = I{N) H IN. It is readily observed that, restricted to TV, Hamiltonian vector fields dj for elements / of I'{N) with respect to the symplectic form Q on Z take their values into TN nOrthnTJV [97]. Then the condition (3.6.2) can be written in the form

{H,I'{N)}CIN, (3.6.3)

whereas

{H,IN}£IN (3.6.4)

v e TZN, O/v — ^TV^J Hpj = i*NH,

3.6. DIRAC CONSTRAINT SYSTEMS 131

in general. A glance at the relations (3.6.3) and (3.6.4) shows that, though the Dirac constraint system on N has a solution, the Hamiltonian vector field dn for the Hamiltonian H with respect to the symplectic form fi on Z is not tangent to N, and its restriction to N is not such a solution in general.

Example 3.6.1. Let the final constraint space N be a coisotropic submanifold of the symplectic manifold (Z, fi). Then IN = I'{N), i.e., there are only first-class constraints. In this case, the Hamiltonian vector fields of the Hamiltonian H and of other Hamiltonians H + / , / € IN, provide solutions of the Dirac constraint system on N. Note that, if a primary constraint space is coisotropic, we arrive at a Dirac Hamiltonian system, where the relation (3.6.3) is exactly the condition of Proposition 3.5.1. •

The goal of constructing a generalized Hamiltonian system is to find a constraint / € IN such that the modified Hamiltonian H + f would satisfy both the conditions

{H + f,I'(N)}cIN, {H + / , IN} C IN-

(3.6.5) (3.6.6)

The condition (3.6.5) is fulfilled for any / e IN, while (3.6.6) is an equation for a second-class constraint / .

Remark 3.6.2. Since 1^ is the ideal of functions vanishing on a manifold N, the normalizer of IN coincides with the normalizer of J'(JV) [97]. Therefore, the conditions (3.6.5) and (3.6.6) may be combined into

{H + f,I'(N)}cI'(N).

•

The equation (3.6.6) is solved in many concrete models, that implies separating first- and second-class constraints. The general problem lies in the fact that the collection of elements which generate If, C I'(N) is necessarily infinite reducible [97]. At the same time, the Hamiltonian vector fields for elements of IN vanish on the constraint space N. Therefore, one can apply the following procedure [137].

Since N is a (2m—n)-dimensional closed imbedded submanifold of Z, the ideal IN is locally generated by a finite basis {/„}, a = 1 , . . . , n, while N is defined locally by the equations (3.5.1). Let the presymplectic form Qn be of constant rank 2m-n — k, i.e., Kerzfl<2 has the dimension k < n at all points z £ Q. It defines a fc-dimensional


characteristic foliation on N. Since N C Z is closed, there exist locally k linearly independent vector fields ub on Z which, restricted to N, are tangent to the leaves of this foliation. They read

n

« 6 = E &"*/.. 6 = 1 , . . . , * , o=l

where g% are functions on Z and #/0 are Hamiltonian vector fields for constraints fa. Then one can choose the collection of constraints fa, b = 1 , . . . ,n, where the first k functions take the form

n 4>b=Y^9bfa< 6 = l , . . . , f c .

Let ■d^ be the Hamiltonian vector fields for these functions. One can easily justify the fact that

0fc|jv = u»|w, b-l,...,k.

It follows that the constraints fa belong to I'(N) \ I%- Therefore, they are called the first-class constraints, while the remaining fa.+\, ■ ■ ■, fa are the second-class constraints. We have the relations

n

{fa, fa} = J2 Cbc<t>a, b=l,...,k, c= l , . . . , n ,

where C£. are local functions on Z. It should be emphasized that the constraints fa, • • •, fa do not constitute any

local basis for I'{N), though fa,- ■ ■ ,fa make up a local basis for the ideal IN-Now let us consider a local Hamiltonian on Z

n %' = Ti + 2_, ^a0a, (3.6.7)

where A" are functions on Z. Since Ti obeys the condition (3.6.5), we find

{H,fa)=YJBabfa, 6 = 1 . . . , A ,

a=l

where B% are functions on Z. Then, the equation (3.6.6) takes the form

{H,fa}+ Y. A"{^ ,0c}= Y,D%, c=k+l,...n, o=t+ l 6=1

(3.6.8)

3.7. HAMILTONIAN SYSTEMS WITH SYMMETRIES 133

where Dbc are functions on Z. This is the system of linear algebraic equations for

the coefficients A°, a = k + 1 , . . . n, of second-class constraints. These coefficients are defined uniquely by the equations (3.6.8), while the coefficients \a, a = 1 , . . . , k, of first-class constraints in the Hamiltonian (3.6.7) remain arbitrary.

Then, being restricted to the constraint space N, the Hamiltonian vector field of the Hamiltonian H' (3.6.7) on Z provides a local solution of the Dirac constraint system on N.

We refer the reader to [137] for a global variant of the above procedure.

Example 3.6.3. If AT is a symplectic submanifold of Z, then I'(N) = 1%. Therefore, we have only second-class constraints, and the Hamiltonian (3.6.7) of a generalized Hamiltonian system is defined uniquely. •

R e m a r k 3.6.4. In concrete models, the final constraint space N fails to be given in advance. Therefore, a different procedure of separating first- and second-class constraints is usually applied (see, e.g., [39, 58, 172]), but its global treatment is under discussion. •

3.7 Hamiltonian systems with symmetries

In this Section, we summarize the well-known results connected with a group action on symplectic and Poisson manifolds [2, 116, 126, 130, 181]. By G throughout is meant a real connected Lie group.

We start from a symplectic manifold case. Let (Z, Q) be a symplectic manifold on which a real connected Lie group G acts on the left so that, whenever g € G, the mapping z *—> gz is a symplectic isomorphism of Z. Such an action of a group G is called a symplectic action.

Remark 3.7.1. The classification of transitive symplectic actions of connected Lie groups has been done (see [69, 72, 103]). •

Since G is connected, its action on a manifold Z is symplectic if and only if, for any element e of the right Lie algebra gr of a the group G, the corresponding vector field £c on Z is locally Hamiltonian, i.e.,

LCtfi = d(&jn)=0.


Let us suppose that it is a Hamiltonian vector field, i.e.,

££Jfi=-d/£, where Jc is a real function on Z.

DEFINITION 3.7.1. A symplectic action of a connected Lie group G on a symplectic manifold Z is called a ffamiitonian action if, whenever e 6 QT, there exists a function Jc on Z such that f£ is the Hamiltonian vector field for Jc. □

PROPOSITION 3.7.2. An action of a connected Lie group G on a symplectic manifold Z is Hamiltonian if and only if there exists a mapping, called a momentum mapping, from Z to the Lie coalgebra

J ■ Z - 0*

such that, whenever e € 0 r, the function

Jt(z) = (J{z),e), Veef l r , (3.7.1)

is that in Definition 3.7.1. □

Proof. If J exists, we obviously have a Hamiltonian action. Conversely, if £c for any e € gr is the Hamiltonian vector field for a function Je on a manifold Z, then the momentum mapping J is defined by the relation (3.7.1) applied to the basis {em} of the Lie algebra gT. QED QED

It is readily observed that, if j and J' are different momentum mapping for the same action of G on Z, then

d{J(z)-J'(z),e)=0,

Example 3.7.2. Let a symplectic form on Z be exact, i.e., ft = dO, and let 6 be G-invariant, i.e.,

Then the map J : Z —» 0*, given by the relation

<J(z),£) = (&j0)(z),

Le,0 = d(&J*) + & jn = o, Ve e a,..

Ve 6 flr,

i.e., J — J' = const, on Z.


is a momentum mapping. •

PROPOSITION 3.7.3. If a Hamiltonian H o n a symplectic manifold Z is invariant under a group G acting on Z, a momentum mapping J defines a family of first integrals of motion of the Hamiltonian system for H. □

Proof. We have

LfcW = Ze\dM = 0, Ve € flr,

and then

{H, J£) = 0, Ve € flr.

QED

Thus, the functions Je{z) (3.7.1) are first integrals of motion. However, they an not pairwise in involution in general. Let us find their Poisson brackets.

DEFINITION 3.7.4. A momentum mapping J is called equivariant if

J(gz)= Ad'g(J(z)), Vp e G,

where Ad'g is the coadjoint representation (1.5.3). □

Example 3.7.3. The momentum mapping in Example 3.7.2 is equivariant. In accordance with the relation (1.5.3), it suffices to show that

Je{gz) = JMg-ne)(z),

i.e.,

(^\8)(gz) = (£Ad,-i(«)J*)(*).

The latter results follow from the relation (1.5.2). •

Example 3.7.4. Let T'M be a symplectic manifold equipped with the canonical symplectic form fi (2.4.2). Let a group G act on M on the left by the generators

U = 4,(9)*-


The canonical lift of this action onto T'M has the generators

lm = e'mdi - PAe3md\ (3.7.2)

and preserves the canonical Liouville form 6 on T'M. In accordance with Example 2.4.4, the vector fields (3.7.2) are Hamiltonian vector fields for the functions

Jm = e'm(q)pi (3.7.3)

and, as follows from Example 3.7.3, there exists the equivariant momentum mapping

J = e\n{q)piem.

In the case of a G-invariant Hamiltonian on T'M, the functions (3.7.3) are first integrals of motion. •

In the case of non-equivariant momentum mappings, let us consider the difference

a(g) = J(gz) - Ad'g(J(z)). (3.7.4)

We refer the reader to [2] for proof that this difference is constant on a symplectic manifold Z, and fulfills the equality

o(gg') = o(g) +Ad'g(o(g')). (3.7.5)

DEFINITION 3.7.5. A map a : G —> g* is called the cocycle on a group G if the condition (3.7.5) holds. A cocycle a is said to be a coboundary on a group G, if there exists an element p, € g* such that

a(g) = p- Ad'g(p,). (3.7.6)

The equivalence classes of cocycles modulo coboundaries make up the cohomology group of the group G. □

By this definition, the difference (3.7.4) is the cocycle of the group G associated with the action of G on Z.

PROPOSITION 3.7.6. Each symplectic action of a group G on a symplectic manifold Z defines a cohomological class [a] of G. □


Proof. Let J and J' be different momentum mappings associated with a symplectic action of a group G on a symplectic manifold Z. Since the difference J— J' is constant on Z, then the difference of the corresponding cocycles a — a' is the coboundary (3.7.6) where n= J- J'. QED

In particular, an equivariant momentum mapping defines the zero cohomological class of the group G.

T H E O R E M 3.7.7. Given a momentum mapping J associated with a symplectic action of a group G on a symplectic manifold Z, the following relation

{Je,Jc,} = J[c,c>]-{Tea{e'),e) (3.7.7)

takes place (we refer the reader again to [2], where, however, the left Lie algebra is utilized and Hamiltonian vector fields differ in the minus sign from those we use). □

In the case of an equivariant momentum mapping as in Example 3.7.4, the relation (3.7.7) leads to the homomorphism

{Je ,Je'} = J\t,e'\ (3.7.8)

of the Lie algebra gr to the Lie algebra of functions on a symplectic manifold Z with respect to the Poisson bracket. These are the desired Poisson brackets of first integrals of motion for a G-invariant Hamiltonian system.

We will refer to this result in connection with the conservation laws in time-dependent mechanics (see Section 5.8).

Let now (Z, w) be a Poisson manifold, on which a connected Lie group G acts on the left so that, whenever g e G, the mapping z —> gz is a Poisson automorphism of Z. Such an action of a group G is called a Poisson action. Since G is connected, its action on a manifold Z is a Poisson action if and only if, for any element e of the right Lie algebra gr of a group G, the corresponding vector field £t on Z is an infinitesimal Poisson automorphism, i.e., the condition (2.3.3) holds. The equivalent conditions are

&({/,»}) = {&(/).$} + {/.&($)}. &({/,$}) = [&,*/](»)-fc.W). [&.*/] = *«.</).


where / , j £ D°(Z), and by dt is meant the Hamiltonian vector field (2.3.4) for a function / . Note that, by very definition of a Hamiltonian action, generators of a Hamiltonian action of a group G on a Poisson manifold (Z, w) are tangent to the leaves of the symplectic foliation on Z, and there is a Hamiltonian action of G on every symplectic leaf in the sense of symplectic geometry.

Definition 3.7.1, Propositions 3.7.2, 3.7.3, and Definition 3.7.4 are naturally extended to a Poisson action of a connected Lie group G on a Poisson manifold. Let us consider the case of an equivariant momentum mapping.

PROPOSITION 3.7.8. [181] An equivariant momentum mapping J : Z —> 0* is a Poisson morphism if the coalgebra a* is provided with the Lie-Poisson structure (2.3.11). a

There is the following reduction theorem.

THEOREM 3.7.9. [181]. Let (Z,w) be a Poisson manifold endowed with a Hamiltonian action of a connected Lie group G and the equivariant momentum map J. Let q be a point of g* such that

• q is a non-critical value for all the restrictions of J to the symplectic leaves FL of Z, i.e., the level set Zq = J~l(q) is a submanifold of Z and, for any symplectic leaf Ft, Fuj = ZqV\ FL is a submanifold of FL;

• intersections F^ of the submanifold Zq with symplectic leaves of Z are clean, i.e., TFu, = TFL f~l TZq, and so are its intersections with the orbits of G in Z.

Let a subgroup Gq C G be the stabilizer of the point q. Then intersections F^ make up a regular foliation on Z,, and its leaves are the orbits of the connected component of the unit in Gq for the action of Gq on Zq. Furthermore, if this foliation is defined by a submersion Zq —> P, the base P has a well-defined Poisson structure whose characteristic distribution is the projection of TC\TZq, where T is the characteristic distribution on (Z, w). This is exactly the reduced Poisson manifold of {Z,w) via (Z„ E = T(orbits G)) (see Definition 2.6.3). a

Note that if the momentum mapping J of a Hamiltonian action on a Poisson manifold is not equivariant, we have the relation

J(gz)=Ad'g(J(z))+<j(g,z), (3.7.9)

3.8. APPENDIX. HAMILTONIAN FIELD THEORY 139

similar to (3.7.4) in the symplectic case, where the function a(g, z) is constant on the leaves of the symplectic foliation on Z. Since now a(g, z) depends on z, we have different stabilizers of q for the action given by (3.7.9) on g* for every symplectic leaf. Therefore, the foliation on Zq is no longer regular.

We end this Section with the theorem which shows the possibility of reducing a Poisson Hamiltonian system possessing symmetries.

T H E O R E M 3.7.10. [129, 181]. Let (Z, N, E) be a Poisson reduction and P a reduced manifold of Z via (N, E). Let H be a Hamiltonian on Z such that the flow $ t of its Hamiltonian vector field dH preserves the submanifold N and the bundle E, and dH is an annihilator of E. Then $ , induces a flow $£ of Poisson automorphisms of P (see Proposition 2.6.5) along the Hamiltonian vector field dq for the Hamiltonian H on P, defined uniquely by the relation H \N= n'H. Moreover, we have

■dq o 7r = TIT O T3H.

a

Thus, one can say that, under the hypotheses of Proposition 3.7.10, the Poisson Hamiltonian system on a Poisson manifold Z reduces to a Poisson system on the reduced Poisson manifold P.

Example 3.7.5. Let (Z, Zq, E = T(orbits G)) be a Poisson reduction as in Theorem 3.7.9. If H is a G-invariant Hamiltonian on the Poisson manifold Z, the Poisson Hamiltonian system reduces to a Poisson system on the reduced Poisson manifold of (Z, w) via (Z„ E = T(orbits G)). •

3.8 Appendix. Hamiltonian field theory

As is well-known, applied to a field theory on a fibre bundle Y -+ X, the familiar symplectic techniques take the form of instantaneous Hamiltonian formalism on an infinite-dimensional momentum phase space [64], in contrast with the finite-dimensional velocity phase space JlY. The Hamiltonian counterpart of the La-grangian formulation of field theory is polysymplectic Hamiltonian formalism on the Legendre bundle n (2.9.7) provided with the polysymplectic form A (2.9.9) (see [57, 159] for a survey). Here we aim to give a brief exposition of this formalism


because its restriction to fibre bundles Q -> K leads in a straightforward manner to the Hamiltonian formulation of time-dependent mechanics [57, 161).

Let Y —> X be a fibre bundle over an n-dimensional manifold X, and

n = VY ACA'T'X) (3.8.1)

the Legendre bundle (2.9.7). It admits the composite fibration

TnX = 7T O 1TUY ■ fl —► V —> X. (3.8.2)

For the sake of convenience, we will call n —> Y the Legendre vector bundle, while the term Legendre bundle stands for the fibration 17 —» X.

Given fibred coordinates ( x \ y') on the fibre bundle Y —» X, the Legendre bundle (3.8.1) is equipped with the holonomic coordinates (xx,y',p*) together with the transition functions (2.9.8). These coordinates are compatible with the composite fibration (3.8.2) and are linear bundle coordinates on the vector bundle U —> Y. We will call them the canonical coordinates.

There is the canonical bundle monomorphism over Y

G : IT -^n/\T'Y®TX, Y Y

9 : (x\ y\p$) ~ -pfdy' Acj®dx,

(3.8.3)

which defines the tangent-valued Liouville form on the Legendre bundle IT Then the polysymplectic form

A = dp,A A dy{ A u <g> dx (3.8.4)

on II is defined as a unique TX-valued (n + 2)-form on II such that the relation

A\cf> =-d(G]<j>)

holds for any exterior 1-form 4> on X. Given the vector Legendre bundle II over a fibre bundle Y —» X, we have the

exact sequence

o — > U X A T " X ^-> zY —>n —>o, x

(3.8.5)

where Zy is the homogeneous Legendre bundle (2.9.3) and

Tzn : ZY —» II (3.8.6)


is an affine bundle over II with 1-dimensional fibres.

DEFINITION 3.8.1. Let h be a section of the fibre bundle (3.8.6). Then the pull-back

H = h*Z : II-> AT*Y,

H = pxdy* AUJ\- Hu,

(3.8.7)

of the canonical form B (2.9.4) on Zy by h is called the polysymplectic Hamiltonian form (or simply the Hamiltonian form). □

The exterior differential of the Hamiltonian form (3.8.7) is the pull-back

dH = h'Qz

on II of the multisymplectic form (2.9.5).

Example 3.8.1. Let

r = dxx <8> (dx + T\di)

be a connection on the fibre bundle Y —> X. Hence, we have the splitting

T : V'Y -» T'Y, T.dtf^ dyl - T\dxx,

of the exact sequence (1.1.8a). Then T also yields the splitting

hr-. n -» ZY, hr : pfdy' ® wA <-> pXdy{ A wx - pXT\u),

of the exact sequence (3.8.5). It follows that every connection T on the fibre bundle Y —» X defines the Hamiltonian form

Hr = h'rE,

Hr = pxdy{ ALJX- p , A I> , (3.8.8)

on the Legendre bundle n . •

PROPOSITION 3.8.2. [57, 159]. Hamiltonian forms on n constitute an affine space modelled over the linear space of horizontal densities

H = Hw:Yl^ AT'X (3.8.9)


on the Legendre bundle II —► X. They axe called Hamiltonian densities. □

This means that, if H is a Hamiltonian form and H is a horizontal density (3.8.9), then H - H is also a Hamiltonian form. Conversely, if H and H' are Hamiltonian forms, their difference H - H' is a Hamiltonian density (3.8.9).

Example 3.8.1 and Proposition 3.8.2 combine into the following.

COROLLARY 3.8.3. Every Hamiltonian form on the Legendre bundle n admits the decomposition

H = Hr- Hr= pxdyi A w A - PXT\UJ - Hru>, (3.8.10)

where T is a connection on Y —» X. O

Furthermore, every Hamiltonian form admits a canonical decomposition as follows. We mean by a Hamiltonian map any fibred morphism

$ : II -> JlY, Y

(3.8.11)

over Y. Its composition with the canonical morphism A (1.3.4) yields the bundle morphism

A o $ : n ^T'X®TY Y Y

represented by the TY-valued 1-form

* = dxx 9 (dx + $\(q)dt) (3.8.12)

on the vector Legendre bundle n —> Y.

Example 3.8.2. Let T be a connection on Y —» X. Then, the composition

f = ro7rny : n — y -> J1Y, f = dxx ® (dx + rA6ts),

is a Hamiltonian map. Conversely, every Hamiltonian map $ : n —» JlY yields the associated connection

T* = $ o 0

y[ o $ = &x{q), < ? e n ,


on Y —» X, where 6 is the global zero section of the vector Legendre bundle II —» Y. In particular, we have

r~ = r 1 r 1 •

•

PROPOSITION 3.8.4. [57, 159] Every Hamiltonian form H on the Legendre bundle II defines the associated Hamiltonian map

H : U -» J 1 ^

(3.8.13)

D

COROLLARY 3.8.5. Every Hamiltonian form H on the Legendre bundle n determines the associated connection

TH = HoO

on Y -> X. a

In particular, we have

T//J. — r ,

where HY is the Hamiltonian form (3.8.8) associated with the connection T on Y -> X.

COROLLARY 3.8.6. Every Hamiltonian form (3.8.10) admits the canonical splitting

H = HYH — H.

a

The following assertion generalizes Example 3.8.1.

PROPOSITION 3.8.7. Every Hamiltonian map (3.8.11) represented by the form (3.8.12) on n defines the associated Hamiltonian form

H* = $ j e = pfdy* Aw»-p*$'Aw,

y\oH = d{H.


where 0 is the tangent-valued Liouville form (3.8.3). □

In particular, if

HS = H,

then H = Hr for some connection r on Y. Hamilton equations in conservative mechanics are the equations of integral curves

of Hamiltonian vector fields. Hamilton equations in polysymplectic Hamiltonian formalism are the equations of integral sections for Hamiltonian connections as follows.

Let J ' n be the first order jet manifold of the Legendre bundle II —> X. It is equipped with the adapted fibred coordinates

(xA ,3/ , ,p,A ,^,pV).

We have the commutative diagram

I I n 52. Y

y]t ° J^ar = vl-

DEFINITION 3.8.8. A connection

y = dx*®(dx + y\d, + rtidl)

on the Legendre bundle n —► X is said to be a locally Hamiltonian connection if the exterior form

7J A = drf A dyl Aux- (7 |dpA - y^dy') A u (3.8.14)

is closed. D

Example 3.8.3. Every connection T on a fibre bundle Y —> X gives rise to the connection

f = dxx® [dx + rx(y)di + (3.8.15) (-d}rxtf + KX\P] - Kx\f})^\

j^J'j^Xjly


on the Legendre bundle II —» X, where A" is a symmetric linear connection (1.4.9) on TX. The connection f (3.8.15) obeys the relation

fjA = d ( r j 6 ) .

It follows that T is a locally Hamiltonian connection. •

Thus, locally Hamiltonian connections always exist on the Legendre bundle n —* X, and every connection r on Y —> X gives rise to a locally Hamiltonian connection on n -> X.

It is easily observed that a connection 7 on the fibre bundle n —> X is a locally Hamiltonian connection if and only if 7 fulfills the conditions

dW, ~ dill = 0, d.7^ - %"!& = 0, d/7i + « = 0 .

(3.8.16) (3.8.17) (3.8.18)

Using the relation (3.8.18), we find that the second term in the right-hand side of the expression (3.8.14) is a closed form. Then, in accordance with the relative Poincare lemma, this expression is brought locally into the form

7JA = d(pxdy* A u j - Hyu) = dH7, (3.8.19)

where H-, is a local function on n such that

l l = W , 7AA, = -dtUr

Given a connection T on the fibre bundle Y —> X, the local form H^ in the expression (3.8.19) can be written as

H1 = H r - HVUJ,

where Hpui is a local horizontal density on n —» X. In accordance with Proposition 3.8.2, it follows that / / 7 defines the local section

h1:(xx,y\p?)~(xx,y\plp=-HJ

of the fibre bundle ZY -» n, i.e., H^ is a local Hamiltonian form. Thus, we have proved the following.


PROPOSITION 3.8.9. For every locally Hamiltonian connection 7 on the Legendre bundle n —► X, there exists a local Hamiltonian form H in a neighbourhood of each point q € fl such that

7JA = dH.

a

Let us formulate the converse assertion.

DEFINITION 3.8.10. The Hamilton operator £H for a Hamiltonian form H on the Legendre bundle n —» X is denned to be the first order differential operator

EH :J1n^nA1T'U,

£H = dH-A = [(y\ - d\H)dp* - (pk + djTi)^] A u, (3.8.20)

where

A = dpxx A dy' AuA + px

Mdy' Au - y\dp* A u

is the pull-back of the polysymplectic form A (3.8.4) on J ' n . D

A glance at the expression (3.8.20) shows that the Hamilton operator EH is an afEne morphism over n of constant rank. Thereby, its kernel is an affine subbundle

£H = Ker £H -* n (3.8.21)

of the affine jet bundle J ' n —> U which is given by the coordinate relations

v\ = d\n,

This affine bundle is modelled over the vector subbundle of the vector bundle

r*x®vn->n n

(3.8.22)

which is defined by the coordinate relations

yl = o, PA, = 0

with respect to the fibre coordinates (FA.PAI) o n (3-8.22).

(3.8.23)

Pi = -diH.


Since <£« (3.8.21) is an affine subbundle, it is a closed imbedded submanifold of the jet bundle JlU —> X and, therefore, is a system of first order differential equations on II —» X in accordance with Definition 1.3.3.

DEFINITION 3.8.11. The first order differential equations € w (3.8.21) are called the Hamilton equations for the Hamiltonian form H on the Legendre bundle II. □

Since the subbundle <BH (3.8.21) is affine, it always admits a global section 7. Any such section is a connection on II —» X which meets the condition

£H o 7 = 0.

This condition takes the form

7JA = dH. (3.8.24)

It follows that every connection on II —» X which takes its values into the Hamilton equations £# is a locally Hamiltonian connection.

DEFINITION 3.8.12. A locally Hamiltonian connection 7 on the Legendre bundle n —> X is said to be a Hamiltonian connection associated with a Hamiltonian form H if 7 obeys the relation (3.8.24). □

Thus, we have proved the following.

PROPOSITION 3.8.13. Every Hamiltonian form on the Legendre bundle n has an associated Hamiltonian connection. D

We have the equations of a Hamiltonian connection associated with a given Hamiltonian form:

7i = d\H, 7AAi = - m

(3.8.25) (3.8.26)

By the equation (3.8.25), every Hamiltonian connection 7 for a Hamiltonian form H satisfies the relation

Jlirny 0 7 = / / , 7 = dxx ® (dx + diHdi + T&S*) ,

(3.8.27)


where H is the Hamiltonian map (3.8.13). It projects over the connection YH on Y —> X associated with the Hamiltonian form H. We have the following commutative diagram

j ' n

n ■£-> J1Y

o \ / * r„ Y

A glance at the equations (3.8.26) shows that there is a set of Hamiltonian connections associated with the same Hamiltonian form H. They differ from each other in soldering forms a on n —» X which obey the equations

5JQ = 0, 31 = 0,

and take their values into the constraint subbundle (3.8.23) of the vector bundle (3.8.22).

A classical solution of the Hamilton equations (3.8.21), by definition, is a section r of the Legendre bundle n —» X such that its jet prolongation J1r takes its values into the kernel of the Hamilton operator ZH (3.8.20). Then, r satisfies the differential equations

8^ = diU or, (3.8.28a) (3.8.28b)

Every integral section Jlr = 7 o r of a Hamiltonian connection 7 associated with a Hamiltonian form H is a classical solution of the corresponding Hamilton equations. Conversely, if r is a global solution of the Hamilton equations (3.8.28a) - (3.8.28b) for a Hamiltonian form H, there exists an extension of this solution

Jh-.riX) — j ' n

to a Hamiltonian connection which has r as an integral section. Substituting Jlr in (3.8.27), we obtain the identity

J'(7Tny or) = H or

J'*nr

si = o,

aAr,A = -diH o r.


for every classical solution r of the Hamilton equations (3.8.28a - (3.8.28b).

Remark 3.8.4. Note that the Hamilton equations (3.8.28a) - (3.8.28b) can be introduced without appealing to the Hamilton operator. They are equivalent to the relation

r'{u\dH) = Q

which is assumed to hold for any vertical vector field u on the Legendre bundle n-> X. •

Chapter 4

Lagrangian time-dependent mechanics

This Chapter is devoted to the formulation of time-dependent mechanics on a phase space of coordinates and velocities. This phase space is the first order jet manifold JlQ of a configuration bundle Q —► R. The familiar case of the direct product Q = R x M corresponds to the choice of a certain reference frame. For the sake of brevity, the above-mentioned formulation is called Lagrangian time-dependent mechanics although equations of motion are not necessarily Lagrange equations.

Connections provide the main ingredients in the Lagrangian formulation of time-dependent mechanics. These are the reference frames seen as connections on a configuration bundle Q —» R, the dynamic equations represented by connections on the jet bundle JlQ —» R, the dynamic connections on the affine jet bundle JlQ —» Q, the connections on the tangent bundles TQ —> Q and TJlQ —» JlQ, and the Lagrangian connections. For instance, we show that every non-relativistic dynamic equation on a configuration space Q can be seen as a geodesic equation with respect to some connection on the tangent bundle TQ —> Q.

Reference frames are necessarily involved when we deal with dynamic equations, forces, accelerations, and conservation laws in time-dependent mechanics.

The mass metric is another interesting object of Lagrangian and Newtonian systems, of the inverse problem, and of systems with holonomic and non-holonomic constraints.

Throughout this Chapter,

T T : Q — R (4.0.1)

151

152 CHAPTER 4. LAGRANGIAN TIME-DEPENDENT MECHANICS

is a fibre bundle whose typical fibre M is an m-dimensional manifold, while its base R is parameterized by the Cartesian coordinates t possessing the transition functions t' = t+ const. In the universal unit system, the physical dimension of t is equal to [length]. A fibre bundle Q —► R is endowed with bundle coordinates {t,q'}. For the sake of convenience, we also use the compact notation qx where q° = t.

4.1 Fibre bundles over R

In this Section, we point out the most important peculiarities of fibre bundles over R, which we refer to in the sequel.

1.

The base R of a fibre bundle (4.0.1) is provided with the standard vector field 3, and the standard 1-form dt which are invariant under the coordinate transformations f = t+ const. The same symbol dt also stands for any pull-back of the standard 1-form dt onto a fibre bundle over R. Note on one-to-one correspondence between the vector fields fdt and the real functions / on R. The similar correspondence between the horizontal densities 4>dt and the real functions 4> on a, fibre bundle Q — R takes place. Roughly speaking, we may neglect the contribution of TTR and T*R to several expressions. At the same time, one should be careful with such simplification in the framework of the universal unit system. For instance, the coefficient 0 of a horizontal density <j>dt has the physical dimension [length]-1, whereas a function (j> is physically dimensionless.

2.

Since R is contractible, any fibre bundle over R is obviously trivial. Different trivializations

rj) : Q = R x M (4.1.1)

differ from each other in the fibrations Q —> M, while the fibration Q —* R is once for all.

Let J ' Q be the first order jet manifold of a fibre bundle Q —» R (4.0.1). It is provided with the adapted coordinates (t,q',q't). Given a direct product

Q = R x M - > 1

4.1. FIBRE BUNDLES OVER K 153

coordinated by (f.,if), there is the canonical isomorphism

Jl{RxM) = R x T M , —i —■

(4.1.2)

that one can justify by inspection of the transition functions of the coordinates q[ and q , when the transition functions of qx are independent of t. Due to the isomorphism (4.1.2), every trivialization (4.1.1) yields the corresponding trivialization of the jet manifold

JlQ = 1 x TM. (4.1.3)

3.

Given the jet manifold JlQ of a fibre bundle Q —> R, the canonical imbedding (1.3.4) takes the form

A: J'Q^TQ,

X:(t,q\q't)^(t,q',i = l,q' = ql).

(4.1.4)

For brevity, we will write

A = ck = dt + q\di, (4.1.5)

where by dt is meant the total derivative. From now on, we will identify the jet manifold JlQ with its image in TQ.

Remark 4.1.1. Following precisely the expression (1.3.4), we should write the morphism A (4.1.5) in the form

A = dt ® (dt + qldi). (4.1.6)

With respect to the universal unit system, the physical dimension of A (4.1.5) is [length]"1, while A (4.1.6) is dimensionless. •

Using the morphism (4.1.4), one can define the contraction

JlQxT'Q - > Q x R , Q Q

(«|; t, 9.) •-» ^J fat + q,dq') = i + q\q„ (4.1.7)

where (i, g', t, q{) are the coordinates on the cotangent bundle T'Q.


4.

A glance at the expression (4.1.4) shows that the affine jet bundle J ' Q —> Q is modelled over the vertical tangent bundle VQ of the fibre bundle of Q —> R. As a consequence, we have the following canonical splitting (1.1.6) of the vertical tangent bundle VQ J ' Q of the affine jet bundle JXQ -» Q:

a : VQJlQ Si JlQ x VQ,

"(flf) = ft,

(4.1.8)

together with the corresponding splitting of the vertical cotangent bundle VqJlQ of JlQ -» Q:

a* : V<V'Q S J ' Q x VQ,

a*(a7j)=39i!

(4.1.9)

where dq\ and dq' are the holonomic bases for VQ JlQ and V'Q, respectively. Then the exact sequence (1.6.9a) of vertical bundles over the composite fibre bundle

JXQ -* Q - R (4.1.10)

reads

I ""' 1 0 —*VQJlQ <^VJlQ ^*JlQxVQ —*0.

Hence, we obtain the following linear endomorphism over J1Q of the vertical tangent bundle V JlQ of the jet bundle JXQ — R:

v^ioa^ony : VJ'Q^ VJlQ,

v(dj) = dl jjjgj = 0-This endomorphism obeys the nilpotency rule

v o v = 0.

(4.1.11)

(4.1.12)

Combining the horizontal splitting (1.3.11), the corresponding projection

pr2 : JXQ x T Q ^ JlQ xVQ^ VQJ'Q, Q Q

dt ~ -q\d\, dx -> d\,

4.1. FIBRE BUNDLES OVER R 155

and the inclusion VJlQ «-» TJlQ, one can extend the endomorphism (4.1.11) to the tangent bundle TJlQ:

v:TJlQ^TJlQ, v(dt) = -qldj vjdi) ■ g, g(g) = 0. (4.1.13)

This is called the verticaJ endomorphism. It inherits the nilpotency property (4.1.12). The transpose of the vertical endomorphism v (4.1.13) is

v' : T'JlQ -> T V ' Q , v'(dt) = 0, v'(dq') = 0, v*(dql) = 6\ (4.1.14)

where

9l = dqi - q\dt

are the contact forms (1.3.6). The nilpotency rule

v" o v* = 0

is also fulfilled. The homomorphisms v and v' are associated with the tangent-valued 1-form

v = ^ ® a?

in accordance with the relations (1.2.36) - (1.2.37). With the endomorphism {?*, one can introduce the vertical exterior differential

dy = v* o d,

acting on the algebra D'iJ^) of exterior forms on the jet manifold JlQ. For example, if / is a function on JlQ, we have

d.f = %f#.

5.

In view of the morphism A (4.1.4), any connection

r = dt ® (dt + ra<) (4.1.15)


on a fibre bundle Q -» R can be identified with a nowhere vanishing horizontal vector field

T = dt + r<9, (4.1.16)

on Q which is the horizontal lift rdt (1.4.7) of the standard vector field d, on R by means of the connection (4.1.15). Conversely, any vector field T on Q such that dt\T = 1 defines a connection on Q —► R. Of course, the integral curves

c : R D ( ) ^ Q , i(r) = l, ci(T) = roc(r)< r e ( ) ,

of the vector field (4.1.16) coincide with the integral sections

dtc\t) = (roc)'(t)

of the connection (4.1.15). Connections on a fibre bundle Q —* R constitute an affine space modelled over

the vector space of vertical vector fields on Q —» R. Accordingly, the covariant differential (1.4.5) associated with a connection r on Q —> R takes its values into the vertical tangent bundle VQ of Q —> R:

Dr: JlQ -+VQ,

q'oDr = q\ - P .

(4.1.17)

A connection T on a fibre bundle Q —> R is obviously flat. It yields a horizontal distribution on Q. The integral manifolds of this distribution are integral curves of the vector field (4.1.16) which are transversal to the fibres of the fibre bundle Q~*R.

COROLLARY 4.1.1. By virtue of Proposition 1.4.1, each connection T on a fibre bundle Q —» R defines an atlas of local constant trivializations of Q —» R such that the associated bundle coordinates (t,q') on Q possess the transition function q' —> q"(q3) independent of (, and

r = dt (4.1.18)

with respect to these coordinates. Conversely, every atlas of local constant trivializations of the fibre bundle Q —> R determines a connection on Q —> R which is equal to (4.1.18) relative to this atlas. D

4.1. FIBRE BUNDLES OVER R 157

A connection T on a fibre bundle Q —» R is said to be complete if the horizontal vector field (4.1.16) is complete.

PROPOSITION 4.1.2. Every trivialization of a fibre bundle Q - > R yields a complete connection on this fibre bundle. Conversely, every complete connection r on Q —» R defines its trivialization (4.1.1) such that the vector field (4.1.16) equals dt relative to the bundle coordinates associated with this trivialization. □

Proof. Every trivialization of Q —• R defines the horizontal lift of the standard vector field dt on R to the vector field r = dt on Q which is obviously complete. Then the corresponding connection on Q —» R is also complete. Conversely, let T be a complete connection on a fibre bundle Q —> R. The horizontal vector field T (4.1.16) is the generator of a 1-parameter group Gr acting freely on Q. The orbits of this action are integral curves of the vector field F. Hence, we obtain a projection

Q -» Q/Gr = M (4.1.19)

of Q along the above-mentioned integral curves onto some fibre of Q, e.g., Qt=o — M. Combining the projection (4.1.19) and the projection Q —> R gives a desired trivialization (4.1.1) of Q. QED QED

6.

Let J1J1Q be the repeated jet manifold of a fibre bundle Q —> R, provided with the adapted coordinates

(t,q\qlq\t),qlt)-

For a fibre bundle Q —* R, we have the canonical isomorphism k between the affine fibrations 7rn (1.3.13) and Jln\ (1.3.15) of JXJ[Q over JlQ, i.e.,

7rn ok = J07T01, kok^ld^J'Q,

where

q't0k = q\t), q\t) °k = q\, qtt °k = q'u- (4.1.20)

In particular, the affine bundle 7ru (1.3.13) is modelled over the vertical tangent bundle V JlQ of JlQ -» R which is canonically isomorphic to the underlying vector bundle JlVQ — JlQ of the affine bundle J 1 ^ (1.3.15).


By JQJ1Q throughout is meant the first order jet manifold of the affine jet bundle JlQ —► Q. The adapted coordinates on JQJlQ are (g \ q\, q\t).

For a fibre bundle Q -> 1 , the sesquiholonomic jet manifold PQ coincides with the second order jet manifold J2Q, coordinated by

(<.«*. «{,««)■

The affine bundle J2Q —» J ' Q is modelled over the vertical tangent bundle

VQ J ' Q £ ^ Q x VQ -» J ' Q (4.1.21)

of the affine jet bundle JXQ —► Q. There are the imbeddings

J 2 Q ^ T V ' Q " VQTQ * T2Q C TTQ,

>2 : (t, ?', «{, qtt) >-» (*, ?', $> * = 1,9* = <?!> ?l = ?«), TA o A2 : (t, q\ q\, g't) w (t, <?\ t = t = 1, q* = q< = g*, t = 0, <j* = q'tt),

(4.1.22)

(4.1.23)

where

(i,g\i,g',t,d/,i',<?)

are the coordinates on the double tangent bundle TTQ, by VQTQ is meant the vertical tangent bundle of TQ —» Q, and T2Q C TTQ is a second order tangent space, given by the coordinate relation i = i.

Due to the morphism (4.1.22), any connection £ on the jet bundle JlQ —> R is represented by a horizontal vector field on JlQ such that £\dt = 1.

Example 4.1.2. A connection on the jet bundle J*Q —» R is defined as a section of the affine bundle irn (1-3.13). A connection T (4.1.16) on a fibre bundle Q —» R has the jet prolongation to the section J1F of the affine bundle J ' T Q . By virtue of the isomorphism k (4.1.20), every connection r on Q —» R gives rise to the connection

JTd=ko J T : JlQ^ J ' J ' Q ,

JT = dt + Pd , + dtrdl (4.1.24)

on the jet bundle J^Q -* R. •

A connection on the jet bundle J1Q —» R is said to be holonomic if it is a section

£ = 4®(A + flfo+fflf)


of the holonomic subbundle J2Q —> JlQ of J1 JlQ -* JlQ. In view of the morphism (4.1.22), a holonomic connection is represented by a horizontal vector field

(, = a, + q\d{ + g g (4.1.25) on J ' Q . Conversely, every vector field f on JlQ which fulfills the conditions

dt\t. = 1, v(0 = 0,

where v is the vertical endomorphism (4.1.13), is a holonomic connection on the jet bundle JlQ -> R.

Holonomic connections (4.1.25) make up an affine space modelled over the linear space of vertical vector fields on the affine jet bundle JlQ —► Q, i.e., which live in VQJ'Q.

A holonomic connection £ defines the corresponding covariant differential (4.1.17) on the jet manifold ^Q:

(4.1.25)

Ds : J ' J ' Q —^VQJlQ C VJlQ,

q'oDi = 0, $ o D( = q\t - f,

which takes its values into the vertical tangent bundle VQ.J1Q of the jet bundle JlQ —y Q. Then, by virtue of Proposition 1.3.1, any integral section c : () —» JlQ for a holonomic connection £ is holonomic, i.e., c = c where c is a curve in Q.

4.2 Dynamic equations

We start our exposition of Lagrangian time-dependent mechanics from the notion of a second order dynamic equation on a configuration bundle Q —» R.

Recall that a dynamic equation, by definition, is a differential equation which can be algebraically solved for the highest order derivatives.

DEFINITION 4.2.1. Let

r = dt + F'di

be a connection on a fibre bundle Q —> M. The corresponding covariant differential DT (4.1.17) is a first order differential operator on Q. Its kernel, given by the coordinate relations

fl* = !*(*,«'). (4.2.1)


is a closed subbundle of the jet bundle JlQ -» R. Therefore, in accordance with Definition 1.3.3, this is a first order differential equation on the fibre bundle Q —> R, called the first order dynamic equation on Q —► R. □

Solutions of the first order dynamic equation (4.2.1) are integral sections (geodesic curves) for the connection I\

DEFINITION 4.2.2. Let us consider the first order dynamic equation (4.2.1) on the jet bundle J1Q —► R, which is associated with a holonomic connection f (4.1.25) on J ' Q —► R. This is a closed subbundle of the second order jet bundle J2Q —» R, given by the coordinate relations

& = ?«,<!>,<d). (4.2.2)

Consequently, it is a second order differential equation on the fibre bundle Q —► R in accordance with Definition 1.3.3. This equation is called a second order dynamic equation, or simply a dynamic equation, if there is no danger of confusion. The corresponding horizontal vector field f (4.1.25) is also termed a dynamic equation. a

A solution of the dynamic equation (4.2.2), called a motion, is a curve c in Q whose second order jet prolongation c lives in (4.2.2). It is clear that any integral section c for the holonomic connection £ is the jet prolongation c of a solution c of the dynamic equation (4.2.2), i.e.,

c' = e ° c, (4.2.3)

and vice versa.

Remark 4.2.1. By very definition, the second order dynamic equation (4.2.2) on a fibre bundle Q —> R is equivalent to the system of first order differential equations

9(0 = 9t. Qlt = C(t,qj,ql), (4.2.4)

on the jet bundle JlQ —> R. Any solution c of these equations takes its values into J2Q and, by virtue of Proposition 1.3.1, is holonomic, i.e., c = c. Therefore, the equations (4.2.2) and (4.2.4) are equivalent. The equation (4.2.4) is said to be the first order reduction of the second order dynamic equation (4.2.2) . •


One can easily find the transformation law of a second order dynamic equation under bundle coordinate transformations ql —> q'^t^g3). This transformation law reads

<& = ?, e = (?dj + gjgfodk + 24djdt + 9?W% ?)■ (4.2.5)

Example 4.2.2. From the physical viewpoint, the most interesting dynamic equations are the quadratic dynamic equations, i.e., those which are polynomials in the coordinates q\ of degree < 2. This property is global due to the transformation law (4.2.5). The dynamic equation of a relativistic particle exemplifies a non-polynomial dynamic equation. •

A dynamic equation ( o n a fibre bundle Q —» K is said to be conservative if there exists a trivialization (4.1.1) of Q whose corresponding trivialization (4.1.3) of J*Q is such that the vector field f (4.1.25) on JlQ is projectable over M. Then this projection

Hf = <j,'3 + e(<rW)£<

is a second order dynamic equation on the typical fibre M of Q —* K in accordance with Definition 3.1.2. Conversely, every autonomous second order dynamic equation E on a manifold M can be seen as a conservative dynamic equation

is = dt + 4% + u'di (4.2.6)

on the fibre bundle K x M —> K in accordance with the isomorphism (4.1.3). We conclude this Section with the following theorem.

T H E O R E M 4.2.3. Any dynamic equation on a fibre bundle Q —► K is equivalent to an autonomous second order dynamic equation on a manifold Q. □

Proof. Given a dynamic equation £ on a fibre bundle Q —> R, let us consider the diagram

J*Q -^T2Q

' I Is JIQ J u TQ

(4.2.7)

where 5 is a holonomic vector field on the tangent bundle TQ, and we use the morphism (4.1.23). A glance at the expression (4.1.23) shows that the diagram


(4.2.7) can be commutative only if the component 5° of a vector field S vanishes. Since the transition functions t —> t' are independent of g\ such a vector field may exist on TQ. Now, the diagram (4.2.7) becomes commutative if the dynamic equation f and a vector field E fulfill the relation

C = S%q>,i = l,q>=q>). (4.2.8)

It is easily seen that this relation holds globally because the substitution of ql = q\ in the transformation law of a vector field H restates the transformation law (4.2.5) of the holonomic connection £. In accordance with the relation (4.2.8), the desired vector field S is an extension of the section TXo\2o£ of the fibre bundle T*Q —> TQ over the closed submanifold JlQ C TQ to a global section. Such an extension always exists by virtue of Theorem 1.1.2, but is not unique. Then the dynamic equation (4.2.2) can be written in the form

Itt - - I ( = 1 I ^ = ? J • (4.2.9)

This is equivalent to the autonomous second order dynamic equation

t = Q, f = l, ? = S*', (4.2.10)

on Q. Being a solution of the equation (4.2.10), a curve c in Q also satisfies the equation (4.2.9), and vice versa. QED

It should be emphasized that, written in the bundle coordinates (£, q'), the second order equation (4.2.10) is well defined with respect to any coordinates on Q.

QED

4.3 Dynamic connections

In order to say more than was said in Theorem 4.2.3, we will consider the relationship between the holonomic connections on the jet bundle JlQ —> R and the connections on the affine jet bundle JlQ —> Q (see Propositions 4.3.1 and 4.3.2 below).

Let

7 : J'Q -» JlQJlQ

be a connection on the affine jet bundle J*Q —» Q. It takes the coordinate form

7 = <V®(dA + 7ldf), (4.3.1)

4.3. DYNAMIC CONNECTIONS 163

with the transformation law

Tj = (VTi + *tf)f£. (4.3.2)

Remark 4.3.1. In view of the canonical splitting (4.1.8), the curvature (1.4.8) of a connection 7 (4.3.1) reads

R: ^Q-^AT'Q ® VQ, JlQ

R = 2^dqX A dq» ® 5, = {h?kidt}k A dcf + R^dt A dq>) ® du

R\» = 9A7" - d^\ + 7ia j 7 ; - yid^. (4.3.3)

Using the contraction (4.1.7), we obtain the soldering form

AJ.R = [(«*,«? + R^)dql - R^tfdt] ® a,

on the affine jet bundle JlQ —► Q. Its image by the canonical projection T'Q —► V'Q is the tensor field

R:J1Q-*V*Q®VQ, Q

R=(Ry^ + RlJW^di, (4.3.4)

and then we come to the scalar field

R:JlQ^ R,

R = R-liit + Ron (4.3.5)

on the jet manifold JlQ. •

PROPOSITION 4.3.1. Any connection 7 (4.3.1) on the affine jet bundle JlQ -* Q defines the holonomic connection

£7 = p o 7 : JlQ^ J^JXQ^J2Q, (4.3.6)

on the jet bundle JlQ -> R. D

e, = ^ + q^ + (7i + qf7j)af.


Proof. Let us consider the composite fibre bundle (4.1.10) and the morphism p (1.6.6) which reads

p : J^Q 3 (q\ q\, q\t) - . (q\ q\, q\t) = q\, £ = ^ + 4<?jt) ^ J'Q- (4-3-7)

A connection 7 (4.3.1) and the morphism p (4.3.7) combine into a desired holonomic connection f7 (4.3.6) on the jet bundle JlQ -> R. QED

It follows that every connection 7 (4.3.1) on the affine jet bundle J1Q —» Q yields the dynamic equation

on the configuration bundle Q —> R. This is precisely the restriction to J2Q of the kernel KerD^ of the vertical covariant differential D7 (1.6.11) defined by the connection 7:

Dy : J'J'Q -> VQJ'Q,

q't°D-, = q'tt - 7o - <Al)- (4.3.9)

(4.3.8)

(4.3.7)

Qu = 7o + <dYj

p : J'QJlQ 3 (q\qlvq\t) ~ (q\ql,q\t) = q\,qlt = 4 + 4<?jt) ^ J*Q-

Therefore, connections on the jet bundle JlQ —> <5 are called the dynamic connections. The corresponding equation (4.2.3) can be written in the form

C1 = p o 7 o c,

where p is the morphism (4.3.7). Of course, different dynamic connections can lead to the same dynamic equation

(4.3.8).

PROPOSITION 4.3.2. Any holonomic connection £ (4.1.25) on the jet bundle J ' Q —* R defines the dynamic connection

7j = dt ® \dt + (g - yt^)dj] + dq>® [d, + jgjgg] (4.3.10)

on the affine jet bundle JlQ —* Q. □

Proof. Let £ be a holonomic vector field (4.1.25). Given an arbitrary vertical vector field

u = a% + Vd',


on the jet bundle J1Q —> R, let us put

h(u)d=[SMu)]-mM),

where v is the vertical endomorphism (4.1.13). Then, there is the endomorphism

h ■ 4% + 9 $ - . -q% + (q\ - cfd]V)d\.

It is readily observed that this endomorphism obeys the condition

h°h = h-Then, one can construct the projection

J< ■?% + &! "(il-l?%?)%■ Recall that a holonomic connection £ on JlQ —> R defines the projection (1.4.3) which reads

£ : TJlQ 3 tdt + q% + q\d\ -> (ql - %)% + (41 ~ i?)$ € VJlQ.

Let us consider the composition of morphisms

idt + q% + gjd? ~

This corresponds to the connection 75 (4.3.10) on the affine jet bundle JlQ —> Q. QED

It is readily observed that the dynamic connection 7^ (4.3.10), denned by a dynamic equation, possesses the property

7* = grj + gjgj (4.3.11)

which implies the relation

atf = af-tf.

It(u) = -a«at + (b{ - aiq&di

h-VJlQ ^VJ'Q,

J ^ ^ + ldVJ'Q) :VJlQ^QVQJlQ,

[Q\-m~\4^)-\^d]m-/? o £ : TJlQ -» VJ'Q - VQJlQ,


Therefore, a dynamic connection 7, obeying the condition (4.3.11), is said to be symmetric. The torsion of a dynamic connection 7 is defined as the tensor field

T:JlQ^V'Q®VQ, Q

T = J*3j* ® dk, (4.3.12)

It follows at once that a dynamic connection is symmetric if and only if its torsion vanishes.

Let 7 be a dynamic connection (4.3.1) and £7 the corresponding dynamic equation (4.3.6). Then the dynamic connection (4.3.10) associated with the dynamic equation £7 takes the form

^! = 5(7? + ahJ + «&•#), 7«,S = £" - ftSc?-It is readily observed that 7 = 7* if and only if the torsion T (4.3.12) of the dynamic connection 7 vanishes.

Example 4.3.2. Since the jet bundle J1Q —> Q is affine, it admits an affine connection

7 = dqx ® [9, + ( 7 y O + 7 y O t f ) d i ] - (4-3.13)

This connection is symmetric if and only if 7^ = f'x. One can easily justify that an affine dynamic connection generates a quadratic dynamic equation, and vice versa. Nevertheless, a non-affine dynamic connection, whose symmetric part is affine, also define a quadratic dynamic equation.

An affine connection (4.3.13) on the affine jet bundle J1Q —* Q yields the linear connection

7 = dqX®[dX+lU<f)q(di}

on the vertical tangent bundle VQ —► Q. •

Using the notion of a dynamic connection, we may modify Theorem 3.1.4 as follows. Let H be an autonomous second order dynamic equation on a manifold M, and f= (4.2.6) the corresponding conservative dynamic equation on the bundle R x M —> R. The latter yields the dynamic connection 7 (4.3.10) on the fibre bundle

R x TM - » R x M. R x TM ~*RxM.

7 = dqX®[dX+lU<f)q(di}

7=dqx ® [d, + ( 7 yo+7yo<?n^] . (4.3.13)

Tt = 7* - aho* - d"#tf ■


Its components 7! are exactly those of the connection (3.1.7) on the tangent bundle TM —> M in Theorem 3.1.4, while 7J make up a vertical vector field

e = T $ = (H1 - i^S^ja, (4.3.14)

on TM —» M. Thus, we have shown the following.

PROPOSITION 4.3.3. Every autonomous second order dynamic equation 5 (3.1.4) on a manifold M admits the decomposition

H1 = Kjq* + e{

where K is the connection (3.1.7) on the tangent bundle TM —» M, and e is the vertical vector field (4.3.14) on TM -» M. □

Remark 4.3.3. With a dynamic connection 75 (4.3.10), we also restate the well-known linear connection on the tangent bundle TJXQ —> J ' Q , associated with a dynamic equation £ on Q [132], and can write it with respect to holonomic bases.

Given a holonomic connection f (4.1.25), we have the corresponding horizontal splitting (1.4.1) of the tangent bundle TJlQ of JlQ:

TJlQ = Hc® VJlQ, JlQ

tit + q% + $% = tf + (<?' - t&dt + (4$ - f )flf,

(4.3.15)

(4.3.16)

and the corresponding horizontal splitting (1.4.2) of the cotangent bundle T'JlQ of

T"J1Q = R ® W ' Q ,

tdt + qM + #M = (t + 9t9i + {'$<« + <?/ + 9i(<*2f " Cdt), (4.3.17)

Using these spUttings, one can introduce the non-holonomic bases (£, d„ d\) for the tangent bundle TJXQ, and the non-holonomic bases

(dt,6',dql-Cdt)

for the cotangent bundle T'^Q of JlQ.


Furthermore, the dynamic connection 7{ (4.3.10), associated with the dynamic equation £, provides the corresponding splitting (1.6.10a) of the vertical tangent bundle VJlQ of JlQ:

VJlQ = VQJlQ © //-,.,

q% + q\d\ = (q\ - i^fljf1)^ + ?'(a, + \&£d[). (4.3.18)

The splittings (4.3.16) and (4.3.18) combine into the splitting

TJ'Q = H,@ Hlt ® VQJXQ (4.3.19)

together with the non-holonomic bases

(£, h, = d} + \d)Cdl a?) (4.3.20)

for the tangent bundle TJlQ [55]. Note that the fibre bundle H^ in the splitting (4.3.19), like VQJ 'Q , is isomorphic to the pull-back

Hlt = JlQ xVQ^ J 'Q, h, <—> dr (4.3.21)

Accordingly, the dynamic connection 7{ (4.3.10) determines the corresponding splitting (1.6.10b) of the vertical cotangent bundle V'JlQ of JlQ. Combining this splitting with the decomposition (4.3.17) gives the splitting

T'JlQ = R ® V'Q ® VAJlQ JlQ JlQ

together with the non-holonomic bases

{dt,e\dq\-^dqx)

for the cotangent bundle T'JlQ of JlQ. Every connection 7 (4.3.1) on the affine jet bundle JlQ —* Q gives rise to the

connection V7 (1.6.12) on the composite vertical tangent bundle

VQJlQ - JlQ -» Q.

This connection reads

Vy = dqx® (dx + 7iaf + ^7i«J'af).


Since JlQ —» Q is an affine bundle, the connection Vj, in turn, can be seen as the composite connection (1.6.8) generated by the connection 7 on the affine jet bundle J1Q —> Q, and the linear connection (1.6.14):

7 = dqx ® (dx + d)i\cttd\) + dqk ® dl (4.3.22)

on the vertical tangent bundle VQJ1Q —> JlQ. Due to the splitting (4.1.8), the connection 7 (4.3.22) in fact is a linear connection

7 = dqx ® (dx + d'-yitfdi) + dqkt ® 6k (4.3.23)

on the pull-back (4.3.21). In particular, if 7 = 75 is the connection (4.3.10), the connection 7^ (4.3.23) yields a connection on the pull-back bundle H-,( —> JlQ in the splitting (4.3.19).

Due to the splitting (4.3.19), the tangent bundle TJlQ —* JlQ can be provided with a linear connection A [132] which is the direct sum of

• the trivial connection on the linear bundle H^ —> J1Q,

• the connection 7^ (4.3.23) on the pull-back bundle H1( —> J ' Q ,

• and the similar connection (4.3.22) on the vertical tangent bundle VQJ1Q —> JlQ.

With respect to the non-holonomic basis (4.3.20), where

idt + cfdi + q\d\ = it, + u% + v*dl (4.3.24)

the above-mentioned linear connection A reads

A = dqx ® (dx + d ;7> J ^ + df^J^\ + ^ ® a'■ ( 4 3 2 5 )

Recall the expression (4.3.10) for the components 7^ of the dynamic connection 7?. Substituting

qi = iq\+ui, q\ = ie + ukYk+vi

from (4.3.24) in (4.3.25), we obtain the linear connection A written with respect to the holonomic bases (dx,df) for TJlQ:

A = dqx® (dx + Ax) + dq\® (dl + A\), Ax = i[-4dn\d, + (dxC - <£drfk - 4l\^k

x + <&~&drfx - ^ D S f ] + fld^A + (dxl) + 7&7* - I'daDdW + fariM,

A\ = ifa + (d*? - q?dH)dt] + q>dhkdi.

7 = dq* ® (dx + dftqid!) + dqk ® dl (4.3.22)

(4.3.23) 7 = dc? ® (dx + d'jxtfdi) + dqk ® dk

(4.3.24)

(4.3.25)

qi = iq\+ui, q\ = t f + Ukfk + v*

A = dqx®(dx + d^W-^ + W ^ ) +d4® dl

A = dqx® (dx + Ax) + dq\® (dl + A\),

Ax = i[-4dn\d, + (dxC - rf^j " <7?7^7Afc + H ^ \ ~ S*af7i)4f] +

fldrfA + (dxl) + 7 & 7 * - 7^*7l)9 |] + # 9 , 7 ^ ,


The covariant derivatives V relative to the connection A fulfill the conditions

VA(0 = VKO = o, V,(5j) = -dtfA, VA(^) = -CSTJ^,

(4.3.26)

In particular, by virtue of the equality (4.3.26), every solution c of the dynamic equation £ is also a geodesic of the linear connection A. •

4.4 Non-relativistic geodesic equations

In this Section, we aim to show that every dynamic equation on a configuration bundle Q —> M is equivalent to a geodesic equation on the tangent bundle TQ —► Q.

We start from the relation between the dynamic connections 7 on the affine jet bundle JlQ —* Q and the connections

K = dqx® (dx + Kfa) (4.4.1)

on the tangent bundle TQ —» Q of the configuration space Q. We will use the notation (1.2.4).

Let us consider the diagram

JlQJlQ J-±Jl

QTQ

, I I . JlQ -±-> TQ

(4.4.2)

where JQTQ is the first order jet manifold of the tangent bundle TQ —» Q, coordinated by

(t,q\i,q\(i)^ (q%).

The jet prolongation over Q of the canonical imbedding A (4.1.4) reads

•/'A : (t,q\ql,qU) - (t,q',i = l,«f = g|, (t)„ = 0,(q% = <£,).

Then we have

JJA o 7 ; (t, g', gj) ,_ (<, g\ t = l,g« = q', (*)„ = 0, (g% = -£),

v|(5j) = V|(A,) = o, v*(a3) = -flftflj.

X o A : (t, q\ ql) » (t, e?\ « = ! , ? = flj, (t)„ = Kj, («% = /if*).

4.4. NON-RELATIVISTIC GEODESIC EQUATIONS 171

It follows that the diagram (4.4.2) can be commutative only if the components K° of the connection K (4.4.1) on the tangent bundle TQ —> Q vanish.

Since the transition functions t —* t* are independent of q', a connection

K = dqx ® (3A + iq«3i) (4.4.3)

with A"° = 0 may exist on the tangent bundle TQ —♦ Q in accordance with the transformation law

K'i-idjfKi + drfl^-y (4.4.4)

Now the diagram (4.4.2) becomes commutative if the connections 7 and K fulfill the relation

^ = K^o X = Kl(t,q',t = l,q' = q^).\ (4.4.5)

It is easily seen that this relation holds globally because the substitution of ql = qlt

in (4.4.4) restates the transformation law (4.3.2) of a connection on the affine jet bundle JlQ —» Q. In accordance with the relation (4.4.5), the desired connection K is an extension of the section J'A o 7 of the affine jet bundle JQTQ —* TQ over the closed submanifold JlQ C TQ to a global section. Such an extension always exists by virtue of Theorem 1.1.2, but is not unique. Thus, we have proved the following.

PROPOSITION 4.4.1. In accordance with the relation (4.4.5), every dynamic equation on a configuration bundle Q —> R can be written in the form

qlt = K'0°\ + q{K>o\,\ (4.4.6)

where 7f is a connection (4.4.3) on the tangent bundle TQ —» Q. Conversely, each connection K (4.4.3) on TQ —> Q defines the dynamic connection 7 (4.4.5) on the affine jet bundle JXQ —> Q and the dynamic equation (4.4.6) on a configuration bundle Q -» R. □

Then we come to the following theorem.

T H E O R E M 4.4.2. Every dynamic equation (4.2.2) on a configuration bundle Q -» R is equivalent to the geodesic equation

g0 = 0, q° = 1, i? = K\{<f,q»)q\ (4.4.7)

K"x = (dJq«Ki + d»q*)^-x.

K = dqx® (dx + K\di)

(4.4.4)

(4.4.5)

(4.4.3)

(4.4.7) if = 0, q° = 1,

? = Ki{(f,q»)e,


on the tangent bundle TQ relative to a connection K with the components K° = 0 and K\ (4.4.5). Its solution is a geodesic curve in Q which also obeys the dynamic equation (4.4.6), and vice versa. □

In accordance with this theorem, the second order equation (4.2.10) in Theorem 4.2.3 can be chosen as a geodesic equation. It should be emphasized that, written in the bundle coordinates (£,<?'), the geodesic equation (4.4.7) and the connection K (4.4.5) are well defined with respect to any coordinates on Q.

As was mentioned, from the physical viewpoint, the most interesting dynamic equations are the quadratic ones

? = ^(Qniti + b)(qnii + f'(q")- (4.4.8)

This property is global due to the transformation law (4.2.5). Then one can use the following two facts.

PROPOSITION 4.4.3. There is one-to-one correspondence between the affine connections 7 on the affine jet bundle J ' Q —» Q and the linear connections K (4.4.3) on the tangent bundle TQ —» Q. D

Proof. This correspondence is given by the relation (4.4.5), written in the form

< = 7Jo + lU<& = V o ( ? " ) t + KJiWW]^^* = K;0{q") + K*i{<f)4t

i.e.,

7 > — KJ\.

QED

In particular, if an affine dynamic connection 7 is symmetric, so is the corresponding linear connection K.

COROLLARY 4.4.4. Every quadratic dynamic equation (4.4.8) on a configuration bundle Q —» R of non-relativistic mechanics gives rise to the geodesic equation

if = 0, «7° = 1,

9" = Jjk(qt'W<ik + l>)(q''W<i0 + /V)«V (4.4.9)

4.4. NON-RELATIVISTIC GEODESIC EQUATIONS 173

on the tangent bundle TQ with respect to the symmetric linear connection

Kx\ = 0, ^o'o = / ' i KOJ = -b), Kkj = a'kj (4.4.10)

on the tangent bundle TQ —» Q. □

The geodesic equation (4.4.9), however, is not unique for the dynamic equation (4.4.8).

PROPOSITION 4.4.5. Any quadratic dynamic equation (4.4.8), being equivalent to the geodesic equation with respect to the symmetric linear connection K (4.4.10), is also equivalent to the geodesic equation with respect to an affine connection K' on TQ —> Q which differs from K (4.4.10) in a soldering form a on TQ —> Q with the components

"S*o, 4 = A* + (s - l)h'kq°, a'0 = -shlq* - htf + hi

where s and h\ are local functions on Q. D

Proof. The proof follows from direct computation.

Proposition 4.4.5 can also be deduced from the following lemma.

LEMMA 4.4.6. Every affine vertical vector field

QED

a = irm+bWWttf (4.4.11)

on the affine jet bundle JlQ —► Q is extended to the soldering form

a = {f'dt + b\dqk) ® di (4.4.12)

on the tangent bundle TQ —> Q. □

Proof. Similarly to Proposition 4.4.3, one can show that there is one-to-one correspondence between the VQ J'Q-valued affine vector fields (4.4.11) on the jet manifold J'<3 and the VgTQ-valued linear vertical vector fields

a = [§{?)# + ftf)?\di on the tangent bundle TQ. This linear vertical vector field determines the desired soldering form (4.4.12). QED


In Section 6.4, we will use Theorem 4.4.2, Corollary 4.4.4 and Proposition 4.4.5 in order to study the relationship between non-relativistic and relativistic equations of motion.

Now let us extend our inspection of dynamic equations to connections on the tangent bundle TM —» M of the typical fibre M of a configuration bundle Q —> R. In this case, the relationship fails to be canonical, but depends on a trivialization (4.1.1) of Q-*R.

Given such a trivialization, let (t.g*) be the associated coordinates on Q, where if are coordinates on M with transition functions independent of t. The corresponding trivialization (4.1.3) of JXQ —» R takes place in the coordinates (t,q',q ), where q are coordinates on TM. With respect to these coordinates, the transformation law (4.3.2) of a dynamic connection 7 on the affine jet bundle JiQ —► Q reads

-v" 1 0 dq'% H

( dq'ln + at*-dq"

It follows that, given a trivialization of Q —> R, a connection 7 on J ' Q —> Q defines the time-dependent vertical vector field

dq R x TM -* VTM

and the time-dependent connection

dqk ® (W+* (t,<?,if) ) dq'

d RxTM JXTM C TTM (4.4.13)

on the tangent bundle TM M. Conversely, let us consider a connection

K dqh ® (- + Ktf,$) d

d? ) on the tangent bundle TM —> M. Given the above-mentioned trivialization of the configuration bundle Q —> R, the connection K defines the connection K (4.4.3), with the components

K 0. KI Kk,

on the tangent bundle TQ —> Q. The corresponding dynamic connection 7 on the affine jet bundle J1Q —> Q reads

% 0, 7i rfc. (4.4.14)

4.5. REFERENCE FRAMES 175

Using the transformation law (4.3.2), one can extend the expression (4.4.14) to arbitrary bundle coordinates (t, q') on the configuration space Q as follows:

it ^K&(qr),it(Qr,qrt)) + dq^dq^ + an

dkT,

% dtr ±9^4- 7ir*, where

P $«*(*,?)

(4.4.15)

is the connection on Q -» R, corresponding to a given trivialization of Q, i.e., P = 0 relative to (t,(f). The dynamic equation on Q defined by the dynamic connection (4.4.15) takes the form

<?« = dtr + qid^r + iM - rA). (4.4.16)

By construction, it is a conservative dynamic equation. Thus, we have proved the following.

PROPOSITION 4.4.7. Any connection K on the typical fibre M of a configuration bundle Q —► K yields a conservative dynamic equation (4.4.16) on Q. O

4.5 Reference frames

From the physical viewpoint, a reference frame in non-relativistic mechanics determines a tangent vector at each point of a configuration space Q, which characterizes the velocity of an "observer" at this point. This speculation leads us to suggest the following mathematical definition of a reference frame in non-relativistic mechanics [57, 132, 161].

DEFINITION 4.5.1. In non-relativistic mechanics, a reference frame is a connection T on the configuration bundle Q —* K. □

In accordance with this definition, one can think of the horizontal vector field (4.1.16), associated with a connection r on Q —> R, as being a family of "observers", while the corresponding covariant differential

q^Dr(qt) = ql-r


determines the relative velocities with respect to the reference frame I \ In particular, given a motion c : R —» Q, its covariant derivative V r c (1.4.6)

with respect to a connection T is the velocity of this motion relative to the reference frame T. For instance, if c is an integral section for the connection T, the velocity of the motion c relative to the reference frame T is equal to 0. Conversely, every motion c : R —> Q, defines a reference frame Tc such that the velocity of c relative to Tc vanishes. This reference frame Tc is an extension of the section c(R) —> JlQ of the affine jet bundle JlQ —> Q over the closed submanifold c(R) G Q to a global section in accordance with Theorem 1.1.2.

Remark 4.5.1. It should be emphasized that the vertical tangent bundle VQ of a configuration bundle Q, but not the jet manifold J:Q plays the role of the "space of coordinates and velocities", while elements of J1Q may be termed the absolute velocities. In the universal unit system, elements of VQ, however, have the same physical dimension [q] as elements of Q, whereas absolute velocities are of physical dimension [g] — 1. •

By virtue of Corollary 4.1.1, any reference frame T on a configuration bundle Q —> R is associated with an atlas of local constant trivializations, and vice versa. The connection T reduces to

r = dt (4.5.1)

with respect to the corresponding coordinates (t,?1), whose transition functions q1 —> q" are independent of time. One can think of these coordinates as being also the reference frame, corresponding to the connection (4.5.1). They are called the adapted coordinates to the reference frame I\ Thus, we come to the following definition, equivalent to Definition 4.5.1.

DEFINITION 4.5.2. In non-relativistic mechanics, a reference frame is an atlas of local constant trivializations of a configuration bundle Q —> R. □

In particular, with respect to the coordinates if adapted to a reference frame T, the velocities relative to this reference frame are equal to the absolute ones

Dr(q't) = fr = 5'c

A reference frame is said to be compJete if the associated connection F is complete. By virtue of Proposition 4.1.2, every complete reference frame defines a trivialization of a bundle Q —> R, and vice versa.

Remark 4.5.2. Given a reference frame T, one should solve the equations

r'M(t,f)) = « , (4.5.2a)

a^a^v, + 56 (4.5.2b)

in order to find the coordinates {t,qa) adapted to T. Let (i, g") and ((, g2) be the adapted coordinates for reference frames Tj and r2 ,

respectively. In accordance with the equality (4.5.2b), the components r \ of the connection 1^ with respect to the coordinates (t, g2) and the components r 2 of the connection T2 with respect to the coordinates (t, g") fulfill the relation

M r - + ra - 0

•

Using the relations (4.5.2a) - (4.5.2b), one can rewrite the coordinate transformation law (4.2.5) of dynamic equations as follows. Let

?u=T (4.5.3)

be a dynamic equation on a configuration space Q, written with respect to a reference frame (t,if1). Then, relative to arbitrary bundle coordinates (t,ql) on Q —> R, the dynamic equation (4.5.3) takes the form

qlt = dtri + djr{qi-P) -g^-r'«-r,>+f^ (4.5.4)

where T is the connection corresponding to the reference frame (t,q"). The dynamic equation (4.5.4) can be expressed in the relative velocities <j{. = q\ — T' with respect to the initial reference frame (t,qa). We have

*t-V4-gij£?**+&M-*>- (4.5.5)

Accordingly, any dynamic equation (4.2.2) can be expressed in the relative velocities <jf, = q\ — P with respect to an arbitrary reference frame T as follows:

Mi = (£ - Jryt = e- ckr, (4.5.6)

4.5. REFERENCE FRAMES 177


where jr is the prolongation (4.1.24) of the connection T onto the jet bundle JlQ —* R.

For instance, let us consider the following particular reference frame T for a dynamic equation f. The covariant derivative of a reference frame T with respect to the corresponding dynamic connection 7^ (4.3.10) reads

V 7 r = Q -» T'Q x VQJ'Q,

wr = 'v}rkdqx®dk, vjr* = aAr*-7$or.

(4.5.7)

A connection T is called a geodesic reference frame for the dynamic equation f if

rjv^r rA(dArk - 7* ° r) (diC - c o r)di o. (4.5.8)

PROPOSITION 4.5.3. Integral sections c of a reference frame T are solutions of a dynamic equation f if and only if T is a geodesic reference frame for f. □

Proof. The proof follows at once from substitution of the equality (4.5.8) in the dynamic equation (4.5.6). QED

Remark 4.5.3. The left- and right-hand sides of the equation (4.5.6) separately are not well-behaved objects. This equation will be brought below into the covariant form (4.7.6). •

Reference frames play a prominent role in many constructions of time-dependent mechanics. In particular, we obtain a converse of Theorem 4.4.2.

THEOREM 4.5.4. Given a reference frame T, any connection K (4.4.1) on the tangent bundle TQ —> Q defines a dynamic equation

e = (K\ - rK°)q* \.0=l.,=q, . D

This theorem is a corollary of Proposition 4.4.1 and the following lemma.

LEMMA 4.5.5. Given a connection T on the fibre bundle Q - > R and a connection K on the tangent bundle TQ -* Q, there is the connection K on TQ -* Q with the components

*A° = 0, K\ = K\ - VKl

4.6. FREE MOTION EQUATIONS 179

□ Proof. The proof follows from the inspection of transition functions. QED

4.6 Free motion equations

Let us point out the following interesting class of dynamic equations which we agree to call the free motion equations.

DEFINITION 4.6.1. We say that the dynamic equation (4.2.2) is a free motion equation if there exists a reference frame ((,?*) on the configuration space Q such that this equation reads

Tu = 0- (4.6.1)

□ With respect to arbitrary bundle coordinates (t, <?'), a free motion equation takes

the form

* = 4P + djr(<d - V) - $^-k(4 - *)<<* - n (4.6.2)

where P = dtgt(t,'^) is the connection associated with the initial frame (t,q') (cf. (4.5.4)). One can think of the right-hand side of the equation (4.6.2) as being the general coordinate expression for an inertial force in non-relativistic mechanics. The corresponding dynamic connection 7^ on the affine jet bundle J ' Q —► Q reads

7i = dkr - dqmdq3dqk{ql >' 75 = ^ + ^ - 7 * 1 * .

(4.6.3)

It is affine. By virtue of Proposition 4.4.3, this dynamic connection defines a linear connection K on the tangent bundle TQ —> Q, whose curvature necessarily vanishes. Thus, we come to the following criterion of a dynamic equation to be a free motion equation.

PROPOSITION 4.6.2. If £ is a free motion equation on a configuration space Q, it is quadratic, and the corresponding symmetric linear connection (4.4.10) on the tangent bundle TQ —» Q is a curvature-free connection. □


This criterion is not a sufficient condition because it may happen that the components of a curvature-free symmetric linear connection on TQ —► Q vanish with respect to the coordinates on Q which are not compatible with the fibration Q —> R.

The similar criterion involves the curvature of a dynamic connection (4.6.3) of a free motion equation.

PROPOSITION 4.6.3. If f is a free motion equation, then the curvature R (4.3.3) of the corresponding dynamic connection 7^ is equal to 0, and so are the tensor field R (4.3.4) and the scalar field R (4.3.5). D

Proposition 4.6.3 also fails to be a sufficient condition. If the curvature R (4.3.3) of a dynamic connection 7 vanishes, it may happen that components of 7^ are equal to 0 with respect to non-holonomic bundle coordinates on the affine jet bundle JlQ-^Q.

Nevertheless, we can formulate the necessary and sufficient condition of the existence of a free motion equation on a configuration space Q.

PROPOSITION 4.6.4. A free motion equation on a fibre bundle Q —> R exists if and only if the typical fibre M of Q admits a curvature-free symmetric linear connection. D

Proof. Let a free motion equation take the form (4.6.1) with respect to some atlas of local constant trivializations of a fibre bundle Q —> R. By virtue of Proposition 4.3.2, there exists an affine dynamic connection 7 on the affine jet bundle JlQ —» Q whose components relative to this atlas are equal to 0. Given a trivialization chart of this atlas, the connection 7 defines the curvature-free symmetric linear connection (4.4.13) on M. The converse statement follows at once from Proposition 4.4.7. QED

The free motion equation (4.6.2) is simplified if the coordinate transition functions q1 —* q1 are affine in the coordinates <p. Then we have

qxtt = dtP - PdjP + 2q1

td]r. (4.6.4)

Example 4 .6 .1 . Let us consider a free motion on a plane R2. The corresponding configuration bundle is R3 -» R, coordinated by (t,r). The dynamic equation of this motion is

f = 0. (4.6.5)

4.6. FREE MOTION EQUATIONS 181

Let us choose the rotatory reference frame with the adapted coordinates

r = AT, A _ / c o s w t - s i n w A

\ sin art cos art ) (4.6.6)

Relative to these coordinates, the connection T corresponding to the initial reference frame reads

r = dtr = dtA ■ A-XT.

Then the free motion equation (4.6.5) with respect to the rotatory reference frame (4.6.6) takes the familiar form

rM = ■M")* (4.6.7)

The first term in the right-hand side of the equation (4.6.7) is the centrifugal force (—PojjT'), while the second one is the Coriolis force (2q\djYx). •

The following lemma shows that the free motion equation (4.6.4) is affine in the coordinates q' and q\.

LEMMA 4.6.5. Let (£,<f) be a reference frame on a configuration bundle Q —► R and T the corresponding connection. Components P of this connection with respect to another coordinate system (i, q%) are affine functions in the coordinates q' if and only if the transition functions between the coordinates q° and q' are affine. □

Proof. If

q* = o£(t)g» + b'(t),

then we have

r H O ^ a ^ - f r O + d . Conversely, let

r = c)(t)q> + s'(t)

such that the homogeneous linear system of differential equations

aJ<?i(«) = c'(ty(«) + s'W (4.6.8)


has a solution on R. Let a)(t) be the mxm matrix whose columns are m fundamental solutions qi{t),--,qm(t) of the equations (4.6.8) (see, e.g., [22]). This matrix is invertible for all t 6 R. Then the connection T is associated with the coordinates ((,5") such that

9i=oi(*)r+6'(o, where functions bl(t) fulfill the equations

dtV = ( 0 £ W " + *'■ QED

One can easily find the geodesic reference frames for the free motion equation

lit = o. (4.6.9)

They are F* = V* = const. By virtue of Lemma 4.6.5, these reference frames define the adapted coordinates

5* = Vjq> - v't - a\ k'j = const., vx = const., a' = const. (4.6.10)

The equation (4.6.9) obviously keeps its free motion form under the transformations (4.6.10) between the geodesic reference frames. It is readily observed that these transformations are precisely the elements of the Galilei group.

4.7 Relative acceleration

In comparison with the notion of a relative velocity, that of a relative acceleration is more intricate.

To consider a relative acceleration with respect to a reference frame T, one should prolong the connection T on the configuration bundle Q —* R to a holonomic connection £ r on the jet bundle JlQ -> R. Note that the jet prolongation JV (4.1.24) of r onto JlQ —» R is not holonomic. We can construct the desired prolongation by means of a dynamic connection 7 on the affine jet bundle JlQ —> Q.

LEMMA 4.7.1. Let us consider the composite bundle (4.1.10). Given a frame T on Q —> R and a dynamic connections 7 on J ' Q -» Q, there exists a dynamic connection 7 on J ' Q —► Q with the components

7i = 7i, 7; = tkr - Ykrk. (4.7.1)

4.7. RELATIVE ACCELERATION 183

D

Proof. Combining the connection r on Q -> R and the connection 7 on JlQ -> Q gives the composite connection (1.6.8) on JlQ -> R which reads

B = dt ® (ft + Pft + (7jp + 7*)ft).

Let j r be the jet prolongation (4.1.24) of the connection T on JlQ -* R. Then the difference

JT-B = dt® {dtV - 7 ' r * - 7J)9f

is a VQJ'Q-valued soldering form on the jet bundle J1Q -» R, which is also a soldering form on the affine jet bundle JlQ -+ Q. The desired connection (4.7.1) is

7 = 7 + j r - B = dt ® (ft + (d(r - 7*r*)ft) + A?* ® (ft + 7j$).

QED

Now, we construct a certain soldering form on the affine jet bundle JlQ —^ Q and add it to this connection. Let us apply the canonical projection T'Q —> V'Q and then the imbedding T : V'Q -> T'Q to the covariant derivative (4.5.7) of the reference frame T with respect to the dynamic connection 7. We obtain the VQJ1Q- valued 1-form

a = [-P(ftP - 7* o r)dt + (ftp - 7* o r)dq'} ® ft

on Q whose pull-back onto JlQ is the desired soldering form. The sum

def~ . 7r = 7 + °,

called the frame connection, reads

7r{, = *r* - 7Jr* - p(ftp - 7j o r), 7ri = 7l + dkr - 7* o r.

(4.7.2)

This connection yields the desired holonomic connection

& = ckv + (ftp + yt - 7 ' o r)(q* - r")

on the jet bundle JXQ -» R.


Let £ be a dynamic equation and 7 = 7^ the connection (4.3.10) associated with £• Then one can think of the vertical vector field

ar=g-fr = (f-fr)ff (4.7.3)

on the affine jet bundle JlQ —» Q as being a relative acceleration with respect to the reference frame T in comparison with the absolute acceleration £.

For instance, let us consider a reference frame which is geodesic for the dynamic equation £, i.e., the relation (4.5.8) holds. Then the relative acceleration of a motion c with respect to the reference frame T is

tt - fr) ° r = 0. Let £ now be an arbitrary dynamic equation, written with respect to coordinates

(t, ql) adapted to the reference frame T, i.e., P = 0. In these coordinates, the relative acceleration with respect to the reference frame T is

4 = e W . <A) - \<}t(dkc - dkc i^=0). (4.7.4)

Given another bundle coordinates {t,q") on Q —* R, this dynamic equation takes the form (4.5.5), while the relative acceleration (4.7.4) with respect to the reference frame T reads

4 = S,g''4. Then we can write a dynamic equation (4.2.2) in the form which is covariant under coordinate transformations:

■P-rrgl = ^91 ~ £r = a r , (4.7.5)

where D 7 r is the vertical covariant differential (4.3.9) with respect to the frame connection 7r (4.7.2) on the affine jet bundle J ' Q —► Q.

In particular, if £ is a free motion equation which takes the form (4.6.1) with respect to a reference frame V, then

D^q\ = 0

relative to arbitrary bundle coordinates on the configuration bundle Q —> R.

4.7. RELATIVE ACCELERATION 185

The left-hand side of the dynamic equation (4.7.5) can also be expressed in the relative velocities such that this dynamic equation takes the form

^ 9 r - yr'kQr = ar (4.7.6)

which is the covariant form of the equation (4.5.6). The concept of a relative acceleration is understood better when we deal with

the quadratic dynamic equation £, and the corresponding dynamic connection 7 is affine.

LEMMA 4.7.2. If a dynamic connection 7 is affine, i.e.,

7A — 7AO + 7Ak9t!

so is a frame connection 7 r for any frame I\ □

Proof. The proof follows from direct computation. We have

1rl = dtr + (dJri-1'k]rk)(<ti-n, 7ri = a t r + 7 i J (^-P)

or

7r}k 7j*>

TTofc & r - 7 - f c F , 7rio = dkT*-T^P, (4.7.7)

-rrloo = dtr-PdJr + Y}krrk. QED

In particular, we obtain

7r}/t = Y]k, 7rok = 7rlM) = 7rw = 0

relative to the coordinates adapted to a reference frame T. A glance at the expression (4.7.7) shows that, if a dynamic connection 7 is

symmetric, so is a frame connection jp.

COROLLARY 4.7.3. If a dynamic equation £ is quadratic, the relative acceleration ar (4.7.3) is always affine, and it admits the decomposition

a'r = -(rAvir + 2^vir), (4.7.8)


where 7 = 7c is the dynamic connection (4.3.10), and

9r = 9t - T \ 9? = 1, r° = i,

is the relative velocity with respect to the reference frame T. □

Note that the splitting (4.7.8) gives a generalized Coriolis theorem. In particular, the well-known analogy between inertia! and electromagnetic forces is restated. Corollary 4.7.3 shows that this analogy can be extended to arbitrary quadratic dynamic equation.

4.8 Lagrangian systems

A velocity phase space JlQ of time-dependent mechanics does not admit any canonical structure mentioned in the previous Chapter. Since J1Q is an odd-dimensional manifold, it has no symplectic structure, whereas presymplectic and Poisson structures are defined only for Lagrangian systems, and depend on a Lagrangian. By a Lagrangian system is meant a mechanical system whose motions are solutions of Lagrange equations for some Lagrangian on a velocity phase space J ' Q . Obviously, a mechanical system is not necessarily Lagrangian, whereas Lagrange equations are not necessarily dynamic equations. Moreover, it may happen that Lagrange equations fail to be differential equations in a strict sense (see Definition 1.3.3). Note that, in the framework of Lagrangian formalism, one also meets Cartan and Hamilton-De Donder equations, besides the Lagrange one. These equations are equivalent to each other and to Lagrange equations in the case of a regular Lagrangian.

A Lagrangian of a mechanical system is defined as a horizontal density

L = £dt, C : JlQ -> R,

(4.8.1)

on the velocity phase space JlQ, where £ is called a Lagrangian function or simply a Lagrangian if there is no danger of confusion. With respect to the universal unit system, a Lagrangian (4.8.1) is physically dimensionless.

Here, we do not study the calculus of variations in depth, but apply in a straightforward manner the first variational formula [57].

Let us consider a projectable vector field

u = uldt + u'di, u« = 0,1, (4.8.2)

4.8. LAGRANGIAN SYSTEMS 187

on a configuration bundle Q —> M and calculate the Lie derivative of a Lagrangian (4.8.1) along the jet prolongation

u = tfdt + u{d, + dtu% dt = dt + q\di + q\tdl

(1.3.7) of u. We obtain

l^L = (u\dC)dt = (u'dt + u% + dtu'dDCdt. (4.8.3)

The first variational formula provides the following canonical decomposition of the Lie derivative (4.8.3) in accordance with the variational problem:

u\dC = (u' - u'qpS, + dt{u\HL), (4.8.4)

where

HL = v'idL) + L = ntdq{ - {iriq\ - C)dt (4.8.5)

is the Poincare-Cartan form (see (4.1.14)), and

£L : J2Q -» VQ, £L = S.dq' = (d, - dtdPCdtf (4.8.6)

is the Euler-La.gra.nge operator for a Lagrangian L. The latter can be seen as a 2-form

Si = (d< - dtdl)Cdq% A dt.

Its coefficients Si are called variational derivatives. We will use the notation

iTi = d\C, 7rJt = d\d\C

The kernel Ker£L C J2Q of the Euler-Lagrange operator (4.8.6) defines the system of second order differential equations on Q

(ft - dtdj)C = 0, (4.8.7)

caUed the Lagrange equations. Its solutions are (local) sections c of the fibre bundle Q —► R whose second order jet prolongations c five in (4.8.7). They obey the equations

dtC o c — -y{^i o c) = 0. dt (4.8.8)

http://Euler-La.gra.nge


Remark 4.8.1. The kernel of an Euler-Lagrange operator Ker£j, fails to be a subbundle of the fibre bundle J2Q —► R in general. Therefore, it may happen that, as was mentioned above, the Lagrange equations (4.8.7) are not differential equations in a strict sense. In particular, by virtue of Proposition 1.3.4, it is a differential equation when the morphism (4.8.6) is of constant rank, e.g., if a Lagrangian L is regular. Recall that a Lagrangian L is called regular (non-degenerate), if

det 7r ^ 0

everywhere on the velocity phase space J1Q. If a Lagrangian L is non-degenerate, the Lagrange equations can be solved algebraically for second order derivatives, and are equivalent to a dynamic equation. •

Example 4.8.2. Let Q = R2 —► K be a configuration space, coordinated by {t,q). The corresponding velocity phase space JlQ is equipped with the adapted coordinates (t,q,qt). The Lagrangian

L = -q2q2dt

on JlQ leads to the Euler-Lagrange operator

£L = \qq2t ~ dt(q2qt)]dq

whose kernel is not a submanifold at the point q = 0. •

Every Lagrangian L on the jet manifold JlQ yields the Legendre map

L:JlQ^ V'Q, P , O L = 7T,,

(4.8.9)

where (t,g',Pi) are coordinates on the vertical cotangent bundle V'Q. Indeed, due to the vertical splitting (4.1.8), the vertical tangent morphism VL to L yields the linear morphism

VL: JlQx VQ-+R

and, consequently, the morphism (4.8.9). In time-dependent mechanics on a configuration space Q —» R, the vertical

cotangent bundle V'Q plays the role of a momentum phase space (see Chapter 5).


Remark 4.8.3. The Legendre map (4.8.9) is a local diffeomorphism, i.e., it is of maximal constant rank if and only if a Lagrangian L is regular. A Lagrangian L is called hypenegular if the Legendre map L is a diffeomorphism. •

In Section 5.1, we will show that the vertical cotangent bundle V'Q is provided with the canonical 3-form

ft d= dPi A dq' A dt, (4.8.10)

derived from the polysymplectic form (2.9.9). Let us consider the pull-back

ilL = L'U = d-Ki A dqi A dt (4.8.11)

on JlQ of the canonical form (4.8.10) by means of the Legendre map L (4.8.9). The form fi^ provides Lagrangian formalism with the construction similar to Ha-miltonian mechanics, but which depends on the choice of a Lagrangian L. In the framework of this construction, the Lagrangian counterpart of a Hamiltonian form is the Poincare-Cartan form (4.8.5).

If a Lagrangian L is regular, the form Q,L (4.8.11) defines a Poisson structure on the jet manifold JiQ as follows. By means of f2L, every vertical vector field

d = tf% + &%

on the jet bundle JlQ —> R corresponds to the 2-form

tfjftz, = {[<5%, + ^{djin - BkKjffl ~ $%dqi} A dt.

This is one-to-one correspondence if a Lagrangian L is regular. Indeed, given an arbitrary 2-form

<t> = [4>idqi + fadql) A dt

on JlQ, the algebraic equations

&nji+#>(dini- dtirj) = <k,

-#Kfi = <Pj

have a unique solution

& = -{*-ifL, $ = (T-i)*»[^ + ( j r 1 ) * " ^ * * - ft»r*)].


In particular, every function / on JlQ determines a vertical vector field

#f = fr-yfye, - (n-ly%f + (Tr-^&jid^ - di^d) (4.8.12)

on J1Q —> K in accordance with the relation

■d,\UL = -df Adt.

Then the Poisson bracket

{/,$}x.*=*fJ*/jni. f,9eO°(JlQ), {f,9}i = (*-lY3(difdJg-dt

igd]f) +

(ft.»r i k-Slkirn)(ir-1)«(ir-1)^/ajff,

(4.8.13)

can be defined on the space D°(JlQ) of functions on the jet manifold JlQ. In particular, the vertical vector field d} (4.8.12) is the Hamiltonian vector field for the function / with respect to the Poisson structure (4.8.13). The Poisson structure (4.8.13) defines the corresponding symplectic foliation on the jet manifold J1Q, which coincides with the fibration JlQ —>R. The symplectic form on the leaf J]Q of this foliation is

f2t = dirx A dq'

[57, 182].

Example 4.8.4. If a Lagrangian L is hyperregular, the Poisson structure (4.8.13) is isomorphic to the canonical Poisson structure on the momentum phase space V'Q (see Section 5.1). Indeed, the Poisson bracket (4.8.13) reads

{Wj} = {q',qi} = 0, {7r,,g,} = ff.

•

Remark 4.8.5. If a Lagrangian is degenerate, one may try to introduce a Poisson structure on JlQ corresponding to the presymplectic form dHL (see (4.8.23) below) in accordance with the procedure suggested in [49] (see Remark 2.5.1). •

DEFINITION 4.8.1. A connection

ZL = dt + f%+edl


on the jet bundle J ' Q —» H is said to be a Lagrangian connection for the Lagrangian L if it obeys the equation

q j n L = d//L (4.8.14)

which takes the coordinate form

(/' - #"* = o, a,£ - a(7r, - fd^i - e*it + {p - 4)d^ = o.

(4.8.15a) (4.8.15b)

□

Example 4.8.6. A glance at the equations (4.8.15a) - (4.8.15b) shows that, for a regular Lagrangian L, a Lagrangian connection is necessarily holonomic and unique.

In order to clarify the meaning of the equation (4.8.14), let us consider the Lagrangian

Cd^C(t,q',qi) + (qlt)-qi)nl(t,q\ql)

on the repeated jet manifold J1JiQ. The corresponding Euler-Lagrange operator, called the Euler-Lagrange-Cartan operator, reads

£z : JlJ'Q -* T'J'Q, £T = [(diC + diTTjiq!^ -qi)- dtn,)dq' + 7r0(^() - q})dq\} A dt, (4.8.16)

dt = dt + q\t)di + q\td\.

Then the equation (4.8.14) is equivalent to the condition

UiJ'Q) C K e r £ r

which is the first order differential equation on the jet manifold J ' Q , called the Cartan equations. They read

*v(4t) ~ d) = °- (4.8.17a)

diC - dtiTi + {qj{t) - qi)diir} = 0.

Remark 4.8.7. In Section 5.2, we will show that the Euler-Lagrange-Cartan operator (4.8.16) is the Lagrangian counterpart of a Hamilton operator. •

(4.8.17b)


Solutions of the Cartan equations (4.8.17a) - (4.8.17b) are (local) integral sections £:(}—» JlQ of Lagrangian connections &, for a Lagrangian L, i.e., Jlc = £L°C. Furthermore, the equation (4.8.14) for these integral sections is equivalent to the relation

?{d\dHL) = 0 (4.8.18)

which is assumed to hold for any vertical vector field d on the jet bundle JlQ —» R. It is easily seen that the restriction

Ker£L = J2Q n K e r £ r (4.8.19)

of the Euler-Lagrange-Cartan operator £ j (4.8.16) to the second order jet manifold J2Q C JlJlQ recovers the Euler-Lagrange operator £L (4.8.6) for the Lagrangian L. Therefore, the Lagrange equations (4.8.7) are equivalent to the Cartan equations (4.8.17a) - (4.8.17b) on the holonomic sections c = Jlc of the jet bundle JlQ -> R. It follows that solutions of the Lagrange equations (4.8.7) are integral sections of holonomic Lagrangian connections for the Lagrangian L, which take their values into the kernel of the Euler-Lagrange operator (4.8.19). Different holonomic Lagrangian connections lead to different dynamic equations associated with the same system of Lagrange equations.

In the case of a regular Lagrangian L, the Cartan equations (4.8.17a) - (4.8.17b) are equivalent to the Lagrange equations (4.8.7), and lead to the unique dynamic equation

q3tt = (n-1Y3[-diC + dtnl + qtdkirl} (4.8.20)

on the configuration space Q. By very definition, the Poincare-Cartan form Hi (4.8.5) defines the fibred mor-

phism

HL-.J'Q -^T'Q, (4.8.21)

{pt,p)oHL = (TT,, C-ntqi),

from the affine jet bundle JXQ —> Q to the cotangent bundle T'Q —» Q of Q, which plays the role of the homogeneous Legendre bundle (2.9.3), equipped with the coordinates (t,q',p,pi). The morphism (4.8.21) is termed the Legendre morphism


associated with Hi. There is the following relation between the Legendre map L (4.8.9) and the Legendre morphism HL (4.8.21):

L = C o HL, (4.8.22)

where C is the canonical projection T'Q —> V'Q (1.1.7). Furthermore, it is readily observed that the Poincare—Cartan form Hi is the

pull-back of the canonical Liouville form

E = pdt + pidqx

on the cotangent bundle T'Q by the associated Legendre morphism (4.8.21). Accordingly, the presymplectic form

dHL = d7r, A dq* - d^tf -C)Adt (4.8.23)

on the jet manifold J1Q is the pull-back of the canonical symplectic form

dz. = dp Adt + dpi A dql

on the cotangent bundle T'Q of the configuration space Q.

Remark 4.8.8. The presymplectic form dHL (4.8.23), together with the 1-form dt, define a copresymplectic structure on the configuration space JlQ [37, 86]. Let us take the exterior product

(dHL)m Adt = det(7Ty)( ]T dql A dq*)™ A dt. I

(4.8.24)

If it is nowhere vanishing, the pair (dHi,dt) is called a cosymplectic structure on the (2m + l)-dimensional manifold J1Q. A glance at the expression (4.8.24) shows that the pair (dHi, dt) is cosymplectic if and only if the Lagrangian L is regular. A cosymplectic structure (dHi,dt) on JlQ defines the isomorphism

TJlQ 9 » H v\dHL + {v\dt)dt e T'JlQ.

In particular, let T be a subbundle of the tangent bundle TJlQ —» J1Q. One can define the orthocomplement

T 1 = {v € TJlQ : v\dHL + (v\dt)dt € Ann (T)}

of T in TJlQ with respect to (dHt, dt).


Note that the classification of Lagrangians on the product R x M by the properties of the intersection KerdHi n Kei dt has been suggested [86]. This classification remains also true for Lagrangians on an arbitrary fibre bundle Q —» R, though not all its constructions (e.g., the tangent-valued forms S and ~S (see [86])) are maintained under time-dependent transformations. •

Given a Lagrangian L, let the image ZL of the configuration space JlQ by the Legendre morphism Hi (4.8.21) be an imbedded subbundle

k ■ ZL *-* T'Q

of the cotangent bundle T'Q. It is provided with the pull-back De Dondei form

=.L = i'LE.

By analogy with the Cartan equations (4.8.18), the corresponding Hamilton-De Donder equations for sections r of the fibre bundle Zi —» R are written as the condition

T'{u\dEL) = 0 (4.8.25)

where u is an arbitrary vertical vector field on ZL —> R. To obtain these equations in an explicit form, one should substitute the solutions

q,t{.t,qi,pj), £{t,q>,pj,p)

of the equations

Pi = *,{t,qj,q{),

p = C{t,q>,4) - ni(t,q>,qi)q\ (4.8.26)

in the Cartan equations (4.8.18). If a Lagrangian L is regular, the equations (4.8.26) have a unique solution. Then

the Hamilton-De Donder equations take the coordinate form

dtr = -dlr, dtT{ = dir,

r = C(t,q\g't{t,g],pJ))-plq't(t,qi,p1), (4.8.27)

and are equivalent to the Cartan equations. If a Lagrangian L is degenerate, the equations (4.8.26) may admit different solutions, or no solutions at all. More can be said in the following case.

4.9. NEWTONIAN SYSTEMS 195

PROPOSITION 4.8.2. Let the Legendre morphism HL : JlQ —► ZL be a submersion. Then a section c of the jet bundle JlQ —> K is a solution of the Cartan equations (4.8.18) if and only if HL ° c is a solution of the Hamilton-De Donder equations (4.8.25) [63]. a

Remark 4.8.9. In the case of a regular Lagrangian L, the Hamilton-De Donder equations (4.8.27) can be seen as the equations of motion in variant (B) of the Dirac system on the symplectic manifold T'Q in the case of: (i) a zero Hamiltonian, (ii) the primary constraint space N = ZL (4.8.26), and (iii) with the additional condition that a solution v of the equation (3.6.1) has the component v° = 1. The latter condition guarantees that the motion parameter is t. •

In conclusion, let us touch briefly on the case of conservative Lagrangians. Let us suppose that, given a trivialization

ip:Q = R x M

in the coordinates (£,<f), a Lagrangian L is independent of t. With respect to these coordinates, the configuration space JlQ admits the presymplectic form

u/L = ctn, A dqx

(cf. (3.3.5)), and the equation (4.8.14) for a Lagrangian connection fx, may be written in the form

(,L\UL = -d{-Kiq\ - C).

Thus, a conservative Lagrangian system reduces to the presymplectic Hamiltonian system whose Hamiltonian is the energy function (^gj — C) (see Examples 3.3.2 and 3.4.2). In particular, if a Lagrangian is degenerate, one can apply the procedure from Section 3.6 to investigate it [138]. We refer the reader to [31, 37, 111] for extension of this procedure to time-dependent degenerate Lagrangian systems on the product R x M. We will investigate degenerate Lagrangian systems in the framework of Hamiltonian formalism (see Section 5.5).

4.9 Newtonian systems

Let L be a Lagrangian on a velocity phase space J1Q and L the Legendre map (4.8.9). Due to the vertical splitting (1.1.6) of VV'Q, the vertical tangent map VL


to L reads

VL : VQJXQ - . V'Q x V'Q. Q

It yields the linear fibred morphism

bd5= f(IdJ1Q,pr2oVL) : VQJlQ - V ^ ' Q ,

b : d[ i-> ffyd^,

(4.9.1)

where {dg?} are bases for the fibres of the vertical tangent bundle VA JlQ —> J ' Q . The morphism (4.9.1) defines the mapping

JlQ -* VAJ'Q 9 VAJlQ W J'Q *

and, due to the splitting (4.1.9), also the mapping

m:JlQ — > V Q ® V Q ,

"*ij = P,j o m = 7Ty,

where (£,(?', p tJ) are holonomic coordinates on V'Q®V'Q. Thus, n^ = mtJ are 2 ___

components of the W'Q-valued field m on the velocity phase space J Q. It is called the mass tensor.

Let a Lagrangian L be regular. Then the mass tensor is non-degenerate, and defines a fibre metric, called mass metric, in the vertical tangent bundle VQJ^Q —» JlQ. Let us recall that, if a Lagrangian L is regular, there exists a unique Lagrangian connection £& for L which is holonomic in accordance with the equation (4.8.15a), and obeys the equation

mutt* = -dtn - djTTi^t + d,£. (4.9.2)

This holonomic connection defines the dynamic equation (4.8.20). At the same time, the equation (4.9.2) leads to the commutative diagram

VQJ'Q -*-» VAJ'Q

DKL \/eL

J2Q

SL = \>ODU,

£> = mxk{q'!t-(,kL), (4.9.3)


where D^L is the covariant differential (4.1.17) relative to the connection fa. Furthermore, the derivation of (4.9.2) with respect to q\ results in the relation

fa\dmij + 771^7* + mjknff = 0, (4.9.4)

where

li — 2 °«SL

are coefficients of the symmetric dynamic connection 7 ^ (4.3.10) corresponding to the dynamic equation fa.

Thus, each regular Lagrangian L defines the dynamic equation fa, related to the Euler-Lagrange operator £L by means of the equality (4.9.3), and the non-degenerate mass tensor m^, related to the dynamic equation fa by means of the relation (4.9.4). This is a Newtonian system in accordance with the following definition.

DEFINITION 4.9.1. Let Q —> R be a fibre bundle together with

• a (non-degenerate) fibre metric m in the fibre bundle VQJ1Q —> JYQ:

m:JlQ-+VQ®VQ, Q

m = -mijdq' V dq3,

satisfying the symmetry condition

dlrriij = d'rriik, (4.9.5)

• and a holonomic connection f (4.1.25) on the jet bundle JlQ —► K, related to the fibre metric m by the compatibility condition (4.9.4).

The triple (Q, m, ?) is called a Newtonian system. This is not the terminology of [31]. □

Note that the compatibility condition (4.9.4) can also be introduced in an intrinsic way as

V ?m = 0,


where by V is the covariant derivative with respect to the connection 7^ on the vertical cotangent bundle VqJlQ —* J1Q, which is dual of the connection \ (4.3.23) on the vertical tangent bundle VQ JlQ —> JlQ.

Definition 4.9.1 generalizes the second Newton law of point mechanics. Indeed, the dynamic equation for a Newtonian system is equivalent to the equation

m,k(qu - £*) = 0. (4.9.6)

There are two main reasons for considering Newtonian systems. From the physical viewpoint, with a mass tensor, we can introduce the notion

of an external force. Note that, in the universal unit system, the mass tensor m is dimensional. For instance, the dimension of a mass tensor of a point mass with respect to Cartesian coordinates q' is [length]-1, while that with respect to the angle coordinates is [length].

DEFINITION 4.9.2. An external force is defined as a section of the vertical cotangent bundle VQJ1Q —> JlQ. Let us also bear in mind the isomorphism (4.1.9). □

Note that there are no canonical isomorphisms between the vertical cotangent bundle VQ JlQ and the vertical tangent bundle VQ J ' Q of JlQ. Therefore, one should distinguish forces and accelerations which are related by means of a mass metric (see also Remark 4.9.2 below).

Let (Q, m, £) be a Newtonian system and / an external force. Then

o=r+(m-n (4.9.7)

is a dynamic equation, but the triple (Q, m, £/) is not a Newtonian system in general. As follows from direct computation, if and only if an external force possesses the property

af/i + fl}/i = o, (4.9.8)

then £/ (4.9.7) fulfills the relation (4.9.4), and (Q, m, £/) is also a Newtonian system.

Example 4.9.1. For instance, the Lorentz force

}i = eF^q?, 9? = 1,

where

(4.9.9)

F\» = dxA^ - d^Ax (4.9.10)


is the electromagnetic strength, obeys the condition (4.9.8). Note that the Lorentz force (4.9.9), just as other forces, can be expressed in the relative velocities qr with respect to an arbitrary reference frame T:

*-£($*-*+4 where q are the coordinates adapted to the reference frame T, and JP is the electromagnetic strength, written with respect to these coordinates. •

Remark 4.9.2. The contribution of an external force / to a dynamic equation

qit-Sl = (m-l)ikfk

of a Newtonian system obviously depends on a mass tensor. It should be emphasized that, besides external forces, we have a universai force which is a holonomic connection

C = ^ A < A , A - Q? = 1,

associated with a symmetric linear connection K (4.4.3) on the tangent bundle TQ —> Q. From the physical viewpoint, this is a non-relativistic gravitational force, including an inertial force, whose contribution to a dynamic equation is independent of a mass tensor. •

From the mathematical viewpoint, the equation (4.9.6) is the kernel of an Euler-Lagrange-type operator (see (4.9.25) below). By an appropriate choice of a mass tensor, one may hope to bring it into Lagrange equations. We have seen that a non-degenerate Lagrangian system is necessarily a Newtonian one, while a converse statement is not generally true.

Example 4.9.3. Let us consider a non-degenerate quadratic Lagrangian

£ = ^ ( ^ ) * r f + w ) ? ; + <K<n> (4.9.11)

where the mass tensor m tJ is a Riemannian metric in the vertical tangent bundle VQ —» Q (see the isomorphism (4.1.8)). Then the Lagrangian L (4.9.11) can be written as

£ = -^Sa^gfgf, <?? = !, (4.9.12)


where g is the metric

Poo = -2<t>, 9oi — ~k<> 9i3 = - " > ■ ) (4.9.13)

on the tangent bundle TQ. The corresponding Lagrange equations take the form

q\t = -{m-l),kUMtll 9? = 1, (4.9.14)

where

{\nv} = —Adxg^u + d^g^x - d^gx,,)

are the Christoffel symbols of the metric (4.9.13). Let us assume that this metric is non-degenerate. By virtue of Corollary 4.4.4, the dynamic equation (4.9.14) gives rise to the geodesic equation (4.4.9) on the tangent bundle TQ, which reads

9fl = 0, 9° = 1,

«* = { A V H V - <AAO,}?V.

Let us now bring the Lagrangian function (4.9.11) into the form

£ = y(O(?;-r)te-r') + 0'(9"), (4.9.15)

where T is a Lagrangian frame connection o n Q - > R which takes its values into the kernel of the Legendre map L (see Section 5.6). This connection defines an atlas of local constant trivializations of the fibre bundle Q —» R and the corresponding coordinates (t,5*) on Q such that the transition functions q1 —► q" are independent oft, and P = 0 with respect to (£,<f). In these coordinates, the Lagrangian (4.9.15) reads

£■=^m+wof ))• (4.9.16)

Let us assume that <f>' is a nowhere vanishing function on Q. Then the Lagrange equations (4.9.14) take the form

Qu = {x\}t?t, 3? = l, where {>'„} are the Christoffel symbols of the metric (4.9.13), whose components with respect to the coordinates ((,?") read

goo = - 2 0 ' , 9oi = 0, 9x] = - ?Ry . (4.9.17)


Then the spatial part of the corresponding geodesic equation

5° = 0, 9 = 1 , Q ={x ■/}? q (4.9.18)

on the tangent bundle TQ is precisely the spatial part of the geodesic equation with respect to the Levi-Civita connection for the metric (4.9.17) on TQ. •

This example shows that a mass tensor may be treated sometimes as a field variable.

A Newtonian system (Q, rh, £) is said to be standard, if rh is the pull-back on VQJ1Q of a fibre metric in the vertical tangent bundle VQ —► Q in accordance with the isomorphisms (4.1.8) and (4.1.9), i.e., the mass tensor rh is independent of the velocity coordinates q\.

It is readily observed that any fibre metric rh in VQ —» Q can be seen as a mass metric of a standard Newtonian system, given by the Lagrangian

c = \nrnWM - n(4 - n, (4.9.19)

where T is a reference frame. If rh is a Riemannian metric, one can think of the Lagrangian function (4.9.19) as being a kinetic energy with respect to the reference frame T.

Example 4.9.4. Let us consider a system of n distinguishable particles with masses ( m i , . . . , mn) in a 3-dimensional Euclidean space R3. Their positions ( r i , . . . , r„) span the configuration space R3n. The total kinetic energy is

Z A=l

that corresponds to the mass tensor

TnABtj = S/igSiftrtAi A,B = l , . . . n , i,j = 1,2,3,

on the configuration space R3n. To separate the translation degrees of freedom, one performs a linear coordinate transformation

( r i , . . . , r n ) (-► (7»i , . . . ,p„_ i ,R) ,


where R is the centre of mass, while the n - 1 vectors (pu.. ., p„-i) are mass-weighted Jacobi vectors (see their definition below) [119, 120). The Jacobi vectors PA are chosen so that the kinetic energy about the centre of mass has the form

i n - l

Z A=l (4.9.20)

that corresponds to the Euclidean mass tensor

TKABi] = ^AB^ij, A,B = l , . . . n - l , i,j = 1,2,3,

on the translation-reduced configuration space R3n~3. The usual procedure for defining Jacobi vectors involves organizing the particles into a hierarchy of clusters, in which every cluster consists of one or more particles, and where each Jacobi vector joins the centres of mass of two clusters, thereby creating a larger cluster. A Jacobi vector, weighted by the square root of the reduced mass of the two clusters it joins, is the above-mentioned mass-weighted Jacobi vector. For example, in the four-body problem, one can use the following clustering of the particles:

Pi = y/fH(T2-n),

P 2 = \ / ^2 ( r 4 - r 3 ) , - .— / m 3 r 3 + m4r4 mir1 + m 2 r 2 \ P3 = V^3 I I ,

\ m3 + m4 mi + m2 / Hi mi m2'

_L J_ J_ p.2 m3 m4'

J_ 1 1 u.3 m\ + m.2 m 3 + m 4

Different clusterings lead to different collections of Jacobi vectors, which are related by linear transformations. Since these transformations maintain the Euclidean form (4.9.20) of the kinetic energy, they are elements of the group 0(n — 1), called the "democracy group". •

Now let us turn to the conditions for a Newtonian system to be a Lagrangian one. This is the well-known inverse problem formulated for time-dependent mechamcs. We will investigate it as the particular inverse problem for dynamic systems on fibre bundles [57].

The equation (4.9.6) is the kernel of the second order differential Euler-Lagrange type operator

£:J2Q^ V'Q, £ = mik(Zk - qk

tt)dqx. (4.9.21)


One can write such an operator as a 2-form

£ = eiei A dt, (4.9.22)

where 6l are contact forms (1.3.6). Obviously, the class of Euler-Lagrange type operators includes the Euler-Lagrange operators (4.8.6). Thus, the inverse problem reduces to the conditions for the Euler-Lagrange type operators to be the Euler-Lagrange ones.

Given a fibre bundle Q —» R, we have the so called variationaJ sequence

0 ^ R ^ O ° ( Q ) ^ D ° ' 1 ( J 1 Q ) ^Qll(J2Q) 6-^0%l(J3Q) ^ • • ■ , (4.9.23)

where O0,l(J1Q) is the space of horizontal densities Cdt on JlQ, while D1,l(J2Q) is the space of 2-forms £id8' A dt on J2Q, and so on [46, 57, 179]. In particular, the elements of D0,1(JlQ) are the Lagrangians Cdt, while those of Dl,1(J2Q) are the Euler-Lagrange-type operators written as 2-forms (4.9.22). The key point lies in the fact that the variationaJ sequence (4.9.23) is a complex. It implies that

S o dH = 0, <52 o 8 = 0,

(4.9.24) (4.9.25)

where

dH(f) = dtfdt, feO°{JlQ),

is the horizontal exterior differential,

6(Cdt) = (di - dtdDLP A dt (4.9.26)

is the Euler-Lagrange map (or the variationai operator), and

^ ( ^ A dt) = [(2d; - djdj + dldf^P A 6' + {d)8i + dlSj - 2dtdf£i)6\ A 9j + {df£{ - df^)^ A 6*} A dt = 0

is the Helmholtz-Sonin map. A glance at the variationai operator (4.9.26) shows that the image Im<5 of the

variationai operator (4.9.26) consists of the Euler-Lagrange operators, and Im<5 C Ker<52 in accordance with the equality (4.9.25). It follows that

62(£) = 0 (4.9.27)


is the necessary condition for an Euler-Lagrange type operator £ to be an Euler-Lagrange one

£ = 6(L) (4.9.28)

for some Lagrangian L. The condition (4.9.27) takes the coordinate form

d3£, - di£} + fkiBtSj - Sfa) = 0,

d)£i + d\£j - 2dtdf£x = 0, df£i - df£i = 0.

(4.9.29a)

(4.9.29b) (4.9.29c)

The obstruction preventing the condition (4.9.27) from implying the equality (4.9.28) is topological [46, 179). If

Q = Rm + 1 -> R

(in particular, locally), the variational sequence (4.9.23) is exact, i.e., Im6 = Ker62. and the equality (4.9.27) is also a sufficient condition for an Euler-Lagrange type operator £ to be an Euler-Lagrange one.

Remark 4.9.5. A glance at the equality (4.9.24) shows that the Lagrangian L in the equality (4.9.28) is defined modulo the variationally trivial Lagrangians

L = ckfdt (4.9.30)

where / is a function on Q. •

Applying the condition (4.9.27) to the operator (4.9.21), we restate the well-known Helmholtz conditions (see [34, 78, 81, 143, 156] and references therein). It is readily observed, that the condition (4.9.29c) is satisfied since the mass tensor is symmetric. The condition (4.9.29b) holds due to the equality (4.9.4) and the property (4.9.5). Thus, it suffices to verify the condition (4.9.29a) for a Newtonian system to be a Lagrangian one.

Example 4.9.6. Let £ be a free motion equation which takes the form (4.6.9) with respect to a reference frame (i.g1), and let m be a mass tensor which depends only on the velocity coordinates q\. Such a mass tensor may exist in accordance with affine coordinate transformations (4.6.10) which maintain the equation (4.6.9). Then £ and m make up a Newtonian system. This system is a Lagrangian one if


m is constant with respect to the above-mentioned reference frame (t,if). Relative to arbitrary coordinates on a configuration space Q, the corresponding Lagrangian takes the form (4.9.19), where T is the connection associated with the reference frame {t,qi). •

Example 4.9.7. Let us consider the 1-dimensional motion of a point mass mo subject to friction. It is described by the equation

77io«jH = -kqt, fc>0, (4.9.31)

on the configuration space R2 —» K, coordinated by [t, q). This mechanical system is characterized by the mass function m = mo and the holonomic connection

k f = 8t + qtdq qtfr m (4.9.32)

but it is neither a Newtonian nor a Lagrangian system. The conditions (4.9.29a) and (4.9.29c) are satisfied for an arbitrary mass function m(t,q,qt), whereas the conditions (4.9.4) and (4.9.29b) take the form

—kqtd^m — km + dtm + qtdqm = 0. (4.9.33)

The mass function m = const, fails to satisfy this relation. Nevertheless, the equation (4.9.33) has a solution

' Jfc m = mo exp — t .

[mo (4.9.34)

The mechanical system characterized by the mass function (4.9.34) and the holonomic connection (4.9.32) is both a Newtonian and a Lagrangian system with the Havas Lagrangian

, 1 [ k 1 2 C = -mo exp — t q,

2 [m0 J (4.9.35)

[151]. The corresponding Lagrange equations are equivalent to the equation of motion (4.9.31). •

Example 4.9.8. [143]. The dynamic equation

Ztt - Vt = 0, Vtt - y = 0

on the configuration space R3 —► K is not equivalent to any system of Lagrange equations. •


4.10 Holonomic constraints

Let (Q, m, £) be a Newtonian system, where m is a Riemannian metric in the vertical tangent bundle VQJ1Q —» JlQ. This Section is devoted to the restriction of (<5, m, £) to a closed imbedded fibred submanifold iN : N «-» Q of the configuration bundle Q —» R, which is treated as a holonomic constraint. We refer the reader to [24] and references therein for the geometric description of holonomic constraints in conservative mechanics.

With a holonomic constraint N C Q, we have the following imbedding diagrams

VN <-. VQ

I I N «-+ Q .

\ / R

y2 /v -> J 2 < 2

I I I I

N ^-> Q \ /

R

Furthermore, there exists an open tubular neighbourhood U of the submanifold N, which is a fibred manifold over TV [109]. Therefore, one can provide the configuration space Q with the atlas of bundle coordinates (t, aT, ql) such that

<?' \N= 0, ql |JIJV= o, 9M \J*N= 0.

We will continue to use the notation qx for the whole coordinate collection (t, <rr, <j'). Let n be the induced Riemannian fibre metric in the vertical tangent bundle

VNJ1N —> J*N. Its components are

nrs = mTS \jiN .

Since

d ^ , = d^rrir, \jiN,

the Riemannian metric n satisfies the symmetry condition (4.9.5). Using the Riemannian metric m in VQJ1Q —> J ' Q , we can define the orthocom-

plement V -» JlN of the subbundle VNJlN C VQJ1Q \JIN and the corresponding splitting

VQJ'Q \JIN= VNJ'N © V (4.10.1)

4.10. HOLONOMIC CONSTRAINTS 207

together with the canonical projections

Prj : VQJlQ \fiN-> VNJ'N,

P^i(dtr)=dtr, ptx(flf) = n"msidl (4.10.2)

and

pr2 : VQJ'Q | J I A f - V,

pr2(5*) = 0, pr2(9j) = d\ - nT'm3ld'r = tft.

Since the restriction J2Q \j\N—> JYN is an affine bundle modelled over the vector bundle VQJ1Q \JIN, the splitting (4.10.1) defines the corresponding decomposition

J2Q U A T = J2N 0 V. J*N

Then the dynamic equation £ : JlQ -» J2Q of our Newtonian system splits in the following way:

f | j 'w= (IN + r, (4.10.3)

where

&v ■ JlN ~* J2N, CN = (t;T+nr'msie)\jiN, (4.10.4)

and

r = Z% : JlN — V. (4.10.5)

It follows that the dynamic equation f on the configuration space Q induces the dynamic equation £N (4.10.3) on the holonomic constraint N. In an equivalent way, the induced dynamic equation £^ is characterized by the relation

KTSCN = ("V.£' + "hi?) Ij'AT ■ (4.10.6)

PROPOSITION 4.10.1. Let (<2,m,£) be a Newtonian system on a configuration space Q, and N C Q a holonomic constraint. Then (Af, n,£jv) is a Newtonian system. □

Proof. The proof of the compatibility condition (4.9.4) for £N and n follows from direct computation. QED


Let 7? be the dynamic connection (4.3.10) associated with a dynamic equation £ of the Newtonian system in question. It can be represented by the tangent valued form (1.4.4) which reads

7 { : JlQ -» T'J'Q ® VQJlQ,

7« = (dq\ - 7 iV) ® 9, + (d< - 7XV) ® &• Let us consider the map

7« : J*N «-. J ' Q IJIJV ^ T ' J ' Q ® VQJ 1 *? | 7 , W —>

rv'jv ® vw1^, where the dual morphism (1.1.3)

( . / % ) • : T V ' Q | j .w -> TV'TV

and the morphism pr, (4.10.2) are utilized. We obtain

% = {dort-%dt-r3do°)®dT, (4.10.7)

lo = (7o + ™r'"i5.7o) \j*N, 7p = (7p + nTSmstjlp) \jiN

It follows that the map 7^ is a connection on the affine jet bundle J N —» N.

PROPOSITION 4.10.2. The dynamic connection 7 (4.10.7), induced on the jet bundle JlN —► N of the holonomic constraint N by the dynamic connection 7^, defines a dynamic equation on N which is precisely £N (4.10.4), induced on N by the dynamic equation {, □

Proof. The proof is straightforward. QED

It is readily observed that the dynamic connection 75 differs in a torsion from the symmetric dynamic connection -yN, associated with the dynamic equation £#. We have the relation

% = -\TZ + INI r, = \Ti+yNi

77 = -d^n^m^e ~ nrhdlmhtC- (4.10.8)

Let now a Newtonian system (Q, m,£) be a Lagrangian system for a regular Lagrangian L on the velocity phase space J*Q. The pull-back Ls = i*NL of L by

4.10. HOLONOMIC CONSTRAINTS 209

the imbedding JliN is a Lagrangian on the holonomic constraint N. We come to the following assertion.

PROPOSITION 4.10.3. The dynamic equation £LN, associated with the induced Lagrangian L^ on N, coincides with the dynamic equation £/,# (4.10.4) induced on N by the dynamic equation £L, associated with the Lagrangian L. □

Proof. We obtain the relation

( m r . a + mrt&) \jxN= -dtdlrLN - a°td,drLN + d'rLN = n r 3£l„

(see (4.9.2)), where

nTS = mrs \jiN= dffiLrf.

This is precisely the relation (4.10.6). QED

It is readily observed that the Lagrange equations for a Lagrangian L^ on the holonomic constraint N are locally equivalent to those of the local Lagrangian L + Xiy' on Q, where X{ are the Lagrange multipliers. Consequently, the dynamic equation £LM = £LH describes a motion on an ideal holonomic constraint N. This fact also remains true for an arbitrary Newtonian system. Indeed, the splitting (4.10.3) shows that one can think of the dynamic equation £# as being the dynamic equation £ at the points of N plus the additional acceleration —r (4.10.5), treated as a constraint reaction acceleration. Then, bearing in mind the isomorphism (4.1.8), it is readily observed that this acceleration is orthogonal to the holonomic constraint N with respect to the mass metric m, i.e.,

m(u, s) = m^v!1^ — m/lrii''nrsmSJ£'7 = 0

for any local vertical tangent vector field uhdh on N —» K. Thus, we generalize the D'Alembert principle (the principle of virtual displace

ments) for an arbitrary Newtonian system. In accordance with this principle, a constraint reaction acceleration of a Newtonian system on an ideal holonomic constraint is orthogonal to the constraint manifold with respect to the mass metric of this system.

One can say more when (Q,m,£) is a standard Newtonian system, i.e., m is a Riemannian metric in the vertical tangent bundle VQ —> Q. This metric induces


the Riemannian metric n on the vertical tangent bundle VN —► N. Its components are

TV, = m r s \N .

We have the splitting

TQ\N=TN®V N

with the canonical projections

prt(ar) = ar, Wi(9i) = nr'msidr, pr2(dr) = 0, pr2(9i) = d\ - rfm^dr = du

and the corresponding splitting of the first order jet manifold

JXQ\N= JlN®V. N

In particular, any reference frame T on a configuration space Q defines a reference frame r \ on the holonomic constraint TV in accordance with the decomposition

r \N= rN + av, nrsr% = ( m r s r S + TTViP) I AT . (4.10.9)

Moreover, the expression (4.10.8) shows that, in the case of a standard Newtonian system, the dynamic connection 74 (4.10.7) is precisely the dynamic connection y^N, associated with the dynamic equation fa on the holonomic constraint N.

Example 4.10.1. Let (Q,m,£) be a Newtonian system for a free motion equation £. Then, fa fails to be a free motion equation on a holonomic constraint N in general. In particular, if L is a Lagrangian (4.9.19), its restriction to a holonomic. constraint N reads

LN = 2nT-(°l ~ rN)(o! ~ Tsw) + ^n{ov,ov),

where I \ is the induced frame (4.10.9). •

4.11. NON-HOLONOMIC CONSTRAINTS 211

4.11 Non-holonomic constraints

Here we are concerned with the geometric theory of non-holonomic constraints in time-dependent mechanics. We refer the reader to [25, 30, 112, 113, 124, 183, 184] and references therein for the geometric description of non-holonomic constraints in conservative mechanics. The key problem is that the method of the Lagrange multipliers is not appropriate to non-holonomic constraints in general (see [7, 25, 112, 114]).

Let the jet manifold JlQ be a velocity phase space of time-dependent mechanics on a configuration bundle Q —» R. The most general non-holonomic constraints considered in the literature are given by codistributions S or, accordingly, by distributions Ann(S) on the jet manifold JlQ [55, 132]. In conservative mechanics on a configuration space M, this is the case of a codistribution on TM [183]. Submani-folds of the jet manifold JlQ [88, 106] and distributions on a configuration space Q [114] can also be seen as non-holonomic constraints.

In connection with non-holonomic constraints in time-dependent mechanics, one usually studies the following problem. Given a configuration space Q, let £ be a dynamic equation on Q, and S a codistribution on JlQ whose annihilator Ann (S) is treated as a non-holonomic constraint. Similarly to the case of holonomic constraints, the goal is a decomposition

£ = I + r, (4.11.1)

where £ is a dynamic equation obeying the condition

| c A n n ( S ) .

Then one can think of the dynamic equation £ as describing a mechanical system subject to the non-holonomic constraint S, while (— r) is said to be the constraint reaction acceleration.

Let us assume that the codistribution S has dimension n and is locally spanned by the 1-forms

s" = sa0dt + s°dq' + s°dq\

on the jet manifold JlQ. Then a dynamic equation £ is compatible with the non-holonomic constraint S if

Sa(£)=Z]sa = Sao + S°ql + S°e = 0.


This equation is algebraically solvable for n components of £ if and only if the nxm matrix s° (g \g | ) has everywhere maximal rank n < m. Therefore, we will restrict our consideration to the non-holonomic constraints, called admissible, such that

dimS = dim?(S),

where v* is the vertical endomorphism (4.1.14)

Example 4.11.1. Note that holonomic constraints can also be seen as the non-holonomic ones which, however, are not admissible. •

If a non-holonomic constraint is admissible, there exists a local m x n matrix sJ,(gA,<?!) such that

4*J = <t (4.11.2)

Then, the local decomposition (4.11.1) of a dynamic equation £ can be written in the form

g = g + JJafjQ. (4.11.3)

The global decomposition (4.11.1) exists by virtue of the following lemma.

LEMMA 4.11.1. [55]. The intersection

F= J 2 Q n A n n ( S )

is an affine bundle over J ' Q , modelled over the vector bundle

F = VQJlQn Ann (S).

n

Proof. The intersection F consists of the vertical vectors vxd\ 6 VQJ1Q which fulfill the conditions

*?(«*,«*)«* = 0.

Since the non-holonomic constraint S is admissible, every fibre of F is of dimension m-n, i.e., F is a vector bundle, while F is an affine bundle. QED

Since the intersection J2Q n Ann (S) —> J1Q is an affine bundle, it has always a global section f by virtue of Theorem 1.1.2.


To construct the global decomposition (4.11.1), one should perform a splitting of the vertical tangent bundle

VQJlQ = F © V (4.11.4)

in order to obtain the corresponding splitting of the second order jet manifold

J2Q = F © V (4.11.5)

which is modelled over (4.11.4). Here, V —» J ' Q should be interpreted as the bundle of possible constraint reaction accelerations.

If an admissible non-holonomic constraint S is of dimension n = m, a dynamic equation £ is decomposed in a unique fashion. If n < m, the decomposition (4.11.1) is indeterminate. Different variants of this decomposition lead to different constraint reaction accelerations which, from the physical viewpoint, characterize different types of non-holonomic constraints.

Let us turn now to some important particular cases of non-holonomic constraints.

Example 4.11.2. Let N be a closed imbedded submanifold of the velocity phase space JlQ, defined locally by the equations

/a(g\<?l) = o, a = 1 , . . . ,n < m. (4.11.6)

One can treat AT as a non-holonomic constraint at points N C J ' Q , given by the codistribution S = Ann(TN) on JlQ \s- This codistribution is locally spanned by the 1-forms

sa = dfa = dtfadt + d]}adq> + aj/°d^.

The non-holonomic constraint ,/V is admissible if and only if the matrix (9| /°) is of maximal rank n. It follows that N is a fibred submanifold of the affine jet bundle JiQ —» Q. Moreover, it is possible to separate locally some velocities <£, a = 1 , . . . , n, expressed as functions of the remaining coordinates (qx, qf , • • •, <?[") such that the equations (4.11.6) are brought into the form

Qt ~9 {Q ,Qt i • • • . It ) = 0- (4.11.7)

Then we have

sg = -dtga, «? = ~dwa, S? = 6f - dig", (4.11.8)


where dlbga = 0 for all indices a, b = 1 , . . . , n. Now, applying the above procedure to

the codistribution (4.11.8), one can obtain a dynamic equation on a submanifold N, compatible with the constraint Ann (TN). For instance, one can use the particular solution s\ = 6l

a of the equations (4.11.2), where the matrix s? is given by the expression (4.11.8) [106]. Then the (local) dynamic equation £ (4.11.3), compatible with the constraint (4.11.7), reads

£ , /a = e, £° = {• _ s*(£) = dkgae + dtga + qfrg*. (4.11.9)

•

Example 4.11.3. A non-holonomic constraint N is said to be linear if it is an affine subbundle of the affine jet bundle JlQ —> Q, which is given by the local equations

r=/oV)+/f(<?A)<?; = o, (4.11.10)

where the matrix / ta is of maximal rank. A linear non-holonomic constraint is always

admissible. Since N is an affine subbundle of JlQ —> Q, it has a global section V which is a connection on the configuration bundle Q —> R, called the constraint reference frame. With this connection F, the constraints (4.11.10) take the form

f°(qxM - n = o. (4.11.11)

We may say that the linear constraint is stationary with respect to the constraint reference frame T. Then, one can think of

<t ~ it - r ,

satisfying the equation (4.11.11), as virtual velocities relative to the linear constraint

Example 4.11.4. Let a configuration space Q admit a composite fibration Q —» E -> R, where

7TQE : Q — E

is a fibre bundle, and let (t, aT, ga) be coordinates on Q, compatible with this fibration. Given a connection

B = dt ® (dt + Bada) + daT ® (dT + B?da) (4.11.12)


on the fibre bundle Q —» E, we obtain the corresponding horizontal splitting (1.4.1) of the tangent bundle TQ. Restricted to the jet manifold JXQ C TQ, this splitting reads

J1Q = B(n'QEJ1Yi)®VEQ,

dt + c\dT + q?da = [(dt + Bada) + art{dT + Ba

Tda)] + [tf - Ba - o\B?\da,

where 7TQE J l E is the pull-back of the affine jet bundle J ' E —> E onto Q. It is readily observed that

N = B(n'QSJlZ)

is an affine subbundle of the affine jet bundle JlQ —► Q, defined locally by the equations

qf - oTtB°(qX) ~ Ba(qx) = 0

(cf. (4.11.7)). Then this subbundle yields a linear non-holonomic constraint, given by the codistribution S = Ann (TN) [162, 163]. This codistribution is locally spanned by the 1-forms

sa = -{dtBa + o\dtB«)dt - (daBa + ortdsB*)do" -

(dbBa + artdbBa

T)dqb + dqf - B«do\. (4.11.13)

With the connection (4.11.12), we also have the splitting (1.6.10a) of the vertical tangent bundle VQ of Q —> R and the corresponding splitting of the vertical tangent bundle VQJ1Q (see the canonical isomorphism (4.1.21)). The latter splitting reads

VQJlQ = F 0 V ,

a[dl + qfdl = art(% + B^a) + (q? - B r V[)^ . (4.11.14)

It is readily observed that F |/v consists of vertical vectors which are the annihilators of the codistribution (4.11.13). The splitting (4.11.14) yields the corresponding splitting (4.11.5) of the second order jet manifold J2Q. Then, given a dynamic equation f on JlQ, we obtain the decomposition (4.11.1) which reads

f = c, e=r - s°(o


(cf. (4.11.9)). •

Now we consider Newtonian systems because they provide the vertical tangent bundle VQJ1Q with a non-degenerate fibre metric m. Let us assume that m is a Riemannian metric. With this metric, we immediately obtain the splitting (4.11.4), where V is the orthocomplement of F. Then the decomposition (4.11.3) takes the form

f = | i + matm"5fot(0. (4.11.15)

where m,^ is the inverse matrix of

mab = s's^mv

[55] By definition, the decomposition (4.11.15) satisfies the generalized D'AIembert principle. The constraint reaction acceleration

- r ' = -mabmlJs°sb(Z) (4.11.16)

is orthogonal to every element of VQjiQnArm (S) with respect to the mass metric m. Since elements of VQ J 'QflAnn (S) can be treated as the virtual accelerations relative to the non-holonomic constraint S, the constraint reaction acceleration (4.11.16) characterizes S as an ideai non-holonomic constraint.

The Gauss principle is also fulfilled as follows. Given a dynamic equation £ and the above-mentioned fibre metric m, let us define a positive function G(w) on J^Q as

G(w) = rh (£(ir?(uO) - U/,£(JT?(IU)) - w) ,

G(q\ q\, qltt) = mi}(q\ <g)(?(q\ <£) - q\t)(?(q\ <jtfc) - q}t).

We say that

IMI = ^ G H

is a norm of w 6 J2Q.

PROPOSITION 4.11.2. [55]. Among all dynamic equations compatible with a non-holonomic constraint, the dynamic equation £ defined by the decomposition (4.11.15) is that of least norm. D


Proof. Let £ be another dynamic equation which takes its values into F. Then

Then we obtain

K l l - « a - f + € - C . € - € + € - 0 = llCll + S(C-C.C-0. i.e., ||Cn > llltl- QED

Now we will show that, in the case of non-degenerate Lagrangian system and linear non-holonomic constraints, the decomposition (4.11.15) satisfies the traditional D'Alembert principle.

Let S be an admissible non-holonomic constraint on the velocity phase space JXQ, and L a regular Lagrangian on JlQ with a Riemannian mass metric ml} = nl}. Since this is a particular Newtonian system, we obtain the dynamic equation

& = ft - fw '^^g i + jjq* + «&] (4.H.17)

which is compatible with the constraint S, treated as an ideal non-holonomic constraint. This is the system of Lagrange equations in the presence of a non-Lagrangian external force

F, = -rH^Afe) (see the equation (4.12.9) below). It is a constraint reaction force. Let us consider the energy-momenturn conservation law in the presence of this force. It reads

LpL - frft = -d(Tr,

where LfL is the Lie derivative of the Lagrangian L along a reference frame T and Tr is the energy function relative to this reference frame (see (4.12.15), (4.12.16), and (4.12.19) below). In particular, if a non-holonomic constraint is linear and T is a constraint reference frame (see Example 4.11.3), the constraint reaction force does not contribute to the energy conservation law. It follows that, in this case, the standard D'Alembert principle holds, while the equation (4.11.17) describes a motion in the presence of an ideal non-holonomic constraint in the sense of this D'Alembert principle.


The constrained motion equation (4.11.17) on a configuration space Q is neither a system of Lagrange equations nor a dynamic equation of a Newtonian system. In Section 5.11, we will show that this can be seen as Hamilton equations in the framework of the Hamiltonian formalism extended to the configuration space VQ.

4.12 Lagrangian conservation laws

Given a Lagrangian system ( J 'Q , L), its integrals of motion can be found if a Lagrangian L possesses symmetries. In time-dependent mechanics, these integrals of motion should be covariant under time-dependent transformations. For instance, the canonical energy function

EL = mqxt - £ (4.12.1)

fails to be such an integral of motion. There are different approaches in order to obtain the conservation laws in Lagrangian dynamics (see [1, 29, 51, 52, 57, 160] for the geometric methods). As in field theory, we will use the first variational formula of the calculus of variations [57, 160].

Let L be a Lagrangian (4.8.1) on the velocity phase space JlQ and

u = uldt + u'dx, ul = 0,1, (4.12.2)

a projectable vector field (4.8.2) on the configuration bundle Q —► R. By virtue of Theorem 1.2.1, this vector field may be treated as the generator of a 1-parameter group of local automorphisms, called gauge transformations, of the fibre bundle Q —> R. In particular, if ul = 0, the vertical vector field (4.12.2) is the generator of vertical automorphisms of the fibre bundle Q —> R projected over the identity transformation of the base R. If ul = 1, the vector field u (4.12.2) is projected over the standard vector field dt on the base R, which is the generator of the group of translations of R.

Let us apply the first variational formula (4.8.4) to a Lagrangian L and to a vector field u (4.12.2). On the shell (4.8.7), the identity (4.8.4) is brought into the weak identity

u\dC » -dtl,

(u'dt + u% + dtu'dl)C w -(f,(7r,(u'g; - u') - u'C),

(4.12.3)

4.12. LAGRANGIAN CONSERVATION LAWS 219

where, by analogy with field theory,

T = -u\HL = ■Ki{utq\ - u{) - ulC (4.12.4)

is said to be the current along the vector field u. The symbol " ~" stands throughout for weak equalities fulfilled on-shell.

If the Lie derivative L^L of a Lagrangian L along a vector field u vanishes, i.e., L is invariant under the 1-parameter group generated by u, we have the weak conservation law

0 « -d,[7T,(u'qJ - u*) - «'£]. (4.12.5)

It is brought into the differential conservation law

0 ~ —T[(W> ° c)(u'3[c' - u ' o c ) - u'C o c] at

on solutions c of the Lagrange equations (4.8.8). A glance at this expression shows that, in time-dependent mechanics, the conserved current (4.12.4) plays the role of an integral of motion.

Remark 4.12.1. Let L be a Lagrangian on the velocity phase space J ' Q . Every integral of motion (4.12.4) defines a non-holonomic constraint dT = 0 on JlQ such that any Lagrangian connection fx, for L is a dynamic equation compatible with this constraint. •

Remark 4.12.2. The conservation law (4.12.5) is broken if a Lagrangian depends on variable parameters which do not live in the dynamic shell (4.8.7) (see Section 5.9). In order to obtain a conservation law in the presence of such parameters, let us consider a bundle product

Qtot = Q x Y (4.12.6)

of a configuration bundle Q —> R with coordinates [t, q') and a fibre bundle Y —> R with coordinates (t, yA), whose sections are the variable parameters which take the background values

yB = <PB(x), „f = d,4>B{x).

A Lagrangian L is defined on the total velocity phase space ./'Qtot-


Let u be a projectable vector field (4.12.2) on Qtot which projects also onto Y because the transformations of parameters do not depend on the dynamic variables. This vector field takes the coordinate form

u = uldt + uA(t, yB)dA + u'{t,yB, <?)%. (4.12.7)

Substitution of the vector field (4.12.7) into the first variational formula (4.8.4) leads to the first variationai formula in the presence of parameters

[u'd, + uAdA + u'd, + dtuAdA + dtu'dflC = (uA - ytul)dAC + (4.12.8)

vAdt(v.A - j / t V ) + (W - qlu^Si - dt[«i(uWt ~ «*) - «*£]•

Then the weak identity

[u% + uAdA + u'd, + dtUAdlA + dtUl%]C » {uA - VS)9A^ +

■nAdt(uA - yAul) - MM^Ql ~ " ' ) - u ' £ ]

is fulfilled on the dynamic shell (4.8.7). If a Lagrangian L is invariant under gauge transformations of the product (4.12.6)

whose generator is the vector field u (4.12.7), we obtain the weaic conservation Jaw in the presence of parameters

(uA - y^u{)dAC + nAdt(uA - yAul) « dt{K,{ulq't - u') - u*£].

•

Remark 4.12.3. The first variational formula (4.8.4) can also be utilized when a Lagrangian possesses symmetries, but a motion equation is seen as Lagrange equations plus additional non-Lagrangian external forces and reads

[di-dtdf)C + Fi{t,qi,<ft) = 0. (4.12.9)

Let us substitute £ = —F, from this equality in the first variational formula (4.8.4) and assume that the Lie derivative of the Lagrangian L along a vector field u vanishes. Then, we have the conservation law

(it1 - q\)F, g - d t M u ' q j - u') - u '£]. (4.12.10)

•


It is easy to see that the weak identity (4.12.3) is linear in the vector field u. Therefore, one can consider superposition of the weak identities (4.12.3) associated with different vector fields.

For instance, if u and v! are projectable vector fields (4.12.2), projected onto the standard vector field dt on R, the difference of the corresponding weak identities (4.12.3) results in the weak identity (4.12.3) associated with the vertical vector field u — v!. Conversely, every vector field u (4.12.2), projected onto dt, can be written as the sum

u = Y + ti (4.12.11)

of some reference frame

r = dt + rdt (4.12.12)

and a vertical vector field $ on Q. It follows that the weak identity (4.12.3) associated with an arbitrary vector

field u (4.12.2) can be represented as the superposition of those associated with a reference frame V (4.12.12) and some vertical vector field d.

If u = d is a vertical field, the weak identity (4.12.3) reads

(^di + dt^dDC^dtin^).

If the Lie derivative of L along -d vanishes, we obtain from (4.12.5) the weak conservation law

0 w Mviti') (4.12.13)

and the integral of motion

1 = -TTjg. (4.12.14)

By analogy with field theory, (4.12.13) is called the Noether conservation law for the Noether current (4.12.14).

Example 4.12.4. Let assume that, given a trivialization Q = R x M in coordinates (t, q'), a Lagrangian L is independent of a coordinate g1. Then the Lie derivative of L along the vertical vector field d = d\ equals zero, and we have the conserved Noether current (4.12.14) which reduces to the momentum X = —TTI.. With respect to arbitrary coordinates (t,^'), this conserved Noether current takes the form

1 = - ^


In particular, the free motion Lagrangian admits m conserved Noether currents. •

In the case of a reference frame T (4.12.12), where u' = 1, the weak identity (4.12.3) reads

{dt + rdi + d(raj)£ « -Mniqt - V) - c), (4.12.15)

where

X r = 7r,(gt' - P ) - C (4.12.16)

is said to be the energy function relative to the reference frame T [51, 57, 161). With respect to the coordinates adapted to the reference frame T, the weak

identity (4.12.15) takes the form of the familiar energy conservation law

dtC ss -dt(niq't - £) , (4.12.17)

and Xr coincides with the canonical energy function Ei (4.12.1). It follows that the canonical energy function Ei is not a unique existent energy function. Each reference frame defines an energy function.

Example 4.12.5. Let us consider a free motion on a configuration space Q. It is described by the Lagrangian

£ = 2m 'j9t9t- m = const., (4.12.18)

written with respect to a reference frame (£,<?") such that the free motion dynamic equation takes the form (4.6.1). Let T be the associated connection. Then the conserved energy function Tr (4.12.16) relative to this reference frame T is precisely the kinetic energy of this free motion. Relative to arbitrary coordinates ((, q') on Q, it takes the form

xr = m(q\ -r)-C = ImjWM-n(qi - P).

Now we generalize this example for a motion described by the equation (4.12.9), where £ is the free motion Lagrangian (4.12.18) and F is an external force. The Lie derivative of the Lagrangian (4.12.18) along the reference frame T vanishes, and we have the weak equality (4.12.10) which reads

q\.Ft « dtlr, (4.12.19)


where q'r is the relative velocity. This is the well-known physical law whose left-hand side is the power of an external force. •

Example 4.12.6. Let us consider a 1-dimensional motion of a point mass mo subject to friction on the configuration space R2 —► R, coordinated by (t,q) (see Example 4.9.7). It is described by the dynamic equation (4.9.31) which is the system of Lagrange equations for the Lagrangian L (4.9.35). It is readily observed that the Lie derivative (4.8.3) of this Lagrangian along the vector field

T = dt-~qdq 2 mo (4.12.20)

vanishes. Hence, we have the conserved energy function (4.12.16) with respect to the reference frame T (4.12.20). This energy function reads

_ 1 [ k , , k . 1 ., mk? o 2 T = ^wi0exp — ' Qt(Qt + —Q) = ^rnqr - — j g , 2 [m0 J mo 2 8mo

where m is the mass function (4.9.34). •

Since any vector field u (4.12.2) can be represented as the sum (4.12.11) of a reference frame T (4.12.12) and a vertical vector field $, each current (4.12.4) along a vector field u (4.12.2) is the sum of a Noether current (4.12.14) along the vertical vector field t? and the energy function (4.12.16) relative to the reference frame T [51, 57, 161]. Conversely, energy functions relative to different reference frames V and T' differ from each other in the Noether current along the vertical vector field

r-r. In conclusion, we touch on gauge transformations with a generator u (4.12.2),

which preserve the Euler-Lagrange operator SL, but not necessarily a Lagrangian L. They are called generalized invariant transformations [57, 105, 174]. We use the formula

Lj2u£ t — £L_L

[54, 57], where

J2u = u'dt + u% + dtu'dj + dtdttfd?

is the second order jet prolongation of a vector field (4.12.2). The following two assertions are immediate corollaries of this formula.


COROLLARY 4.12.1. If a Lagrangian L is invariant under a 1-parameter gauge group whose generator is a vector field u (4.12.2), so is the associated Euler-Lagrange operator EL. D

COROLLARY 4.12.2. If an Euler-Lagrange operator £/,, associated with a Lagrangian L, is invariant under a 1-parameter gauge group whose generator is a vector field u (4.12.2), the Lie derivative h^L is a variationally trivial Lagrangian (4.9.30). D

Let us consider conservation laws in the case of generalized invariant transformations. Let L be a Lagrangian and EL the associated Euler-Lagrange operator. Let u be a vector field (4.12.2) which is the generator of a local 1-parameter group of gauge transformations such that

L/2u£z, = 0. (4.12.21)

Then we have the equality

LuL = d,/,

where / is a function on Q. In this case, the weak identity (4.12.3) reads

dtf^dti^iiS-qb+u'C),

and we obtain the weak equality

O w d e M u ' - < ? ; ) + « ' £ - / ) . (4.12.22)

Example 4.12.7. Let L be the free motion Lagrangian (4.12.18). The corresponding Euler-Lagrange operator

EL = -rriijqltdq'

is invariant under the Galilei transformations with the generator

ui = vit + ai, v' = const., a1 = const., (4.12.23)

(see (4.6.10)). At the same time, the Lie derivative of the free motion Lagrangian (4.12.18) along the vector field (4.12.23) does not vanish, and we have

LjjL = mijVl(ft = d t (m l ; vV + c), c = const.,

□

□


[148]. Then the weak equality (4.12.22) shows that (q\t — q%) is a constant of motion.

Remark 4.12.8. The invariance condition (4.12.21) is generalized as

if one deals with symmetry transformations of the differential equation Si = 0 [93, 94]. For instance, the Galilei transformations (4.6.10) are the symmetry transformations of the free motion equation. •

L^u^L ~ 0

•

•

Chapter 5

Hamiltonian time-dependent mechanics

This Chapter is devoted to the Hamiltonian formulation of time-dependent mechanics with respect to an arbitrary reference frame. Let us recall that a configuration bundle Q —> R of time-dependent mechanics is isomorphic to the product R x M, where M is a typical fibre of Q, but this isomorphism is not canonical in general. Different trivializations Q = R x M correspond to different reference frames. For this reason, many constructions of Hamiltonian conservative mechanics on a sym-plectic manifold Z and its extension to the product I x Z can not be applied to mechanical systems which are subject to time-dependent transformations, including canonical transformations and reference-frame transformations.

Let us summarize the main peculiarities of Hamiltonian time-dependent mechanics.

• A momentum phase space of time-dependent mechanics is provided with the canonical degenerate Poisson structure. However, Hamiltonian time-dependent mechanics does not reduce to a Poisson Hamiltonian system, since a Hamiltonian on a momentum phase space of time-dependent mechanics is not a scalar function under time-dependent transformations (see Remark 5.1.1 below).

• As a consequence, the evolution equation of time-dependent mechanics is not expressed in terms of a Poisson bracket, and integrals of motion cannot be defined as functions in involution with a Hamiltonian. For the same reason, the familiar procedure of describing constraint Hamiltonian systems in Section

227

228 CHAPTER 5. HAMILTONIAN TIME-DEPENDENT MECHANICS

3.6 cannot be applied to time-dependent mechanics.

• Hamiltonian and Lagrangian formulations of time-dependent mechanics are equivalent in the case of hyperregular Lagrangians. A degenerate Lagrangian requires a set of associated Hamiltonians in order to exhaust all solutions of the Lagrange equations.

We follow the notation of the previous Chapter, where Q —» K is a fibre bundle (4.0.1), coordinated by (t, ql), whose typical fibre M is an m-dimensional manifold, while its base R is equipped with the Cartesian coordinate t possessing the transition functions t' = t+ const.

5.1 Canonical Poisson structure

As was mentioned, the momentum phase space of time-dependent mechanics is the vertical cotangent bundle

7rn : V'Q - Q (5.1.1)

of a configuration bundle Q —> R. This is the particular Legendre bundle II (2.9.7) over a fibre bundle Q —> X when X = R. The vertical cotangent bundle (5.1.1) is equipped with the coordinates (£,<?', p< = q,).

The momentum phase space V'Q is provided with the canonical Poisson structure as follows (cf. Example 2.3.8). Let us consider the cotangent bundle T'Q of the configuration space Q, equipped with the coordinates (t,<?',p,Pi). This is the homogeneous Legendre bundle (2.9.3) over a fibre bundle Q —> X when X = R. The cotangent bundle T'Q admits the canonical Liouville form

3 = pdt + p,dq' (5.1.2)

and the canonical symplectic form

dE = dp A dt 4- dpi A dq'. (5.1.3)

The corresponding Poisson bracket on the vector space D°(T'Q) of functions on T'Q reads

{/, 9} = Vfdtg - Vgdtf + d'fdl9 - d'gdj. (5.1.4)

5.1. CANONICAL POISSON STRUCTURE 229

Let us consider the subspace of D°(T'Q) which comprises the pull-backs £*/ on T'Q of functions / on the vertical cotangent bundle V'Q by the canonical projection T'Q —> V'Q (1.1.7). It is easily seen that this subspace is closed under the Poisson bracket (5.1.4). By virtue of Proposition 2.3.1, there exists the canonical Poisson structure

{!,9}v = difdig-digdJ (5.1.5)

on momentum phase space V'Q induced by (5.1.4), i.e.,

C{f,9}v = {Cf,Cg}-

The corresponding Poisson bivector field

w{df,dg) = {f,g}v

on V'Q —> R is vertical with respect to the fibration V'Q —► M, and reads

w'J = 0, wi} - 0, W*j = 1. (5.1.6)

A glance at this expression shows that the holonomic coordinates of V'Q are canonical for the Poisson structure (5.1.5). Since the rank of the bivector field w (5.1.6) is constant, the Poisson structure (5.1.5) is regular. This Poisson structure is obviously degenerate.

Given the Poisson bracket (5.1.5), the Hamiltonian vector field ■Of for a function / on the momentum phase space V'Q is defined by the relation

{f,9h VP 6 D°(V'Q).

It is the vertical vector field

0/ = #fdj - (5.1.7)

on the fibre bundle V'Q —► R. Hence, the characteristic distribution of the Poisson structure (5.1.5) is precisely the vertical tangent bundle VV'Q C TV'Q of the fibre bundle V'Q — R.

In accordance with Theorem 2.3.3, the Poisson structure (5.1.5) defines the sym-plectic foliation on the momentum phase space V'Q, which coincides with the fibration V'Q —* R. The symplectic forms on the fibres of V'Q -> R are the pull-backs

Qt = dpi ^ dq'

= 4f\dg,

d,fd'


of the canonical symplectic form on the typical fibre T'M of the fibre bundle V'Q —♦ R with respect to trivialization morphisms [27, 74, 161]. Given such a trivialization

V'Q & R x T'M, (5.1.8)

the Poisson structure (5.1.5) is isomorphic to the direct product of the zero Poisson structure on R and the canonical symplectic structure on T'M (see Example 2.3.7).

The Poisson structure (5.1.5) can be introduced in a different way [57, 161]. The polysymplectic form (2.9.9) on the Legendre bundle V'Q -* Q (5.1.1) reads

A = dp, Adq{ Adt®dt,

and reduces to the canonical exterior 3-form

n = dp, A dq' A dt. (5.1.9)

This is the exterior differential of the canonical 2-form

0 = Pidq' A dt (5.1.10)

on the momentum phase space V'Q. The canonical forms (5.1.9) and (5.1.10) are maintained under any holonomic coordinate transformation of V'Q.

Given the canonical form (5.1.9), every function / on the momentum phase space V'Q determines the corresponding Hamiltonian vector field $/ (5.1.7) by the relation

tf,Jft = -dfAdt, (5.1.11)

while the Poisson bracket (5.1.5) is defined by the condition

{f,g}vdt = 4g\4f\n.

Remark 5.1.1. It should be emphasized that, though the momentum phase space V'Q of time-dependent mechanics is provided with the canonical Poisson structure, non-conservative mechanical systems are not reduced to the Poisson Hamiltonian systems of Section 3.2. Given a trivialization (5.1.8), i.e., a reference frame, one can write the Hamilton equations (5.2.18a) - (5.2.18b) (see below) as the equations of the Hamiltonian vector field d-n (5.1.7) for a Hamiltonian Ti with respect to the Poisson structure (5.1.5). Moreover, by very definition of the canonical form ft (5.1.9), the reference frame transformations are canonical transformations for the

5.2. HAMILTONIAN CONNECTIONS AND HAMILTONIAN FORMS 231

Poisson structure (5.1.5) (see Section 5.3). However, a Hamiltonian H (see (5.2.10) below) is not a scalar function under reference frame transformations in general. Therefore these transformations are not the equivalence transformation of a Poisson Hamiltonian system. •

5.2 Hamiltonian connections and Hamiltonian forms

The dynamics of time-dependent mechanics on a phase phase space V*Q is described by first order dynamic equation.

DEFINITION 5.2.1. Let

7 = dt + j% +<*# (5.2.1)

be a connection on the fibre bundle V'Q —> M. In accordance with Definition 4.2.1, it defines the first order dynamic equation on the momentum phase space V'Q —► K written as

q\ = V, Pu = 7i

with respect to the adapted coordinates

{t,q\Pi,q't,Pti)

on the jet manifold J1 V'Q. a

Example 5.2.1. Let us consider the product

(J = R x M - t I ,

coordinated by (t, ql). We have the canonical isomorphism

V'Q = R x TM,

with the coordinates p< = Qj, and the corresponding isomorphism

JW'Q^RXTT'M, (5.2.2)

with the coordinates q\ = <?\p« = j>i- Every vector field

u = u'dt + Uid'


on the cotangent bundle T'M defines the first order dynamic equation

7 = dt + u% + u,9*

on V'Q (5.2.2). Its projection

qx = u\ p, = ux

onto T'M is an autonomous first order dynamic equation in conservative mechanics (see Definition 3.1.1). •

Let J2Q be the second order jet manifold of a configuration space Q —» K provided with the adapted coordinates

(*,?', «*.««)•

Recall that it coincides with the sesquiholonomic jet manifold T*Q. Following the general scheme of polysymplectic formalism in field theory (see

Section 3.8), we say that a connection (5.2.1) on the Legendre bundle V'Q —» R is locally Hamiltonian if the exterior form 7jfi is closed, i.e.,

L,n = d(7jn) = o. (5.2.3)

It is readily observed that a connection 7 (5.2.1) on V'Q —» R is locally Hamiltonian if and only if it obeys the conditions

ay - tvy = 0, drf] - djli = 0, dj-y' + a y = 0.

(5.2.4a) (5.2.4b)

(5.2.4c)

Example 5.2.2. Every connection

r : Q -» J'Q C TQ,

r = 9, + Fdit (5.2.5)

on the configuration bundle Q —» R gives rise to the locally Hamiltonian connection

r = KT = a( + riai-p,aJr,aj (5.2.6)

(1.6.13) on the momentum phase space V'Q —► R such that

T\n = dHr, Hr = pjdtf - pjFdt. (5.2.7)


•

Locally Hamiltonian connections constitute an affine space modelled over the linear space of vertical vector fields fl on the Legendre bundle V'Q —> R, which fulfill the same condition

d(tfjn) = 0 (5.2.8)

as (5.2.3). These vector fields are locally the Hamiltonian vector fields in accordance with the following lemma.

LEMMA 5.2.2. Every closed form -y\fl on the fibre bundle V'Q -» R is exact. □

Proof. Let us consider the decomposition

7=r+tf, (5.2.9)

where T is a connection on Q —> R and f is its lift (5.2.6) onto the fibre bundle V'Q —> R, while -d is a vertical vector field on V'Q satisfying the relation (5.2.8). It is easily seen that

■d\n = aAdt,

where a is a 1-form on V'Q. Using the properties of the De Rham cohomology of a manifold product, one can show that every closed 2-form a A dt on a manifold V'Q, diffeomorphic to the product R x T'M, is exact, and so is 7JO [57]. Moreover, in accordance with the relative Poincare lemma, we can write locally

ti\n = df Adt,

where / is a local function on V'Q. QBD

DEFINITION 5.2.3. An exterior 1-form H on the momentum phase space V'Q is said to be a locally Hamiltonian form if

7 j n = dH

for a connection 7 on the fibre bundle V'Q —> R. □


By virtue of Proposition 5.2.2, there is one-to-one correspondence between the locally Hamiltonian connections and locally Hamiltonian forms, considered throughout modulo closed forms. In particular, the exterior form Hr (5.2.7) is a locally Hamiltonian form.

DEFINITION 5.2.4. An exterior 1-form H on the momentum phase space V'Q is called a Hamiltonian form if it is the pull-back

H = h*S - pidq* - Hit (5.2.10)

of the Liouville form S (5.1.2) on the homogeneous Legendre bundle T'Q by a section h of the fibre bundle

C : TQ -» V'Q. (5.2.11)

a

Remark 5.2.3. Note that, with respect to the universal unit system, a Hamiltonian form is physically dimensionless. •

Remark 5.2.4. Given a trivialization Q = R x M, the Hamiltonian form (5.2.10) is the well-known integral invariant of Poincare-Cartan [6]. Therefore, the coefficient H of the horizontal part of the Hamiltonian form (5.2.10) is said to be a Hamiltonian. A glance at the expression (5.2.10) shows that Hamiltonians fail to be functions, i.e., H $. £)°(V'Q), but make up an affine space modelled over the linear space of functions on V'Q. *

For instance, every connection T on a configuration bundle Q —» R is an affine section

p o r = -p.r of the fibre bundle (5.2.11) (see (1.3.12)), and defines the Hamiltonian form Hr

(5.2.7) on the momentum phase space V'Q, where its Hamiltonian is

H = PiF.

It follows that any Hamiltonian form on the momentum phase space V'Q admits the splitting

H = HT- Hrdt = Pldql - {p,F + Hr)dt, (5.2.12)


where T is a connection on Q -» R, and Hr is a real function on V'Q, called the Hamiltonian function.

Since a connection T on a configuration space Q —» K is treated as a reference frame, i.e., as a kinematic object, one may say that, in accordance with the decomposition (5.2.12), every Hamiltonian form H is split into kinematic and dynamic parts, and so is its Hamiltonian H. In Section 5.7, we will show that the Hamiltonian function HT in the splitting (5.2.12) is the Hamiltonian energy function relative to the reference frame I\ Of course, the splitting (5.2.12) is not unique. Nevertheless, every Hamiltonian form H admits the canonical splitting (5.2.12) as follows.

We mean by a Hamiltonian map any fibred morphism

*:VQfJlQ, 5 t ° $ = &(p), P e V'Q, (5.2.13)

over Q from the momentum phase space V'Q to the velocity phase space J1Q. Its composition with the canonical morphism A (4.1.4) yields the fibred morphism

9: V'Q ^TQ,

$ = dt + Vd{. (5.2.14)

In particular, every connection T on a configuration bundle Q —> R defines the Hamiltonian map

f = T o Trn : V'Q - Q -» JxQt

T = dt + r%. Conversely, every Hamiltonian map $ (5.2.13) yields the associated connection

r* = $ o 6 on the configuration bundle Q —» R, where 0 is the global zero section of the vertical cotangent bundle V'Q —» Q. For instance, we have

r~ = r

PROPOSITION 5.2.5. Every Hamiltonian map (5.2.14) defines the associated Hamiltonian form

H* = - $ J 0 = (A o $)*H = pidq1 - Pi&dt


on V'Q, where 0 is the canonical 2-form (5.1.10). □

PROPOSITION 5.2.6. Every Hamiltonian form H on the momentum phase space V'Q defines the associated Hamiltonian map

H : V'Q -> JlQ,

q]oH = dxU.

□

Proof. Let h : V'Q --> T'Q be a section of the fibre bundle (5.2.11), which corresponds to the Hamiltonian form H, i.e., H = h'E. The vertical tangent map Vh of the morphism h defines the linear fibred morphism

Vh : VV'Q -> V'Q x T'Q Q

over V'Q. Therefore, it can be represented by the section

Vh = dp, ® {dq* - d'Hdt)

of the fibre bundle

VV'Q ® T'Q -> V'Q.

After natural contractions, this section is brought into the section

Vh = {dq* - d'Hdt) ® ft

of the pull-back

VQx(T'Q®VQ)->VQ, Q

and takes its values into the image of the jet manifold J1Q by its canonical imbedding (1.3.5) into T*Q®VQ. QED

COROLLARY 5.2.7. Every Hamiltonian form H on the momentum phase space V'Q determines the associated connection

TH = H o 0


on the configuration bundle Q —► R. It is called a Hamiltonian frame connectioi □

In particular, we have

rWr = r, where Hr is the Hamiltonian form (5.2.7) associated with the connection T on the fibre bundle Q -» R.

COROLLARY 5.2.8. Every Hamiltonian form H (5.2.10) on the momentum phase space V'Q admits the canonical splitting

H = HrH — Hdt.

a

Let us turn now to the relationship between the locally Hamiltonian forms and the Hamiltonian ones.

PROPOSITION 5.2.9. Every locally Hamiltonian form on the momentum phase space V'Q is locally a Hamiltonian form. □

Proof. Given two locally Hamiltonian forms H*, and Hy on the momentum phase space V'Q, their difference

(7 = fly — Hy,

Ar = ( 7 - V ) j n ,

is a 1-form on V'Q such that the 2-form a A dt is closed and, consequently, exact by virtue of Lemma 5.2.2. Hence, in accordance with the relative Poincare lemma, we have

<T = fdt + dg,

where / and g are local functions on V'Q. Then, we deduce from the splitting (5.2.9) that, in a neighbourhood of every point p € V'Q, a locally Hamiltonian form H~! can be written as

H^ = Hr + fdt,


and coincides with the pull-back of the Liouville form E on the cotangent bundle T*Q by the local section

(t,q',Pi) -» (t,q\Pi,P = -P,F + f)

of the fibre bundle (5.2.11). QED

A converse of Proposition 5.2.9 is the following.

PROPOSITION 5.2.10. Given a Hamiltonian form H on the momentum phase space V'Q, there exists a unique connection JH on the Legendre bundle V'Q —» R, called the Hamiltonian connection, such that

y„\n = dH. (5.2.15)

a

Proof. As in the polysymplectic case, let us introduce the first order Hamilton operator on the phase space V'Q, which is associated with the Hamiltonian form H. This Hamilton operator reads

£H : J'V'Q ^ AT'V'Q,

£H d=dH - AJfi = [(<?; - d'H)dPl - (pti + dtTVjdq*) A dt, (5.2.16)

where A is the canonical monomorphism (4.1.4) and J 1 V'Q is the jet manifold of the fibre bundle V'Q —► R, coordinated by

{t,q\p„q],Pti)

It is readily observed that the kernel of the Hamilton operator EH (5.2.16) is an image of the global section

7w = dt + d'Hdt - dindi (5.2.17)

of the affine jet bundle JlV"Q —» V'Q, which is precisely the desired Hamiltonian connection (see Remark 5.2.6 below for a different proof). QED

Let us recall that Hamiltonian forms H on the momentum phase space V'Q make up an affine space modelled over the linear space of horizontal densities fdt


on the fibre bundle V'Q -> R. Then, as follows from the relation (5.2.15), Hamiltonian connections 7# constitute an affine space modelled over the linear space of Hamiltonian vector fields (5.1.11):

H-H' = df, 7 « - 7tf' = - $ / ■

The kernel of the covariant differential D1H, associated with a Hamiltonian connection (5.2.17), is a closed imbedded subbundle of the jet bundle J ' V Q -» R, and defines the system of first order differential equations on the momentum phase space V 'Q. These are called the Hamilton equations

g| = d{n, Pti = -diH

(5.2.18a) (5.2.18b)

for the Hamiltonian form H. Note that, in comparison with the Lagrange equations, the Hamilton ones are always well defined differential equations. The integral sections r : R 3 () - • V'Q of the Hamiltonian connection (5.2.17) (or equivalently, the integral curves of the horizontal vector field (5.2.17)) are solutions of the Hamilton equations (5.2.18a) - (5.2.18b). Moreover, a glance at the equation (5.2.18a) shows that the relation

J1{TTU or) = H or (5.2.19)

is fulfilled for any solution r of the Hamilton equations (5.2.18a) - (5.2.18b).

Remark 5.2.5. The Hamilton equations (5.2.18a) - (5.2.18b) are the particular case of the first order dynamic equations

qt = Y, Pti = 7. (5.2.20)

on the momentum phase space V'Q —» R, where 7 is a connection (5.2.1) on the fibre bundle V'Q — R.

The first order reduction of the equation of a motion of a point mass m subject to friction in Example 4.9.7 exemplifies first order dynamic equations which are not Hamilton ones. These equations read

1 Qt = —P, m0

k Pt = P

m0

(5.2.21)

The connection 1 k

7 = dt + —pdq pdp m m


does not obey the condition (5.2.4c) for a locally Hamiltonian connection. At the same time, the equations (5.2.21) are equivalent to the Hamilton equations for the Hamiltonian associated with the Lagrangian (4.9.35). •

Note that the Hamilton equations (5.2.18a) - (5.2.18b) can be introduced without appealing to the Hamilton operator. On sections r of the fibre bundle V'Q —» R, these equations

r ' = &H o r, r< = -d,H o r (5.2.22)

are equivalent to the relation

r'{u\dH) = 0 (5.2.23)

which is assumed to hold for any vertical vector field u on V'Q —► R.

Remark 5.2.6. Every Hamiltonian form H on a. momentum phase space V'Q defines a presymplectic structure. A Hamiltonian form H = h'E, by definition, is a pull-back of the canonical Liouville form 5 (5.1.2) by means of a section h of the fibre bundle T'Q —> V'Q. Accordingly, its exterior differential

dH = h'dE = {dpi + diHdt) A {dq? - d'Tidt) (5.2.24)

is the pull-back of the canonical symplectic form (5.1.3), and is a presymplectic form. The presymplectic form (5.2.24) has constant rank 2m since the form

{dH)m = {dPi A dg')m - m{dpx A dq')m-1 AdH/\dt (5.2.25)

is nowhere vanishing. It is also easily seen that

{dH)m Adty^O.

It follows that the pair (dH, dt) defines a cosymplectic structure on V'Q (see Remark 4.8.8). Since the presymplectic form dH is of constant rank 2m, its kernel Ker dH is a 1-dimensional distribution which is spanned by the vector field ^H (5.2.17). Hence, there is a unique vector field u = 7// on V'Q such that

u\dH = 0, u\dt = 1

(see [37]). This is a different proof of Proposition 5.2.10. •


PROPOSITION 5.2.11. The Hamiltonian form (5.2.6) is a contact form on the momentum phase space V'Q if the function

jH\H = pidiH-H=[H] (5.2.26)

nowhere vanishes [116]. □

Proof. Since a Hamiltonian connection 7# (5.2.17) is a nowhere vanishing vector field, the condition

H A {dH)m ± 0

for a Hamiltonian form H to be a contact form is equivalent to the condition

lH\{H A (dH)m) = {lH\H){dH)m = [H\(dHr ± 0.

The result follows because the form (dH)m (5.2.25) is nowhere vanishing. QED

Note that one may try to add some exact form, e.g., the form cdt, c = const., to a Hamiltonian form H in order to make the function \H] nowhere vanishing. For instance, the Hamiltonian form i / r (5.2.7) fails to be a contact one since [Hr] = 0, but the equivalent Hamiltonian form Hr - dt, where [Hr — dt] = 1, is a contact form.

If a Hamiltonian form H is a contact one, the corresponding Reeb vector field (2.2.5) reads

EH = [WTV (5.2.27)

By virtue of Proposition 2.2.6, one can then introduce the Jacobi bracket defined by the vector field (5.2.27) and by the bivector field wH on V'Q derived from the relations (2.2.8) which read

wH(<P,.)\H = 0, wH(<t>,.)}dH=-(4>-(EH\<t>)H),

whenever 0 is a 1-form on V'Q. We find

wH{<P, tf) = 0Vi - ol4>i + Pia'EH\(p - p{(j>'EH\a,


where 4>, a axe arbitrary 1-forms on V'Q. The corresponding Jacobi bracket on the momentum phase space V'Q is

{J,9}H = wH(4>,a) + EH\(fdg-gdf) =

V,9}v + [HT'fobtfJtf - lfhH\dg), where {/, g}v is the Poisson bracket (5.1.5) and

[ / ] = ? . £ > ' / - / , [g]=Pidig-g.

Remark 5.2.7. Given the Poisson bracket (5.1.5) on the momentum phase space V'Q, one can introduce the generalized Poisson bracket {., .}w (2.8.3) on the exterior algebra Q'(V'Q), and the bracket {., .}d (2.8.5) on the quotient

0'(VQ)/dD'(VQ).

In particular, the generalized Poisson bracket (2.8.3) of Hamiltonian forms H and H' reads

{H, H'}w = Pi(d"H' - d'H)dt.

•

5.3 C a n o n i c a l t rans format ions

Canonical transformations in time-dependent mechanics are not compatible with the fibration V'Q —► Q of the momentum phase space in general.

DEFINITION 5.3.1. By a canonical automorphism is meant a vertical automorphism p of the fibre bundle V'Q —» R, which preserves the canonical Poisson structure (5.1.5) on the momentum phase space V'Q, i.e.,

{f°P,g°p}v = {f,g}v°P-

D

It is easily seen that an automorphism p of V'Q —» R is canonical if and only if it preserves the canonical form SI (5.1.9) on V'Q, i.e.,

n = p'n.

5.3. CANONICAL TRANSFORMATIONS 243

DEFINITION 5.3.2. The bundle coordinates of the fibre bundle V'Q -» R are called canonical if they are canonical for the Poisson structure (5.1.5). □

It is readily observed that canonical coordinate transformations satisfy the relation

dp, dpk dpk dp, dpUW dqi dqk dqk dqi '

dpk dqi dq> dpk 3'

By very definition of the canonical form il, the holonomic coordinates of the vertical cotangent bundle V'Q —► Q are the canonical coordinates. Accordingly, holonomic automorphisms

(<?\P.)~ (<?",?: = f^Pj) (5.3.1)

of V'Q —► Q, induced by the vertical automorphisms of the configuration bundle Q —► R are also canonical.

PROPOSITION 5.3.3. Locally Hamiltonian connections are transformed into each other by canonical automorphisms, and so are locally Hamiltonian forms. □

Proof. If 7 is a locally Hamiltonian connection for H, we have

7>(7)jn = (p-'rwn) = d[fjrlYH\ and Tp(y) is also a locally Hamiltonian connection. QED

PROPOSITION 5.3.4. Let 7 be a complete locally Hamiltonian connection on V'Q —> R, i.e., the vector field (5.2.1) is complete. There exist canonical coordinates on V'Q such that 7 = dt. □

Proof. A glance at the relation (5.2.3) shows that each locally Hamiltonian connection 7 is the generator of a local 1-parameter group of canonical automorphisms of the fibre bundle V'Q -» R. Let V0'Q be the fibre of V'Q -> R at 0 G R. Then

= 0,

= 0,

= *

i—+

D


canonical coordinates of the symplectic manifold V£Q = T'M dragged along integral curves of the complete vector field 7 determine the desired canonical coordinates on V'Q. QED

In other words, a complete locally Hamiltonian connection 7 on the momentum phase space V'Q in accordance with Proposition 4.1.2 defines a trivialization

QED

0 : V'Q - R x V0'Q (5.3.2)

of the fibre bundle V'Q —> R such that the corresponding coordinates of V'Q, compatible with this trivialization, are canonical. However, it should be emphasized that, although the fibre V0*Q is diffeomorphic to T'M, the trivialization (5.3.2) fails to be a trivialization of the type V'Q = R x T'M since the trivialization morphism rp is not a bundle morphism of the fibre bundle V'Q —» Q.

In particular, let H be a Hamiltonian form (5.2.12) such that the corresponding Hamiltonian connection -yH (5.2.17) is complete. By virtue of Proposition 5.3.4, there exists a trivialization of the phase space V'Q with respect to the global canonical coordinates {qA,PA) (where qA are not coordinates on Q in general) such that

ft = dpA A dqA A dt, 1H = dt, dH = dpA A dqA,

and H reduces to the Hamiltonian form

H = pAdqA

with the Hamiltonian H = 0. Then the corresponding Hamilton equations take the form of the equilibrium equations

qt = 0, PtA = 0 (5.3.3)

such that qA(t,qt,pl) and PA(t,ql,Pi) are constants of motion. Accordingly, any Hamiltonian Ti can be locally brought into zero, and the cor

responding Hamilton equations are reduced to the equilibrium ones (5.3.3) by local canonical coordinate transformations.

Example 5.3.1. Let us consider the 1-dimensional motion with constant acceleration a with respect to the coordinates (t,q). Its Hamiltonian on the momentum phase space R3 —» R reads

P2

5.3. CANONICAL TRANSFORMATIONS 245

The associated Hamiltonian connection is

1H = dt+ pdq + adp.

This Hamiltonian connection is complete. The canonical coordinate transformations

at2

q =q-pt + —, p' =p — at (5.3.4)

bring it into fn = dt. Then, the functions q'(t, q%p) and p'(t,p) (5.3.4) are constants of motion. •

Example 5.3.2. Let us consider the 1-dimensional oscillator with respect to the same coordinates as in the previous Example. Its Hamiltonian on the momentum phase space R3 —* K reads

K=\{p2 + q2)-

The associated Hamiltonian connection is

lH = dt+ pdq - qdp,

which is complete. The canonical coordinate transformations

q' = qcost — psint, p' = p cos t + q sin t (5.3.5)

bring it into JH = dt. Then, the functions q^(i, q,p) and p'(t,q,p) (5.3.5) are constants of motion. •

It should be emphasized that, in general, canonical automorphisms do not transform Hamiltonian forms into Hamiltonian forms, but only locally.

Let H be a Hamiltonian form (5.2.10) on a momentum phase space V'Q equipped with the coordinates (t,qx,Pi). Given a canonical automorphism p, we have

d(p*H - H) = 0,

where

p'H = pidp1 -Ho pdt. (5.3.6)

Therefore, we can write locally

H -p'H = dS, (5.3.7)


where S(t, q',Pi) is a local function on a momentum phase space V'Q. It should be emphasized that, if a configuration space Q is a contractible mani

fold, the equality (5.3.7) is fulfilled globally, and the function S is defined everywhere on V'Q.

Substituting (5.3.6) in (5.3.7), we obtain the relations

diS = Pi- Pjdtp',

dlS = -pjdx(P,

H-H' = p,dtp' - dtS. Taken on the graph

A„ = {(p,p(p)) 6 V'Q x V'Q, p e V'Q}

of the canonical automorphism, the function S plays the role of a generating function of canonical transformations.

Example 5.3.3. Given a canonical automorphism p, let

d e t ( d V ) ^ 0 . (5.3.8)

Then, if the graph A„ can be coordinated by

(t,q',qn = qtop),

we obtain the familiar relations dS

Pi=dq~'' , _ dS

p'~ do"' H-H' = dtS(t,q\q").

• (5.3.9)

Example 5.3.4. For instance, the generating function of a holonomic automorphism (5.3.1), where det(<9i<j'J) ^ 0, is

S(t,q'3,Pi) = -ql(t, q'j)Pl.

•

Let H be a Hamiltonian form whose associated Hamiltonian connection is complete. By virtue of Proposition 5.3.4, there exists a canonical automorphism p which bring its Hamiltonian into zero. If the condition (5.3.8) holds, then (5.3.9) is the Hamilton-Jacobi equation.

5.4. THE EVOLUTION EQUATION 247

5.4 The evolution equation

Given a Hamiltonian form H (5.2.12) and the corresponding Hamiltonian connection 7 H (5.2.17) on the momentum phase space V'Q, let us consider the Lie derivative of a function / e D°(V'Q) along the horizontal vector field 7/f, which reads

!*»»/ = 7«J4f = (dt + VHdi - W#)J. (5.4.1)

This equality is the evolution equation in time-dependent mechanics. Substituting a solution r of the Hamilton equations (5.2.22) in (5.4.1), we obtain the time evolution of / along this solution:

L-m/or = JtU°r)-It should be emphasized that, in comparison with the evolution equation (3.2.5)

in symplectic mechanics, the right-hand side of the evolution equation (5.4.1) is not reduced to the Poisson bracket since a Hamiltonian H o n a momentum phase space of time-dependent mechanics is not a function, i.e., 7i £ D°(V'Q). Of course, one can define locally the Poisson bracket {H,f}v of a Hamiltonian H with a function / on V'Q. However, being equal to zero with respect to some coordinates, this Poisson bracket does not necessarily vanish with respect to other coordinates.

Given the splitting (5.2.12) of a Hamiltonian form H, the evolution equation (5.4.1) is written as

L7„/ = dtf+(r% - diVp^f+iHrJh- (5.4.2)

In particular, the following two consequences of this form of the evolution equation should be pointed out.

• Since the evolution equation (5.4.2) is not reduced to the Poisson bracket, the integrals of motion in time-dependent mechanics cannot be defined as functions on a momentum phase space, which are in involution with a Hamiltonian. Therefore, to obtain conservation laws in Hamiltonian time-dependent mechanics, we follow the methods of field theory (see Section 5.7).

• The second (kinematic) term in the right-hand side of this equality plays an essential role under quantization. It makes the quantization procedure dependent on a reference frame. In Appendix B, we will show that quantizations under different reference frames fail to be equivalent in general.


Note that the kinematic term in the evolution equation (5.4.2) can be eliminated at least locally by means of canonical transformations. Let a connection T in the splitting of a Hamiltonian form (5.2.12) be complete. With respect to the coordinate system (t, q') adapted to the reference frame T, the configuration bundle Q is trivialized, and the corresponding holonomic coordinates (t, q',pi) on the momentum phase space V'Q are canonical. With respect to these coordinates, the evolution equation (5.4.2) takes the familiar form

L^Hf = dtf + {H,f}v.

5.5 Degenerate systems

This Section is devoted to the relationship between Lagrangian and Hamiltonian formalisms of time-dependent mechanics. This relationship is characterized by the diagram

V'Q ■£* JIQ

z\ _ j . JXQ ^-V'Q

which fails to commute in general. Its jet extension is

jiy'Q J^JlJlQ

J'L | I PL

JlJlQ i^-JlV'Q

where the jet prolongations of Hamiltonian and Legendre maps read

JlH:JlV'Q^JlJlQ, (<?\ q\, it), lit) oj'H = (q\ d'H, q\, dt&H), Jxl:JlJlQ^JlV'Q, (<?\P.. q't.Pti) °J1L = (g\7r„ q\ty d,7r,).

As we will show below, in the case of a hyperregular Lagrangian when the Legendre map L (4.8.9) is a diffeomorphism, Lagrangian and Hamiltonian formalisms are equivalent. Conversely, let H be a Hamiltonian form and JH the corresponding

5.5. DEGENERATE SYSTEMS 249

Hamiltonian connection (5.2.17) on the momentum phase space V'Q —» K. Let us consider the composition of morphisms

JlHolH:V'Q^J2QdJlJlQ, (<M 0 . <4Wtf °7/f = (d'Kd'KjHlMd'H)).

(5.5.1)

If the Hamiltonian map H is a diffeomorphism, then JlH 0 7 ^ 0 H~l is a holonomic connection on the jet bundle JlQ —► M, which defines a dynamic equation on the velocity configuration space J1Q. This dynamic equation is equivalent to the system of Lagrange equations for the Lagrangian function

£ = H-U[H]

which is the pull-back on JXQ of the function [H\ (5.2.26) by the morphism / / " ' . Let us consider more general conditions for solutions of Hamilton equations on

a momentum phase space to be solutions of Lagrange equations and second order dynamic equations on a velocity phase space, and vice versa.

Following the general scheme of polysymplectic Hamiltonian formalism [57, 158, 159], we say that a Hamiltonian form H on the momentum phase space V'Q is associated with a Lagrangian L on the velocity phase space JlQ if H obeys the conditions

L°HoL = L,

H = H& + LoH (5.5.2a) (5.5.2b)

[161]. A glance at the condition (5.5.2a) shows that the composition L o H is the projection operator

p,(p) = d , £ ( t , ^ , c ^ ( p ) ) , | peQ, (5.5.3)

from V'Q onto a submanifold

N = L{JXQ) C V'Q, (5.5.4)

called the Lagrangian constraint space. Given a Hamiltonian form H associated wit! the Lagrangian L, this constraint space is defined by the relation (5.5.3). Accordingly, the composition HoL is the projection operator from JiQ onto a submanifold H(N) C JlQ. The relation (5.5.2b) takes the form

CoH = [H] (5.5.5)


everywhere on V'Q.

Remark 5.5.1. Unless otherwise stated, we regard N as a subset of the momentum phase space V'Q, without a manifold structure. All objects are defined on the whole phase space V'Q, and their restriction to N means only that their values at the points of N C V'Q are considered. •

PROPOSITION 5.5.1. A Hamiltonian form H associated with a Lagrangian L obeys the relation

H \s= H'HL \N, (5.5.6)

where Hi is the Poincare-Cartan form (4.8.5). □

Proof. The proof follows from direct computation. QED

Acting on both sides of the equality (5.5.5) by the exterior differential, we obtain the relations

QED

dtH(p) = -(dt£)(t,q>,d>H(p)), peN,

dlH{p) = -{diC){t^,&H{p)), peN,

(P, - (a^)(t,^,^H))a'd°w = o.

(5.5.7)

(5.5.8) (5.5.9)

A glance at the relation (5.5.9) shows that:

• the condition (5.5.2a) is a corollary of the condition (5.5.2b) if the Hamiltonian map H is regular, i.e.,

det (a '^W) yt 0

at all points of the Lagrangian constraint space N;

• the Hamiltonian map H is always degenerate outside the Lagrangian constraint space N.

Example 5.5.2. Let L = 0 be the zero Lagrangian. In this case, the Lagrangian constraint space is N = 0(Q), where 0 is the canonical zero section of the Legendre bundle V'Q —» Q. The condition (5.5.2a) is fulfilled trivially for every Hamiltonian map, while the condition (5.5.2b) takes the coordinate form

H = Pid{H.


Any Hamiltonian form Hr (5.2.7) obeys this condition, and is associated with the Lagrangian L = 0. •

If a Lagrangian L is hyperregular, there exists a unique associated Hamiltonian form

H = #£_, + (L-l)'L (5.5.10)

such that

H = L~\ Pi=nl(q\d^H(qX,Pk)), g{ saw,*,(**,rf}). (5.5.11)

As an immediate consequence of (5.5.11), we have

JlH = (J1L)-\

PROPOSITION 5.5.2. Let L be a hyperregular Lagrangian, and H the associated Hamiltonian form. The following relations hold:

HL = L'H, (5.5.12) eI = (j1ly£H, (5.5.13) £B = {JlH)'£T, (5.5.14)

where EH is the Hamilton operator (5.2.16) for H, and £j is the Euler-Lagrange-Cartan operator (4.8.16) for L. O


A glance at the relations (5.5.6) and (5.5.12) shows that the Poincare-Cartan form is the Lagrangian counterpart of a Hamiltonian form, whereas it follows from (5.5.13) and (5.5.14) that the Lagrangian counterpart of the Hamilton operator is the Euler-Lagrange-Cartan operator £ j (4.8.16).

In particular, if 7// is a Hamiltonian connection for the associated Hamiltonian form H (5.5.10), then, by virtue of the equality (5.5.14), the composition JlH o^H

(5.5.1) takes its values into the kernel of the Euler-Lagrange-Cartan operator £^ or. more exactly, in the kernel of the Euler-Lagrange operator £L- Then the composition

QED

J 1 / f o 7 „ o L : ; ' ( j 3 ( t , l /i , g ; ) M

(t, q\ q\ = VH o L, q\t) = d'H o L, q\t = lH\d(d'H) o L) =

(t, q\ q\, q\t) = <?|, q\t = JH\d(d'H) o L) € J2Q e


is a holonomic Lagrangian connection for L. Conversely, if £L is a Lagrangian connection for L, then

yH = JlL°£Lo H

is a Hamiltonian connection for H. This proves the following assertion.

PROPOSITION 5.5.3. Let L be a hyperregular Lagrangian and H the associated Hamiltonian form (5.5.10).

(i) Let a section r of the fibre bundle V'Q -» 1 be a solution of the Hamilton equations (5.2.18a) - (5.2.18b) for the Hamiltonian form H. Then the section

c = 7rn o r

of the fibre bundle Q —» R is a solution of the Lagrange equations (4.8.7) for the Lagrangian L, while its first order jet prolongation c satisfies the Cartan equations (4.8.17a) - (4.8.17b).

(ii) Conversely, if a section c of the jet bundle J1Q —► R is a solution of the Cartan equations for the Lagrangian L, the section

r = L o c

of the fibre bundle V'Q —> R satisfies the Hamilton equations (5.2.18a) - (5.2.18b) for the Hamiltonian form H. O

It follows that, given a hyperregular Lagrangian, there is one-to-one correspondence between the solutions of the Lagrange equations (and, consequently, of the Cartan equations) and the solutions of the Hamilton equations for the associated Hamiltonian form.

In the case of regular Lagrangian L, the Lagrangian constraint space N is an open subbundle of the Legendre bundle V'Q —» Q. U N jt V'Q, an associated Hamiltonian form fails to be defined everywhere on V'Q in general. At the same time, an open constraint subbundle N can be provided with the appropriate pull-back structure with respect to the imbedding N •—» V'Q so that we may restrict our consideration to Hamiltonian forms on N. If a regular Lagrangian is additionally semiregular (see Definition 5.5.4 below), the associated Legendre morphism is a diffeomorphism of JlQ onto the Lagrangian constraint space N and, on N, we can restate all results true for hyperregular Lagrangians.


Example 5.5.3. Let Q be the fibre bundle R 2 - » R with coordinates (t,q). Its jet manifold JlQ 3 K3 and its Legendre bundle V'Q = R3 are coordinated by (t,q,qt) and (tf,g,p), respectively. Put

L = exp(qt)dt. (5.5.15)

This Lagrangian is regular, but not hyperregular. The corresponding Legendre map reads

po L = expift.

It follows that the Lagrangian constraint space N is given by the coordinate relation p > 0. This is an open subbundle of the Legendre bundle, and L is a diffeomorphism of JXQ onto N. Hence, there is a unique Hamiltonian form H on N which is associated with the Lagrangian (5.5.15). Its Hamiltonian function reads

H = p{\np- 1).

This Hamiltonian form, however, fails to be smoothly extended to V'Q. •

Hereafter, we will restrict our consideration of degenerate systems described by semiregular Lagrangians. Most physically interesting Lagrangians, including the quadratic ones, are of this type. On the other hand, there is a comprehensive relationship between Lagrangian and Hamiltonian formalisms in this case [57, 158, 159, 190].

DEFINITION 5.5.4. A Lagrangian L is said to be semiregular, if the pre-image L'1(p) of any point p € N is a connected submanifold of the velocity phase space JlQ. □

PROPOSITION 5.5.5. All Hamiltonian forms H associated with a semiregular Lagrangian L coincide with each other at the points of the Lagrangian constraint space N, i.e.,

H \N— H' |/V .

Moreover, the Poincare—Cartan form Hi for the Lagrangian L is the pull-back

HL = L'H, faqi - C)dt = H(t,qi,n)dt, (5.5.16)


of any associated Hamiltonian form H by the Legendre map L. □

Proof. Let u be a vertical vector field on the jet bundle J ' Q -» Q. If u takes its values into the kernel Ker TL of the tangent morphism to L, it is easy to see that

LUH£ = 0.

Hence, the Poincare-Cartan form HL for a semiregular Lagrangian L is constant on the connected pre-image £~'(p) of each point p € N. Then results follow from (5.5.6). QED

Note that the Hamilton operators for Hamiltonian forms in Proposition 5.5.5 do not necessarily coincide at points of the Lagrangian constraint N because of the derivatives of these forms.

Let H be a Hamiltonian form associated with a semiregular Lagrangian L. Acting by the exterior differential on the relation (5.5.16), we obtain the equality

QED

{q\ - d"H o L)d-nx A dt (5.5.17) - (diC + di(H o L))dqi Adt = 0

or, equivalently, the equalities

7 r t > ( ^ -3>WoL) = 0, Bi*M ~ &H o £) - (diC + (diH) o L) = 0.

Using the equality (5.5.17), one can extend the relation

£z = (jlly£H (5.5.18)

(5.5.13), but not necessarily the relation (5.5.14) to semiregular Lagrangians. The relation (5.5.18) enables us to generalize item (i) of Proposition 5.5.3 for Hamiltonian forms associated with semiregular Lagrangians.

PROPOSITION 5.5.6. Let a section r of the fibre bundle V'Q —> R be a solution of the Hamilton equations for a Hamiltonian form H associated with a semiregular Lagrangian L If r lives in the Lagrangian constraint space ./V, the section

c = nnor

of the fibre bundle Q —> R satisfies the Lagrange equations for L, while its first order jet prolongation

c = f f o r = c


obeys the corresponding Cartan equations (4.8.17a) - (4.8.17b). □

Proof. Put c = H o r. Since r(R) C N, then

r = L o c, f= J ' L o Jlc.

If r is a solution of the Hamilton equations, the exterior form £H vanishes at points of r(R). Hence, the pull-back form

£T = (j1ly£H

vanishes at points of c(R). It follows that the section c of the fibre bundle J ' Q —> R obeys the Cartan equations. By virtue of the equation (5.2.18a), we have c = c, and the section c is a solution of the Lagrange equations, which lives in the submanifold H(N). QED

In the case of semiregular Lagrangians, item (ii) of Proposition 5.5.3 can be modified as follows.

PROPOSITION 5.5.7. Given a semiregular Lagrangian L, let a section c of the jet bundle JlQ —* R be a solution of the Cartan equations (4.8.17a) - (4.8.17b). Let H be a Hamiltonian form associated with L so that the corresponding Hamiltonian map satisfies the condition

QED

HoLo-5= J l ( f o ° 4 (5.5.19)

Then the section

r = L o c,

r, =TTi(t,Cj,4), r* = c\

of the fibre bundle V'Q —» R is a solution of the Hamilton equations (5.2.18a) -(5.2.18b) for H, which lives in the Lagrangian constraint space TV. D

Proof. The Hamilton equations (5.2.18a) hold by virtue of the condition (5.5.19). Using the relations (5.5.17) and (5.5.19), the Hamilton equation (5.2.18b) is brought into the Cartan equation (4.8.17b):

dtTCi o c = -{diH)oloc =

(dtct — cDdiKj o c + d{C o c.

-(3-&H°l°c)dtn3 oc + diCoc =


QED

Proposition 5.5.6 shows that, if H is a Hamiltonian form associated with a semiregular Lagrangian L, every solution of the corresponding Hamilton equations, which lives in the Lagrangian constraint space N, yields a solution of the Cartan equations and of the Lagrange equations for L. Thus, the counterpart of degenerate Lagrangian systems are the constraint Hamiltonian systems.

We will restrict our consideration to the case of a Lagrangian constraint space N. This plays the role of a primary constraint space. In order that local solutions of the Hamilton equations for a Hamiltonian form H exists on N, the Hamiltonian connection 7# must be tangent to TV. This condition is given by the equations

Pi = %C(t,<j>,9iH), (9, + d>Hd, - djH&)\d{pt - d\C{t, q>,d>H)) = 0,

(5.5.20a) (5.5.20b)

where (5.5.20a) is the equation (5.5.3) of a Lagrangian constraint space, while (5.5.20b) requires that the horizontal vector field JH is tangent to N at the point (t, g1, Pi). In contrast with the case of autonomous constraint systems, the left-hand side of the equation (5.5.20b) is not reduced to the Poisson bracket. Therefore, we cannot follow the familiar procedure of constructing the final constraint space in Section 3.5.

There is another possibility. Given a solution of the Lagrange equations, one can try to find an associated Hamiltonian form H such that this solution is a solution of the Hamilton equations for H. It may happen that different solutions of the Lagrange equations require different Hamiltonian forms in general. Thus, degenerate Lagrangian systems are described as multi-Hamiltonian systems.

Note that, in the case of semiregular Lagrangians, the condition (5.5.19) is the obstruction for a solution c of the Cartan equations to be a solution of the Hamilton equations and the Lagrange equations. At the same time, one can try to find a family of associated Hamiltonian forms such that each solution of the Lagrange equations is a solution of the Hamilton equations for some Hamiltonian form from this family.

DEFINITION 5.5.8. We will say that a family of Hamiltonian forms H, associated with a Lagrangian L, is complete if, for each solution c of the Lagrange equations, there exists a solution r of the Hamilton equations for a Hamiltonian form H from


this family so that

c = 7Tn°r, r = Lo c (5.5.21)

(see the relation (5.2.19)). □

Let L be a semiregular Lagrangian. Then, by virtue of Proposition 5.5.7, such a complete family of associated Hamiltonian forms exists if and only if, for every solution c of the Lagrange equations for L, there is a Hamiltonian form H from this family such that

H o L o c = c. (5.5.22)

Example 5.5.4. Let L = 0. This Lagrangian is semiregular. Its Lagrange equations reduce to the identity 0 = 0. Every section c of the configuration bundle Q —► R is a solution of this equation. Given a section c, let T be a connection on the fibre bundle Q —> R such that c is an integral section of T. The Hamiltonian form Hr (5.2.7) is associated with L = 0, and the Hamiltonian map H^ obeys the relation (5.5.22). The corresponding Hamilton equations read

?! = r1, Pu = -PjdiT3.

They have a solution

r = L o c,

r ' = c \ n = o, which lives in the Lagrangian constraint space p, = 0. •

The example below shows that a complete family of associated Hamiltonian forms may exist when a Lagrangian is not necessarily semiregular.

Example 5.5.5. Let Q be the fibre bundle R2 —♦ R1 in Example 5.5.3 with coordinates (t,q). Put

£ = i(*)3-The associated Legendre map reads

poL = <fi. (5.5.23)


The corresponding Lagrangian constraint space N is given by the coordinate relation p > 0. It fails to be a submanifold of the momentum phase space V'Q. There exist two associated Hamiltonian forms

H+ = pdq - -p3'2dt,

H-=pdq+-p3/2dt,

on N, which correspond to the two different solutions

<7t = VP, Qt = -y/p

of the equation (5.5.23). The Hamiltonian forms H+ and # _ make up a complete family. •

Remark 5.5.6. Let 7# be a Hamiltonian connection for a Hamiltonian form H associated with a semiregular Lagrangian L. By virtue of the relation (5.5.18), the composition JlH o -yH (5.5.1) takes its values into the kernel of the Euler-Lagrange operator £{,. Since

HoL\S(Q) = UH(Q),

the morphism

j'Ho-yol: j ' Q g f t / o i l H (t, q\ qt = d'H o L, q\t) = Shi ° L, q\t = yH\d(d>H) o L)

restricted to H(Q) is the holonomic section over H(Q) C JlQ of the affine jet bundle J2Q —> JlQ. Let H(Q) be a closed submanifold, e.g., when a Lagrangian L is almost regular (see Definition 5.5.9 below). Then the section JlH o^HoL can be extended to a holonomic connection on the jet bundle JlQ —» R, which defines a dynamic equation on the configuration space Q. In this case, given a solution r of the Hamilton equations for H which lives in the Lagrangian constraint space ,/V, its projection 7rn o r is a solution of both the Lagrange equations and the dynamic one. •

DEFINITION 5.5.9. A semiregular Lagrangian L is called almost regular if: (i) the Lagrangian constraint space N is a closed imbedded subbundle i^ : N ^-> V'Q of

5.5. DEGENERATE SYSTEMS ZiO\y

the Legendre bundle V'Q —» Q and (ii) the Legendre map L : JXQ —» N is a fibre bundle. D

PROPOSITION 5.5.10. [57, 190]. Let L be an almost regular Lagrangian. On an open neighbourhood in V*Q of each point q € N, there exists a complete family of local Hamiltonian forms associated with L. O

Let L be an almost regular Lagrangian L. Given an associated Hamiltonian form H, the Hamiltonian map

H\N:N^ JlQ

restricted to N is a section of the fibre bundle L : JlQ —» N, and its image H(N) is a closed imbedded submanifold of the configuration space JlQ. Therefore, it follows from Remark 5.5.6 that each solution of the Lagrange equations for L, which is a solution of Hamilton equations, is a solution of a dynamic equation.

Since solutions of the Lagrange equations for a degenerate Lagrangian may correspond to solutions of different Hamilton equations, we can conclude that, roughly speaking, the Hamilton equations involve some additional conditions in comparison with the Lagrange equations. Therefore, let us separate a part of the Hamilton equations which are defined on the Lagrangian constraint space Q in the case of an almost regular Lagrangian.

Let

IT def .» Tf "N —lN™

be the restriction of a Hamiltonian form H associated with an almost regular Lagrangian L to the Lagrangian constraint space N. By virtue of Proposition 5.5.5, this restriction, called the constrained Hamiltonian form, is uniquely defined. Moreover, the Poincare—Cartan form is the pull-back He = L'Hs of the constrained Hamiltonian form HN. On sections r of the fibre bundle Q —» R, we can write the constrained Hamilton equations

r'{uN\dHN) = 0, (5.5.24)

where us is an arbitrary vertical vector field on the fibre bundle N —> R [56, 57, 161]. For the sake of simplicity, we can identify a vertical vector field uN on TV —> R with


its image Tz^(u^) and can bring the constrained Hamilton equations (5.5.24) into the form

r'{uN\dH) = 0, (5.5.25)

where r is a section of the fibre bundle TV —> R, and u^ is an arbitrary vertical vector field on N —» R. It should be emphasized that these equations fail to be equivalent to the Hamilton equations (5.2.23) restricted to the Lagrangian constraint space N.

If an almost regular Lagrangian admits associated Hamiltonian forms (see Proposition 5.5.10), the following three assertions together with Proposition 5.5.7 give the relationship between Cartan, Hamilton-De Donder, Hamilton, and constrained Hamilton equations [57, 161].

PROPOSITION 5.5.11. For any Hamiltonian form H associated with an almost regular Lagrangian L, every solution r of the Hamilton equations which lives in the Lagrangian constraint space N is a solution of the constrained Hamilton equations (5.5.25). □

Proof. For any vertical vector field Ufj on the fibre bundle N —* K, the vector field Tiw(ww) is obviously a local vertical vector field on the fibre bundle V'Q —» R. Then we have

r'(uN\dHN) = r'(uN\i%dH) = T'[TiN{uN)\dH) = 0

if r is a solution of the Hamilton equations (5.2.23) for the Hamiltonian form H. QED

PROPOSITION 5.5.12. A section c of the jet bundle JlQ -> R is a solution of the Cartan equations (4.8.18) if and only if Loc is a solution of the constrained Hamilton equations (5.5.25). □

Proof. Let uN be a vertical vector field on the fibre bundle N —► R. Since L is a submersion, there exists a vertical vector field v on the jet bundle JlQ —> R such that

TL ov = us-


For instance, v is the horizontal lift of u by means of a connection on the affine jet bundle JlQ —► Q. Let a section c : M «—► JlQ be a solution of the Cartan equations (4.8.18). Then we have

(Loc)*(uN\dHN) = c*{v\dHL) = 0. (5.5.26)

It follows that the section L oc : R ^* N is a. solution of the equations (5.5.25). The converse is obtained by running (5.5.26) in reverse, bearing in mind that the restriction of any vector field v on JlQ to c(R) is projectable by L. QED

PROPOSITION 5.5.13. The constrained Hamilton equations (5.5.24) are equivalent to the Hamilton-De Donder equations (4.8.25). □

Proof. Let H be a Hamiltonian form associated with an almost regular Lagrangian L, and let h be the corresponding section of the fibre bundle

C : T*Q -» VQ

(see Definition 5.2.4). This section yields the morphism

hN = h o iN : N -» T*Q

which does not depend on the choice of H. This is a local section of the fibre bundle T*Q -» V*Q over N C VQ, i.e,

C,ohN = lAN. (5.5.27)

Moreover, we have

HN = i*NH = i*N(h*E) = h*NE,

whenever H is a Hamiltonian form associated with the Lagrangian L. In accordance with the relation (5.5.16),

HL = hN o L, (5.5.28)

where HL is the Legendre morphism (4.8.21) associated with the Poincare-Cartan form Hi. Substituting (4.8.22) in (5.5.28), we obtain

HL = hNoC,o HL.


It follows that

hN°C \zL Wii(ZL ) , (5.5.29)

where iL(ZL) = HL(JlQ) is the image of the Legendre morphism HL. A glance at the relations (5.5.27) and (5.5.29) shows that there is the bundle isomorphism

ZL • N

over Q. Since H^ h*NE and EL ilH, we have

HN {iLl o hNyzL, ~L {c°hyHN.

Hence, the Hamilton-De Donder equations (4.8.25) are equivalent to the constrained Hamilton equations (5.5.24). QED

5.6 Quadratic degenerate systems

As an important illustration of Propositions 5.5.6 and 5.5.7, let us describe the complete families of Hamiltoman forms associated with almost regular quadratic Lagrangians. The Hamiltonians of these Hamiltonian forms are quadratic in momenta.

Remark 5.6.1. Since Hamiltonians in time-dependent mechanics are not scalar functions on a momentum phase space, one cannot apply to them the well-known analysis of the normal forms [18] (e.g., quadratic Hamiltonians [6] in symplectic mechanics). •

Given a configuration bundle Q —► R, let us consider a quadratic Lagrangian

C -aijq\4, + biql + c, (5.6.1)

where a, b and c are local functions on Q. This property is global due to the transformation law of the velocity coordinates q\. The associated Legendre map reads

Pi o L = a^ql + bi. (5.6.2)

5.6. QUADRATIC DEGENERATE SYSTEMS 263

LEMMA 5.6.1. The Lagrangian (5.6.1) is semiregular. □

Proof. For any point p of the Lagrangian constraint space N (5.5.4), the system of linear algebraic equations (5.6.2) for q\ has solutions which make up an afHne space modelled over the linear space of solutions of the homogeneous linear algebraic equations

0 = Oy^,

where q> are the coordinates on the vertical tangent bundle VQ. This affine space is obviously connected. QED

Let us assume that the Lagrangian L (5.6.1) is almost regular, i.e., the matrix a„ is of constant rank.

The Legendre map (5.6.2) is an affine morphism over Q. It defines the corresponding linear morphism

L:VQ ^VQ,

p{oL = ai:jq>,

whose image N is a linear subbundle of the Legendre bundle V*Q ^ Q. Accordingly, the Lagrangian constraint space N, given by the equations (5.6.2), is an affine subbundle, modelled over N, of the momentum phase space VQ —» Q. Hence, the constraint fibre bundle N —> Q has a global section. For the sake of simplicity, let us assume that this is the canonical zero section 0(Q) of the vertical cotangent bundle VQ -> Q. Then N = N.

The kernel

KerL = L-1(0(Q))

of the Legendre map is an affine subbundle of the affine jet bundle JXQ —> Q, which is modelled over the vector bundle

KeiL = L~\0(Q))cVQ.

Then there exists a connection

T : Q -» Ker L, (5.6.3)

aijP + h = 0, (5.6.4)


on the configuration bundle Q - » R , which takes its values into KerL. It is called a Lagrangian frame connection. With this connection, the quadratic Lagrangian (5.6.1) can be brought into the form

c = 2ay9r?f + c'>

* = 4 - r'-For instance, if the quadratic Lagrangian (5.6.1) is regular, there is a unique solution (5.6.3) of the algebraic equations (5.6.4).

PROPOSITION 5.6.2. There exists a linear map

<r : V*Q -> VQ, (5.6.5)

qloa = at]pi,

over Q such that

L o a o iN = i^.

D

Proof. The map (5.6.5) is a solution of the algebraic equations

dija^akb = (lib- (5.6.6)

The matrix ay is symmetric. After diagonalization, let this matrix have non-vanishing components aAA, A & I. Then a solution of the equations (5.6.6) takes the form

1 °AA = ^X' OAA' = 0, A, A1 6 / , A? A',

while the remaining components are arbitrary. In particular, there is a solution

1 °AA = ^ u - OAA' = 0, OCB = 0, B*I, (5.6.7)

which satisfies the relation

a = a o L o a, (5.6.8)

<T'J = aikakb^b.


QED

From now on, by a is meant the solution (5.6.7). If the Lagrangian (5.6.1) is regular, the map (5.6.5) is uniquely determined by the equation (5.6.6).

The Lagrangian frame connection (5.6.3) and the map (5.6.5) play a prominent role in the construction below.

PROPOSITION 5.6.3. The matrix a defines the splitting

J 1 Q = K e r Z e I m ( a o Z ) , Q

(5.6.9)

4 = [qj - <?k{a.kd + h)} + \oik{akj<& + Ml-

of the velocity phase space JlQ. D

In particular, it follows that, since L and a are linear morphisms, their composition L o a is a fibration of the momentum phase space V*Q over the Lagrangian constraint space N.

PROPOSITION 5.6.4. There is also the following splitting of the momentum phase space

V*Q Ker a®N, Q

Pi \pi - aijcr^pk] + [aija3kpk].

a

COROLLARY 5.6.5. Every vertical vector field

u = u'di + Uid'

on the momentum phase space V*Q —> M admits the decomposition

U — (u — Upf) + UN, (5.6.10)

m = [Ui - aija3kuk] + [atja^Uk],

where

uN = uldi + aijt7jkukd'


is a vertical vector field on the Lagrangian constraint space N —> R. □

Given the linear map a (5.6.5) and the Lagrangian frame connection T (5.6.3), let us consider the affine Hamiltonian map

$ = T + a : V'Q -> J'Q, (5.6.11)

$ ' = F ( g ) + < 7 yP i ,

and the Hamiltonian form

H = H* + Ldt,

H = ntf - [r(Pi - -h) + -O»KPS - c]dt. (5.6.12)

It is readily observed that H = $ and, by virtue of (5.6.8), the Hamiltonian form H is associated with the quadratic Lagrangian (5.6.1).

We aim to show that the Hamiltonian forms (5.6.12), parameterized by the Lagrangian frame connections T (5.6.3), constitute a complete family.

Given the Hamiltonian form (5.6.12), let us consider the Hamilton equations (5.2.18a) for sections r of the fibre bundle V'Q -» R. They read

c = (f + a) o r, c = 7Tn o r, (5.6.13)

or

V (r i = aijrjt

where

y t r ' = <V - ( T o c)1

is the covariant derivative relative to the connection T. With the splitting (5.6.9), we have the surjections

S = prl: JlQ-*KetL, S : q\ — ql - oik{akjqi + bk),

and

T = pr2 : JlQ -» Im(cr o Z),

F:qi->aik(akjqi + bk).


With respect to these surjections, the Hamilton equations (5.6.13) are split into the following two parts:

S o c = T o c,

Vtr{ = aik(akjdtri + bk), (5.6.14)

and

T o c = a o r, (5.6.15) aik{akjdtTJ + h) = aikrk.

The Hamilton equations (5.6.14) are independent of the canonical momenta rk

and play the role of gauge conditions. Moreover, for every section c of the fibre bundle Q —► R (in particular, for every solution of the Lagrange equations for the Lagrangian L), there exists a Lagrangian frame connection T (5.6.3) such that the equation (5.6.14) holds. Indeed, let F* be a connection on the fibre bundle Q —> R whose integral section is c. Put

T^SoT1, r = r' i-<7 i lc(afcJr' i + 6fc).

In this case, the Hamiltonian map (5.6.11) satisfies the relation (5.5.22) for c, i.e.,

$ o L o c = c.

Hence, the Hamiltonian forms (5.6.12) constitute a complete family. The Hamiltonian forms from this family differ from each other only in the Lagrangian frame connections T (5.6.3) which lead to the different gauge conditions (5.6.14). It is readily observed that a Lagrangian frame connection V is also a Hamiltonian frame connection for the corresponding Hamiltonian form (5.6.12).

PROPOSITION 5.6.6. For every Hamiltonian form H (5.6.12), the Hamilton equations (5.2.18b) and (5.6.15) on sections of the constraint fibre bundle N —> R are equivalent to the constrained Hamilton equations (5.5.25). □

Proof. In accordance with the decomposition (5.6.10) of a vertical vector field u on the momentum phase space V*Q —» R, the constrained Hamilton equations (5.5.25) take the form

r*(aija:>kdi\dH)=0, (5.6.16a)

r*{dt\dH) = 0. (5.6.16b)


The equations (5.6.16b) are obviously the Hamilton equations (5.2.18b) for H. Bearing in mind the relations (5.6.4) and (5.6.8), one can easily bring the equations (5.6.16a) into the form (5.6.15). QED

It follows that the equations (5.6.14) are the additional conditions which make the Hamilton equations differ from the constrained Hamilton equations (5.5.25) and the Lagrange equations in the case of a quadratic Lagrangian.

COROLLARY 5.6.7. By virtue of Proposition 5.5.12, a section c of the jet bundle JlQ -» 1 is a solution of the Cartan equations (4.8.17a) - (4.8.17b) for the almost regular Lagrangian (5.6.1) if and only if Loc is a solution of the constrained Hamilton equations (5.2.18b) and (5.6.15). □

It follows that the equations (5.6.14) are responsible for the obstruction condition (5.5.19) for solutions c of the Cartan equations to be solutions of the Hamilton equations and of the Lagrange equations for the Lagrangian (5.6.1). It is readily seen that the equations (5.6.15) do not contribute to the obstruction condition (5.5.19). If r = L o c, these equations hold for solutions c of the Cartan equations (4.8.17a).

PROPOSITION 5.6.8. Let c be a solution of the Cartan equations for the almost regular quadratic Lagrangian L (5.6.1). Let Co be a section of the fibre bundle VQ —► R, which takes its values into Ker L and projects onto c = 7TQ O C. Then the sum c + Co over Q is also a solution of the Cartan equations. □

Proof. The proof is an immediate consequence of the relation

r = L o c = L o ( c + Co).

QED

Since Hamiltonian forms associated with an almost regular quadratic Lagrangian (5.6.1) constitute a complete family, that is, each solution of the Lagrange equations is also a solution of some Hamilton equations, we can conclude that each solution of the Lagrange equations is a solution of some dynamic equation.

5.7. HAMILTONIAN CONSERVATION LAWS 269

5.7 Hamiltonian conservation laws

As was mentioned above, the notion of integrals of motion as functions in involution with a Hamiltonian can not be extended to time-dependent mechanics because its Hamiltonian is not a scalar function under time-dependent transformations.

In Section 4.12, we have studied the conservation laws in Lagrangian time-dependent mechanics. Therefore, in order to discover the conservation laws in Hamiltonian time-dependent mechanics, let us consider the following construction.

Given a Hamiltonian form H (5.2.10) on a momentum phase space V*Q, let us consider the Lagrangian

LHd=(Piqi-H)dt (5.7.1)

(cf. (4.4.6)) on the jet manifold JlV*Q of the momentum phase space V*Q —* M. Note that the Lagrangian (5.7.1) plays a prominent role in the functional integral approach to mechanics (see Remark 5.11.4 below). Here, our utilization of this Lagrangian is based on the following two assertions.

LEMMA 5.7.1. The Poincare-Cartan form for the Lagrangian LH (5.7.1) coincides with the pull-back onto JXV*Q of the Hamiltonian form H by the projection J1V*Q ~* V*Q, while the Euler-Lagrange operator for LH coincides with the pull-back onto J2V*Q of the Hamilton operator for H by projection J2V*Q —> JlV*Q. a


It follows from this Lemma that the Lagrange equations for the Lagrangian LH

(5.7.1) are equivalent to the Hamilton equations for the Hamiltonian form H.

LEMMA 5.7.2. Let u be a vector field (4.12.2) on a configuration space Q, and

u = uldt + u'di — diUjpjd' (5.7.2)

its lift onto the momentum phase space V*Q in accordance with the formula (1.6.13). Then, there is the equality

L ; H = Lji~LH,

where Jlu is the jet prolongation (1.3.7) of the vector field u onto JlVQ. □



Thus, we can use the general procedure of constructing Lagrangian conservation laws in Section 4.12 for the Lagrangian LH in order to obtain the Hamiltonian conservation laws.

Let us apply the first variational formula (4.8.4) to the Lagrangian (5.7.1). Then the weak identity (4.12.3) reads

-u%H - u'dtH + Pidttf ~ -dt%, (5.7.3)

X = -pitf + u 'H. (5.7.4)

In the case of a vertical vector field u, when u( = 0, the weak identity (5.7.3) leads to the weak equality

L J I ^ H ~ dt{piU%).

If \JJI~LH = 0, we have the weak conservation law

0 pa dt(piu')

of the Noether current

% = -piU*. (5.7.5)

For a reference frame Y (5.2.5) on the configuration bundle Q —> K, the weak identity (5.7.3) takes the form

-dtH - rdiH+PidtV « -dtHT,

where H? = Ji — P iP is the Hamiltonian function in the splitting (5.2.12). In the case of semiregular Lagrangians, there is the following relationship between

Lagrangian and Hamiltonian conserved currents.

PROPOSITION 5.7.3. Let H be a Hamiltonian form on the momentum phase space V*Q, which is associated with a semiregular Lagrangian L on the velocity phase space J 1 ^ . Let r be a solution of the Hamilton equations (5.2.18a) - (5.2.18b) for H, which lives in the Lagrangian constraint space N, and c the associated solution of the Lagrange equations for the Lagrangian L so that the conditions (5.5.21) are

5.8. TIME-DEPENDENT SYSTEMS WITH SYMMETRIES 271

satisfied. Let u be a vector field (4.12.2) on the configuration space Q —> R, Then we have

T{r) = 1{Hor), X(Zoc)=X(c),

where X is the current (4.12.4) on the configuration space JXQ, and X is the current (5.7.4) on the phase space V*Q. □


In accordance with Proposition 5.7.3, the Hamiltonian counterpart of the Lagran-gian energy function Xr (4.12.16) relative to a reference frame T is the Hamiltonian function HT in the splitting (5.2.12). Therefore, one can think of the Hamiltonian function Hr as being the Hamiltonian energy function relative to the reference frame T. In particular, if P = 0, we obtain the well-known energy conservation law

dtH « dtH

relative to the coordinates adapted to the reference frame V. This is the Hamiltonian variant of the Lagrangian energy conservation law (4.12.17).

5.8 Time-dependent systems with symmetries

In mechanics, just as in field theory, systems with symmetries can be described in terms of bundles with structure groups.

Let 7Tp : P —► R be a principal bundle with a structure Lie group G. Let

Q = (P x M)/G (5.8.1)

be a P-associated fibre bundle. By gauge transformations of a principal bundle P are meant its vertical auto

morphisms $ which are equivariant with respect to the canonical action (1.5.4) of the structure group G on P on the right, i.e.,

$ o rg — rg o $ , VgeG. (5.8.2)

Due to the property (5.8.2), gauge transformations $ of a principal bundle yield the gauge transformations of a P-associated fibre bundle

$ Q : (P x M)/G -» ($(P) x M)/G.


Thus, one can treat a fibre bundle Q (5.8.1) as a configuration space of a mechanical system with the gauge symmetry group G.

Every gauge transformation of the configuration bundle Q - » l gives rise canoni-cally to the holonomic vertical automorphisms (5.3.1) of the momentum phase space VQ —> M. They are canonical automorphisms for the canonical Poisson structure (5.1.5) on VQ. Moreover, this group action is Hamiltonian as can be seen in the following way (see Definition 3.7.1).

Any 1-parameter group of gauge transformations of a principal bundle P defines a vector field (its generator) on P which is invariant under the canonical action of the group G on P. There is one-to-one correspondence between these right-invariant vector fields on P, called principal vector fields, and the sections of the gauge algebra bundle VQP (1.5.6), whose typical fibre is the right Lie algebra jj r .

Accordingly, any 1-parameter group of gauge transformations of a configuration space Q, associated with P , defines a principal vertical vector field

C = C(g)d,

on the configuration space Q. This vector field gives rise to the vector field

Z^cat-atfpjtf on the momentum phase space (see (5.7.2)). It is readily observed that this is the Hamiltonian vector field for the Noether current

J = Pi€(q) (5.8.3)

(5.7.5) which is defined globally on VQ (see Example 3.7.4). It follows that, given a finite dimensional Lie group of gauge transformations, one may apply the familiar theorems of reduction, e.g., Theorem 3.7.9 to time-dependent mechanics, but not Theorem 3.7.10 and other results which involve the condition of an invariance of a Hamiltonian under a group action. Hamiltonians in time-dependent mechanics fail to be invariant under time-dependent transformations, including gauge transformations.

Remark 5.8.1. It should be emphasized that gauge transformations make up a group whose suitable Sobolev completion is a Banach Lie group. The possible extension of the above-mentioned theorems to this infinite-dimensional Lie group in the spirit of [36] seems promising. At the same time, we will show below that a Hamiltonian form may possess a finite-dimensional Lie group of gauge transformations. •

5.8. TIME-DEPENDENT SYSTEMS WITH SYMMETRIES 273

Let us restrict our consideration to the case of a vector bundle Q —> R. A principal vector field on this vector bundle reads

£ = ar(t)emiiq'ai, (5.8.4)

where em ' j are generators of the action of the group G on the typical fibre M, and am(t) are local functions on R. The vector field (5.8.4) gives rise to the vector field

| = a m ( t )E m y9 , - am(t)emijPld>

on the momentum phase space VQ. It is readily observed that this vector field is the Hamiltonian vector field for the Noether current p;£' (5.8.3) relative to the canonical Poisson structure on VQ. Moreover, these Noether currents constitute the Lie algebra with respect to the Poisson bracket

{pie,pd'j}v=Pi[^']i. (5.8.5)

In particular, let pf£* and Pi£" be conserved Noether currents for a Hamiltonian form H on the momentum phase space VQ. Since

K,e] = [f,r], Jl[U'\ = [J1lJ1Z] and

Lj i j i / f = 0, Lji^Lff = 0,

we obtain

L J M « ' 1 L H = °'

i.e., the Noether current (5.8.5) is also conserved. Thus, conserved Noether currents in time-dependent mechanics constitute a Lie algebra (cf. Example 3.7.4). This algebra obviously is finite-dimensional, and corresponds to some finite-dimensional Lie group of gauge symmetries of a Hamiltonian form H. As was shown, an action of this group on a momentum phase space VQ is Hamiltonian, with an equivariant momentum mapping.

This result remains true for Noether currents (4.12.14) in Lagrangian mechanics. Given a Lagrangian L, the Noether currents (4.12.14) [conserved Noether currents (4.12.14)] along vector fields (5.8.4) constitute a Lie algebra with respect to the Poisson bracket (4.8.13).


5.9 Systems with time-dependent parameters

Let us consider a configuration space which is a composite fibre bundle

Q ^ E - > R , (5.9.1)

coordinated by (£, am, q') where (t, am) are coordinates of the fibre bundle E —> R. We will treat sections h of the fibre bundle £ —► R as time-dependent parameters, and say that the configuration space (5.9.1) describes a mechanical system with time-dependent parameters. Let us call E -» R the parameter bundle. Note that the fibre bundle Q —» E is not necessarily trivial.

Let us recall that, by virtue of Proposition 1.6.2, every section h of the parameter bundle E —> R defines the restriction

Qh = h'Q (5.9.2)

of the fibre bundle Q —> E to h(R) C E, which is a subbundle ih : QH *-» Q of the fibre bundle Q —► R. One can think of the fibre bundle Qh —> R as being a configuration space of a mechanical system with the background parameter function h(t).

The velocity momentum space of a mechanical system with parameters is the jet manifold JXQ of the composite fibre bundle (5.9.1) which is equipped with the adapted coordinates

{t,o ,q ,at ,qt).

Let the fibre bundle Q —> E be provided with a connection

A = dt ® (dt + A\d{) + dam <g> (dm + <,<%). (5.9.3)

Then the corresponding vertical covariant differential

D:JXQ^ VZQ, D = (qi-A\- A^o?)*, (5.9.4)

(1.6.11) is defined on the configuration space Q. Given a section h of the parameter bundle E —> R, its restriction to Pihi^Qh) C JlQ is precisely the familiar covariant differential on Qh corresponding to the restriction

Ah=dt + {Aimdthm + (Aoh)\)di (5.9.5)

5.9. SYSTEMS WITH TIME-DEPENDENT PARAMETERS 275

of the connection A to h(R) C E. Therefore, one may use the vertical covari-ant differential D in order to construct Lagrangians for a mechanical system with parameters on the configuration space Q (5.9.1).

We will suppose that a Lagrangian L depends on derivatives of parameters a™ only through the vertical covariant differential D (5.9.4), i.e.,

L = C{t, am) q\q\ ~ A* - Aô?)*. (5.9.6)

Such a Lagrangian is obviously degenerate because of the constraint condition

nm + Aî = 0.

As a consequence, the total system of the Lagrange equations

(di - dt%)C = 0, (5.9.7)

(dm - dt&JC = 0 (5.9.8)

admits a solution only if the very particular relation

(dm + Aimdi)C + TTidtAin = 0 (5.9.9)

holds. However, we believe that parameter functions are background, i.e., independent of a motion. Therefore, only the Lagrange equations (5.9.7) should be considered.

In particular, let us apply the first variational formula (4.12.8) in order to obtain conservation laws of a mechanical system with time-dependent parameters. Let

u = %% + um{t, ah)dm + u{(t, ak, qj)di, u* = 0,1, (5.9.10)

be a vector field which is projectable with respect to both the fibration Q - * B and Q —» E. Then the Lie derivative L^L of a Lagrangian L along the vector field u (5.9.10) reads

[u% + umdm + v.% + dtiTdl + dtu'dl}L = (um - aûl)dmC + 7rmd,(um - a j V ) +

(«' - <?>% - *[»ri(»*flj - u*) - u'£], dt = dt + <r?dm + qtdi + a^dl

m + &%.

On the shell (5.9.7), the weak identity

[u% + umdm + u% + dtumdlm + dtu'dfiC «

(um - ofu^drnC + ^mdt{um - ffjV) - *[«&% ~ « ' ) ~ ^C]


is fulfilled. If the Lie derivative L^L vanishes, we obtain the conservation law

0 w (um - aT^)dmC + nmdt(um - a?ul) - ^ [^(u 'g j - u*) - u«£] (5.9.11)

for a system with time-dependent parameters. Now, let us describe such a system in the framework of Hamiltonian formalism.

Its momentum phase space is the vertical cotangent bundle V*Q. Given a connection (5.9.3), we have the splitting (1.6.10b) which reads

V*Q = A{VZQ)®{QxV*Y,),

Pidq1 + pmdam = Pi(dq' - A^da"1) + (pm + A^da™.

Then VQ can be provided with the coordinates

Pi=Pi, Pm=Pm+Alrnpi,

compatible with this splitting. However, these coordinates fail to be canonical in general. Given a section h of the parameter bundle S - * K , the submanifold

{a = h(t), Pm = Q}

of the momentum phase space V*Q is isomorphic to the Legendre bundle V*Qh of the subbundle (5.9.2) of the fibre bundle Q —> R, which is the configuration space of a mechanical system with the parameter function h(t).

Let us consider Hamiltonian forms on the momentum phase space VQ which are associated with the Lagrangian L (5.9.6). Given a connection

r = dt + rmdm (5.9.12)

on the parameter bundle E —* K, the desired Hamiltonian form

H = {pxdq{ + Pmdam) - faA" + pmTm + H(t, am, q^p^dt (5.9.13)

can be found, where

B = dt + rmdm + B'd{, B1 = Ai + y^I™,

is the composite connection (1.6.8) on the fibre bundle Q —> K, which is defined by the connection A (5.9.3) on Q —» E and the connection F (5.9.12) on E —► R. The Hamiltonian function H satisfies the conditions

ni(t,qi,am,diH(t,am,qi,ni))=7Th

Pid'n-n^c^q^^^d'n)


which axe obtained by substitution of the expression (5.9.13) in the conditions (5.5.2a) - (5.5.2b).

The Hamilton equations for the Hamiltonian form (5.9.13) read

ql = A' + A^r™ + dm (5.9.14a)

Pu = -Pj{diAj + ^ r ) - W, (5.9.14b) _m run (5.9.14c) Ptm = -Pi(dmA{ + r n a m 4 ) - dmn, (5.9.14d)

whereas the Lagrangian constraint space is given by the equations

pi = TTi(t,q\(Tm,diH(t,CTm,q\pi)), (5.9.15) pm + AIJH = 0. (5.9.16)

The system of equations (5.9.14a) - (5.9.16) are related in the sense of Section 4.5 to the Lagrange equations (5.9.7) - (5.9.8). Because of the equations (5.9.14d) and (5.9.16), this system is overdetermined. Therefore, the Hamilton equations (5.9.14a) - (5.9.14d) admit a solution living in the Lagrangian constraint space (5.9.15) - (5.9.16) if the very particular condition, similar to the condition (5.9.9), holds. Since the Lagrange equations (5.9.8) axe not considered, we also can ignore the equation (5.9.14d). Note that the equations (5.9.14d) and (5.9.16) determine only the momenta pm and do not influence other equations.

Therefore, let us consider the system of equations (5.9.14a) - (5.9.14c) and (5.9.15) - (5.9.16). Let the connection T in the equation (5.9.14c) be complete and admit the integral section h(t). This equation together with the equation (5.9.16) defines a submanifold V'Qh of the momentum phase space V*Q, which is the momentum phase space of a mechanical system with the parameter function h(t). The remaining equations (5.9.14a) - (5.9.14b) and (5.9.15) are the equations of this system on the momentum phase space V*Qh, which correspond to the Lagrange equations (5.9.7) in the presence of the background parameter function h(t).

Conversely, whenever h(t) is a parameter function, there exists a connection T on the parameter bundle E —» R such that h(t) is an integral section T. Then the system of equations (5.9.14a) - (5.9.14b) and (5.9.15) - (5.9.16) describes a mechanical system with the background parameter function h(t). Moreover, we can locally restrict our consideration to the equations (5.9.14a) - (5.9.14b) and (5.9.15).

The following examples illustrate the above construction.


Example 5.9.1. Let us consider the 1-dimensional motion of a probe particle in the presence of a force field whose centre moves. The configuration space of this system is the composite fibre bundle

M3 -> M2 -> E,

coordinated by (t, a, q) where a is a coordinate of the field centre with respect to some inertial frame and q is a coordinate of the probe particle with respect to the field centre. There is the natural inclusion

QxTY,B {t,a,q,i,a) ^ {t,o,i,a,y = -a) eTQ

which defines the connection

A = dt ® dt + da <g> (d„ - dq)

on the fibre bundle Q —> E. The corresponding vertical covariant differential (5.9.4) reads

D = (qt + at)d„.

This is precisely the velocity of the probe particle with respect to the inertial frame. Then the Lagrangian of this particle takes the form

L=[^(qt + at)2-V(q)}dt. (5.9.17)

In particular, we can obtain the energy conservation law for this system. Let us consider the vector field u = dt. The Lie derivative of the Lagrangian (5.9.17) along this vector field vanishes. Using the formula (5.9.11), we obtain

0 « —TTq(7u — dt[irqqt — £],

or

0 ~ dqCat ~ dt[irq{qt + ot) - £],

where

X = 7r,(ft+ < r t ) - £

is the energy function of the probe particle with respect to the inertial reference frame. •


Example 5.9.2. Let us consider the 1-dimensional oscillator whose frequency is a time-dependent parameter. In accordance with the adiabatic hypothesis, there exists a coordinate q such that this oscillator differs from the standard one only in kinetic energy.

Let us take the configuration space

Q = R x R + x R - » R + x R - + R ,

provided with the coordinates (t, a > 0, q), where a is the frequency parameter. The corresponding momentum phase space V*Q is coordinated by {t,cr,q,pa,pg). The oscillator is characterized by the Hamiltonian form

H = pada + pqdy — Hdt,

H=paT° + \{oyq + q\ (5.9.18)

The corresponding Hamiltonian frame connection

TH = dt + T°tda (5.9.19)

on the configuration bundle Q —► R is the composite connection (1.6.8) generated by some linear connection

r = dt ® (dt + T(t)oda)

on the parameter bundle

S = K + x l - . R

and by the trivial connection

A = dt ® dt + da ® da

on the fibre bundle Q —> E. The Hamiltonian form (5.9.18) is associated with the Lagrangian

L=\{o-2c?t-qi)dt

which describes the oscillator with time-dependent frequency a = h{t) in accordance with the adiabatic hypothesis.


The Hamilton equations (5.9.14a) - (5.9.14c) for the Hamiltonian form (5.9.18) read

qt = ^2P„ (5.9.20a)

Pt = -9, (5.9.20b)

ct = T{t)a. (5.9.20c)

They also describe the oscillator with the variable frequency a = h{t) if we put r = h~ldth. The Lagrangian constraint space equations (5.9.15) are trivial.

Let us consider the canonical transformation

q = aq', 1 ,

, 1 i , Po=Va QPq-

a Relative to the new coordinates (t,a, ^ .p '^ .p ' , ) , the connection (5.9.19) is written

rH = dt + T°{da - q'dq,),

while the Hamiltonian form (5.9.18) reads

H = p'ada + p'qdq' - Hdt,

n = p^-p'qq'T + \(pf + a^).

Accordingly, the Hamilton equations (5.9.20a) - (5.9.20c) take the form

Pty = \PT - -V, (5.9.21a)

q't = -±q>T°+p'q, (5.9.21b)

<>t = 1 7 - (5.9.21c)

Substitution of a = h(t) and the equation (5.9.21c) in the equations (5.9.21a) and (5.9.21b) leads to the Hamilton equations corresponding to the Hamiltonian form

H = p'qdq' - Hdt,

H = -i<Wp;+^(p',2+*v2).


This Hamiltonian form describes the well-known Berry oscillator where the connection (5.9.5) is the classical variant of the Berry connection

Ah = dt dtaq'dq/ a

[135, 144]. •

Example 5.9.3. Let us consider an n-body as in Example 4.9.4. Let R3™-3 be the translation-reduced configuration space of the mass-weight Jacobi vectors {pA}-Two configurations {pA} and {pA} are said to define the same shape of the n-body if pA = RpA for some rotation R £ 50(3) . This introduces the equivalence relation between configurations, and the shape space S of the n-body is defined as the quotient

S = R 3 n - 3 / 5 0 ( 3 )

[119, 120]. Then we have the composite fibre bundle

1 x R3 n _ 3 -» R x S -» R, (5.9.22)

where the fibre bundle

K3""3 _> S (5.9.23)

has the structure group 50(3) . The composite fibre bundle (5.9.22) is provided with the bundle coordinates(t, am,ql), where q', i = 1,2,3, are some angle coordinates, e.g., the Eulerian angles, while am, m = 1 , . . . ,3n — 6, are said to be the shape coordinates on S. A section QA{om) of the fibre bundle (5.9.23), called a gauge convention, determines an orientation of the n-body in a space. Given such a section, any point {pa} of the translation-reduced configuration space R3™-3 is written as

PA = R(qTQA{°m)-

This relation yields the splitting

~pA = diRcfQji + dm~eAdm

of the tangent bundle TR3 n 3, which determines a connection A on the fibre bundle (5.9.23). This is also a connection on the fibre bundle

R x R 3 " - 3 -► R x 5 ,


with the components At = 0. Then the corresponding vertical covariant differential (5.9.4) reads

D = (4 - /&?)*. With this vertical covariant differential, the total angular velocity of the n-body takes the form

0 = a^D* = u + Am<77\ (5.9.24)

where &i are certain kinematic coefficients and 3 is the angular velocity of the n-body as a rigid one. In particular, we obtain the phenomenon of a faffing cat if 0 = 0 so that

u = -Am<rtm

The coefficients A m in the expression (5.9.24) are called gauge potentials, and the mechanics of deformable bodies is sometimes termed gauge mechanics. •

5.10 Unified Lagrangian and Hamiltonian formalism

The relationship between Lagrangian and Hamiltonian formalisms in Section 5.5 is broken by canonical transformations if the transition functions q' —♦ <r" depend on momenta. The following construction enables us to overcome this difficulty.

Given a configuration bundle Q —► M, let V*JlQ be the vertical cotangent bundle of the velocity phase space JlQ —» R, with coordinates

(*,?',?!, ft,?'),

and let JXV*Q be the jet manifold of the phase space V*Q —» M, with coordinates

{t,q\Pi,q\,Pti)-

PROPOSITION 5.10.1. There is the isomorphism

n = V* JXQ & JlV"Q, ft *—> Pu, 4t <—► P i , (5.10.1)

over JlQ. □

5.10. UNIFIED LAGRANGIAN AND HAMILTONIAN FORMALISM 283

Proof. The isomorphism (5.10.1) can be proved by an inspection of transition functions of the coordinates {%,$) and (puPti)- QED

Due to the isomorphism (5.10.1), one can think of II as being both the momentum phase space over the velocity phase space JlQ and the velocity phase space over the momentum phase space V*Q. Hence, the space II can be utilized as the unified phase space of joint Lagrangian and Hamiltonian formalism.

Remark 5.10.1. In connection with this, note that, according to [10, 127, 180], the dynamics of autonomous mechanical system described by a degenerate Lagrangian L on TM is governed by a differential equation on T*M generated by dL, due to the canonical diffeomorphism between TT'M and T*TM. •

The unified phase space I I is equipped with the holonomic coordinates

{t,q\qlt,PtuPi), (5.10.2)

where {q',pu) and {q\,Pi) are canonically conjugate pairs. It is endowed with the jet prolongation of the canonical 3-form (4.8.10), which with respect to the coordinates (5.10.2) reads

ft = (dpti A dq' + dpi A dq\) A dt = dt{dpi A dq' A dt), (5.10.3)

where

dt = dt + q\di + ptid'

is the total derivative on II . The corresponding Poisson bracket (5.1.5) takes the form

{/,ff}n df_ 0£ df_ dg_ _ dg_dl _ dg_dl dptidqi dpidqt dpudq* dpidq\

(5.10.4)

It is readily observed that the canonical form (5.10.3) and, consequently, the Poisson bracket (5.10.4) are invariant under the transformations of the unified phase space II , which are the jet prolongation of the canonical automorphisms of the phase space V*Q.

Let

H Ptidq'+ Pidq\ - Hu(t,q',q\,pu,Pi)dt


be a Hamiltonian form on II. The corresponding Hamilton equations (5.2.18a) (5.2.18b) read

OPti (5.10.5a)

, , dHn dpi

(5.10.5b)

, dHn (5.10.5c)

dHn (5.10.5d)

Substitution of (5.10.5a) in (5.10.5b) and of (5.10.5c) in (5.10.5d) leads to the equations

mn = &Hn dpu dpi '

(5.10.6a)

&Hn _ dHn * dq\ " dq'

(5.10.6b)

which look like the Lagrange equations for the "Lagrangian" Tin- Though Hn is not a true Lagrangian, one can put

H„ = -C + dt(Pir), (5.10.7)

whenever L is a Lagrangian on the velocity phase space J Q. Then the equations (5.10.6a) - (5.10.6b) are equivalent to the Lagrange equations for a Lagrangian L on JlQ. However, their solutions fail to be solutions of the corresponding Ha.mil-ton equations (5.10.5a) - (5.10.5d) for the Hamiltonian (5.10.7) in general. This illustrates the fact that solutions of the Hamilton equations (5.10.5a) - (5.10.5d) are necessarily solutions of the Lagrange equations (5.10.6a) - (5.10.6b), but the converse statement is not true.

To construct the joint Lagrangian-Hamiltonian formalism, let us consider the Hamiltonian form

H = pudg' + Pidgj - (dtH + {ml -H)- C)dt (5.10.8)

on the unified phase space II, where L is a semiregular Lagrangian on the velocity phase space JlQ and H is an associated Hamiltonian form on the momentum phase

http://Ha.mil

5.11. VERTICAL EXTENSION OF HAMILTONIAN FORMALISM 285

space V*Q. The Hamilton equations (5.10.5a) - (5.10.5d) for H (5.10.8) read

<w &H dpi'

(5.10.9a)

dtq] 2m OPi on

+ dpi (5.10.9b)

dtPi dq* dC (5.10.9c)

dtPu -dt an an dc dqi + dqi + dtf

(5.10.9d)

Using the relations (5.5.2a) and (5.5.7), one can show that solutions of the Hamilton equations (5.2.18a) - (5.2.18b) for the Hamiltonian form H, which live in the Lagrangian constraint space L^Q) C V*Q are solutions of the equations (5.10.9a) - (5.10.9d).

Now let us consider the Lagrange equations (5.10.6a) - (5.10.6b) for the Hamiltonian form (5.10.8). They read

<k<t dpi o, (5.10.10a)

dtPi _dU _

dqi d£ dqr (5.10.10b)

In accordance with Proposition 5.5.7, every solution of the Lagrange equations for the Lagrangian L such that the relation (5.5.22) holds is a solution of the equations (5.10.10a) - (5.10.10b).

In particular, if the Lagrangian L is hyperregular, the equations (5.10.10a) -(5.10.10b) and the equations (5.10.9a) - (5.10.9d) are equivalent to the Lagrange equations for L and the Hamilton equations for the associated Hamiltonian form.

5.11 Vertical extension of Hamiltonian formalism

Let Q —► R, coordinated by {t,ql), be a configuration space of time-dependent mechanics. Let us consider the vertical tangent bundle VQ of the fibre bundle Q —> R, coordinated by (t,q',q'). It can be seen as the new configuration space, called the vertical configuration space. Here, we will be concerned with the following three applications of Hamiltonian mechanics on this configuration space:

• mechanical systems with linear deviations (see Example 5.11.1 below),


• Hamiltonian description of non-holonomic constraints (see Example 5.11-3 below),

• Hamiltonian BRST mechanics (see Appendix A).

Given VQ, the corresponding velocity phase space is the first order jet manifold JlVQ of the fibre bundle VQ -» R. It is canonically isomorphic (see (1.3.8)) to the vertical tangent bundle VJlQ of the ordinary velocity phase space JXQ -» M, coordinated by

{%<?,$ AA)-We call VJlQ the vertical velocity phase space.

The Legendre bundle of the vertical configuration space VQ is the vertical cotangent bundle V'VQ of the fibre bundle VQ —> M. It plays the role of a momentum phase space, called the vertical momentum phase space, of mechanical systems on the configuration space VQ. The vertical momentum phase space is canonically isomorphic (see (1.1.10)) to the vertical tangent bundle VV'Q of the ordinary momentum phase space V*Q —> K, coordinated by

(t,q',Pi,q',Pi).

It is easily seen from the transformation laws

Pi = djlPv

., dp\ . dp[ dq> . dtf

that (q\fii) and {q',Pi) are canonically conjugate pairs. Thus, we have observed that the vertical velocity and momentum phase spaces,

corresponding to the configuration space VQ, are the vertical tangent bundles of ordinary velocity and momentum phase spaces, respectively.

The momentum phase space VV'Q is endowed with the canonical 3-form

Qv = [dpi A dql + dpi A dq^] A dt. (5.11.1)

For brevity, one can write

flv = dy£l,


where fi is the canonical 3-form (5.1.9) on the momentum phase space V*Q, and the formal operator of the vertical derivative

dy = q'di + pjd'

(1.1.4) is utilized. This operator possesses the properties

dv(4> ACT) = dv(<f>) ACT + </)A9V((T), ^ae&iV'Q), dv{dcj>) = d{dv{cj>)).

The canonical 3-form (5.11.1) provides VVQ with the canonical Poisson structure, given by the Poisson bracket

{/, g}vv = frfdjg + d'fdjg - VgdJ - frgdj.

Recall the notation (1.2.4). The notions of a Hamiltonian vector field, a Hamiltonian connection, a Hamil-

tonian form and others on the vertical momentum phase space VV*Q = V*VQ are introduced in accordance with the general formalism of Hamiltonian time-dependent mechanics, expressed in Section 5.2. In particular, any Hamiltonian form on VV*Q reads

Hy = Pidq1 + Pidq' — Hydt.

Since a Hamiltonian form is defined modulo exact forms and the function p{<? is globally defined on VV*Q, we will write a Hamiltonian form on the momentum phase space VVQ as

Hy = pidq1 — cfdpi — Ttydt. (5.11.2)

The corresponding Hamilton equations read

V = ql = d'Hv, (5.11.3a)

li =Pti = -diHv, (5.11.3b)

f = 4 = d'Hv, (5.11.3c)

7i =Pti = -diHv, (5.11.3d)

where

7 = dt + J% + "fid* + j % + jid*


is a Hamiltonian connection on the momentum phase space VV'Q —> R. We have the following relationship between Hamiltonian formalism on the mo

mentum phase space V'Q and that on VV'Q.

PROPOSITION 5.11.1. Let 7# be a Hamiltonian connection on the ordinary momentum phase space V*Q —► R for a Hamiltonian form

H = pidq* - Hdt. (5.11.4)

Then the connection VjH (1.6.12) on the vertical momentum phase space VV'Q —► R is a Hamiltonian connection for the Hamiltonian form

Hv = Ptdq' — q'dpi — dyTidt, (5.11.5)

dvH = (q% + pj&yH,

called the vertical extension of the Hamiltonian form H. D

Proof. A direct computation shows that, given a Hamiltonian connection

lH = dt + 'yidi + 'yid',

f = d'H, 7« = -diH,

on the fibre bundle V'Q —> R, the connection

V-TH = dt + j'di + 7idl + fdi + j{d\ 71 = dv-y', 7i = dvlu

on the fibre bundle VV'Q -» R satisfies the Hamilton equations (5.11.3a) - (5.11.3d) for the Hamiltonian form (5.11.5), i.e.,

y = titty = d'H, (5.11.6a) 7i = -mv = -diH, (5.11.6b) f = d'Hv = dvdln, (5.11.6c) 7i = -diHv = -dvdiH. (5.11.6d)

QED

In particular, given a splitting

H = p,-r + HT


of a Hamiltonian Tt with respect to a connection Y on Q —> R, there is the corresponding splitting

Hv = ftP - tfi-pidjF) + dvHT

of the Hamiltonian Hv (5.11.5) with respect to the connection T (5.2.6) on the fibre bundle V*Q -^ R.

It is easily seen that the Hamilton equations (5.11.6a) - (5.11.6b) for the Hamiltonian form Hv (5.11.5) are precisely the Hamilton equations (5.2.18a) - (5.2.18b) for the Hamiltonian form H.

Example 5.11.1. In order to clarify the physical meaning of the Hamilton equations (5.11.6c) - (5.11.6d), let us suppose that Q —> R is a vector bundle. Due to the splitting

VQ^Q®Q,

we can think of the vertical configuration space VQ as the configuration space of the initial mechanical system and its small linear deviations

3* + eg*, £ € R ,

where £ is a small parameter. Accordingly, the vertical momentum phase space

VVQ = VQ®VQ

can be seen in a similar way as the momentum phase space of the initial Hamiltonian mechanical system and its small linear deviations

qi + eq\ Pi + epi-

Let if be a Hamiltonian form on the momentum phase space V*Q of the initial Hamiltonian system, and let r{t) be a solution of the Hamilton equations (5.11.6a) - (5.11.6b). Let r(£) be a Jacobi field, i.e.,

r(t) + ef(t)

is also a solution of the same Hamilton equations modulo terms of order two in e. Then it is readily observed that the Jacobi field f(t) satisfies the Hamilton equations (5.11.6c) - (5.11.6d) for the Hamiltonian form Hv (5.11.5). Substituting a solution r(t) of the Hamilton equations (5.11.6a) - (5.11.6b) in the Hamilton equations (5.11.6c) - (5.11.6d), we obtain the system of homogeneous linear differential


equations for the Jacobi field F(t), which are actually the differential equations of linear deviations. •

Returning to the definition of a Hamiltonian form Hv (5.11.5), let us consider the vertical tangent bundle VT*Q of the cotangent bundle T*Q -» E. It is equipped with holonomic coordinates

{t,q',Pi,P,q',Pi,p)

and endowed with the canonical form

Ey = pdt + pidq1 — qldpi.

One may also utilize another canonical form

Ev + d{q'pi)

since the exact form d(<j*p,*) is globally defined. By very definition, a Hamiltonian form H (5.11.4) on the ordinary momentum phase space V*Q is the pull-back H = h'E of the Liouville form E (5.1.2) on the cotangent bundle T*Q by a section h of the fibre bundle T'Q -> V'Q. Then we have

Hv = (VhYEv,

where

Vh : VVQ -» VT'Q

is the vertical tangent map to h. We can also generalize Proposition 5.11.1 as follows.

PROPOSITION 5.11.2. Any connection 7 on the momentum phase space V'Q —» M gives rise to a Hamiltonian connection on the vertical momentum phase space VVQ - > R . D

Proof. Let us consider the Hamiltonian form

Hv = pijdq1 - fdt) - ^(dpi - -gdi) = pidgi - tfdpj - (pg* - tf-y^dt

on the vertical momentum phase space VV*Q. The corresponding Hamiltonian connection on VVQ —> R is given by the Hamilton equations (5.11.3a) - (5.11.3d) and reads

f = 7\ 7i = 7t, V-fcay-a'aS, 7i = -Pj-8i7 J+?39i7j. (5.11.7)


In particular, if 7 is a locally Hamiltonian connection on the fibre bundle V*Q —> R, i.e., satisfies the relations (5.2.4a) - (5.2.4c), then the Hamiltonian connection (5.11.7) is precisely the connection V7 (1.6.12). QED

It follows that every first order dynamic equation (5.2.20) on the momentum phase space VQ can be seen as the Hamilton equations (5.11.3a) - (5.11.3b) for a suitable Hamiltonian form on the vertical momentum phase space.

Example 5.11.2. The equations of a motion (5.2.21) of a point mass m subject to friction in Example 5.2.5 are the Hamilton equations (5.11.3a) - (5.11.3b) for the Hamiltonian

Wv =p{—p-\ q) m m

on the vertical momentum phase space. •

Similarly to the relationship between Hamiltonian formalisms on ordinary and vertical momentum phase spaces, that between Lagrangian formalisms on ordinary and vertical velocity phase spaces can also be considered.

Given a Lagrangian L = Cdt (4.8.1) on an ordinary velocity phase space JlQ, let us consider the vertical tangent morphism

VC : JlQ -* R x R

to the Lagrangian function C, and the Lagrangian

Ly = (pr2 ° V£)dt = Cydt, (5.11.8)

Cy=dyC= (fdj + q\d\)C,

on the vertical velocity phase space VJlQ. It is called the vertical extension of the Lagrangian L. The corresponding Lagrange equations read

(4 - dtdt)cv = 0, (5.11.9a)

{d, - dtdt)£v = 0. (5.11.9b)

The Lagrange equations (5.11.9a) are precisely the Lagrange equations for the initial Lagrangian L. If a configuration space Q —> M is a vector bundle, the Lagrange equations (5.11.9b), just as the Hamilton equations (5.11.6c) - (5.11.6d) do, describe small linear deviations of the initial mechanical system.


Let us turn now to the relationship between Lagrangian and Hamiltonian formalisms on the vertical configuration space.

Each Lagrangian Lv (5.11.8) on the vertical velocity phase space VJlQ yields the Legendre map

LV = VL:VJ1Q ^VV'Q, VQ

(5.11.10)

Pi = d\CV = TTi, Pi = dy^i, (5.11.11)

which is the vertical tangent map to the Legendre map L. Each Hamiltonian form Hv (5.11.5) on the vertical momentum phase space

VV*Q yields the Hamiltonian map

HV = VH : VV'Q -> VJXQ, VQ

(5.11.12)

qi = ffnv = am, q\ = dvd'H,

which is the vertical tangent map to the Hamiltonian map H.

PROPOSITION 5.11.3. If a Hamiltonian form H on the momentum phase space VQ is associated with a Lagrangian L on the velocity phase space JlQ, then its vertical extension Hv (5.11.5) on the vertical momentum phase space VV'Q is associated with the vertical extension Lv (5.11.8) of the Lagrangian L on the vertical velocity phase space V JXQ. D

Proof. By virtue of the relations (5.11.10) and (5.11.12), the condition (5.5.2a) for the Hamiltonian form Hv reads

VL o VH o VL = VL,

and follows at once from the condition (5.5.2a) for the Hamiltonian form H. The condition (5.5.2b) for the Hamiltonian form Hv, written in the coordinate form (5.5.5), follows immediately from the condition (5.5.5) for the Hamiltonian form H. QED

The equalities (5.11.11) show that, if a Lagrangian L on the velocity phase space JlQ is semiregular, so is its vertical extension Lv (5.11.8) on the vertical velocity phase space VJlQ. Therefore, Proposition 5.11.3 can also be formulated for a semiregular Lagrangian. In particular, if the relation (5.5.16) is fulfilled for H and L, it is true for the vertical extensions Hv and Ly-


Example 5.11.3. Let us consider the Hamiltonian counterpart of the constrained motion equation (4.11.15) in the case of a hyperregular Lagrangian and a non-holonomic constraint. By virtue of Proposition 5.11.2, it can be represented as the Hamilton equations (5.11.3a) - (5.11.3b) for a suitable Hamiltonian form on the vertical phase space VV*Q.

Let L be a hyperregular Lagrangian with a Riemannian mass metric m and S a codistribution on the velocity phase space JXQ, whose annihilator Ann (S) is an admissible non-holonomic constraint on JlQ. In the case of a hyperregular Lagrangian, Hamiltonian and Lagrangian formalisms of time-dependent mechanics are equivalent, and we have

7 H = J ' i o ^ o f f .

Let introduce the notation M , J = d'&>H. There are the relations

Mik(mkj o H) = 8}, mkj(Mik o I) = 6), TTlij = Tty.

It follows that M is a fibre metric in the vertical tangent bundle VQV'Q of the fibre bundle VQ ~* Q.

Given the above-mentioned codistribution S on JXQ, let us consider the pull-back codistribution H*S on V*Q spanned locally by the 1-forms

/?» = H*sa = (sg + s^dtd>n)dt + (s? + s'di&HW + s°MijdPj = ffidt + (3fdq' + $aidPi.

This codistribution defines a non-holonomic constraint on the momentum phase space V*Q.

Given a Hamiltonian connection 7// (5.2.17), let us find its splitting

1H = 7 + 0 (5.11.13)

where 7 is a connection on V*Q —► R which satisfies the condition

7 C Ann( iTS) . (5.11.14)

The connection 7 (5.11.14) obviously defines a first order dynamic equation on the momentum phase space V*Q, which is compatible with the non-holonomic constraint H*S. The decomposition (5.11.13) is not unique. Let us construct it as follows.


Given a Hamiltonian connection 7//, we consider the codistribution SH on V'Q, spanned locally by the 1-forms dq' - i\jdt. Its annihilator Ann (SH) is an affine subbundle of the affine jet bundle JlV'Q -> V'Q modelled over the vertical tangent bundle VQV'Q. The Hamiltonian connection j H is obviously a section of this subbundle. Let us consider the intersection

W = A n n ( S H ) n A n n ( # * S ) . (5.11.15)

LEMMA 5.11.4. The intersection W (5.11.15) is an affine bundle over V'Q, modelled over the vector bundle

W = VQV*Q nAnn(H'S).

a

Proof. The intersection W consists of elements v of VQV'Q which fulfill the conditions

v{pa' = 0.

Since the non-holonomic constraint S is admissible and the matrix M%i is non-degenerate, every fibre of W is of dimension m — n, i.e., W is a vector bundle, while W is an affine bundle. QED

Then, using the fibre metric M in VQV'Q, we obtain the splitting

VQV'Q = W e V,

where V is the orthocomplement of W, and the associated splitting

Ann (SH) = W ® V.

The corresponding decomposition (5.11.13) reads

7 = lH-MabMtJpaipb(lH)d>, (5.11.16)

where Mat, is the inverse matrix of

Mab = fripMMij.


The splitting (5.11.16) is the Hamiltonian counterpart of the splitting (4.11.15). We have the relations

~ab = Mab Q ^ /3"(7tf) = s Q ( a ) ° H,

and as a consequence

7 = J ^ o f o f f .

Note that the above procedure can be extended in a straightforward manner to any standard Newtonian system seen as a Lagrangian system with the Lagrangian (4.9.19) and an external force. Following this procedure, one may also study a non-holonomic Hamiltonian system, without appealing to its Lagrangian counterpart.

The connection (5.11.16) defines the system of first order dynamic equations

«{= d*n, Pu = -W - M o 6 M y ^ /3»( 7 f f ) (5.11.17)

on the momentum phase space V*Q, which are not Hamilton equations. Nevertheless, in accordance with Proposition 5.11.2, one can restate the constrained motion equations (5.11.17) as the Hamilton equations (5.11.6a) - (5.11.6b) for the Hamiltonian form

Hv = fad? - tfdpi - dvHdt - qiMabMij$a:i(3b('yH)dt

on the vertical momentum phase space VV'Q. Its last term can be written in brief as (—<9v/J?9J$7). To exclude the remaining Hamilton equations (5.11.6c) - (5.11.6d), one can choose their solutions q' = pi = 0. •

Remark 5.11.4. Let Hy be a Hamiltonian form (5.11.2) on the vertical momentum phase space VV'Q. Similarly to the Lagrangian (5.7.1), let us consider the Lagrangian

LH = Piq\ - qlPu - 'Hv (5.11.18)

on the first order jet manifold JlVV*Q of the fibre bundle VV*Q —> M. It is readily observed that its Lagrange equations are precisely the Hamilton equations (5.11.3a) - (5.11.3d) for the Hamiltonian form Hy- In particular, let H be a Hamiltonian form on an ordinary momentum phase space VQ and "Hv = dy'H- In this case, the Lagrangian (5.11.18) reads

LH=Pi(qi-dlH)-qi(Pti + diH).


It is easily seen that this Lagrangian vanishes on solutions of the Hamilton equations for the Hamiltonian form H. For this reason, it plays a prominent role for the functional integral formulation of mechanics [65, 66] •

5.12 Appendix. Time-reparametrized mechanics

We have assumed above that the base M of a configuration space of time-dependent mechanics is parameterized by the standard coordinate t with the transition functions t —> t' = t+ const. Here, we consider an arbitrary reparametrization of time

t -» a = m (5.12.1)

which is discussed in some models of quantum mechanics [75, 153, 154]. In this case, K is not only an affine space as before, but a 1-dimensional manifold. Then the standard vector field dt and the standard 1-form dt on R are not invariant. Therefore, we should follow the polysymplectic Hamiltonian formalism of Section 3.8. Nevertheless, Hamiltonian formalism of time-reparametrized mechanics possesses some peculiarities due to R.

1. There exists the invariant tangent-valued 1-form

eR = dt® dt

on the base R of a configuration space of time-reparametrized mechanics.

2. The velocity phase space J1Q of time-reparametrized mechanics is no more an affine subbundle of the tangent bundle TQ under time reparametrizations. Nevertheless, the momentum phase space U —> Q (3.8.1) is isomorphic to the vertical cotangent bundle V*Q —> Q. It follows that the momentum phase space of time-reparametrized mechanics is provided with the canonical Poisson structure (5.1.5). Moreover, this Poisson structure is invariant under time reparametrization (5.12.1) which, consequently, is a canonical transformation.

3. The momentum phase space V*Q is endowed with the canonical polysymplectic form

A = dpi A dq' Adt® dt.

Then the notions of a Hamiltonian connection and a Hamiltonian form are the repetitions of those in Section 3.8. At the same time, since the homogeneous Legendre

5.12. APPENDIX. TIME-REPARAMETRIZED MECHANICS 297

bundle (2.9.3) of time-reparametrized mechanics is the cotangent bundle T*Q of Q, Hamiltonian forms and Hamilton equations of time-reparametrized mechanics have the same form as those in time-dependent mechanics. The difference is only that a Hamiltonian function Tir in the splitting (5.2.12) is a density, but not a function under the transformations (5.12.1).

4. Since Lagrangians and Hamiltonians of time-reparametrized mechanics are densities under the transformations (5.12.1), one should introduce a volume element on the base M in order to construct them in explicit form. A key problem of models with time reparametrization lies in the fact that the volume dt on the time axis M is not invariant under time reparametrization transformations. Another problem is concerned with mass tensors which also fail to be invariant under time reparametrizations.

Chapter 6

Relativistic mechanics

Let us note on the following two main peculiarities of relativistic mechanics in comparison with non-relativistic one.

• A configuration space Q of relativistic mechanics has no fibration Q —> R in general or, at least, it has no preferable fibration over K. Therefore, one cannot use the familiar formalism of jets of sections of fibre bundles, but must generalize it for jets of submanifolds.

• Hamiltonian relativistic mechanics is described as an autonomous constraint Dirac system on the hyperboloids of relativistic momenta in the momentum phase space T*Q.

Relativistic mechanics can be formulated on an arbitrary configuration space Q. If Q is a 4-dimensional manifold provided with a pseudo-Riemannian metric g, we come to relativistic mechanics in the presence of a gravitational field g. If Q = R4

and g is the Minkowski metric, this is the case of Special Relativity.

6.1 Jets of submanifolds

We start from the general notion of jets of submanifolds [57, 104].

DEFINITION 6.1.1. Let Z be a manifold of dimension m + n. The first order jet manifold J\Z of n-dimensional submanifolds of a manifold Z comprises the equivalence classes [S]l

z, z € Z, of n-dimensional imbedded submanifolds of Z which

299

300 CHAPTER 6. RELATIVISTIC MECHANICS

pass through z € Z and which are tangent to each other at z. There is the natural fibration J\Z -> Z. D

Remark 6.1.1. In fact, the definition of the jet [S}1 of submanifolds involves only local properties of submanifolds around the point z € Z. It is easily extended to higher order jets of submanifolds. By definition, J°Z = Z. •

The set of jets J\Z is provided with a manifold structure as follows. Let Y —> X be an (m + n)-dimensional fibre bundle over an n-dimensional base

X and let $ be an imbedding of Y into Z. Then there is the natural injection of the jet manifold JlY of the fibre bundle Y —> X into J\Z such that

7 1 * : JlY -> J\Z, (6.1.1)

J l S »-» [Sjsfgfa;))! S = I m ( $ o s ) ,

where s are sections of Y —» X.

PROPOSITION 6.1.2. The injection (6.1.1) defines a chart on J^Z. Such charts cover the set J^Z, and transition functions between these charts are differentiable. They provide the set J\Z with the structure of a finite-dimensional manifold. □

Proof. The proof is based on the fact that, given a submanifold S C Z which belongs to the jet [S\l, there exists a neighbourhood Uz of the point z and the tubular neighbourhood Us of Uz D S so that the fibration

ua->u,ns takes place [109]. This means that every jet [S]l belongs to a chart of the above-mentioned type. We will describe these charts and the corresponding transition functions in explicit form. QED

It is convenient to use the following coordinate atlases of the jet manifold J\Z of n-dimensional submanifolds of Z.

Let Z be equipped with a coordinate atlas

Mz*)}, A= 1,... ,n + m. (6.1.2)

6.1. JETS OF SUBMANIFOLDS 301

Though J°Z, by definition, is diffeomorphic to Z, let us provide J°Z with the atlas where every chart (U; zA) on a domain U C Z is replaced with the

( n + m\ _ (n + m)\ m J n\m\

charts on the same domain U which correspond to different partitions of the collection (z1 • • • zA) in the collections of n and m coordinates, denoted by

(zV), A = 1, , . . , !» , i = 1 , . . . ,m. (6.1.3)

The transition functions between the coordinate charts (6.1.3) of J°Z, associated with the same coordinate chart (6.1.2) of Z, are reduced simply to an exchange between coordinates xx and yl. Transition functions between arbitrary coordinate charts (6.1.3) of the manifold J°Z read

2A=ffV*,tf0. yi = fi(x",yj), (6.1.4)

x* = g*{x?,yl)1 f = fi^y3). Given the coordinate atlas (6.1.3) of the manifold J°Z, the jet manifold J\Z is provided with the adapted coordinates

(x\y\yi), X = l , . . . , n , i = 1 , . . . , » ? i . (6.1.5)

R e m a r k 6.1.2. If S C Z is an imbedded submanifold which belongs to the jet [S]l, there exists an open neighbourhood Uz of z £ Z and a coordinate chart (xx,yl) (6.1.3) which covers Uz so that SfMJz\s given by the coordinate relations

y{ = 5{(xA)

and

(x\y\yi)([S]l) = (xx(z),Si(^(z)),dxSi(x"(z))).

•

Using the operators of total derivatives

d\ = dx + y\di,


one can obtain the transition functions of the coordinates (6.1.5) under coordinate transformations (6.1.4) in explicit form. Given coordinate transformations (6.1.4), one can easily find that

d~x = [d~,ga{x^^)]dxa. (6.1.6)

Then we have

t-Kw+K&ypM] ( ^ + ^ ) / w ) - (6.1.7)

It is readily observed that the transition functions (1.3.1) are a particular case of the coordinate transformations (6.1.7) when the transition functions ga (6.1.4) are independent of coordinates y1. Note that, in contrast with (1.3.1), the coordinate transformations (6.1.7) are not affine. It follows that the fibration J\Z —> Z is not an affine bundle.

There is one-to-one correspondence

A :[5]^x^+y-([5U)a,) (6.1.8)

between the jets [S]\ at a point z € Z and n-dimensional vector subspaces of the tangent space TZZ. It follows that the jet bundle J\Z -> Z has the structure group

GL(n, m; M) C GL(m + n, R)

of linear transformations of the vector space Mm+n which preserve its subspace Rn. Its typical fibre is the Grassman manifold

<d{m + n,n) = GL(n + m\ R)/GL{n, m; R)

of n-dimensional subspaces of the vector space Rm + n . Thus, the jet bundle J\Z —> Z is a Grassman bundle [45, 70]. In particular, if n = 1, the fibre coordinates y'0 on J\Z —> Z with the transition functions (6.1.7) are precisely the standard coordinates of the projective space RP m .

Example 6.1.3. Let Y —> X be an (m + n)-dimensional fibre bundle over an n-dimensional base X, and JlY the first order jet manifold of its sections. Let J„Y be the first order jet manifold of n-dimensional subbundles of Y. Then the injection JlY <-> J^Y (6.1.1) is an affine subbundle of the jet bundle J\Y -+ Y. Its fibre at a point y &Y consists of the n-dimensional subspaces of the tangent space TyY whose intersections with the vertical subspace VVY of TyY are the zero vector. •

6.2. RELATIVISTIC VELOCITY AND MOMENTUM PHASE SPACES 303

Remark 6.1.4. Generalizing the notion of a connection on a bundle, one may think of global sections of the jet bundle J\Z - > Z a s being preconnections on the manifold Z [134]. By virtue of the well-known theorem [80], if such a preconnection T exists, its image T(Z) in the tangent bundle TZ —> Z by the correspondence A (6.1.8) is a vector subbundle of TZ with the structure group GL(n, K). The quotient TZjY(Z) is also a vector subbundle with the structure group GL(m, M). Thus, we have the decomposition

TZ = T{Z) e TZ/T(Z)

of the tangent bundle TZ, which can be treated as a horizontal splitting with respect to the preconnection I\ However, it should be emphasized that, since J\Z —► Z is not an affine bundle, preconnections do not make an affine space. Moreover, it may happen that a preconnection does not exist on a manifold Z. •

6.2 Relativistic velocity and momentum phase spaces

Let us now turn to the case of relativistic mechanics, where Z = Q is an (m + 1)-dimensional configuration space which has not necessarily a fibration or a preferable fibration over the time axis E. We call Q a reiativistic configuration space in comparison with a non-relativistic configuration space Q endowed with a fibration Q —> M. This configuration space is provided with the coordinate atlas (6.1.3) together with the transition functions (6.1.4), i.e.,

(q°,q% i = 1 , . . . , m,

«°-«W), 9 W ( « V ) - (6.2.1)

The coordinates q° in different charts of Z play the role of a temporal coordinate. Note that, given a coordinate chart (U; q°, q'), we have a local fibre bundle

U 5 (q°,qi)^q°£ QCR (6.2.2)

which can be treated as a configuration space of a local non-relativistic mechanical system.

The velocity phase space of relativistic mechanics on the configuration space Q is the first order jet manifold J\Q of 1-dimensional subbundles of Q. It is endowed


with the coordinates (q°, q\qo) (6.1.5). Their transition functions are obtained as follows.

Given coordinate transformations (6.2.1), the total derivative (6.1.6) reads

d~0 djb(q°)dqo [dq°+q°dq«) dqO.

In accordance with the relation (6.1.7), we have

—I WK°(?) \d<p+%d<?) The solution of this equation is

9o I (8? >9?\ [dqO + 9o dq" ) ■

(6.2.3)

A glance at this transformation law shows that the velocity phase space J\Z —* Z of relativistic mechanics is really a projective bundle.

Example 6.2.1. Let consider the configuration space Q = M4, provided with the Cartesian coordinates

(«V). i 1,2,3.

This is the case of Special Relativity. For instance, let

9° g°cha — q1sha,

f -<7°sha + g'cha, (6.2.4)

<?'* q2'3-

be a Lorentz transformation of the plane (q°, q1). Substituting these expressions in the formula (6.2.3), we obtain

ql —sha + ggcha cha — q^sha

^■3

9o

2,3 go

cha — go sha This is precisely the transformation law of 3-velocities in Special Relativity if we put

cha 1 sha v

6.2. RELATIVISTIC VELOCITY AND MOMENTUM PHASE SPACES 305

where v is the velocity of a reference frame moving along the axis q1. •

Thus, one can think of the velocity phase space J\Q of relativistic mechanics as the space of 3-velocities v of a relativistic system. For the sake of convenience, we will call J\Q the 3-velocity phase space, though the dimension of Q is not equal necessarily to 3 + 1. Given a coordinate chart {U;q°,ql), 3-velocities v = (vl

0) of a relativistic system can be seen as velocities of a local non-relativistic system (6.2.2) with respect to the corresponding local reference frame T = dt on U. However, the notion of a reference frame in non-relativistic mechanics has no meaning in relativistic mechanics since the relativistic transformations ql

0 —> <$ and P —> P 1 are not affine and the relative velocity gj — P is not maintained under these transformations.

To introduce the relativistic velocities, let us consider the tangent bundle TQ of the configurations space Q. It is equipped with the holonomic coordinates

(9°,g\9°,<f)-In accordance with (6.1.8), there is the multivalued morphism A from the 3-velocity phase space J\Q to the tangent bundle TQ when a point (q°,q',q'0) € J\Q corresponds to a line

A^j^teV^JKTQ (•6.2.5)

in the tangent space to Q at the point (q°,q')- Conversely, there is the morphism

Q TQ-+JIQ,

Qo°Q 9°' (6.2.6)

such that

go X IdJlQ.

Indeed, it is readily observed that q'0 and ql/q° have the same transformation laws. It should be emphasized that, though the expressions (6.2.5) and (6.2.6) are singular at 5° = 0, this point belongs to another coordinate chart, and the morphisms A and g are well defined.

Thus, one can think of the tangent bundle TQ as being the space of relativistic velocities or 4-velocities of a relativistic system. It is called the 4-velocity phase space.


Remark 6.2.2. Note that, with respect to the universal unit system, the elements of TQ are not physically dimensionless, in contrast with the physical relativistic velocities. •

Since the morphism A of J\Q onto TQ is multivalued and the converse morphism (6.2.6) is a surjection, one may try to find a subbundle W of the tangent bundle TQ such that

Q:W-J%

is an injection. Let us assume that Q is oriented and endowed with a pseudo-Riemannian metric g with the signature (+, — . . . — ) . The pair (Q,g) is called a relativistic system. This metric g defines the subbundle of veJocity hyperboloids

Wg = {qx 6 TQ : g^q)?? = 1}. (6.2.7)

Of course, Wg is neither a vector nor an affine subbundle of TQ. Let Q be time-oriented with respect to the pseudo-Riemannian metric g. By definition, this means that Wg is a disjoint union of two connected subbundles W+ and W~ [77, 155]. Then it is readily observed that the restriction of the morphism g (6.2.6) to each of these subbundles is an injection into J\Q.

Let us consider the image of this injection in a fibre of J\Q at a point q £ Q. There are local coordinates (q°, q') on a neighbourhood of q € Q such that the pseudo-Riemannian metric g(q) at q comes to the Minkowski metric

g(q) =?7 = d i a g ( l , - l , ■ • • , - ! ) . (6.2.8)

With respect to these coordinates, the velocity hyperboloid Wq C TqQ is given by the equation

(90)2 - E(<n2 = i-i

This is the union of the subset W+, where q° > 0, and W~, where q° < 0. Restricted to W+, the morphism (6.2.6) takes the familiar form of the relations between 3- and 4-velocities

yfl ~ Ulb)2

\A-n<?i)2

6.3. KELATTVISTIC DYNAMICS 307

in Special Relativity. The image of each of the hyperboloids W* in the 3-velocity phase space J{Q by the morphism (6.2.6) is the open ball

B * o ) 2 < 1- (6.2.9) i

This relation means that 3-velocities of a relativists system (Q,g) are bounded in accordance with the relativistic principle.

Let us turn now to the momentum phase space of relativistic mechanics. Given a chart of the relativistic configuration space Q, the homogeneous Legendre bundle corresponding to the local non-relativistic system (6.2.2) is the cotangent bundle T'Q. This fact motivates us to choose T'Q as the reJativistic momentum phase space, equipped with the holonomic coordinates ($\pA). The cotangent bundle T'Q is endowed with the canonical symplectic form

n = 4pMAdgM, (6.2.10)

called the reJativistic symplectic form. Given a pseudo-Riemannian metric g on the relativistic configuration space Q,

we have the corresponding isomorphism between relativistic velocity and momentum phase spaces TQ and T'Q. Prom the physical viewpoint, this isomorphism, however, fails to be correct. With respect to the universal unit system, a metric tensor g is dimensionless, whereas 4-velocities and relativistic momenta have different physical dimension. The latter are of dimension [length]"1. Therefore, we should use a mass metric in order to perform the above-mentioned isomorphism.

6.3 Relativistic dynamics

In a straightforward manner, Lagrangian formalism fails to be appropriate to relativistic mechanics because a Lagrangian

L = Cdq°, (6.3.1)

by very definition, is defined only locally on a coordinate chart (6.1.5) of the 3-velocity phase space J\Q. For instance, the Lagrangian

Lm = -mil - E(9J)2V (6.3.2)


of a free relativistic point mass m in Special Relativity, can be defined on each coordinate chart of J\Q, but it is not maintained in a straightforward manner by the Lorentz transformations (6.2.4) because of the term dq°. Given a motion q* = cl(q°) with respect to a coordinate chart (q°, q'), its Lorentz transformation (6.2.4) reads

cf> = q°cha-c1(q0)aha, c1(9°) = -g0(g0)sha + c1(<?

0(g°))cha,

^ V ) = c2'V(9°)). Then we have the Lorentz invariance of the pull-back c*Lm of the Lagrangian (6.3.2), i.e.,

c Lm — c Lm

where dg0 = d?o(5°)dg° and dqo is the total derivative. Therefore, we concentrate our consideration on Hamiltonian relativistic mechan

ics. Let L be a local regular Lagrangian (6.3.1) on a coordinate chart (q°, qx) of the

3-velocity phase space J\Q- In accordance with Remark 4.8.9, it defines a local Dirac constraint system on the relativistic momentum phase space T*Q in the case of a zero Hamiltonian and the primary constraint space given by

dq'0

dC dq'o

(6.3.3)

Since the Lagrangian L is regular, the equations (6.3.3) are solvable for

Po = Po(<?",Pi)-

Then a solution of the Lagrange equations is an integral curve of the vector field

tf = d0 + Pdi + Ad{

on the primary constraint space (6.3.3), which fulfills the equation

tfjnN = 0, tfi = diPo, * = d'p0,

where fi^ is the restriction of the relativistic symplectic form (6.2.10) to the constraint space (6.3.3). For instance, the Lagrangian (6.3.2) (which is regular on the

6.3. RELATIVISTIC DYNAMICS 309

ball (6.2.9)) defines a Dirac constraint system on T*Q in the case of a zero Hamil-tonian and the primary constraint space

2 V ^ 2 2

i (6.3.4)

Therefore, let us describe a relativistic system (Q, g) as an autonomous Hamil-tonian system on the symplectic manifold T*Q, characterized by a Hamiltonian

H : T'Q -» R, (6.3.5)

called a relativistic Hamiltonian.

Remark 6.3.1. It should be emphasized that the coordinate —po, but not the relativistic Hamiltonian H, plays the role of a relativistic energy function. •

Any relativistic Hamiltonian H (6.3.5) defines the Hamiltonian map

H : T*Q -* TQ, q» = d*H, (6.3.6)

(3.3.4) over Q from the relativistic momentum space T*Q to the 4-velocity phase space TQ. Since the 4-velocities of a relativistic system live in the velocity hyper-boloids (6.2.7), we have the constraint subspace

N = WWg, (6.3.7) g ^ ^ H ^ H = 1,

of the relativistic momentum phase space T*Q. It follows that the relativistic system (Q, g) can be described as an autonomous Dirac constraint system on the primary constraint space N (6.3.7) [154]. Its solutions are integral curves of the Hamiltonian vector field

u = u'di + Uid1

on N C T*Q, that obeys the Hamilton equation

u\i*NQ = -i*NdH. (6.3.8)

Its solution does not necessarily exist.


Example 6.3.2. The relativistic Hamiltonian of a free relativistic point mass in Special Relativity is

H = -^V>P„ (6.3.9)

where r) is the Minkowski metric (6.2.8), while the constraint space N (6.3.7) is given by the equation (6.3.4). It is readily observed that the restriction of the Hamiltonian (6.3.9) to this constraint space is a constant function, i.e.,

i"NdH = 0.

Then the Hamilton equation (6.3.8) takes the form

u\i*NSl = 0. (6.3.10)

We obtain its solution

pi = Ui = 0, ,t t u°V'kpk

<? = u = I . = sjm? - v}kPjPk

and then the familiar expression for 3-velocities

i _ mriikpk

^Jm2 - n>kpjpk

Pi = const.

•

Example 6.3.3. Let us consider a point electric charge e in the Minkowski space in the presence of an electromagnetic potential Ax- Its relativistic Hamiltonian reads

H = - ^ " " ( J V - c4 . ) ( f t " eA»)<

while the constraint space TV (6.3.7) is

v""{Pn - eAll){pv -eAv) = m2.

As in the previous Example, the Hamilton equation (6.3.8) takes the form (6.3.10). Its solution is

Pk (6.3.11) uk = u^dkAp,

qk fc== u°rfk(pj - eAj)

Jm2 - rfUjpi - eAi){Pj - eAj) (6.3.12)

6.4. RELATIVISTIC GEODESIC EQUATIONS 311

The equality (6.3.12) leads to the usual expression for the 3-velocities

Pk I : T -rt-k

Substituting this expression in the equality (6.3.11), we obtain the familiar equation of motion of a relativistic charge in an electromagnetic field. •

Example 6.3.4. The relativistic Hamiltonian for a point mass m in a gravitational field g on a 4-dimensional manifold Q reads

H -L9r{-q)*>*» while the constraint space N (6.3.7) is

9*VVvP» = m2-

As in previous Examples, the equation (6.3.8) takes the form (6.3.10). •

Examples (6.3.2) - (6.3.4) illustrate the relativistic systems where the restriction of a relativistic Hamiltonian to the constraint space N (6.3.7) is a constant function. In this case, the Hamilton equation takes the form (6.3.10). In accordance with Proposition 3.4.2, this equation has a solution everywhere on the primary constraint space TV (6.3.7).

Remark 6.3.5. Given a symplectic manifold (T*Q,Q) and its submanifold N (6.3.7), one may apply the reduction procedure described in Section 2.6. For the relativistic systems in Examples 6.3.2 - 6.3.4, this procedure is not, however, useful since the pull-back Hamiltonian i*NJi on the constraint space N is a constant function and the associated Hamiltonian vector fields belong to the kernel of the presymplectic form i*NQ on N. *

6.4 Relativistic geodesic equations

Let H be a relativistic Hamiltonian on the relativistic momentum phase space T*Q and

in d'Hdi - diHd* (6.4.1)


the corresponding Hamiltonian vector field whose integral curves are solutions of the Hamilton equation

frJfi = -dH.

If the Hamiltonian map H (6.3.6) is a diffeomorphism, the Hamiltonian vector field (6.4.1) yields the holonomic vector field

e r H o ^ o H - ^ ^ + f i (6.4.2)

e (dad^H^H - <9ad"HdQH) o H 1 ,

Dn the 4-velocity space 1 Q, where

(«*,«*, 4", $*)oTH {<f, d»H, <f, dad»¥Lqa + 0°9"Hpo)

is the tangent morphism to the Hamiltonian map H. The holonomic vector field (6.4.2) defines a conservative second order dynamic equation

q» (dad»ndan - dQd"HdQH) o H 1 , (6.4.3)

called the relativistic dynamic equation, on the relativistic configuration space Q (see Definition 3.1.2).

For instance, the relativistic dynamic equation (6.4.3) for a point electric charge in Example 6.3.3 takes the well-known form

q» = rf"qxF^ (6.4.4)

where F is the electromagnetic strength (4.9.10). The relativistic dynamic equation (6.4.3) for a point mass m in a gravitational field g in Example 6.3.4 reads

r = {AM,}<ZV, (6.4.5)

where

{A" ,} —^9v'a{Q\9oLV + d*9a\ - dag\u)

are the Christoffel symbols of the pseudo-Riemannian metric g. The equations (6.4.4) and (6.4.5) exemplify relativistic dynamic equations which

are geodesic equations

<r W.tfV (6.4.6)


with respect to a connection

K = dqx® {dx + Kfa) (6.4.7)

on the tangent bundle TQ —* Q (see Definition 3.1.6). We call (6.4.6) the relativistic geodesic equation. For instance, a connection K in the equation (6.4.5), is the Levi-Civita connection of the pseudo-Riemannian metric g. In the equation (6.4.4), K is the zero Levi-Civita connection of the Minkowski metric plus the soldering form

o = rrFvXd(?®dll. (6.4.8)

We say that a relativistic geodesic equation (6.4.6) on the 4-velocity phase space TQ describes a relativistic system (Q, g) if its geodesic vector field does not leave the subbundle of velocity hyperboloids (6.2.7). It suffices to require that the condition

(dxg^q" + 2g^Kxl)qxq" = 0 (6.4.9)

holds for all tangent vectors which belong to Wg (6.2.7). Obviously, the Levi-Civita connection {A^I/} of the metric g fulfills the condition (6.4.9). Any connection K on TQ —» Q can be written as

^ = {AM?" + *AV,? A ) ,

where

a = a$dqx ® dx

is a soldering form. Then the condition (6.4.9) takes the form

g ^ g g q V = Q- (6.4.10)

It is readily observed that the soldering form (6.4.8) in the equation (6.4.4) obeys this condition for the Minkowski metric r).

Let now compare relativistic and non-relativistic geodesic equations. In physical applications, one usually thinks of non-relativistic mechanics as being an approximation of small velocities of relativistic theory. At the same time, the 3-velocities in mathematical formalism of non-relativistic mechanics are not bounded. It has long been recognized that the relation between the mathematical schemes of relativistic and non-relativistic mechanics is not trivial.


Let a relativistic configuration space Q admit a fibration Q —> R, where M is a time axis. One can think of the fibre bundle Q —> K as being a configuration space of a non-relativistic mechanical system.

In order to compare relativistic and non-relativistic dynamics, one should consider pseudo-Riemannian metric on TQ, compatible with the fibration Q —» R. Note that M is a time of non-relativistic mechanics. It is one and the same for all non-relativistic observers. In the framework of a relativistic theory, this time can be seen as a cosmological time. Given a fibration Q —* E, a pseudo-Riemannian metric on the tangent bundle TQ is said to be admissible if it is defined by a pair (gR, T) of a Riemannian metric on Q and a non-relativistic reference frame T, i.e.,

2r®r R 9=Jf~f~9 '

(6.4.11)

I r |2= s* FT* = j ^ r , in accordance with the well-known theorem [77]. The vector field T is a time-like vector relative to the pseudo-Riemannian metric g (6.4.11), but not with respect to other admissible pseudo-Riemannian metrics in general.

There is the canonical imbedding (4.1.4) of the velocity phase space JlQ of non-relativistic mechanics into the affine subbundle

4° = 1, ** = «& (6.4.12)

of the 4-velocity phase space TQ. Then one can think of (6.4.12) as the 4-velocities of a non-relativistic system. The relation (6.4.12) differs from the familiar relation (6.2.6) between 4- and 3-velocities of a relativistic system. In particular, the temporal component q° of 4-velocities of a non-relativistic system equals 1 (relative to the universal unit system). It follows that the 4-velocities of relativistic and non-relativistic systems occupy different subbundles of the 4-velocity space TQ. Moreover, Theorem 4.4.2 shows that both relativistic and non-relativistic equations of motion can be seen as the geodesic equations on the same tangent bundle TQ. The difference between them lies in the fact that their solutions live in the different subbundles (6.2.7) and (6.4.12) of TQ. At the same time, relativistic equations, expressed in the 3-velocities q'/q° of a relativistic system, tend exactly to the non-relativistic equations on the subbundle (6.4.12) when q° —» 1, floo —* 1, ie . , where non-relativistic mechanics and the non-relativistic approximation of a relativistic theory only coincide.


Let (q°, q') be a non-relativistic reference frame on Q compatible with the fibra-tion of Q —» R. Given a non-relativistic geodesic equation (4.4.7), we will say that the relativistic geodesic equation (6.4.4) is the relativization of (4.4.7) if the spatial parts of these equations are the same.

In accordance with Lemma (4.5.5), any relativistic geodesic equation with respect to a connection K (6.4.7) is a relativization of a non-relativistic geodesic equation with respect to the connection

K = dqx®{dx + {K\-TiKi)di),

where P = 0 is the connection corresponding to the reference frame (q°,q'). Of course, for different reference frames, we have different non-relativistic limits of the same relativistic equation. The converse procedure is more intricate.

Example 4.9.3 shows that Lagrange equations for any non-relativistic quadratic Lagrangian (4.9.11) admit a relativization with respect to a Lagrangian frame connection. It follows that inertial forces do not admit relativization. Conversely, let us consider a geodesic motion

r = {AM?V (6.4.13)

in the presence of a pseudo-Riemannian metric g on a relativistic configuration space Q. Let (q0,^) be local hyperbolic coordinates such that poo = 1, 9ot = 0. These coordinates make up a non-relativistic reference frame for a local fibration Q —* K. Then, by virtue of Lemma (4.5.5), the equation (6.4.13) has the non-relativistic limit (4.9.18) which is the Lagrange equations for the Lagrangian (4.9.16) where $ = 0. This Lagrangian describes a free non-relativistic mechanical system with the mass tensor ffiij = —gij.

In view of Proposition 4.4.5, the relativization procedure, however, fails to be unique. Therefore, the relativization (4.9.12) of an arbitrary quadratic Lagrangian (4.9.11) may lead to confusion. In fact, this relativization procedure is appropriate to a gravitational Lagrangian (4.9.15), where <jf is a gravitational potential. An arbitrary quadratic dynamic equation can be written in the form

?So = -(m-1)* Wflfoff + W t f . 9o° = 1-

where {\k^} are the Christoffel symbols of some pseudo-Riemannian metric g, whose spatial part is the mass tensor (—rn^), while

WteS+W) (6.4.14)


is an external force. With respect to the coordinates where <?oi = 0, one may construct the relativistic equation

9* = { A K V + ^ g \ (6.4.15)

where the soldering form a must fulfill the condition (6.4.10). It takes place only if

QikVj + 9ijVk = 0,

i.e., the external force (6.4.14) is the Lorentz-type force plus some potential one. Then, we have

ffg = 0, 4 = -</%>&&, 4 = K-The relativization (6.4.15) exhausts almost all familiar examples. To complete

our exposition, we will point out also another relativization procedure. Let a non-relativistic equation ^'(x^) be a spatial part of a 4-vector £x in the Minkowski space. Then one can write the relativistic equation

ax=e - va0e<iaqx- (6.4.16)

This is the case, e.g., for the relativistic hydrodynamics in gravitation theory. However, the non-relativistic limit q° = 1 of (6.4.16) does not coincide with the initial non-relativistic equation. There are also other variants of relativistic hydrodynamic equations [107].

Appendix A

Geometry of B R S T mechanics

As is well-known, the BRST formalism is one of the corner stones of contemporary quantum field theory. This formalism has been extended to mechanics [5, 65, 66, 140, 141]. One can think of BRST mechanics as being a particular case of supersymmetric mechanics [3, 42, 108], where supercharges are the BRST charges, or mechanics on supermanifolds [4, 33, 41, 95, 165]. We aim to show that Hamiltonian time-dependent mechanics on a vertical configuration space VV'Q admits the natural extension to BRST mechanics on graded manifolds.

We start from the naive BRST extension of Hamiltonian mechanics where the odd variables are introduced in a formal way.

Let the configuration space Q —> K be a vector bundle. Besides the familiar coordinates (t, ql,Pi, ql,pi) on the vertical configuration space VV'Q, let us consider additionally the odd variables

£ i Cji ^ » Q (A.l)

which are assumed to be anticommutative. Following the physical terminology, we will call c and c (A.l) the ghosts and antighosts, respectively. Let & and c1 with respect to the indices i have the same transition functions as the linear coordinates qx, while c, and Cj have the transition functions of the coordinates pi; i.e.,

c" c, dqj

c" ~< dqi _ dq'iCj'

(A.2)

The formal products of the odd variables (A.l) make up the algebra of polynomials over the ring of complex functions on VV*Q. Provided with the unit element, this is the Z2-graded ring V(c,c) of polynomials of the odd variables. One may say

317

318 APPENDIX A. GEOMETRY OF BRST MECHANICS

that such a polynomial is globally defined if it is invariant under the transformations (A.2). We will follow the convention where the antighosts c are written on the left of the ghosts c.

For the sake of convenience, we will use the compact notation yA for both even and odd variables. Put {y} = 0 for the even variables and {y} = 1 for the odd ones.

Let us consider the left differentiations d/dy of the graded ring V{c,c), which possess the properties

^ | ( * ) * + ( - i ) W { w } * | ; ( " ) . (A.3)

d d dyA dyB K ' dyBdyA'

(A.4)

We introduce the exterior forms dy as the dual of the derivatives d/dy such that

{dy} = {y}, (A.5)

^ J * 8 - * (A.6)

One may define the exterior product of exterior forms dy by the rule

dy A dy1 = {-l)^^+1dy' A dy. (A.7)

With this exterior product, the ring V(c, c) is extended to the (Z, Z2)-bi-graded ring of polynomials V(c, c, dy) which is defined as the algebra of exterior forms dy over the ring V(c,c), where

ydy' = (-1)WMW)V, <M <r = ( - 1 ) I * N + W W ^ A ^, 4>,a e P{c,c,dy). (A.8)

Accordingly, the interior product (A.6) obeys the relation

^ J (0 A a) = ( | - j ^ ) A a + (-l)W+«H»>0 A ( | . j f f ) . (A.9)

The ring V(c, c, dy) is also provided with the exterior differential

*=?V^Wl (A. 10)

where the derivation

> 4>eV(c,c,dy),

319

fulfills the conditions (A.3), (A.4) and

> = °' As follows from (A.4) and (A.7), the exterior differential (A. 10) is a nilpotent operator, i.e.,

dod = 0.

It is readily observed that, putting c = 0, c = 0, we restate the familiar exterior algebra of even variables y.

The main principle of the BRST extension of mechanics is its invariance under the BRST and anti-BRST transformations whose generators read

o id d . . ^ .. d ■& = c1—- + Ci— + iql— + i p ; — ,

oq' dpi oc1 oct (All)

dq' dpi dcl l dci (A.12)

[65, 66]. The generators (A.11) - (A.12) fulfill the nilpotency rule

M = Q, tftf = 0, && + && = 0,

i.e., for any function f(y), we have

tfJd(tfJd/) = 0, 0Jd(0|d/) = O, tfJd(tfJd/)+tfJ(i(tfJd/) = 0.

Following the criterion of BRST and anti-BRST invariance, let us consider a Hamiltonian form H (5.2.10) on the momentum phase space V*Q and its vertical extension Hy (5.11.5) on the vertical momentum phase space VV*Q. We aim to find a 1-form He € V(c,c, dy) such that the sum

Hs = Hy + Hc

is BRST-invariant, i.e., its Lie derivative along the vector field i? (A. 11) vanishes:

U{HV + Hc) = #\d(Hv + Hc) + d($\(Hv + Hc)) = 0.


We obtain a desired Hamiltonian form

Hs = pidq' — q*dpi + i(cidc' - c'dci) - b\Hdt - Hcdt, (A.13)

Uc = i ^ + cjdj)(cjd:' + Jdj)H,

called the BRST extension of the Hamiltonian form H. It is easily justified that this form is also anti-BRST-invariant, i.e., its Lie derivative along the vector field ■d (A. 12) also vanishes.

In particular, given a splitting of the Hamiltonian

H = plTi]q> + H (A. 14)

with respect to some reference frame T on the configuration space Q —> K, there is the corresponding splitting of the BRST extended Hamiltonian

ns p ; i V - ^ ( - P A ) + ztfV - ^(-rjCi) + dvH + i{Zjd> + c>dj)(cJdj + d>d3)H

with respect to the connection f (5.2.6) on the momentum phase space V*Q —> K. This splitting shows that the Hamiltonian form (A.13) is globally defined, and that the connection is given on the odd variables {c',Ci) as on the elements of the fibre bundle VVQ -> K.

We also can write the BRST extension of the canonical 3-form Q (5.1.9) on the momentum phase space V*Q. This extension is deduced from the canonical 3-form flv (5.11.1) on the vertical momentum phase space VV'Q, and reads

fls = [dpi A dq' + dpi A dqx + i{dc\ A dc' - dc1 A dci)} A dt. (A.15)

The form (A.15) is globally defined. By construction, it is BRST- and anti-BRST-invariant. The Hamiltonian connection

7 = dt + ltd< + ya, + 7<ai + fa, + g, A + f-^ + 9xj- + g'-^-

for the Hamiltonian form Hs (A.13) with respect to the canonical form Q$ satisfies the Hamilton equations

7* = q\ = Shi, (A. 16a)

7i = Pa = - W , (A. 16b)

f = ql = dhis, (A. 16c)

321

7« = Pu = -9iHs, (A.16d) g* = c\ = (cjdi + cjdj)diH, (A.16e)

gi = cti = -(cjcV + c>dj)din, (A.16f) g* = 4 = (c,0* + Jd^d'H, (A.16g) 5i = cti = - ( 0 ^ + ^ ) 9 ^ (A.16h)

for the Hamiltonian form Hs-It is readily observed that the equations (A. 16a) - (A. 16b) are the Hamilton

equations for the initial Hamiltonian form H. To clarify the meaning of other equations, let us note that, if y(t) is a solution of the Hamilton equations (A. 16a) -(A.16h), then the deviations

q^t) + eecfit) + ec\t) + ec*(t), Pi(t) + eepi(t) + £Ci(t) + £Ci{t),

with the odd parameters e and e are solutions of the Hamilton equations for the Hamiltonian form H extended to the case of even, but not necessarily real variables Ql,Pi-

The following construction plays a prominent role for the BRST quantization procedure. Let H b e a Hamiltonian function in some decomposition (A. 14) of a Hamiltonian 7i. Such a Hamiltonian function becomes an operator under quantization. Let us consider the operators

n0 = effHodo e-m = tf _ p(Cj&> + (Jdj)n, /3eR, Up = erm o$oeim = d + Pfedt + T?dj)H,

called the BRST and anti-BRST charges, which act on functions f(y) by the law

*(/) = *!#, Hf) = «/■ It easy to see that these operators are nilpotent, i.e.,

HpoH0 = 0, H0oHp = 0. (A.17)

Let Hs be the BRST extension of the Hamiltonian function H, which is also treated as an operator on functions f(y). Then we obtain the relations

"H$ ° 7~Ls — 'Hs ° Up = 0, Hf3oHs-Hs°H0 = 0, ipns = H0OH0 + H0O n0.


These relations together with the relations (A. 17) provide the operators Tip, Tip, and Hs with the structure of a Lie superalgebra [35], similar to what we have in supersymmetric mechanics [42, 108].

Now let us restate the above construction in strict mathematical terms. In general, BRST mechanics can be developed as mechanics on supermanifolds [4, 33]. However, its naive variant shows that it suffices to consider the following particular case of mechanics on supermanifolds.

• First of all, we should restrict our consideration to vector configuration bundles Q —* K in order to introduce the generators of BRST and anti-BRST transformations.

• Secondly, the BRST extension TLS of a Hamiltonian Ti is a polynomial with respect to the odd variables. Therefore, we can narrow the class of superfunc-tions under consideration.

Let us recall a few basic notions. A vector space Q is called a Z2-graded vector space or simply a graded vector space if there is the decomposition

Q = So © Qi,

where Qo is called its even subspace, while Qi is said to be the odd subspace. An element of a graded vector space is called homogeneous if it belongs to <2o or Q\. The degree of a homogeneous element v e Q is denoted by {v}. A graded vector space is said to be (m, n)-dimensional if dim Q0 = m and dim Qx = n.

By a graded commutative algebra B is meant an algebra which is a vector graded space B = B0(& Bi such that

aras = (—l^ascv G BT+S, ar e BT, as G Bs, s,r = 0,1.

A graded commutative Banach algebra is a graded commutative algebra if it is a Banach algebra and the condition

||oo + OiH = |K|| + I M

is fulfilled. Let V be a vector space, and

A = A V = t 9 A V k

323

its exterior algebra. This is a Z-graded commutative algebra provided with Z2-graded structure as follows:

™ 2 m • . A0 = 1 © A V, m 2m+l

Ai = ® A V. m

It is called the Grassman algebra. Given a basis {cl,i € / } for the vector space V, the elements of the Grassman algebra take the form

a= E E ^i-i.e*1 • • • c**, (A. 18)

where the sum is over all the collections of indices (ix • • • ik) such that no two of them are the permutations of each other. For the sake of simplicity, we will omit the symbol of the exterior product of elements of the Grassman algebra A. The Grassman algebra becomes a graded commutative Banach algebra if its elements (A. 18) are endowed with the norm

NI = E E K- iJ-k (ti-i f c)

Remark 1.0.1. There is a different definition of a Grassman algebra [89], which is equivalent to the above one only in the case of an infinite-dimensional vector space V [40]. •

Let Q be a graded vector space and B a graded commutative algebra with a unit element. If Q is both a left and a right B-module such that

a3vT = vTas € Qs+r, vT € Qr, as G Bs,

it is called a graded B-module. Each B-graded module is obviously a Bo-module.

Example 1.0.2. Every graded vector space Q can be extended to a graded B-module which is its graded B-envelope

BQ (BQ)0 © (BQ)! = (B0 ® Qo © B\ ® fii) © (Bi ® Qo © B0 ® Qi).

The graded envelope

gm+n = ( f im 0 jgn) 0 (_gm ffi fin)


of the (m, n)-dimensional graded vector space Qm,n is said to be a vector superspace. Its Bo-submodule

Bm'n = 5J 1 0 B?

is also called a vector superspace. If B is a Banach algebra, the vector superspace Bm+n is a Banach space provided with the norm

Ikcii £11*11. i

a; € B,

where {c1} is a basis for Qm,n . •

The main ingredient in a theory of supermanifolds is the sheaf B of graded commutative algebras on a manifold Z. Let (Z,B{U)) be its canonical presheaf where by B{U) is meant the graded commutative algebra of local sections of the sheaf B over an open subset U C Z. If the sheaf B fulfills some conditions, called the Rothstein axioms, it is said to be an R-supermanifold (we refer the reader to [11, 40, 152] for details). The notion of an .R-supermanifold includes Berezin's graded manifolds, the supermanifolds of A.Rogers and infinite-dimensional supermanifolds described by A.Jadczyk and K.Pilch.

We restrict our consideration to the following class of .R-supermanifolds [33]. Let E —> Z be a vector bundle with a typical fibre V, and E* —> Z the dual of E —> Z. We consider the Grassman fibre bundle

A £ ' = M 0 ( 8 A £ * ) Z k Z

(A.19)

over Z. Its typical fibre is the Grassman algebra A* = AV. The sections of the Grassman fibre bundle make up the sheaf of Grassman algebras B on the manifold Z, which belongs to the class of graded manifolds. Its sections, i.e., sections of the fibre bundle (A.19) are called superfunctions. Given bundle coordinates (zx, y') with respect to the bases {CJ} in E —> Z, superfunctions take the form

/ = £ £ ail...ikci^---c^, k ( i i - t fc)

(A.20)

where {c1} are the dual bases in E*. Let us consider the sheaf der B of graded differentiations of the sheaf B. Let U

be an open subset of Z and B(U) the restriction of the sheaf B to U. By a graded

325

differentiation of the sheaf B(U) is meant its endomorphism ■§ such that

d : B{U) -* B(U), ■&{ab) = d{a)b + (-l)<"><a>atf(6), (A.21)

for the homogeneous elements d 6 der 6 and a, b G B(U). The graded differentiations of B(U) constitute the B-module der B(\J), and the presheaf of such a B-module generates the sheaf der B. One can show that der B is isomorphic to the sheaf of sections of the fibre bundle

TE ® AE -> Z. E

(A.22)

On can think of its elements as being A-valued vector fields, called vector superfields, on the fibre bundle E. They read

d = aXj^+aii- (A.23)

where ax, a1 are superfunctions (A.20) and

_d___d_ dc' dy*

are elements of the holonomic bases in TE. Due to the splitting VE = E x E, these elements can be identified with the basis elements c, in E. It follows that the operators d/dcl act on superfunctions (A.20) by the law of the exterior product of vectors and covectors. In accordance with the relation (A.21), they are odd. The operators d/dzx are even, and act on the superfunctions (A.20) as familiar derivatives.

The dual of the sheaf der B is the sheaf der*S generated by the linear morphisms

der£|u -> B(U). (A.24)

This is the sheaf of sections of the fibre bundle

T*E®AE->Z. E

Its elements take the form

<fi = a.\dzx + aide', (A.25)

where a\, a, are superfunctions (A.20), while dc1 = dyx are elements of the holonomic bases in T*E. One can think of these elements as being the A-valued 1-forms on the


fibre bundle E. Then the morphisms (A.24) can be treated as the interior product of the vector fields (A.23) and the 1-forms (A.25).

One can introduce the exterior differential of superfunctions

d : B -> der*£, df{d) = ( _ l ) W W ^ ( / ) = (_!){/} Wtfjdf,

which is extended to the exterior algebra of A-valued exterior forms on E, called exterior superforms. These superforms are sections of the fibre bundle

AT*E®AE -> Z, E

(A.26)

where f\T*E is the Grassman fibre bundle induced by T*E, The vector superfields, seen as sections of the fibre bundle (A.22), and the ex

terior superforms, represented by sections of the fibre bundle (A.26), are exactly those mathematical objects which we have considered above on the naive level. They satisfy the rules (A.3) - (A.10). To obtain the BRST extension of Hamilto-nian mechanics in terms of vector superfields and exterior superforms, one can put Z = VV'Q, while E is the vertical tangent bundle of the fibre bundle VVQ - » 1 so that V{c,c,dy) = der*B.

Appendix B

On quantum time-dependent mechanics

Here we are concerned only with a few aspects of quantizing time-dependent mechanics.

We will start from the following important remark. Let Q —> M be a configuration space of time-dependent mechanics, V*Q its momentum phase space, and H a Hamiltonian form on V*Q. Let T be a (complete) reference frame together with the adapted coordinates (t, q') on Q. Since V = 0 relative to these coordinates, the energy function with respect to this reference frame coincides with the Hamiltonian H in the decomposition (5.2.10) (see Section 5.7). We have the Hamilton equation (5.2.18a):

4 d{H, (B.l)

where one can think of q\. = q\ as being the relative velocities with respect to the reference frame T, In another reference frame with the adapted coordinates (£, q11), the Hamiltonian

W(t,9*,pJ) W(*,«*(t,flVi),ft(i,9' ',j<)) ft(t,fl",rf)*«'(t, «*,*{) (B.2)

H(ti<?(t,qH,p,i),pi(t,q'i,$) + ptr*&8".i0

coincides with the energy function with respect to this new reference frame, while the energy function with respect to the initial reference frame is

Hr n{t,<t(t,q'\p,%),pi{t,q'i,P'i)) H'(t,q'\p^ rir*(t,9*,?fl (B.3)

327

328 APPENDIX B. ON QUANTUM TIME-DEPENDENT MECHANICS

(see relation (4.5.2a)). Relative to the coordinates (£,</',$)> t n e Hamilton equation (B.l) reads

H_dH'{t,(p,i/i) qt~ dp[ '

where 4t c a n be treated as the relative velocities with respect to the new reference frame. At the same time, we can rewrite these equations as

a dHr(t,cf,pQ dpi

where gf. are the relative velocities with respect to the initial reference frame. Thus, we arrive at the following conclusion. Let we quantize a Hamiltonian system with a Hamiltonian H, written relative to coordinates (t, q'), and perform a coordinate transformation. In new coordinates, we quantize the Hamiltonian system with respect to the new reference frame if we use the Hamiltonian (B.2), and quantize that with respect to the initial reference frame if we use the Hamiltonian function (B.3). It follows that, in quantum mechanics, a passage from one set of coordinates to another is not a technical procedure. This explains why most quantization formulations lead to correct result only when a Cartesian coordinate system is used [47].

Let us point out the two quantization schemes which are invariant under reference frame transformations.

Let Q —* R be a configuration space of time-dependent mechanics and VQ its momentum phase space, provided with the canonical Poisson structure w (5.1.6). Let H be a Hamiltonian form on V'Q. Given a reference frame T together with the adapted coordinates {t,q%) on Q, the Hamiltonian connection 7# (5.2.17) for H can be represented as the sum

1H = dt + 0n,

where $« is the Hamiltonian vector field for the Hamiltonian H in the decomposition (5.2.10) with respect to the above-mentioned canonical Poisson structure w. It follows that, given a reference frame T, a time-dependent Hamiltonian system with a Hamiltonian form H can be seen as a Poisson Hamiltonian system (w,H). This Poisson Hamiltonian system can be quantized by the methods of geometric quantization as follows (we refer the reader to [103, 167, 181, 188] for the geometric quantization technique). We restrict our consideration to the prequantization procedure.

329

Let M be the typical fibre of the configuration space Q and T*M the corresponding typical fibre of V*Q, provided with the canonical symplectic form Q, (2.4.2). This form is exact. Therefore it belongs to the zero integral Chern class 0 € H2(M,Z). Then the bundle product

T"M x C -» T*M

is the prequa.ntiza.tion bundle over M, provided with the connection

A dpj ® & + dq> 0 (dj - 2mpjdc),

where c is a coordinate on C, such that its curvature is precisely the 2-form

R iA = -2-irin. (B.4)

A complete reference frame T define a projection

7rr : V*Q -> T'M.

The pull-back of the canonical symplectic form £3 by this projection is the exact 2-form

7rf fi = dpi A dq'

such that

IU wViW-Then the pull-back bundle

ir*r(T*M x C) = V*Q x C -» K*Q (B.5)

is the prequantization bundle over V*Q, provided with the connection

n^A = dp, ® &+ dq1 0 (dt - 2nipidc)

such that its curvature obeys the relation similar to (B.4). It follows that the Poisson algebra on the Poisson manifold (V*Q,w) is quantized, i.e., any function / C D°(Vr*(5) defines the operator

/ : s t-> -djf&s + &>f(dj + 2iripj)s (B.6)

http://prequa.ntiza.tion

330 APPENDIX B. ON QUANTUM TIME-DEPENDENT MECHANICS

on sections s(t, qj,Pj) of the prequantization bundle (B.5). Since s are simply complex functions on V*Q, we can rewrite the operator (B.6) in the form

f:s>->{f,s}v + 2iripj9' fs. (B.7)

It is readily observed that this operator is invariant under holonomic transformations of the momentum phase space V*Q. It follows that prequantization (B.7) is independent of a reference frame. The result is obvious since the Poisson bivector w (5.1.6) belongs to the zero element of the second LP-cohomology group HlP(V*Q, w) (see Example 2.7.5).

Another quantization procedure invariant under holonomic time-dependent transformations is based on the Schrodinger equation. Given a reference frame T together with the adapted coordinates (t, q') on Q, let H(t, q^Pj) be a Hamiltonian with respect to this reference frame. Then the corresponding Schrodinger equation reads

H(t,ql,-ihj-)V. (B.8)

Relative to another reference frame coordinates {t,q"), the Schrodinger equation (B.8) takes the form

5 * H'(t,q'\-ih^-)V, (B.9)

where H' is the Hamiltonian (B.2). It follows that the Schrodinger equations (B.8) and (B.9) provide quantization with respect to different reference frames. The following example shows that these quantizations are not equivalent in general.

Example 2.0.3. Let us consider a 1-dimensional motion described by the Ha.mil-tonian

H £ + on the momentum phase space R 3 - » R with respect to some inertial reference frame (t,q,p). The corresponding Schrodinger equation is

V(9) + 9m ™{E-U(q))rl>{q) 0.

Let us consider another reference frame given by a connection T (<?'). The energy function with respect to the reference frame F reads

UT H-pT.

http://Ha.mil

331

The corresponding Schrodinger equation is

r(q) + f?(£ - U(q)Mq) - ^Y{qW{q) 0.

Let us substitute

^{q) = p(q)exp(^JT(q)dqSj (B.10)

in this equation. We obtain

p"(q) + [E- U(q) + ihV(q) + | r2(g)] p{q) 0. (B.H)

For instance, let Y = v = const, be an inertia! reference frame. In this case, the equation (B.l l) takes the form

p"{q) + \E + ^V*~U(q)]p{q) 0.

It is readily observed that energy levels of the quantum system with respect to the moving inertial reference frames are the shifts

E + -S

of those of the initial quantum system. Moreover, the transformation ip —> p (B.10) is not an automorphism of the Hilbert space of states of the initial quantum system. This means that quantum systems with respect to different inertial reference frames are not equivalent. •

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Index

acceleration absolute, 184 relative, 184

adjoint of a form, 91 adjoint representation, 46 admissible metric, 314 almost tangent structure, 32 annihilator of a distribution, 33 antighost, 317 anti-BRST charge, 321 anti-BRST transformations, 319 atlas, 10

holonomic, 15 of constant local trivializations, 45

automorphism of fibre bundles, 12 holonomic, 53 vertical, 12

base of a fibre bundle, 10 basic De Rham cohomology, 95 basic exterior form, 95 Berry connection, 281 Berry oscillator, 281 bivector field, 29 BRST charge, 321 BRST extension of the canonical -form,

320 BRST extension of a Hamiltonian form,

320

BRST transformations, 319 bundle, 9

affine, 14 associated, 51 of principal connections, 51 principal, 48 vector, 12

dual, 13

canonical automorphism, 242 canonical complex, 91 canonical form on a frame bundle, 53 canonical homology, 91 canonical horizontal splitting, 38 canonical Liouville form, 74 canonical symplectic form, 74 canonical vector field, 67,74 canonical 3-form, 230

jet prolongation, 283 Cartan equations, 116,191 Casimir function, 67 chain complex, 89 characteristic distribution, 69

of a presymplectic form, 81 characteristic foliation of a presymplec

tic form, 81 coadjoint representation, 47 coboundary on a group, 136 cochain complex, 90

347

348 INDEX

cocycle associated with an action of a group, 136

cocycle on a group, 136 codistribution, 33 cofiag of a codistribution, 34 cohomology group, 90 cohomology of a group, 136 coisotropic ideal, 89 coisotropic imbedding, 83 coisotropic submanifold

of a Jacobi manifold, 60 of a Poisson manifold, 70 of a symplectic manifold, 78

complete family of Hamiltonian forms, 256

composite connection, 55 composite fibre bundle, 53 configuration bundle, 6 configuration space, 3 connection, 42

affine, 44 dual, 44 dynamic, 164

symmetric, 166 complete, 157 curvature-free, 45 flat, 45 holonomic, 158 Lagrangian, 191 linear , 44 principal, 50

associated, 52 connection form, 51

local, 51 conservation law, 219

constraint non-holonomic, 216 constraint reaction acceleration, 209 constraint reaction force, 217 constraint reference frame, 214 constraint space, 124

final, 125,130 Lagrangian, 249 primary, 124

constraints admissible, 212 complete, 125 holonomic, 206

ideal, 209 first-class, 126,132 primary, 124 second class, 126,132 secondary, 125 tertiary, 125

contact automorphism, 64 infinitesimal, 64

contact form, 37,61 contact manifold, 61 contact transformation, 63 contraction, 28 coordinates

affine, 14 adapted, 176 canonical, 70,140,243 canonically conjugate, 70 fibred, 10 holonomic , 16 induced, 16

copresymplectic structure, 193 Coriolis theorem, 186 cosymplectic structure, 193

INDEX 349

cotangent bundle, 16 covariant differential, 43

vertical, 55 current, 219 curvature, 44

D'Alembert principle, 209 generalized, 216

Darboux coordinates, 61 De Donder form, 194 democracy group, 202 De Rham cohomology group, 90 De Rham complex, 90 derivative

linear, 15 total, 36,40 variational, 187 vertical, 287

differential equation, 41 differential ideal, 34 differential operator, 41 differential system, 33 Dirac constraint system, 129 Dirac Hamiltonian system, 126

completely integrable, 128 direct product of Poisson structures,

68 distribution, 32

completely integrable, 33 involutive, 32 regular, 33 totally non-holonomic, 33

dynamic equation autonomous, 106 conservative, 161 first order, 160,239

autonomous, 107 quadratic, 161 second order, 160

autonomous , 108

endomorphism of fibre bundles, 12 vertical, 155

energy function, 116,222 canonical, 218 Hamiltonian, 271

equilibrium equation, 244 Euler-Lagrange map, 203 Euler-Lagrange operator, 187 Euler-Lagrange-Cartan operator, 191 Euler-Lagrange-type operator, 202 even subspace, 322 evolution equation, 112,247 exterior algebra, 26 exterior differential

contravariant, 92 horizontal, 203 vertical, 155

exterior form, 25 exterior product of vector bundles, 14 exterior superform, 326

fibration, 10 fibre basis, 13 first integral of motion, 112 flag of a differential system, 33 flag of a distribution, 33 foliation, 34

of level surfaces, 35 simple, 35 singular, 35

force

350 INDEX

centrifugal, 181 Coriolis, 181 external, 198 inertial, 179 Lorentz, 198 universal, 199

frame, 13 frame bundle, 52 frame connection, 183

Hamiltonian, 237 Lagrangian, 264

free motion equation, 179 fundamental identity, 99

Galilei group, 182 gauge algebra bundle, 49 gauge conditions, 267 gauge convention, 281 gauge fields, 122 gauge freedom, 122 gauge potentials, 51 gauge transformations, 12

of a principal bundle, 271 Gauss principle, 216 general covariant transformations, 53 generalized almost Jacobi manifold, 99 generalized almost Poisson manifold,

99 generalized invariant transformations,

223 generating function, 246

of a foliation, 35 of a Lagrangian submanifold, 79

generator, 21,47 geodesic equation, 109 geodesic vector field, 109

geometric quantization, 328 geometric prequantization, 328 germ of a submanifold, 79 ghost, 317 graded commutative algebra, 322

Banach, 322 graded differentiation, 325 graded Lie algebra, 25 graded manifold, 324 graded space, 322 graded S-envelope, 323 graded B-module, 323 graph of a morphism, 70 Grassman algebra, 323 Grassman bundle, 302

Hamilton equation with respect to a symplectic struc

ture, 114 Hamilton equations, 147,239

constrained, 259 Hamilton operator, 146,238 Hamilton-De Donder equation, 194 Hamilton-Jacobi equation, 246 Hamiltonian, 234

admissible, 126 autonomous, 111

Hamiltonian action, 134 Hamiltonian connection, 147,238

locally, 144,232 Hamiltonian density, 142 Hamiltonian form, 141,234

associated with a Lagrangian, 249 constrained, 259 locally, 233 polysymplectic, 141

INDEX 351

Hamiltonian function, 235 Hamiltonian map, 115,142,235 Hamiltonian system

completely integrable, 115 gauge invariant, 122 generalized, 129 presymplectic, 119

pull-back, 121 Hamiltonian vector field

with respect to a Jacobi structure, 59

with respect to a Poisson structure, 68

with respect to a presymplectic structure, 82

locally, 82 with respect to a symplectic struc

ture, 76 locally, 76

Helmholtz-Sonin map, 203 homology group, 89 horizontal density, 27 horizontal distribution, 45 horizontal foliation, 45 horizontal form, 27 horizontal vector field, 156

infinitesimal symmetry, 112 integral curve, 20

admissible, 32 integral invariant of Poincare-Cartan,

234 integral manifold, 32

maximal, 33 of maximal dimension, 33

integral of motion, 219

integral section, 43 interior product, 28

left, 28 of vector bundles, 13 right, 28

inverse problem, 107,202 involution, 67 isomorphism of fibre bundles, 12 isotropic submanifold, 78

Jacobi bracket, 57 generalized, 99

Jacobi bundle, 72 Jacobi field, 289 Jacobi identity, 57

generalized, 99 Jacobi manifold, 57

generalized, 99 Jacobi morphism, 59

conformal, 60 infinitesimal, 59

Jacobi structure, 57 conformaUy equivalent, 60

Jacobi vectors, 202 mass-weighted, 202

jet bundle, 36 affine, 36

jet manifold, 36 higher order, 40 repeated, 38 second order, 39 sesquiholonomic, 39

jet of sections, 36 jet of submanifolds, 299 jet prolongation

of morphisms, 37

352 INDEX

of sections, 37 second order, 40

of vector fields, 37

kernel of a differential operator, 41 of a fibred morphism, 11 of a form, 30

kinetic energy, 201 Koszul-Brylinski-Poisson homology, 91

Lagrange equations, 187 autonomous, 116

Lagrangian, 186 almost regular, 258 autonomous, 115 generalized, 81 hyperregular, 117,189 non-degenerate, 188 quadratic, 262 regular, 115,188 semiregular, 253

Lagrangian function, 186 Lagrangian submanifold, 78

of a Jacobi manifold, 60 of a Poisson manifold, 70

Lagrangian system, 186 left-invariant form, 47

canonical, 47 Legendre bundle, 102,140

homogeneous, 101 Legendre map, 117,188 Legendre morphism, 193 Legendre submanifold, 63 Legendre vector bundle, 140 Leibniz rule, 66

Lichnerowicz-Jacobi cohomology, 94 Lichnerowicz-Poisson cochain complex,

92 Lichnerowicz-Poisson cohomology, 92 Lie algebra

dual, 47 left, 46 right, 46

Lie bracket, 20 Lie coalgebra, 47 Lie derivative, 31 Lie-Poisson structure, 72 lift of a vector field

canonical, 22 horizontal, 43 vertical, 22

Liouville vector field, 22

mass metric, 196 mass tensor, 196 Maurer-Cartan equation, 47 momentum mapping, 134

equivariant, 135 momentum phase space, 4,108 morphism of fibre bundles, 11

affine, 15 linear, 13

motion, 160 multisymplectic form, 101,102 multivector field, 22

simple, 23

Nambu-Jacobi bracket, 99 Nambu-Jacobi manifold, 99 Nambu-Poisson bracket, 99 Nambu-Poisson manifold, 99

INDEX 353

Newtonian system, 197 standard, 201

Noether conservation law, 221 Noether current, 221 normalizer, 88

odd subspace, 322

parameter bundle, 274 Pfaffian system, 34 physical phase space, 122 Poincare-Cartan form, 115,187 Poisson action, 137 Poisson algebra, 66 Poisson automorphism, 67 Poisson bivector field, 58 Poisson bracket, 66

generalized, 99 Poisson bundle, 72 Poisson codifferential, 90 Poisson manifold, 66

generalized, 99 reduced, 85

Poisson morphism, 67 Poisson reduction, 85 Poisson structure, 66

coinduced, 68 generating, 69 non-degenerate, 66 regular, 66 zero, 66

polysymplectic form, 103 preconnection, 303 prequantization bundle, 329 presheaf, 18

canonical, 19

presymplectic form, 81 presymplectic manifold, 81 projection, 10 pull-back fibre bundle, 12 pull-back form, 26

rank of a bi vector field, 29 rank of a fibred morphism, 11 reduction leafwise, 87 reduction of a Poisson map, 86 reductive structure, 85

of a Poisson manifold, 85 Reeb vector field, 62 reference frame, 175

complete, 176 relativistic configuration space, 303 relativistic dynamic equation, 312 relativistic geodesic equation, 313 relativistic Hamiltonian, 309 relativistic momentum phase space, 307 relativistic principle, 307 relativistic symplectic form, 307 relativistic system, 306 relativistic velocity, 305 restriction of an exterior form, 26 restriction of a fibre bundle, 12 right-invariant form, 47

canonical, 48

Schouten-Nijenhuis bracket, 24 of exterior forms, 97

section, 10 holonomic, 37

shape coordinates, 281 shape space, 281 sheaf, 18

354 INDEX

of continuous functions, 19 soldering form, 43 solution

of a differential equation, 41 of a first order dynamic equation,

107 of a Hamiltonian system, 111 of a second order dynamic equa

tion, 108 spray, 110 stalk, 18 standard vector field, 3 standard 1-form, 3 star operation, 91 subbundle, 11 superfunction, 324 symmetry transformations, 225 symplectic action, 133 symplectic automorphism, 74 symplectic bundle, 72 symplectic foliation, 69 symplectic form, 73 symplectic manifold, 73

reduced, 84 symplectic morphism, 73 symplectic realization, 75 symplectic reduction, 84 symplectic submanifold, 78 symplectomorphism, 74

tangent bundle, 15 tangent lift

of a function, 27 of an exterior form, 27 of a multivector field, 23

tangent map, 16

tangent-valued form, 31 canonical, 32

tensor product of vector bundles, 14 torsion of a dynamic connection, 166 total space, 10 transition functions, 10 translation-reduced configuration space,

202 trivialization, 10 typical fibre, 10

unified phase space, 283 universal unit system, 8

variational operator, 203 variational sequence, 203 vector field, 20

complete, 21 holonomic, 108 left-invariant, 46 principal, 272 projectable, 21 right-invariant, 46 subordinate, 32 vertical, 22

vector superfield, 325 vector superspace, 324 velocity

absolute, 176 relative, 176

velocity hyperboloids, 306 velocity phase space, 3,108 vertical configuration space, 285 vertical cotangent bundle, 17 vertical extension of a Hamiltonian form,

288

INDEX 355

vertical extension of a Lagrangian, 291 vertical momentum phase space, 286 vertical splitting, 17 vertical tangent bundle, 17 vertical tangent map, 17 vertical velocity phase space, 286

Whitney sum of affine bundles, 15 Whitney sum of vector bundles, 14

n-body, 281 n-symplectic form, 102 B-supermanifold, 324

1-parameter group of diffeomorphisms, 21

1-paxameter group of local diffeomorphisms, 20

3-velocity, 305 3-velocity phase space, 305 4-velocity, 305 4-velocity phase space, 305

gauge mechanics

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