Symplectic Geometry and Analytical Mechanics

Mathematics and Its Applications

Managing Editor:

M. HAZE WINKEL

Centre for Mathematics and Computer Science. Amsterdam. The Netherlands

Editorial Board:

F. CALOGERO. Universita deg/i Studi di Roma. Italy Yu. I. MANIN. Stekiov Institute of Mathematics. Moscow. U.S.S.R. A. H. G. RINNOOY KAN. Erasmus Ullh·ersity. Rotterdam. The Netherlands G.-c. ROTA. M.l. T.. Cambridge. Mass .. U.S.A.

Symplectic Geometry and Analytical Mechanics

Paulette Libermann Universite de Paris VIL Paris. France'

and

Charles-Michel Marle Universite Pierre et Marie Curie (Paris VI).

Paris. France

Translated by Bertram Eugene Schwarzbach

D. Reidel Publishing Company A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP

Dordrecht / Boston / Lancaster / Tokyo

Library of Congress Cataloging in PubHcation Data

Libermann, Paulette, 1919-Symplectic geometry and analytical mechanics.

(Mathematics and its applications) Translated from the French. Bibliography: p. Includes index. 1. Geometry, Differential. 2. Symplectic manifolds. 3. Mechanics, An­

alytic. I. Marle, Charles. II. Title. III. Series: Mathematics and its applica­tions (D. Reidel Publishing Company) QA649.L465 1987 516.3'6 86-31568

ISBN-13: 978-90-277-2439-7 e-ISBN-13: 978-94-009-3807-6 DOl: 10.1007/978-94-009-3807-6

Published by D. Reidel Publishing Company, P.O. Box 17,3300 AA Dordrecht, Holland.

Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, MA 02061, U.S.A.

In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322,3300 AH Dordrecht, Holland.

All Rights Reserved

Softcover reprint of the hardcover 1st edition 1987 © 1987 by D. Reidel Publishing Company, Dordrecht, Holland No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner

ala memoire d'Elie Cartan et de Charles Ehresmann

en hommage it Andre Lichnerowicz

Contents

Series Editor's Preface xi

Preface .. xiii

Chapter I. Symplectic vector spaces and symplectic vector bundles 1

Part 1: Symplectic vector spaces . . . . . . 2

1. Properties of exterior forms of arbitrary degree 2 2. Properties of exterior 2-forms ....... 3 3. Symplectic forms and their automorphism groups 6 4. The contravariant approach . . . . . . . . . 8 5. Orthogonality in a symplectic vector space 10 6. Forms induced on a vector subspace of a symplectic vector space 12 7. Additional properties of Lagrangian subspaces 16 8. Reduction of a symplectic vector space. Generalizations 20 9. Decomposition of a symplectic form ....... 23

10. Complex structures adapted to a symplectic structure 26 11. Additional properties of the symplectic group 33

Part 2:

12. 13. 14.

Part 3:

15. 16. 17.

Chapter II.

1. 2. 3. 4.

Symplectic vector bundles

Properties of symplectic vector bundles Orthogonality and the reduction of a symplectic vector bundle Complex structures on symplectic vector bundles . . . . . .

Remarks concerning the operator A and Lepage's decomposition theorem

The decomposition theorem in a symplectic vector space Decomposition theorem for exterior differential forms A first approach to Darboux's theorem .. . . . .

Semi-basic and vertical differential forms in mechanics

Definitions and notations Vector bundles associated with a surjective submersion Semi-basic and vertical differentia! forms The Liouville form on the cotangent bundle

36

36 38 40

43

43 48 51

53

54 54 56 58

viii Contents

5. Symplectic structure on the cotangent bundle 63 6. Semi-basic differential forms of arbitrary degree 67 7. Vector fields and second-order differential equations 72 8. The Legendre transformation on a vector bundle . 73 9. The Legendre transformation on the tangent and cotangent bundles 75

10. Applications to mechanics: Lagrange and Hamilton equations 77 11. Lagrange equations and the calculus of variations 81 12. The Poincare-Cartan integral invariant 83 13. Mechanical systems with time dependent

Hamiltonian or Lagrangian functions 86

Chapter m. Symplectic manifolds and Poisson manifolds 89

90 91 94 96 99

1. Symplectic manifolds; definition and examples 2. .special submanifolds of a symplectic manifold 3. Symplectomorphisms 4. Hamiltonian vector fields 5. The Poisson bracket 6. Hamiltonian systems 7. Presymplectic manifolds 8. Poisson manifolds 9. Poisson morphisms

10. Infinitesimal automorphisms of a Poisson structure 11. The local structure of Poisson m~ifolds 12. The symplectic foliation of a Poisson manifold 13. The local structure of symplectic manifolds 14. Reduction of a symplectic manifold . . . . 15. The Darboux-Weinstein theorems 16. Completely integrable Hamiltonian systems 17. Exercises .............. .

Chapter IV. Action of a Lie group on a symplectic manifold

102 105 107 114 121 125 130 134 141 153 160 18]

185

1. Symplectic and Hamiltonian actions 186 2. Elementary properties of the momentum map 195 3. The equivariance of the momentum map 200 4. Actions of a Lie group on its cotangent bundle 204 5. Momentum maps and Poisson morphisms 213 6. Reduction of a symplectic manifold by the action of a Lie group 217 7. Mutually orthogonal actions and reduction 228 8. Stationary motions of a Hamiltonian system . 238 9. The motion of a rigid body about a fixed point 246

10. Euler's equations . . . . . . . . . 253 11. Special formulae for the group SO(3) . . . . 256 12. ,The Euler-Poinsot problem ........ 260 13. The Euler-Lagrange and Kowalevska problems 265 14. Additional remarks and comments 267 15. Exercises 269

Contents

Chapter V. Contact manifolds

1. Background and notations . 2. Pfaffian equations 3. Principal bundles and projective bundles 4. The class of Pfaffian equations and forms 5. Darboux's theorem for Pfaffian forms and equations 6. Strictly contact structures and Pfaffian structures 7. Project able Pfaffian equations 8. Homogeneous Pfaffian equations 9. Liouville structures . . . . . .

10. Fibered Liouville structures 11. The automorphisms of Liouville structures 12. The infinitesimal automorphisms of Liouville structures 13. The automorphisms of strictly contact structures . . 14. Some contact geometry formulae in local coordinates 15. Homogeneous Hamiltonian systems . . . . 16. Time-dependent Hamiltonian systems . . . 17. The Legendre involution in contact geometry 18. The contravariant point of view .....

Appendix 1. Basic notions of differential geometry

1. 2. 3.

Differentiable maps, immersions, submersions The flow of a vector field ....... . Lie derivatives

ix

275

276 277 279 284 286 289 299 302 306 307 313 315 318 324 327 328 332 336

341

341 346 349

4. Infinitesimal automorphisms and conformal infinitesimal transformations 352 5. Time-dependent vector fields and forms 354 6. Tubular neighborhoods . . . . . . 358 7. Generalizations of Poincare's lemma 359

Appendix 2. Infinitesimal jets

1. 2. 3. 4. 5. 6.

Generalities . . . . . . Velocity spaces . . . . . Second-order differential equations Sprays and the exponential mapping Covelocity spaces . . . . . Liouville forms on jet spaces . . . .

Appendix 3. Distributions, Pfaffian systems and foliations

1. 2. 3. 4.

Distributions and Pfaffian systems ........ . Completely integrable distributions . . . . . . . . . Generalized foliations defined by families of vector fields Differentiable distributions of constant rank . . . . .

365

365 367 371 373 376 379

382

382 384 387 393

x Contents

Appendix 4. Integral invariants

1. 2. 3.

Integral invariants of a vector field Integral invariants of a foliation The characteristic distribution of a differential form

Appendix 5. Lie groups and Lie algebras

1. 2. 3. 4. 5. 6. 7.

Lie groups and Lie algebras; generalities The exponential map . . . . . . . . Action of a Lie group on a manifold The adjoint and coadjoint representations Semi-direct products ........ . Notions regarding the cohomology of Lie groups and Lie algebras Affine actions of Lie groups and Lie algebras . . . . . . . . .

Appendix 6. The Lagrange-Grassmann manifold

1. 2. 3.

The structure of the Lagrange-Grassmann manifold The signature of a Lagrangian triplet . . . . . The fundamental groups of the symplectic group

and of the Lagrange-Grassmann manifold

Appendix 7. Morse families and Lagrangian submanifolds

1. 2. 3. 4. 5. 6.

Lagrangian submanifolds of a cotangent bundle Hamiltonian systems and first-order partial differential equations Contact manifolds and first-order partial differential equations Jacobi's theorem . . . . . . . . . . . . . . . . . . The Hamilton-Jacobi equation for autonomous systems The Hamilton-Jacobi equation for nonautonomous systems

Bibliography

Index ....

395

395 401 404

409

409 415 419 425 429 433 439

448

448 454

458

461

461 473 477 484 490 492

497

519

SERIES EDITOR'S PREFACE

Approach your problems from the right end and begin with the answers. Then one day, perhaps you will find the final question.

'The Hermit Clad in Crane Feathers' in R. van Gulik's The Chinese Maze Murders.

It isn't that they can't see the solution. It is that they can't see the problem.

G. K. Chesterton. The Scandal of Father Brown 'The point of a Pin'.

Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thouglit to be completely disparate are suddenly seen to be related.

Further, the kind and level of sophistication of mathematics applied in various sci­ences has changed drastically in recent years: measure theory is used (non-trivially) in re­gional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homo­topy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics" , "CFD", "completely integrable systems" , "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. This program, Mathe­matics and Its Applications, is devoted to new emerging (sub)disciplines and to such (new) interrelations as exempla gratia:

a central concept which plays an important role in several different mathematical and/or scientific specialized areas; new applications of the results and ideas from one area of scientific endeavour into another; influences which the results, problems and concepts of one field of enquiry have and have had on the development of another.

The Mathematics and Its Applications programme tries to make available a careful selection of books which fit the philosophy outlined above. With such books, which are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope to contribute some­thing towards better communication among the practitioners in diversified fields.

The love affair, or marriage, between geometry and physics, here mainly in the guise of analytical mechanics, is, as is well known, an old one. There seems to have been a period of cooling off. However, after a trial separation period the relations between the two are more vigorous and deeper than ever before.

The branch of geometry chiefly concerned is (generalized) symplectic geometry. Perhaps surprisingly at first sight, given its age (though not really so if one pauses to reflect on the demands made by new developments on the application side), this continues to be a very

xi

xii Series Editor's Preface

active field with many recent new results and concepts. For instance, the notion of a Poisson manifold as a naturally appearing generalization of a symplectic one is quite possibly a more fundamental and more supple tool.

This book specifically aims to present these new developments integrated with the more well-established frameworks of symplectic geometry and, as such, is a timely and, I am convinced, most valuable addition to the existing selection of books on symplectic geometry.

The unreasonable effectiveness of mathematics in science ...

Eugene Wigner

Well, if you know of a better 'ole, go to it.

Bruce Bairnsfather

What is now proved was once only imagined.

William Blake

Bussum, August 1986

As long as algebra and geometry proceeded along separate paths, their advance was slow and their applications limited.

But when these sciences joined company they drew from each other fresh vitality and thence forward marched on at a rapid pace towards perfection.

Joseph Louis Lagrange

Michiel Hazewinkel

Preface

During the last two centuries, analytical mehanics have occupied a prominent place among scientists' interests. The work in this field by such mathematicians as Euler, Lagrange, Laplace, Hamilton, Jacobi, Poisson, Liouville, Poincare, CaratModory, Birkhoff, Lie and E. Cartan has played a major role in the developement of several important branches of mathematics: differential geometry, the calculus of variations, the theory of Lie groups and Lie algebras, and the theory of ordinary and partial differential equations. During the last thirty years, the study of the geometric structures which form the basis of mechanics (symplectic, Poisson and contact structures) has enjoyed renewed vigor. The introduction of modern methods of differential geometry is one of the reasons for this renewal (t) j it has permitted a formulation of global problems and furnished tools with which to solve them.

Even though there are already a number of books that treat this subject (t), the authors believe that it is of value to provide readers with an approach to these methods and to permit them to familiarize themselves with certain recent developments which are not mentioned in the other textbooks in this field, and to acquire the information necessary in order to pursue current research. They have also expounded and employed the methods of exterior algebra which were introduced by E. Cartan.

This work, which is in large part based on lectures given by the authors at the Universities of Paris VI and VII, incorporates numerous points recalled to facilitate study. It was written for students at the end of their 'Second Cycle' program, or at the beginning of their 'Third Cycle' - these are the French designations that correspond, approximately, to American Masters' and Doctoral programs, respectively. It is mainly directed at readers interested in mathematics, but it may be of interest too for physicists, engineers, and for anyone who may be interested in differential geometry and the foundations of mechanics.

This work is composed of five chapters and seven appendices. Chapters I, II and V, as well as appendix 2, were written by the first author (P. L.), while chapters III and IV, as well as the remaining appendices, were written by the second author (C.-M. M.).

The first chapter is composed of three parts. The first part deals with symplectic vector spaces (rank, orthogonality, the linear symplectic group, reduction, the contravariant point of view). These notions are taken up again in the second part which treats symplectic vector bundles. Finally, the third part discusses, in an abbreviated fashion, Lepage's theorems concerning the decomposition and the divisibility of forms on a vector space or a differentiable manifold.

(tl For a more thorough exposition of the reasons of this renewed interest, see the review by S. Sternberg of the book of R. Abraham and J. E. Marsden, published in the Bull. Amer. Math. Soc., March 1980, vol. 2, 318-381. (t) See the books by R. Abraham [1], R. Abraham and J. E. Marsden [1] V. I. Arnold [5], V. I. Arnold and A. Avez [1], Y. Choquet-Bruhat [1], Y. Choquet-Bruhat, C. de Witt-Morette and M. Dillard-Bleik [1], C. Godbillon [1], V. Guillemin and S. Sternberg [1], [5/, R. Hermann [1], G. W. Mackey [1], G. Marmo, E. J. Saletan, A. Simoni and B. Vitale [1], J. K. Moser 4J, C. L. Siegel and J. K. Moser [lJ, J.-M. Souriau [2], S. Sternberg [lJ, W. Thirring [lJ.

xiii

xiv Preface

The second chapter concerns semibasic forms, especially the Liouville form on the co gent bundle. It includes a discussion of the Legendre transformation and its applic2 to mechanics. Though it adopts a different point of view, this chapter has been sugge by J. Klein's formulation of mechanics in terms of the tangent bundle, which has 1 presented in C. Godbillon's book.

The third chapter deals with symplectic structures on a manifold and with their ge alizations: presymplectic structures and, especially, Poisson structures. It has been kn for many years that symplectic manifolds offer a suitable framework for the Hamilto. formulation of classical mechanics. Poisson manifolds, which were discussed by S. Li early as 1890, and, implicitly, recognized by E. Cartan, appear naturally, for example the study of certain foliations of symplectic manifolds. This chapter also includes a t ough discussion of the reduction of a symplectic manifold, and gives a complete acce of Weinstein's generalizations of Darboux's classical theorem. Then it proves Liouvi theorem regarding completely integrable systems, as well as the theorem of Arnold A vez regarding the existence of action-angle coordinates.

The fourth chapter deals with the action of a Lie group on a symplectic manifold. important notion of the momentum map, which is due to J .-M. Souriau and S. Sma! defined and its properties are studied. The special case in which the symplectic mani is the cotangent bundle of the Lie group is of particular interest, and leads quite natu! to the definition of the Lie-Poisson structure on the dual space of the Lie algebra of group. This structure, recognized by S. Lie around 1890, has been recently rediscov< by A. Kirillov, B. Kostant and J.-M. Souriau. The results concerning the reductio a symplectic manifold upon which a Lie group acts are applied to the definition of stationary motions of a Hamiltonian system which possesses a Lie group G of symmet and to the study of their relative stability modulo G. The chapter concludes wit! application of all the notions that were previously introduced to the classical exampl the motion of a rigid body about a fixed point.

The fifth chapter deals with contact structures and, more generally, with Pfa equations of arbitrary class. We prove the theorems of Darboux relative to Pfaflian fc and equations of constant class. A large part of the chapter, which was suggestec the work of A. Lichnerowicz and V. Arnold, concerns the symplectification of a con manifold. These considerations lead to the notion of a Liouville structure on a prine bundle, whose structure group is the multiplicative group R., equipped with a Pfa: form without zeros that is homogeneous of degree 1. Contact geometry has applicat to time-dependent Hamiltonian systems, and to the homogeneous Hamiltonian syst encountered in mechanics. Lastly, the Legendre transformation is studied from the p of view of contact structures.

Appendices 1 (basic notions in differential geometry), 4 (integral invariants) and 5 groups and Lie algebras) deal with basic notions which are used in the book. Rea already familiar with these subjects may skip them or merely refer to them to beci acquainted with our notations. Appendix 3 (distributions, Pfaflian systems and foliati, includes a review of well-known notions, and offers a generalization of Frobenius' theOI due to P. Stefan and H. Sussmann, which is applicable to distributions with noncons1 rank. This is useful in the study of Poisson manifolds. The remaining appendices dev, subjects, already encountered in the body of the book, which are interesting in their' right: the application of the infinitesimal jets of C. Ehresmann to mechanics (appendbc

Preface xv

Lagrange-Grassmann manifolds (appendix 6), the use of Morse families in the definition of Lagrangian submanifolds of a cotangent bundle, first-order partial differential equations, Jacobi's theorem, and the Hamilton-Jacobi equation (appendix 7).

Several important aspects of the mathematical structures of analytical mechanics could not be included in this book, or had to be treated all too briefly. Among them we must mention the Lagrangian formulation of mechanics and the calculus of variations (touched on in chapter II), holonomic and nonholonomic constraints in mechanical systems, the formulation of canonical transformations and of the Hamilton-Jacobi method in the framework of the canonical manifolds of A. Lichnerowicz, the symplectic relations defined by W. M. Tulczyjew (barely mentioned in chapter III), the study of the properties of the Arnold-Leray-Maslov index (whose construction is sketched in appendix 6). In these case, precise references have been supplied to compensate for the gaps.

Other very interesting developments of symplectic geometry, and their applications to mathematical physics, have been left outside the scope of this book: for instance, geometric quantization, for which the reader is referred to the books by D. J. Simms and N. Woodhouse [1] and by N. Woodhouse [1]; deformations of the algebra of differentiable functions on a symplectic manifold and its application to quantization, first introduced by F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer [1], and which is now a very active field of research (see the papers by M. Flato, A. Lichnerowicz and D. Sternheimer [1], A. Lichnerowicz [17], [22]-[24], D. Arnal and J.-C. Cortet [1], M. Cahen and S. Gutt [1], S. Gutt [1], M. De Wilde and P. Lecomte [1], [2], M. De Wilde, S. Gutt and P. Lecomte [1], P. Lecomte [1], [2], C. Moreno and P. Ortega-Navarro [1]); gauge theories and Yang-Mills fields, for which the reader may consult the book by V. Guillemin and S. Sternberg [2]; constrained Hamiltonian systems and Dirac brackets; exponential integrals, their asymptotic behavior, critical exponents and their appearance in various problems in physics, for which the reader may consult the book by V. Guillemin and S. Sternberg [1], and the book and paper by E. Combet [1], [2]. It is our hope that the present book will prepare readers interested in these more advanced developments by providing them with the basic tools.

During the writing of this work Andre Lichnerowicz and Georges Reeb lavished advice and encouragement upon us, and we are pleased to express our gratitude.

We also thank Wlodzimierz Tulczyjew and Sergio Benenti, as well as Claude Albert, Daniel Bernard, Yvonne Choquet-Bruhat, Nicole Desolneux, Claude Godbillon, Joseph Grifone, Joseph Klein, Jean Martinet and Pierre Molino who have shown great interest in this book.

Yvette Kosmann-Schwarzbach and Jean Parizet have graciously permitted us to use their unpublished lectures. Robert Lutz and Ernesto Lacomba have given us the benefit of their knowledge of contact structures and transitive mechanical systems. We are pleased to thank them here, along with Bartholome Coli, Alexandra Cohen, Edmond Combet, Pierre Dazord, Moshe Flato, Jean-Claude Houard, Richard Kerner, Paul Kree, Lucette Losco, Carlos Moreno, Rene Ouzilou, Pham Mau Quan, Guy Pichon and Jean Vaillant, who have frequently discussed with us the various subjects treated here.

Our warmest thanks are extended to the friends who have read parts of our manuscript, suggested improvements and helped us to eliminate some errors: Richard Cushman, Michel Duflo, Thomas Delzant, Jean-Pierre Frant;oise, Daniel Gutkin, Ana Maria Justino, Remy NicolaY, David Wigner, and particularly Yvette Kosmann-Schwarzbach. None of them was

xvi Preface

able to read the entire manuscript, so the authors alone bear the exclusive responsibility for any error which may remain.

We also thank all our colleagues who have encouraged us, and the students who, by their discussion and suggestions, helped us improve upon the material presented here.

Many thanks also to Mrs. Gueguen and Mrs. Arpin, who typed an earlier version of this book in French.

It is a pleasure to thank Bertram Schwarzbach for his very careful translation and typing of the text on a microcomputer, and his invaluable help in proofreading. Our warmest thanks also go to Dominique Foata and Jacques Desarmenien for initiating us in the use of 'lEX, the typesetting system with which was composed this book, to Dominique Bernardi and Robert Pallu de la Barriere who helped us to use microcomputers, to Georges Weil, who printed the final camera-ready sheets on the laser printer of the computer center of the CNRS in Strasbourg, and to Gerard Thomas, who printed again, at the IFP in Rueil­Malmaison, pages on which spelling mistakes were noticed during the production of the book.