CLASSICAL FIELD THEORYAND ANALOGY BETWEEN

NEWTON’S AND MAXWELL’SEQUATIONS

ZBIGNIEW OZIEWICZ∗

Universidad Nacional Autonoma de Mexicooziewicz@redvax1.dgsca.unam.mx

October 1993

Abstract

A bivertical classical field theory include the Newtonian mechanicsand Maxwell’s electromagnetic field theory as the special cases. Thisunification allows to recognize the formal ana Newtonian mechanicsand Maxwell’s electrodynamics.

Dedicated to Professor Constantin Piron

∗On leave of absence from University of Wroc law, Poland. Partially supported byPolish Committee for Scientific Research, Grant 2 2419 92 03/92.

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Contents

Foreword

I. Axiomatique classique

1. Vertical distribution and filtration of forms

2. Classical field theory

3. Submanifolds

4. Hamilton-Lagrange field theory

II. Phenomenology

1. Phenomenological field theory

2. Louville’s differential forms

3. Poincare-Cartan submanifolds

III. Field equations

1. The Hodge’s map

2. Calculus with the splitting

3. Dirac operator and codifferential

4. Legendre’s transforms

5. Second order field equations

References

Foreword

This paper is inspired by Professor Constantin Piron.1. Professor Piron’scredo is the unity of human’s knowledge gained by mathematicians, physi-cists, philosophers, theologians. Piron is known, among other, as the creator

1Professor Piron visited University of Wroc law the first time in 1980. During Mar-tial Law in Poland, when I was in prison in 1982, and again in 1984, Professor Pironbravely keep scientific contact with me, supporting my family and Solidarity, which re-quired courage and a good heart.

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of the New Quantum Mechanics (1983), a realistic theo unifying the classicaland quantum physics.

The main ingredients of the classical physics are Newtonian mechanicsand Maxwell’s electrodynamics. These phenomenological theories are relatedto the diverse phenomena and are presented as disconnected in a separatecourses, textbooks and monographs. The one aim of this paper is to exibitsthe formal analogies among these theories in the framework of the almostmultisymplectic geometry, dΩ 6= 0.

We consider the bundle of the exterior differential forms, in mechanics ofdimension 1 + 4n, in electromagnetism of dimension 4 + 20n, with the phe-nomenological not presymplectic differential form. Our theory is bivertical(definition 5) for arbitrary dimensional space-time manifold. We determinethe four Poincare-Cartan subbundles: hamiltonian, lagrangian and two newnot-named subbundles. This leads to the twelve Legendre’s transforms amongthese subbundles, of which two are the well known.

The field equations of considered bivertical theory for n = 1 reduce to thatof Newton’s equations and for n=4 to Maxwell’s equations. This unificationallows to see analogies among the notions of Newtonian dynamics and ofMaxwell’s theory. In particular, force field↔ current, the London’s equationin electromagnetism is an analogy of the harmonic oscilator force in Newton’sdynamics, one can pose the Kepler’s problem in Maxwell’s electrodynamicsby formal analogy to the Kepler’s problem in mechanics, etc. In the non-linear model we discuss the second order field equations.

The limited space do not allows to include here the discussion of the con-servation laws, Noether currents, energy-momentum tensor, Poynting differ-ential form etc., from the point of view of the presented formal unification.

It is a pleasure to thank Professor Constantin Piron for many inspiringdiscussion during the last 14 years of our friendship. The subject of the paperwas presented in the fall 1992 at the University of Wroc law. Some aspectsrelated to this paper are elaborated by Magdalena Gusiew in the diplomathesis (1993). I am gratefull to Magdalena Gusiew and Grzegorz Jastrzebskifor iluminating conversations.

History. Multisymplectic geometry in classical field theory was initiated byDedecker in 1953 and was developed in Warsaw by W lodzimierz Tulczyjewaround 1968, and by Kijowski (1973), Gawedzki (1972), Szczyrba and Kon-dracki (1979). In Chechia by Krupka since 1975. See Kijowski and Tulczyjew

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(1979).

Notations.Λ ≡ ΛE ≡ ⊕Λk is de Rham complex of the differential forms on a manifoldE, F ≡ Λ0, Z ≡ α ∈ Λ, dα = 0.W ≡ WE ≡ derF is the Lie F -module of the (one)- such that W∧ ≡⊕W∧k is the Grassmann F -algebra and graded Lie IR-algebra of multivectorfields, W∧0TheleftGrassmannmultiplication(analgebramap),e:Λ −→ linΛ,is eαβ ≡ α∧β, e for exterior. Then i denote the interior product, i : W∧ −→linΛ, which is anti-algebra map.|α| ≡ gradeα ∈ IN and ψ denote the automorphism of Grassmann algebras,ψα ≡ (−1)|α|α.

I. Axiomatique classique

Vertical distribution and filtration of forms

Let E be a manifold with the distribution Ver⊂ W which is said to be thevertical distribution.

Definition 1 The F-submodule

Λ(k) ≡ α ∈ Λ, iZα = 0 ∀ Z ∈ Verissaidtobethesubmoduleof

k−verticalforms.ThefactormoduleisdenotedbyΛ[k] ≡ Λ(k)/Λ(k−1) and if α ∈Λ then α/(k) ∈ Λ/Λ(k).

Corollary 1 Λ(k)∧Λ(l) ⊂ Λ(k+l), Λ[k]∧Λ[l] ⊂ Λ[k+l] and we have the filtrationof forms

Λ(0) ⊂ . . . ⊂ Λ(k) ⊂ Λ(k+1) ⊂

Let Ver be the submodule of the differential one-forms anihilating Ver

Ver ≡ α ∈ Λ1; α(Ver) = 0.

Definition 2 The differential form α on E,Ver is said to be vertical ifα ∈ Ver∧. Ver∧0 ≡ FE.

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Let for α ∈ Λ, Distα be the associative distribution of α,

Distα ≡ X ∈W ; iXα = 0.

The form α is decomposable iff

dimE = |α|+ dim Distα.

Lemma 1 .

(i) The form α is 0-vertical iff α is a vertical, Λ(0) = Ver∧,

iVerα = 0 ⇐⇒ α ∈ Ver∧.

(ii) θ ∈ Ver∧

|θ| = dim Ver

Dist θ = Verθ is decomposable

There is the one to one correspondence between the (vertical) distributionsand one-dimensional modules of decomposable forms. We will identify

E, θ ≡ E,Ver ≡ Dist θ,

where the decomposable form θ is defined up to the multiplication b FE. Thedistribution Ver is locally integrable iff dθ is 1-vertical and then E, θ islocally fibered, dθ ∈ Λ(1) ⇐⇒ d(fθ) ∈ Λ(1) for 0 6= f ∈ F .

If E is fibered over oriented manifold M, vol, Eπ−→M, then θ ≡ π∗vol

is a decomposable cocycle.

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Classical field theory

Definition 3 .

(i) The classical field theory is a triple E,Ver,Ω, where Emanifold,VerisadistributionandΩis a differential form on E such that |Ω| = 1 + codimVer > 1.

(ii) The submanifold φ of E is said to be the solution (the space-time manifold)of E,Distθ ≡ Ver,Ω if for every vector field Z on φ∗θ 6= 0 and φ∗iZΩ =0.(1)

(iii) The field theory E,Ver,Ω is said to be regular i integrable distributionHor tangent to the solutions of the the equations (1 is complementary to Ver,

Hor ∩ Ver = 0 and W = Hor ∪Ver.

Comment. The field theory is regular if every solution φ (1) is transversalto Ver and dimφ = codim Ver ≡ |θ|.

The form Ω determine the F -linear map2

Ω : W∧|θ| −→ Λ1.

Proposition 1 Let ker Ω ⊂W∧|θ|. Then

(i) dim ker Ω = 1 =⇒ E,Ver,Ω is re

(ii) Let E,Ver,Ω be regular, codimVer ≡ |θ|Ω be a cocycle (so Ω is sym-plectic). Then dim ker Ω = 1, (=⇒ dimE =odd).

Comment. If |θ| = 1, then a cocycle Ω is regular iff dim ker Ω = 1. The|θ| = 1 refers to mechanics and the property to be regular is said to be theclassical determinism3.

Example. Regular field theory E, θ,Ω need not imply that dim ker Ω =1. Let dimE = 1 + 4n with a chart t, qA, vA, pA, fA and θ ≡ dt. LetΩ ≡ (dpA − fAdt) ∧ (dqA − vAdt), then dΩ 6= 0, dim ker Ω = 1 + 2n and thismechanics E, θ,Ω is regular.

Proof of Proposition 1. The integrable distribution Hor⊂ W tangent tothe solutions of the field equations (1) needs to sa the two conditions

2The form Ω can be viewed as the retrangular matrix

(dimE|θ|

)× (dimE).

3The symplectic mechanics, dΩ = 0 and dim ker Ω = 1, on jet manifolds of arbitraryorder has been presented by Olga Krupkova (1992).

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• θ(Hor∧|θ|) 6= 0 (=⇒ dim Hor ≥ |θ|+ dim(Hor ∩ Ver)),

• Hor∧|θ| ⊂ ker Ω,

(=⇒ dim(Hor∧|θ|) =

(dim Hor|θ|

)≤ dim ker Ω

).

The last condition imply

dim ker Ω = 1 =⇒ dim Hor ≤ |θ|.

It follows that Hor∩Ver= 0 and dim Hor = |θ| ≡ codim Ver, which completethe proof of (i).

Let Ω be a cocycle. Then the associated distribution, ker Ω ⊂ isintegrable.If—θ| =1, then Hor= ker Ω is integrable and the regularity of Ω imply that dim ker Ω =1. 2

Corollary 2 (Gawedzki 1972) Let E,Ver,Ω be regular field theory. Thenit is sufficient to consider the field equations (1) for vertical vector fields only,

φ∗iVerΩ = 0 =⇒ φ∗iWΩ = 0.

Proof. Let the distribution Hor be as in the proof of the Proposition 1 Weneed to show that Ω(Ver∧Hor∧|θ|) = 0 =⇒ Ω(W ∧Hor∧|θ|) = 0. This is thecase if W = Hor ∪Ver and Hor∧(1+|θ|) = 0. 2

Submanifolds

Let 0 ≤ i ≤ |Ω| and 0 ≤ j ≤ 2. Let Ψ3i+j → Ψ3i−2 → E and Ωi ∈Λ(Ψ3i−2), Ω0 ≡ Ω be collections of maximal submanifolds and differentialforms defined by the conditions

Ψ∗θ 6= 0, (Ψ3i)∗dΩi = 0, (Ψ3i+1)∗Ωi = dΩi+1, (Ψ3i+2)∗Ωi = 0.

Therefore dim Ψ3i−1 = dim Ψ3i = dim Ψ3i+1 and

φ → . . . → Ψ4 → (P ≡ Ψ1) → Ψ → E.

Definition 4 .

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(i) The submanifold Ψ → E is said to be the pre-symplectic manifold ofE, θ,Ω if Ψ is a maximal submanifold anihilating dΩ,

Ψ∗θ 6= 0 and Ψ∗dΩ = 0.

The presymplectic manifold Ψ is said to be symplectic if the field theoryΨ,Ψ∗θ,Ψ∗Ω is regular. The regular cocycle Ψ∗Ω is said to be thesymplectic form on Ψ,Ψ∗θ.

(ii) The submanifold P → Ψ → E is said to be Poincare-Cartan submani-fold (exact presymplectic) if P is a maximal submanifold on which Ωis exact,

P∗θ 6= 0 and P∗Ω = dα.

The presymplectic potential α ≡ Ω1 is said to be the Poincare-Cartanform. If the field theory P,P∗θ, dαregularthenα is said to be theregular Poincare-Cartan form.

(iii) The submanifold L ≡ Ψ2 → Ψ → E is said to be manifold of E, θ,Ωif L is a maximal submanifold anihilating Ω,

L∗θ 6= 0 and L∗Ω = 0.

(iv) The submanifold Ψ4 → P → Ψ → E is said to be the Hamilton-Jacobimanifold if Ψ4 is a maximal submanifold on which the Poincare-Cartandifferential form is exact,

Ψ4∗θ 6= 0 and Ψ4

∗α = dΩ2.

The potential Ω2 is said to be the Hamilton-Jacobi differential form,|Ω2| = |θ| − 1. The equations in (iii-iv) are said to be the Hamilton-Jequations.

On presymplectic submanifold Ψ∗Ω is a cocycle, the action integral Oziewicz1992) and the field equations of the definition 3 are the Euler-Lagrang

The (pre)symplectic Ψ and Poincare-Cartan P submanifolds are knownas the phenomenological material relations (p = mv, Kepler problem fA =−q−3qA, D = ε0E, B = µ0H, London equation Jµ = Aµ, etc) and are dis-cussed in the last sections.

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Jacobi in 1838 proved that in mechanics the lagrangian and the Hamilton-Jacobi submanifolds, Ψ2 and Ψ4, are the families of solutions, φ → Ψ4 andφ → (P Ψ4) → Ψ2. The coordinate-free proof is in (Oziewicz and Gruhn1983). The extension of the Jacobi theorem beyond mechanics is not known.

The dimensions of submanifolds are given for the phenomenological fieldtheory (13) and follow from the considerations in the part II, see formulas(17-18).

E ← Ψ← P ← Ψ4 ← φ↓dim ↓dim ↓dim ↓dim

mechanics 1+4n 1+2n 1+n 1strings 2+6n 2+3n 2+n 2

electrostatics 3+8n 3+4n 3+n 3magnetostatics 3+12n 3+6n 3+3n 3

Klein-Gordon fields 4+10n 4+5n 4+n 4electromagnetism 4+20n 4+10n 4+4n 4

God’schoice

materialrelations

“quantumspace”

timespace

space-time

Hamilton-Lagrange field theory

Definition 5 The field theory E,Ver,Ω is said to be k-vertical if 0 6= dΩ ∈Λ(k) and dΩ 6∈ Λ(k−1) or if dΩ = 0, Ω ∈ Λ(k) and Ω 6∈ Λ(k−1).

Comment. For a cocycle Ω this definition was introduced by Kondrack(1978). The notion of the k-vertical field theory is essential for the theory ofthe Poincare-Cartan forms if Ω 6∈ Z and for the Hamilton-Jacobi theory ifΩ ∈ Z.

Note that

Λ|θ|+j(k) /(k − 1) 6= 0 iff j ≤ k ≤ |θ|+ j.

In particular |Ω| = 1 + |θ| =⇒ Ω ∩ Λ(0) = 0.In the section calculus with the splitting we are showing that (see the

definition 10 and corollary 3)

dΛ(k) ⊂

Λ(k+1) if Ver is integrable, dθ ∈ Λ(1)

Λ(k+2) otherwise, dθ 6∈ Λ(1)..

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Let the distribution Ver be integrable. In this case

dΩ ∈ Λ(k)

dΩ 6∈ Λ(k−1)

Ω ∈ Λ(k−1) ⊕ ZΩ 6∈ Λ(k−2)TheformdΩ 6= 0 is k-vertical iff Ω can be decomposed (not uni (

Λ/Zµ−→ Λ,

Λ/Λ(k−1)ν−→ Λ,

µ(Ω/Z) ∈ Λ(k−1), ν(Ω/(k − 1)) ∈ Zand Ω = µ(Ω/Z) + ν(Ω/(k −1)) ∈ Λ(k−1) Suchsplittingsifexistsarenotunique, theyaredetermineduptothe(k-

1)−verticalcocyclesΛ(k−1) ∩ Zloc' dΛ(k−2).

Let the field theory E,Ver,Ω be k-vertical. Then the splitting µ deter-mine ν and vice versa. Locally

ν(Ω/(k − 1))loc' dω,

and Ω = f + dω, where f ≡ Ω− dω ∈ Λ(k−1). (3)

The form ω is determined modulo (k−2)-vertical forms. On Poincare-Cartansubmanifold, P → E, Ω and f are exact,

dF ≡ P∗f, dαF ≡ P∗Ω, (4)

and αF = F+P∗ω mod ZPThereexiststhecorrelationbetweenthedecompositions(2−3)andthePoincare−Cartan

Depending on the choice of f in (3), the potential F could coincide (upto t sign) with the hamiltonian H or with the lagrangian L (see the nextsections freedom in the decomposition (2-3) allows to see more possibilities.

If f is (k − 1)-vertical on E,Ver, then F (4) is (k − 2)-vertical onP,P∗θ,

(Λ|θ|(k−2) ⊕ ZP) 3 F 7−→ dF ıThePoincare− Cartanequation(4),

dF=P∗f, allows to express P∗ω in terms of the partial derivatives of F wrt thebasis of the FP-module Λ

|θ|+1(k−1) ⊂ ΛP . Therefore differential form F determine

the Poincare-Cartan form,

Λ|θ|(k−2)/ZP 3 F 7−→ αF ∈ P

|θ| mod ZP .

This motivate the definition

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Definition 6 Let the distribution Ver be integrable and Ω 6∈ Z.

(i) Let 2 ≤ k ≤ |dΩ| and let the field theory E,Ver,dΩ ∈ Λ(k) and dΩ 6∈

Ω(k−1). Then the FP-module Λ|θ|(k−2) is said to be the module generatingthe Poincare-Cartan forms.

(ii) The field theory E, θ,Ω is said to be the Hamilton-Lagrange fieldtheory, abbreviated by HL, if the FP -dimension of the generating moduleof the Poincare-Cartan forms is 1.

For HL field theory the Poincare-Cartan form is determined by one (pseudo)-scalar function, (lagrangian, hamiltonian,. . .).

Lemma 2 The field theory E,Ver,Ω 6∈ Z is HL iff dΩ is bi-vertical, dΩ ∈Λ(2).

Proof.

dimΛ|θ|(k−2) =min(k−2,|θ|)∑

i=0

(dim Ver

T

)herefore

dimΛ|θ|(k−2) = 1 iff E,Ver,Ω 6 Z is bi-vertical.Comment.InHLfieldtheorythe(local)hamtypetheoriestheanalogoushamiltonianorlagrangiandifferentialformsarenomoreverticalandtJacobitheoryifΩ ∈ Z.

Partial derivatives of vertical forms. Note that

dimΛ|θ|+1(1) = dim Ver.

We will suppose that the modul Λ|θ|+1(1) , on E as well as o submanifold P, is

generated by the differentials of the homogeneous vertical forms (of differentdegrees),

Λ|θ|+1(1) ≡ gendwA, wA ∈ Λ(0).

This means that ∀ α ∈ Λ|θ|+1(1) has the unique de

α = dwA ∧ αA, wA, αA ∈ Λ(0).

In particular the generating set dwA determine the partial derivatives ofthe highest degree vertical forms

Λ|θ|(0) 3 F 7−→ dF = dwA ∧

∂F

∂wA∈ Λ

|θ|+1(1) .

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Example

dL ≡ dqA ∧∂L

∂qA+ dvA ∧

∂L

∂vA.

II. Phenomenology

Phenomenological field theory

Let qA, vA, pA, fA; A ∈ I ⊂ IN be a collection of the vertical differentialforms on E,Ver. Newton’s and Maxwell’s phenomenological equations aswell as of electrostatics and magnetostatics have the form

φ∗(dqA − vA) = 0

φ∗(dpA − fA) = 0. (6)

The differential forms qA, vA, pA, fA in (6) are independent, as they aredetermined by independent experiments. The phenomenological material re-lations among these fields are the consequence of the further independentmeasurements. This was stressed by Newton (1686) and Maxwell. After La-grange it became customary to present the Newton’s equations (as well as ofelectrodynamics, 2A = j) as the second order from the begining, contrary tothe original Newton’s presentation. Also Piron stressed (e.g. in Piron’s Lec-tures on electrodynamics, 1989) that the equations (6) should not presupposethe material relations.

The phenomenological material relations are the equations for the (pre)-symplectic submanifold of the field theory E,Ver,Ω (definition 4 (i)), andare discussed in the next sections.

The aim is to determine the most general regular field theory E,Ver,Ωwhich field equations (1) coincide with the experimental one (6). The diffe-rent field theories with the same set of the first ord equations (6), will leadsto the different (pre)symplectic submanifolds and therefore to the differentsecond order equations considered in the part III. Denote

ϑA ≡ dqA − vA

ωA ≡ dpA − fA. (7)

The solutions φ (6) anihilate the ideal generated by ϑA, ωA, therefore forthe regular field theory we need the equality of the ideals,

geniVerΩ = genϑAIfthedistributionV erisintegrablethen

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ϑA, ωA are 1-vertical. The most general form Ω compatible with (8) needsto be 2-vertical of t

Ω ≡∑A,B

(KAB ∧ ωA ∧ ϑ

B + ΓAB ∧ ϑA ∧ ϑB + χAB ∧ ωA ∧ where

KAB,ΓAB, χ

AB is the collection of a vertical differential forms such that

ΓAB = (−1)|ϑA||ϑB|ΓBA, χAB = (−1)|ωA||ωB|χBA.

Define the dimension of the form as the dimension of the factor module,

dimα ≡ dim Ver− dim(Distα) ∩ Ver. (10)

One of the necessary condition for the implication (9) ⇒ (6) is

dim Ver =∑A

(dim dqA + dim dvA + dim dpA + dim dfA). (11)

Definition 7 The field theory E,Ver,Ω with Ω of the form (9) is said tobe the phenomenological field theory.

Because Ω is IN-homogeneous then

|vA| = 1 + |qA|

|fA| = 1 + |pA|

|qB|+ |fA|+ |KAB | = |θ|

|vB|+ |pA|+ |KAB | = |θ| (12)

In mechanics |K| = |Γ| = |χ| = 0. The conditions

∀ A |ωA| ≥ |ϑA| and |K| = |χ| = 0,

determine the unique grades for mechanics and string theory and n possibil-ities for |θ| = 2n− 1 and 2n. In this case χ can contribute in mechanics andmagnetostatics only.

|θ| |q| |v| |p| |f | |Γ|1 0 1 0 1 0 mechanics2 0 1 1 2 1 strings3 0 1 2 3 2 electrostatics

1 2 1 2 0 magnetostatics4 0 1 3 4 3 Klein-Gordon scalar fields

1 2 2 3 1 electromagnetic field

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If K,Γ, χ are the vertical cocycles then the field theory (13) is HL, dΩ isbivertical, and

Ω = d(KBA ∧ pB + ψΓAB ∧ q

B)− (KBA ∧ fB + ΓBv ) ∧ ϑA + ωA ∧ ωA.

Effectively the “momenta-induction” and “force-current” are rotated andtranslated by “connection Γ”, a natural description of the velocity-dependentforces,

pA 7−→ KBA ∧ pB + ψΓAB ∧ q

B,

fA 7−→ KBA ∧ fB + ΓBA ∧ v

B.

The phenomenological symplectic mechanics (9) without of the χ-terms hasbeen considered by Jadczyk and Modugno (1992).

Consider HL field theory

Ω ≡∑

ωA ∧ ϑA ∈ Λ(1) ⊕ Z ⊂ Λ(2). (13)

The 1-vertical differential forms h, l, s, t are defined by the decomposi-tions (see formulas (2-3)),

Ω = −h+ d(pA ∧ dqA)

= +l + dpA ∧ (dqA − vA)

= +s+ dqA ∧ ψ|θ|(dpA − fA)

= +t+ dqA ∧ ψ|θ|(dpA − fA)− pA ∧ vA. (14)

Explicitely

h ≡ dpA ∧ vA + dqA ∧ ψ|θ|fA,

l ≡ dvA ∧ ψ1+|θ|pA − dqA ∧ ψ|θ|fA,

s ≡ dpA ∧ vA − dfA ∧ (−)|θ|ψqA,

t ≡ dvA ∧ ψ1+|θ|pA + dfA ∧ (−)|θ|ψqA. (15)

Remark.

l + h ≡ t+ s ≡ d( pA ∧ vA),

h− s ≡ t− l ≡ d(ψfA ∧ qA). (16)

If E is fibered and the differential forms qA, vA, pA, fA are the Louville’sforms, considered in the next section, then the differential form Ω (13) isregular and imply the field equations (6).

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Louville’s differential forms

Let M be a manifold and for p ∈M, T ∗pM be the vector IR-space of exterior

forms at p. Let T kMπ−→ M, be the bundle of exterior k-forms, (T kM)p ≡

(T ∗pM)∧k, with (T ∗pM)∧0 ≡ IR.The differential form α ∈ Λk

M determine the section αs ∈ Γ(M,T kM),ΛTkM 3 λ −→ α∗sλ ∈ ΛM .

Definition 8 The differential k-form λ ∈ ΛTkM is said to be the Louville’sform if

α∗sλ = α for every α ∈ ΛkM .

There exists the unique Louville’s differential k-form and has the local form

λ =1

k!

∑λµ1...µkπ

∗(dtµ1 ∧ . . . dtµk).

The Louville’s differential forms of arbitrary degree has been introduced byTulczyjew in 1979. The Louville’s forms are vertical wrt θ ≡ π∗vol

Let the manifold E be the bundle, Eπ−→ M, of the exterior forms of

different degrees on the manifold M,

E ≡⊕A

(T |q

A|M)⊕ (T |vA|M)⊕ (T |pA|M)⊕ (T |fALet

qA, vA, pA, fA be a collection of the Louville’s forms on E.If λ is a Louville’s differential form on E then according to the definition

(14-15),

dim (dλ) =

(dim M|λ|

). (18)

The field theory E (17), π∗volM ,Ω (13) is regular and imply th field equa-tions (6). From formula (11) we get the dimensions, dimE, listed in theTable after the definition 4. The Louville’s forms K,Γ, χ, for simplicity,are not included in E (17). The Louville’s forms K,Γ, χ in (9) contributeto dimE. In mechanics with (9), |θ| = 1, dimE = 1 + 4n+ 2n2 − n.

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Poincare-Cartan submanifolds

Consideration of this section are the same for the case of presymplectic andPoincare-Cartan submanifolds. To be specific we will consider only Poincare-Cartan submanifolds, definition 4 (ii), and only for the case when E is abundle of exterior forms (17) with the conditions (12).

Let the bundle E be splitted with the fiber-preserving projectors πP/C ,

PπP←− E = P ⊕ C

πC−→ C↓ π ↓ π ↓ πM = M = M

Let the subbundles P and C be of equal dimensions, dim Ver (14) on E isThe Poincare-Cartan submanifold P → E is a subbundle of E with dimP =dim P = dim C. Let πP |P be the fiber-preserving isomorphism. Then Pwill be identified with the injection P : P → P ⊂ E, πP P = idP , andϕ ≡ πC P : P −→ C, is a fiber-preserving bundle map. If θ ≡ π∗volM thenP∗θ ≡ θ ∈

Consider the splittings of the bundle E for which Ω have the form (com-pare with the decomposition (2-3))

Ω = ±∑

dπ∗Pα ∧ π∗Cβ + dω. (19)

On Poincare-Cartan submanifold the form∑dπ∗Pα ∧ π

∗Cβ is exact,

P∗(∑

dπ∗Pα ∧ π∗Cβ) =

∑dα ∧ ϕ∗β ≡ dF ∈ ΛP .

Therefore

ϕ∗β ≡∂F

∂α.

The differential forms Ω, h, l, s, t (12-15) are exact on Poincare-Cartsubmanifold P → Ψ → E. In particular the hamiltonian H and the la-grangian L are the differential forms on different Poincare-Cartan subbundlesand are defined as the potentials,

dH ≡ P∗hh, dL ≡ P∗l l, dS ≡ P∗s s, dT ≡ P∗t t. (20)

The compositions, say L ≡ P−1h Pl etc, are sa to be the Legendre’s trans-

forms. With the help of the identities (16) the Legendre’s transforms allow

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to calculate, for example, the lagrangian L for th given hamiltonian H,

dL = P∗l l = (P∗l P−1∗h P∗h)d(pA ∧ v

A)− h

= L∗dP∗h(pA ∧ v (21)

Therefore, modulo cocycles

L = (L∗ P∗h)(pA ∧ vA)− L∗H mod Z

The Poincare-Cartan forms α, dα ≡ P∗Ω, can be expressed in terms ofL,H, S, and T if we identify the decompositions (14-15) with (19-20).

ϕ∗hvA ≡

∂H

∂pA, ϕ∗hfA ≡ ψ|θ|

∂H

∂qA,

αH ≡ −H + pA ∧ dqA modZP ,

dαH =

(dpA − ψ

|θ| ∂H

∂qA

)∧

(dqA −

∂H

∂pA

). (23)

ϕ∗l pA ≡ ψ(1+|θ|) ∂L

∂vA, ϕ∗l fA ≡ −ψ

|θ|

ϕ∗svA ≡

∂S

∂pA, ϕ∗sq

A ≡ (−)1+|θ|ψ∂S

∂fA,

αS ≡ S +∂S

∂fA∧ ψ(1+|θ|)(dpA − fA) modZP ,

dαS = −

d∂S

∂fA− (−)|θ|ψ

∂S

∂pA

∧ ψ(1+|θ|)(dpA − fA). (24)

ϕ∗tpA ≡ ψ(1+|θ|) ∂T

∂vA, ϕ∗t q

A ≡ (−)|θ|ψ

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III. Field equations

The Hodge’s map

We give slight generalization of the Hodge’s map for the manifold with thedistribution Ver⊂ W. When θ = vol (⇔ Distθ ≡ Ver = 0) then this gene-ralization collapse to the usual one. The formulas in the lemma 4 do notsuppose that the scalar product g is symmetric.

By definition the form θ is decomposable therefore

Ver = α ∈ Λ1E, α ∧ θ = 0.

Lemma 3 If α ∈ Ver∧ then for each multivector X ∈W∧

α ∧ iXθ = (−)|α|(1+|X|)iiαXθ.

If α ∈ Ver∧ and |α| = |X| then

α ∧ iXθ = (αX)θ.

Consider the factor module W/Ver as the dual to Ver and let Z ∈(W/Ver)∧|θ| be such that θZ = 1.

Let g, k and their pull-backs (a transpositions) g∗, k∗ be F -linear maps

g, g∗ : Verθ −→ W/Ver,

k, k∗ : W/Ver −→ Ver θ.

For α and β in Ver, g(α⊗β) = g∗(β⊗α), g and k need not to be symmetric(as in Oziewicz 1986). The extension of maps (28) to Grassmann algebramaps (Ver)∧ −→ (W/Ver)∧ is denoted by the same letters.

Definition 9 The F-linear maps ∗ ≡ ∗(g,θ) and ? ≡ ?(k,Z),

(Ver)∧ 3 α 7−→ ∗α ≡ igαθ ∈ Ver∧,

(Ver)∧ 3 α 7−→ ?α ≡ kiαZ ∈ Ver∧,

are said to be the Hodge’s maps with θ ≡ ∗1 and ?θ

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Lemma 4 .

(i) ∗g eα = igα ∗g ψ(ii) ∗g∗ igα = eψα ∗

(iii) ψ ∗ = (−)|θ| ∗ ψ.

(iv) ∗(g,θ) ∗(g∗,θ) = g(θ ⊗ θ) ψ1+|

.(v)Ifα and β ∈ (Ver θ)∧ are of equal grades, then

α ∧ ∗β = g(β ⊗ α)θ = g∗(α⊗ β) = β ∧ ∗g∗α,

g(∗α⊗ ∗β) = g∗(θ ⊗ θ) · g(α⊗ β).

The function g(θ ⊗ θ) ∈ FE is said to be the θ-determinant of g,

detθg ≡ g(θ ⊗ θ)Let

θg ≡ |g(θ ⊗ θ)|−1/2θ, then g(θg ⊗ θg) = sign detθ g.

Calculus with the splitting

Let ∀X ∈ W, ∇X be a zero-grade derivation of the tensor algebra such that∇X |F ≡ X. The derivation ∇X factors to the derivation of exterior algebra,∇X ∈ derΛ. For form α, the composition eα ∇X is a (skew) derivation of Λ,|eα ∇X | = |α|. Let Λ1 ≡ spanεa and W≡ spanXawhereεaXb ≡ δab . Let∇a ≡ ∇Xa , then ∇ ≡ εa ∧ ∇a is a skew derivation, |∇| = +1 and ∇|F ≡ d.The difference ∇− d is said to be the torsion of ∇,

∇ = d+ T ∈ skew derΛ.

The splitting of the Definition 3 (iii) determine the bigradation (IN × IN-grading) of Λ = ⊕Λp,q, with the projectors πp,q, Λp,q ≡ πp,q(Λp+q). Everyderivation of grade +1, is determined by values on generators F , dF, andtherefore is decomposed as the sum of four bigraded derivations.

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Definition 10 Let Ver≡ π1,0Λ1, Hor≡ π0,1Λ1 and bigrade ≡ | · |.

dV |F ≡ π1,0 d, dV |dF ≡ π2,0 d π1,0 + π1,1 d π0,1, |dV | = (+1, 0),dnV |F ≡ 0, dnV |dF ≡ π0,2 d π1,0, |dnV | = (−1,+2),dH |F ≡ π0,1 d, dH|dF ≡ π0,2 d π0,1 + π1,1 d π1,0, |dH | = ( 0,+1),dnH |F ≡ 0, dnH|dF ≡ π2,0 d π0,1, |dnH | = (+2,−1).

Note that dV : F → Ver and dH : F → Hor. The distribution Ver isintegrable iff dnV = 0 and the distribution Hor is integrable iff dnH = 0. Thetensors dnV and dnH measure the noninvolutiveness of these distributions.

Corollary 3d = dnH + dV + dH + dnV , (25)

and d2 = 0 is equivalent to seven conditions:

dnV dnV = 0, bigrade = (−2,+4)dH dnV + dnV dH = 0, bigrade = (−1,+3)

dV dnV + dnV dV + d2H = 0, bigrade = ( 0,+2)

dnV dnH + dnH dnV + dV dH + dH dV = 0, bigrade = (+1,+1)dH dnH + dnH dH + d2

V = 0, bigrade = (+2, 0)dV dnH + dnH dV = 0, bigrade = (+3,−1)

dnH dnH = 0, bigrade = (+4,−2).

If the differential form α is vertical then dV α is vertical, dHα is 1-vertical anddnV α is 2-vertical. From the definition 1 it follows for the phenomenologicalfield theory E,Ver,Ω (13) that

Ω is

2− vertical if Ver is integrable,4− vertical otherwise.

Dirac operator and codifferential

Let g : Ver →Hor, and let γ : Ver∧ −→ lin(Ver∧) be the left Cliffordmultiplication, for ε ∈ Ver and α ∈ (Ver)∧,

γεα ≡ ε ∧ α + igεα, (26)

(e.g. Oziewicz 1986). Denote γµ ≡ γ(εµ).

20

Definition 11 The operator D ≡ γµ∇µ : Λ→ Λ is said to be the splitting-dependent Dirac operator.

When Ver = 0 then D collapse to the usual Dirac operator.Let Ver ≡ spanεµ and Hor≡ spanhµ, where εµhν ≡ δµν . Then

♥ ≡ εµ ∧∇µ ∈ skew der L

♥ = dV + T♥. (27)

Because of (27) we have D = ♥+ δ where δ is a (splitting-dependent) codif-ferential,

δ ≡ igεµ ∇µ, |δ| = −1.

Using Lemma 4 [(i) and (ii)] we calculate

δ ∗g = + ∗g ♥ ψ + igεµ (∇µ∗g),

∗g∗ δ ψ = −♥ ∗g∗ + e∗g∗ ).

Therefore

δ = (sign detθg)(∗(g,θg) ♥ ∗(g∗,θg) (−ψ)|θ| + “∇ ∗ ”),

where“∇ ∗ ” ≡ igεµ (∇µ∗(g,θg)) ∗(g∗,θg) ψ

1+|θ|.

If the Hodge’s map is parallel, ∇∗ = 0, and T♥ = 0 (28), then ♥ = dV andthe codifferential has the form

δ = (sign detθg) ∗(g,θg) dV ∗(g∗,θg) (−ψ)|θ|. (28)

δ2 = −det g∗

| det g|∗g d

2V ∗g ψ

1+|θ|.

If Hor is integrable then d2V = δ2 = 0. The square of the Dirac operator

4g ≡ (dV + δ)2 = δ dV + dV δ is said to be the (splitting-dependent)Laplace-Beltrami-Hodge operator, which commutes with the Hodge’s map ifg is symmetric,

4g ∗ = ∗ 4g∗.

21

Legendre’s transforms

Let qA, qA, . . . be the collection of the vertical forms, |qA| = |qA|, and let

q2 ≡ |g∑

qA ⊗ qA|, l ≡ l(q2). (29)

With the help of formula (v) in the lemma 4 we calculate

d(l∑

qA ∧ ∗qA) = (q2l)′d∑

(qA ∧ ∗qA)− q2l′∑

g(qA ⊗ qA)dθ. (30)

If f, g ∈ F then

d(f ∧ ∗g) = df ∧ ∗d+ dg ∧ ∗f + fgdθ.

This is no more valid for arbitrary grade. The formula (v) of lemma 4 suggestth simplifying assumption

d(α ∧ ∗gβ) ' dα ∧ ∗gβ + dβ ∧ ∗g∗Thiscouldbethecaseif

θ is a cocycle and α and β are the vertical differential forms on which d = dV .Let θ be a cocycle. With the help of (31-32) we get

d(l∑

qA ∧ ∗qA) ' 2(q2l)′∑

dqA ∧ ∗qA. (32)

Every vertical form of highest grade can be expressed in terms of the in-dependent vertical forms. In particular the hamiltonian, H ∈ Λ

|θ|(0), can be

expressed in terms of pA, qA. Consider the example

H ≡1

2k(p2)

∑pA ∧ ∗p

A +1

2l(q2)

∑qA ∧ ∗qA. (33)

More general expressions are possible. Looking at the table after formulas(12) we see, for example, that magnetostatics offers mo possibilities than elec-trostatics for which the above expression (34) for the hamiltonian is uniqueup to the scalar factors.

From (33-34) it follows

∂H

∂pA' (p2k)′ ∗ pA and

∂H

∂qA' (q2l)′ ∗ qA.

With the help of the Legendre’s transforms among different Poincare-Cartansubbundles of E and with the help of the identities (16) we can calculate

22

the vertical differential forms L, S and T (20) on other Poincare-Cartansubbundles. For example

L = L∗(pA ∧∂H

∂pA−H), where L ≡ P−1

h Pl.

We will assume that the Legendre’s transforms commute with the Hodge’smap

L∗ ∗ ' ∗ L∗.

Then it is easy to calculate the Legendre’s transform for monomials,

k(p2) ≡p2k

mand l(q2) ≡ cq2l, where m, c ∈ IR. (34)

Denote

mk ≡ (sign det g)| ∗ θ|−(k+1)/(2k+1) 1 + 2k

1 + k(m

1 + k)1/cl≡

(sign det g)| ∗ θ|(l+1)/(2l+1) 1 + l

1 + 2l[(1 + l)c]

For the given hamiltonian (34-36) we get

H ≡p2k

2m

∑pA ∧ ∗p

A +cq2l

2qA ∧ ∗qA,

L =mk

2v−(2k)/(2k+1)

∑vA ∧ ∗vA −

cq2l

2

∑qA ∧ ∗qA,

S =p2k

2m

∑pA ∧ ∗p

A −1

2clf−(2l)fA∧∗fA,

T =mk

2v−(2k)/(2k+1)

∑vA ∧ ∗vA −

1

2clf−(2l)/(2l+1)

∑fA ∧ ∗f

A.

Assuming that mk and cl are constant we have

∂H∂pA' k+1

mp2k ∗ pA, ∂H

∂qA' (l + 1)cq2l ∗ qA,

∂L∂vA' k+1

2k+1mkv

−2k/(2k+1) ∗ vA,∂L∂qA' −(l + 1)cq2l ∗ qA,

∂S∂pA' k+1

mp2k ∗ pA,

∂S∂fA' − 1

cl

l+12l+1

f−2l/(2l+1) ∗ fA,∂T∂vA' mk

k+12k+1

v−2k/(2k+1) ∗ vA,∂T∂fA' − 1

cl

l+12l+1

f−2l/(2l+1) ∗ fA.

Examples

2l =

0, harmonic oscilator−3, Kepler problem.

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