Classical field theory and analogy between Newton's and Maxwell's equations

Foundations of Physics, Vol. 24, No. 10. 1994

Classical Field Theory and Analogy Between Newton's and Maxwelrs Equations

Zbigniew Oziewicz ~

Received December 30, 1993

A bivertical classicalfieM theory includes the Newtonian mechanics mTd Maxwell's electromagnetic field theory as the special cases. This unification allows one to recogni=e the formal analogies among Newtonian mechmzics and Maxwell's electrodynamics.

F O R E W O R D

This paper is inspired by Professor Constantin Piron. 2 Professor Piron's credo is the unity of human knowledge gained by mathematicians, physicists, philosophers, and theologians. Piron is known, among other, as the creator of the N e w Q u a n t u m M e c h a n i c s (1983), a realistic theory uni fy ing the classical and quantum physics.

The main ingredients of the classical physics are Newtonian mechanics and Maxwell's electrodynamics. These phenomenological theories are related to a diverse phenomena and are presented as disconnected in separate courses, textbooks, and monographs. The aim of this paper is to exhibit the formal analogies among these theories in the framework of the almos t

multisymplectic geometry, dO ~ 0. We consider a vector bundle of exterior differential forms, in

mechanics of dimension 1 + 4n, in electromagnetism of dimension 4 + 20n,

Universidad Nacional Autonoma de M6xico, Facultad de Estudios Superiores Cuautitl~in, Centro de Investigaciones Teoricas, 54700 Cuautitlfin Izcalli, Estado de M6xico. On leave of absence from University of Wroctaw, Poland. Work partially supported by Polish Commit- tee for Scientific Research KBN, Grant #2 P302 023 07.

2 Professor Piron visited University of Wrodaw for the first time in 1980. During martial law in Poland, when I was in prison in 1982, and again in 1984, Professor Piron bravely kept scientific contact with me, supporting my family and Solidarity, which required courage and a big heart.

1379

0015-9018/94/1000-1379507.00/0 �9 1994 Plenum Publishing Corporation

1380 Oziewicz

with a phenomenological not presymplectic differential pseudoform f2. Our theory is bivertical (Definition5) for arbitrary dimensional space-time oriented manifold. We determine four Poincar6-Cartan subbundles: Hamiltonian, Lagrangian, and two new not-named subbundles. This leads to the twelve Legendre's transforms among these subbundles, of which two are well known.

The field equations of the considered bivertical theory for n = 1 reduce to that of Newton's equations and for n = 4 to Maxwell's equations. This unification allows one to see analogies among the notions of Newtonian dynamics and of Maxwell's theory. In particular, force field~--~ current, the London equation in electromagnetism, is an analogy of the harmonic oscillator force in Newton's dynamics, one can pose the Kepler problem in Maxwell's electrodynamics by formal analogy to the Kepler problem in mechanics, etc. In a nonlinear model we discuss second-order field equations.

The limited space do not allow us to include here the discussion of the conservation laws, Noether currents, energy-momentum tensor, Poynting differential form, etc., from the point of view of the presented formal unification.

It is a pleasure to thank Professor Constantin Piron for many inspir- ing discussions during the last 14 years of our friendship. Some aspects related to this paper were elaborated by Magdalena Gusiew-Czud~ak in diploma thesis (1993). I am grateful to Magdalena Gusiew-Czud~ak and Grzegorz JastrzOski for illuminating conversations.

HISTORY

Multisymplectic geometry in classical field theory was initiated by De Donder (1935), Weyl (1935), and by Dedecker in 1953 and was developed in Warsaw by Tulczyjew around 1968, and by Kijowski (1973), GawCdzki (1972), and Szczyrba and Kondracki (1979). It has been developed in Chechia by Krupka since 1975. See Kijowski and Tulczyjew (1979).

NOTATIONS

A - A E - ~ A k is the Rham complex of the differential forms on a manifold E, ~ - - A ~ Z---{otEA, d0t=0}.

W - W e - d e r ~ is the Lie ~-module of the (one)-vector fields on E, such that W ^ - 0 w ^ k is the Grassmann ~-algebra and graded Lie R-algebra of multivector fields, W ̂ ~ - ~ .

Analogy Between Newton's and Maxwell's Equations 1381

An inner product is an algebra map and is denoted by i Ealg( W ^, End A).

I~1 = g r a d e 0c~ N, and ~ denotes the au tomorphism of Grassmann algebras, ~b0~ = ( - 1 )1~1 ~.

1. A X I O M A T I Q U E C L A S S I Q U E

1.1. Vertical Distribution and Filtration of Forms

Let E be a manifold with a distribution Ver c W which is said to be the vertical distribution.

Definition 1. The ~ - s u b m o d u l e

A~kl = {oL ~ A, izO~ = 0 V Z ~ Ver ^~k+ I~}

is said to be the submodule of k-vertical forms. The factor module is denoted by A tkl = A ~k)/A Ik-11, and if cc ~ A, then et/(k) E A /A ~k~.

Corollary 1. A~kl A Ai /~cAr ), Ark J ^ A t / ] C A t k + / 1 and we have the filtration of forms

A~o~C . . . cA~k) c A ~ k + ~ c . . . c A

Let "V~r be the submodule of the differential one-forms annihilating Ver

~Vkr- {~6 A1; ~(Ver) = 0}, "//Er ̂ ~ ~.~ E

Definition 2. A differential form a on {E, Ver} is said to be vertical

if m ~ ~r ^. Let for oc ~ A, Dist ~ be the associative distribution of or,

Dist c~- { X e W; i x ~ = 0 }

A form oc is decomposable iff

dim E = I~l + dim Dist 0~

Lemma 1.

(i) A form 0t is 0-vertical iff oc is a vertical, A(o)= Ybr ̂ ,

iVer~.=O 0 o~E3eEr ̂

1382 Oziewicz

f 0~"Uer ^ Dist 0 = V e r "~ (ii) ~.101 = d i m 3@r} ~ {0 is decomposable J

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

{E, 0} - {E, Ver-- Dist 0}

where the decomposab le form 0 is defined up to the multiplication by a nonzero function from ~E. The distribution Ver is locally integrable iff dO

is 1-vertical and then { E, 0} is locally fibered, dO ~ A II ~ ~ d( fO) ~ A tl ~ for 0 # f ~ .

If E is fibered over oriented manifold E - - - ~ { M , vol}, then 0 - ~ * vol is a decomposable cocycle.

1.2. Classical Field Theory

Definition 3.

(i) A classical field theory is a triple {E, Ver, O}, where E is a manifold, Ver is a distribution, and O is a differential form on E such that IOI = 1 + codim V e r > 1.

(ii) A submanifold ~ of E is said to be a solution of {E, Dist 0 = Ver, 0} if for every vector field Z on E,

~ * 0 # 0 and ~ * i z O = 0 (1)

(iii) A field theory {E, Ver, O} is said to be regular if every integrable distribution Hor tangent to the solutions of Eqs. (1) is complementary to Ver,

H o r c ~ V e r = 0 and W = H o r w V e r

Comment. A feld theory is regular if every solution ~ of (1) is trans- versal to Ver and dim ~ = codim Ver - 10[.

A form O determines the ~- l inear map O: W ^l~ ~ A ~. A form O can t dim e~ (dim E). be viewed as the rectangular ma t r ix , Iol , x

Proposition 1. Let ker 0 c W ^ IOl. Then

(i) dim k e r O = 1 =~ {E, Ver, 0} is regular.


(ii) Let {E, Ver, I2} be regular, c o d i m V e r = 1 0 l = l , and let f2 be a cocycle (so g2 is symplectic). Then dim k e r g 2 = l ( ~ dim E = odd).

Comment . If 101 = 1, then a cocycle I2 is regular iff dim k e r I 2 = 1. The ]01 = 1 refers to mechanics, and the property to be regular is said to be the classical determinism. The symplectic mechanics, d r2=0 and dim ker g2 = 1, on jet manifolds of arbitrary order has been presented by Olga Krupkova (1992).

Example . Regular field theory {E, 0, t2} need not imply that d i m k e r t 2 = l . Let d i m E = l + 4 n with a chart {t, qA, v A,pA,fA} and O=dt. Let I2==-(dpA--fAdt) A(dqA--vAdt); then dg2#0, d imke rg2= I +2n, and this mechanics {E, 0, I2} is regular.

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

0(Hor ^ lot) # 0 ( ~ dim Hor >_- 101 + dim(Hor n Ver))

H ~ 1 7 6 ( = 'd im(H~176176 [0[ ] ~<dimkerg2)

The last condition implies

dim ker f~ = 1 =~ dim Hor ~< 101

It follows that Hor c~ Ver = 0 and dim H o r = 101 =codim Ver, which com- pletes the proof of (i).

Let g2 be a cocycle. Then the associated distribution, ker t-2 c W, is integrable. If [01 = 1, then Hor = ker s'2 is integrable and the regularity of I2 implies that dim ker s = 1. []

Corollary 2 (Gawfdzki, 1972). Let {E, Ver, I2} be regular field theory. Then it is sufficient to consider the field equations (1) for vertical vector fields only,

~b*iver~2 = 0 ~ ~b*iwO = 0

Proof Let the distribution Hor be as in the proof of Proposition 1. We must show that O(Ver ^ Hor "iol) = 0 ~ g2( W A Hor ^ IOl) = 0. This is the case if W = Hor w Ver and Hor ^ ~ + tOl~ = 0. []

82512411o-5

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1.3. Submanifolds

Oziewicz

Definition 4. tion 3(i).

(i)

Let {E, 0, g2} be a classical field theory as in Defini-

A submanifold ~g ~ E is said to be the pre-symplectic manifold of {E, 0, g2} if ~ is a maximal submanifold annihilating dO,

~g*0#0 and ~ * d O = 0

The presymplectic manifold ~ is said to be symplectic if the field theory { ~, ~*0, ~u*O} is regular. The regular cocycle 7t'g2 is said to be the symplectic form on { 7 t, ~*0}.

(ii) A submanifold ~ ~ E is said to be PoincarO-Cartan submanifold (exact presymplectic) if ~ is a maximal submanifold on which 12 is exact,

~ * 0 # 0 and ~ * g 2 = d ~

A presymplectic potential a is said to be the Poincar6-Cartan form. If a field theory {~, ~*0, da} is regular, then a is said to be the regular Poincar6-Cartan form.

(iii) A submanifold ~ ~ ~ ~ E is said to be the Lagrangian manifold of {E, 0, g2} if ~ is a maximal submanifold annihilating O,

&~ and ~ * f 2 = O

(iv) A submanifold J ~ # ~ 7 t ~ E is said to be the Hamilton-Jacobi manifold if J is a maximal submanifold on which the Poincar6- Cartan differential form is exact,

J * O # O and J * ~ = d S "

A potential 5a is said to be the Hamilton-Jacobi differential form, I~1 -101- 1. The equations in (iii)-(iv) are said to be the Hamilton-Jacobi equations.

On the presymplectic submanifold, ~u*12 is a cocycle, the action integral is well defined (see, e.g., Oziewicz, 1992), and the field equations of Definition 3(ii) are the Euler-Lagrange equations.


The (pre)symplectic ~ and Poincar6-Cartan ~ submanifolds are known as the phenomenological material relations (p=mv, Kepler problem fA = -q-3qA, D=eo E, B=/zoH, London equation Jr, =At,, etc.) and are discussed in the last sections.

Jacobi in 1838 proved that in mechanics the Lagrangian and the Hamilton-Jacobi submanifolds, Z~' and J , are the families of solutions, ~b ~ J and ~b ~ ( ~ o J ) ~ c,a. The coordinate-free proof is in Oziewicz and Gruhn (1983). The extension of the Jacobi theorem beyond mechanics is not known.

The following table gives the dimensions of submanifolds for the phenomenological field theory (13) and follows from the considerations in Part 2; see definition (17) and formula (18).

Mechanics Strings Electrostatics Magnetostatics Klein-Gordon fields Electromagnetism

E ~ 7 t ~ # ~ J ~ ~dim ~dim .tdim ~dim

1 + 4 n 1 + 2 n 1 + n 1 2+6n 2+3n 2 + n 2 3+8n 3+4n 3 + n 3 3 + 12n 3 +6n 3+ 3n 3 4+10n 4 + 5 n 4 + n 4 4 + 20n 4 + 1 On 4 + 4n 4 God's material "quantum time space choice relations space" space-time

1.4. Hamilton-Lagrange (De Donder-Weyl) Field Theory

Definition 5. The field theory {E, Ver, I2} is said to be k-vertical if O # dI2 e Atk ~ and dg2 6 Atk_ ,~ or i fdI2=0, Q ~ Atk~ and 12 6 Atk_ ~ ~.

Comment. For a cocycle g2 this definition was introduced by Kondracki (1978). The notion of the k-vertical field theory is essential for the theory of the Poincar6-Cartan forms if t2 r Z and for the Hamilton- Jacobi theory if t2 E Z.

Note that

{AlO'>++}/(k-l)#O iff j < k < l O l + j

In particular, IOI = 1 + 181 =~g2 n A~ol = 0. In the section calculus with the splitting we are showing that (see

Definition 10 and Corollary 3)

d./l(k)C f/l(k+lj) if Verisintegrable, dtg~A~l~ t Atk+21 otherwise, d~r

1386 Oziewicz

Let a distribution Ver be integrable. In this case

dOeAIk) ~ ~ OeAtk-l) [dOr Or

A form dO v~ 0 is k-vertical iff O can be decomposed (not uniquely) as the sum of the (k - 1 )-vertical form and a cocycle. Two fibrations of the de Rham complex A of the differential forms are involved in this decomposition: the first over the factor ~ - m o d u l e A/AIk_~, the second over the factor R-space A/Z. The form dO is k-vertical if there exist the splittings

such that

and

A/Z ~' , A

A/A~k- l)

p (O /Z )~A tk_ , . v ( O / ( k - - 1 ) ) ~ Z

O =p(O/Z) + v(O/(k - 1 )) ~ A~k_ 1~ ~ Z (2)

Such splittings are not unique and are determined up to the ( k - 1 )-vertical cocycles

|oc A t k _ l ) n Z ~- dAo._2~

Let a field theory {E, Ver, O} be k-vertical. Then the splitting p determines v and vice versa. Locally

and

loc v ( O / ( k - 1 ) ) -~ &o

O = f + dr where f - O - d a ; ~ A ~ k _ l l (3)

The form co is determined modulo the (k -2 ) -ve r t i ca l forms. On the Poincar6-Cartan submanifold, # ~ E, O and f are exact,

d F - ~ * f , d~F--~*O (4)

and

O~F.~F-'['-~*s mod Z~


There exists a correlation between the decompositions (2)-(3) and the Poincar6-Cartan submanifolds, and this is shown in Section 2.3.

Depending on the choice o f f in (3), the potential F could coincide (up to the sign) with the Hamiltonian H or with the Lagrangian L (see the next sections); however, the freedom in the decomposition (2)-(3) allows one to see more possibilities.

If f is (k-1)-vertical on {E, Vet}, then F in definition (4)is ( k - 2 ) - vertical on {~', ~*0},

101+1 iAiol @Z~)~F~__,dFsA~k_~ ) (5)

The Poincar6-Cartan equation (4), dF= ~*f, allows one to express ~*o9 in terms of the partial derivatives of F w.r.t, the basis of the o~-module AIOi+l,--A Therefore, the differential form F determines the Poincar6- ( k - I) ~ ~L~.

Cartan form,

AI~ 2) /Z~ F~---~ art A~ I mod Z~,

This motivates the definition.

Definition 6. Let a distribution Ver be integrable and g2 r Z.

(i) Let 2 ~< k ~< [dl2[ and let the field theory { E, Ver, O} be k-vertical, dO e A(~ and dO r 12(t._ i). Then the o~fmodule Al~ said to be the module generating the Poincar6-Cartan forms.

(ii) The field theory {E, 0, 12} is said to be the Hamilton-Lagrange field theory (or De Donder-Weyl), abbreviated by HL, if the ~ , - dimension of the generating module of the Poincar6-Cartan forms is 1.

For the HL field theory the Poincard-Cartan form is determined by one (pseudo)-scalar function, ( Lagrangian, Hamiltonian,...).

Lemma 2. dQ ~ A(2).

Proof

Therefore

A field theory { E, Ver, g2 r Z} is HL iff dO is bi-vertical,

mi"'k-2"~ ) dim{ A I(~ 2,} = E

i = 0

dim{Al~'_2) } = 1 iff {E, Ver, f2r is bi-vertical

1388 Oziewicz

Comment. In HL field theory the (local) Hamiltonian and Lagrangian F in definition (4) are the vert ical differential forms on ~ . For non-HL-type theories the analogous Hamiltonian or Lagrangian differential forms are no longer vertical and therefore cannot be expressed by means of one pseudo- scalar function. Analogous considerations are valid for the Hamil ton- Jacobi theory if s e Z.

Partial Derivatives of Vertical Forms. Note that

dim{ A I~ +1 } = d i m Ver

We will suppose that a module A t01 + I, ..(t~ on E as well as on submanifold ~ , is generated by the differentials of the homogeneous vertical forms (of different degrees),

Al~ +l - gen{ dw A, w A ~ A,o,}

This means that '~ A iol + w e ~ c 1~ has the unique decomposition

ct = d w a A c t A, wA, O~A E A(o}

In particular, a generating set { dw A } determines the partial derivatives of the highest-degree vertical forms

0F AWl ~ F ~--~ d F = dw A ^ ~ ~ A I~11 + 1

10) OW

Example:

0L 0L d L ==_ dq A ^ - - + d r A ^ -

Oq A OV A

2. P H E N O M E N O L O G Y

2.1. Phenomenological Field Theory

Let { qA, vA, PA, fA ; A ~ I c t~ } be a collection of the vertical differential forms on {E, 0} such that { P A , f A } are 0-dependent (pseudo-forms). Newton's and Maxwelrs phenomenological equations as well as the equations of electrostatics and magnetostatics have the form

$*(dq a - v A) = 0 (6)

r - f A ) = 0


The differential forms {qA, v A, PA, )cA} in Eqs. (6) are independent, as they are determined by independent experiments. The phenomenological material relations among these fields are the consequence of further independent measurements. This was stressed by Newton (1686) and Maxwell. After Lagrange it became customary to present the Newton equations (as well as of electrodynamics, V1A = j ) as the second order from the beginning, contrary to the original Newton's presentation. Also Piron stressed (e.g., in Piron's Lectures on electrodynamics, 1989) that Eqs. (6) should not presuppose the material relations.

A phenomenological material relations are the equations for the (pre)symplectic submanifold of the field theory {E, Ver, I2} [Defini- tion 4(i)], and are discussed in the next sections.

The aim is to determine the most general regular field theory {E, Ver, O} in which the field equations (1) coincide with the experimental one (6). The different field theories with the same set of the first-order equations (6) will lead to the different (pre)symplectic submanifolds and therefore to the different second-order equations considered in Part 3. Denote

0 A =_ dq A - v A (7)

~oA - dpA - f ~

The solutions ~b of Eqs. (6) annihilate the ideal generated by {~gA, O~A}; therefore, for the regular field theory we need the equality of ideals,

gen{ iv~rg2} = gen{0 A, O~A} (8)

Eq. (8) is the main equation of the present paper. If a distribution Ver is integrable, then {,gA, o~A} are 1-vertical. The most general form compatible with (8) needs to be 2-vertical of the form

Q - ~ (K~ ^ o~ A ^ OB+FAS ^ 0 A ^ ,gS+Z As ^ O~A ^ COS) (9) A.B

where {K~, FAs, X As} is the collection of vertical differential forms such that

FAB=(_I)I'gAIIOnl I"oA, xAn=(--l)l~176 sA

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

dim 0c = dim Ver - dim{ (Dist ~) c~ Ver} (10)

One of the necessary conditions for the implication (9)=:, (6) is

dim V e r = ~ (dimdqA + d i m d v A + d i m d p A + d i m d f A ) (11) A

1390 Oziewicz

Definition 7. The field theory {E, Ver,/2} wi th /2 of the form (9) is said to be the phenomenological f i eM theory.

Because/2 is M-homogeneous, then

IvAI = 1 + lqAI

ITAI-- 1 + IPAI (12)

Iq"l + If~l + IX~l = IOl

IvSl +IpAI + lX~l=101

In mechanics IKI--IFI = Izl--0. The conditions

VA I~AI/> 10AI and I g l = 12'1 = 0

determine the unique grades for mechanics and string theory and n possibilities for 101 = 2 n - I and 2n. In this case 2' can contribute in mechanics and magnetostatics only.

101 Iql Ivl Ipl I f l IFI

1 0 1 0 1 0 mechanics 2 0 1 1 2 1 strings 3 0 1 2 3 2 electrostatics

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

I 2 2 3 1 electromagnetic field

If { K , F , x } are vertical cocycles, then the field theory (9) is HL, dO is bivertical, and

/ 2= {d(K ] ^ P s + ~FAs ^ qS)--(KSA ^ f s + Fsa ^ vS)} ^ OA + ~oA A CO a

Effectively the "momenta-induction" and "force-current" are rotated and translated by "connection" F, a natural description of a velocity-dependent force,

pA~-"K s ^ p s + ~ b F A s ^ qS,

fAF-~ K s ^ f s + FsA ^ v s

A phenomenological symplectic mechanics (9) without x-terms has been considered by Jadczyk and Modugno (1992).

Consider HL field theory

/ 2 - ~ . c o A ^ O A e A ~ I ) G Z ~ A ( 2 ) (13)


The 1-vertical differential forms {h,/, s, t} are defined by the following decompositions [ like formulas (2)-(3) ]

f 2= - h + d(p A ^ dq A)

= + l + d { p ~ ^ (dqA- -vA)}

= + s + d { q A ^ ~hl~

= + t + d { q A ^ ~hl~ ^ v A} 14)

Explicitly

h = ^ + clq ^ lolf

l - d r A ^ ~k'+l~ ^ Ol~

s - - d p A ^ v A - d f A ^ ( - ) l ~ ~hq A

t - d r A ^ Ol+101pA+dfA ^ (_)lOl ~,qA

l + h = t + s = d ( p A ^ v A)

h _ s _ t _ l _ d ( t p f A ^ qA)

(15)

(16)

If E is fibered and the differential forms {qA, vA, p ,4 , fA} are the Liouville forms, considered in the next section, then the differential form (13) is regular and imply the field equations (6).

2.2. The Liouviile Differential Forms

Let M be a manifold and for p E M, T * M be the vector N-space of exterior forms at p. Let T k M ~ M be the bundle of exterior k-forms, ( T k M ) p = ( T ' M ) ^k, with ( T ~ M ) ^o = fie.

A differential form c( e A M determines unique section c(.~s F ( M , TkM) , and ATkMO ). ~ O~s*). E AM,

A M ~ % ATkM

M ~'~ ' T k M

Definition 8. form if

A differential k-form 2 ~ A r*M is said to be the Liouville

c('2 = ~ for every c~ ~ A k M

1392 Oziewicz

There exists a unique Liouville's differential k-form and has the local form

2 = 1 ~ 2t,,...~,kz~*(dt~' A . . . A dP 'k)

The Liouville differential forms of arbitrary grade have been introduced by Tulczyjew in 1979. The Liouville forms are vertical w.r.t. 0 = n*volM.

Let a manifold E be a vector bundle, E ~ M, of an exterior form of different degrees on a manifold M,

E = ~ { ( T I q A I M ) O ( T I ~ ' A I M ) @ . ( T I P A M ) O ( T I f A I M ) } (17) A

Let {qA, v A, P A , f A } be a collection of the Liouville forms on E. If 2 is a Liouville's differential form on E, then according to definition

(10)-(11),

dim(d2) = (dim M'] (18) \ I).I J

The field theory {E(17), n*volM, ~ (13)} is regular and implies the field equations (6). From formula (11) we get the dimensions, dim E, listed in the table after Definition 4. The Liouville's forms {K,/", X}, for simplicity, are not included in E (17). The Liouville's forms {K, F, X} in (9) contribute to dim E. In mechanics with (9), 101 = 1, dim E = 1 + 4 n + 2 n 2 - n .

2.3. Poincar~--Cartan Submanifolds

The considerations in this section are the same for the case of presymplectic and Poincar6-Cartan submanifolds. To be specific, we will consider Poincar6-Cartan submanifolds, Definition 4(ii), and only the case where E is a bundle of exterior forms (17) with the conditions (12).

Let a bundle E be split with the fiber-preserving projectors zre/c,

P ~ "P E = P O C "c C

M = M = M

Let subbundles P and C be of equal dimensions, and let dim Ver (11) on E be even. A Poincar6-Cartan submanifold ~ ~ E is a subbundle of E with dim ~ - - d i m P = dim C. Let zre]~ be the fiber-preserving isomorphism. Then ~ will be identified with the injection ~: P ~ ~ c E, rre o ~ = idp, and c p - Z r c O . ~ : P ~ C is a fiber-preserving bundle map. If 0-rr*volM, then . .@*O=O~ A e .


Consider splittings of the bundle E for which I2 has the following form [compare with the decomposition (2)-(3)]

(19)

On the Poincard-Cartan submanifold ~ the form Y. dn*0t ^ rc*fl is exact,

~* ( ~ drc*ot ̂ rc*fl) = ~" dot^ ~ o * f l - d F e A e

Therefore

OF ~o * fl =- Oot

The differential forms {12, h , l , s , t } (13)-(15) are exact on the Poincar6-Cartan submanifold ~ ~ ~ E . In particular, a Hamiltonian H and a Lagrangian L are differential forms on different Poincar6-Cartan subbundles and are defined as potentials,

g, d H - ~ h h , d L - ~ ' l , d S - ~ . * s , d T = - ~ * t (20)

The compositions, say ~ e - ~ 7 ~ o4 , etc., are said to be Legendre's transforms. With the help of the identities (16) the Legendre transforms allow one to calculate, for example, the Lagrangian L for the given Hamiltonian H,

dL = ~ * / = (~* ~ ~,7 '* o ~,*){d(pA ^ v A) - h }

= ~*{d~,*(pA ^ v A) - - d H } (21)

Therefore, modulo cocycles

L = (L~ ~ ~ ^ VA)- Z'~ mod Z (22)

The Poincard-Cartan forms {ot, dot--.~*g2}, can be expressed in terms of L, H, S, and T if we identify the decompositions (14)-(15) with (19)-(20).

o tH- - - H + pA ^ dq A m o d Z ~ (23)

OH OH A

q~ h v -- Op A ' ~~ l* f A - O l~ --Oq A

1394 Oziewicz

OL OL cp.pa = r Iol) _ _ ~o?f a - _ r Ov a' Oq A

OL ~L---- L +-~v A ^ q/l +l~ m o d Z ~ (24)

daL = (~d OL-Ov A (-- )lol ~b ~q~OL } ^ ~pr a _va)

OS OS ~o~*vA-op a, ~o*qi-- ( - - ) '+Wl~b~

OS O~s=S+-z-7/^ ~l+l~ mod Z:~ (25)

OJA

does=_ {d_o~_(_OS )lol r J---ppSA} ̂ ~ l + Iot~(dPA--fA)

OT OT q~*PA -= ~,<1+ IOl) Ov A , <,o.q.A __-- ( __ )101 ~// Of A

Ova--f^ -̂---vof A ^ ~v A modZ~. (26)

d ~ r = { ( _ ) OT a ] +lolfa }

The (f, v)-subbundle (26) in electromagnetism was considered by Thirring (1979, p. 109) and therefore one is tempted to call the differential form T the thirringian. See also the last two sections in Part 3.

3. FIELD EQUATIONS

3.1. The Hodge Map

We give a generalization of the Hodge map for a manifold with a distribution V e r c W. When 0=vol (<=>Dist 0=-Ver=0), this generalization collapses to the usual definition. The formulas in Lemma 4 below do not suppose that a scalar product g is symmetric.

By definition, the form 0 is decomposable; therefore

Cer= {oreAd, o~ ̂ 0=0}


Lemma 3. If ~ ~ "Uer ̂ , then for each multivector X~ W ^

ct ^ i x O = ( - )l~lr +lxl) ii, x 0

If ct ~ ~er ^ and I~1 = IXI , then

ct ^ i xO = ( ctX) 0

Consider the factor module W/Ver as the dual to Yrer, and let Z ~ (W/Ver) ^ 101 be such that OZ = 1.

Let g, k and their pull-backs (transpositions) g*, k* be 5~-linear maps

g, g*: :[/'er 0 ~ W/Ver

k, k*: W/Ver ~ :Yer 0

For ~ and fl in ~er, g ( ~ | 1 7 4 g and k need not to be symmetric. The extension of the above maps to Grassmann algebra maps (Cer) ^ ~ ( W / V e r ) ^ is denoted by the same letters.

Definition 9. The ~ - l i n e a r maps �9 = *~g,O~ and * =*~k,z>,

(~//kr)^ ~ - * * ~ = ig=O ~ ~er ^

(3e'er) ^ ~ o~ ~---~ , ct = k i ~ Z ~ Ve r ^

are said to be the Hodge maps with 0 = . 1 and * 0 = 1.

Lemma 4. Let e~fl = ct ^ ft. The following formulas hold,

(i) *gOeo~=ig~O*gO~l~l (ii) *g. Oiga=e~O*g.O~l I~1

(iii) ~ , o . = ( _ ) 1 o l . o~,

(iv) *lg, o)O*~g, o l= g ( O | o~bl +l~

(V) I f ~ and f l ~ ( ~ e r O ) ^ are of equal grades, then

^ , f l = g ( f l | O = g*(o~ | O = f l ^ ,g.Ct

g( ,ct | , f l ) = g*( O | O ) . g ( ~ |

The function g ( O | O)~ ~ e is said to be the 0-determinant of g,

det o g - g ( O | - , 0

Let Og - I g(O | 0)l -i/2 0; then g(Og | Og) = sign deto g.

1396 Oziewicz

3.2. Calculus with the Splitting

Let VX~ W, V x be a zero-grade derivation of the tensor algebra such that V x [ ~ = X . The derivation Vx factors to the derivation of exterior algebra, Vxe der A. For form 0c, the composition e~ o Vx is a (skew) derivation of A, [e~oVx[=[~l. Let A ~ - s p a n { e ~} and W-span{X~}, where e"Xb-d'/,. Let Vo-Vxo; then V - e ~ ^ V, is a skew derivation, IV[ = +1 and V [ J - d. The difference V - d is said to be the torsion of V,

V = d + T~ skew der A

The splitting of Definition3(iii) determines the bigradation ( N x N grading) of A = @ AP'q, with the projectors r~ p'q, A p'q -~ 7zP'q(AP+q). Every derivation is determined by values on generators {o~, d~-}, and therefore a derivation of grade + I is decomposed as the sum of four bigraded derivations.

Definition 10. Let Ver=n l '~ ', H o r = n ~ I, and bigrade = I ' l .

dr. lo~=n~.~ dv ld~-rc2 ,~ odozr'.~ + zrl.~odozr ~ [ d v l = ( + l , 0 )

d,,v[o~ = 0, d,,vld.~=n~176 ld,,vI = ( - 1, +2)

dillon=re ~ d n l d ~ =n~ ~ + n' ' 'odon ~'~ Idu[=(0, +1)

d, ,n l~ =O, d,,nld~-rc'- '~ ~176 ~ [d, nl = ( + 2 , - 1 )

Note that d v : ~ ~fkr and dn:~-*J~or . The distribution Ver is integrable iff dnv= 0 and the distribution Hor is integrable iff dnn = 0. The tensors d~r. and d,,n measure the noninvolutiveness of these distributions.

Corollary 3. d = d n n + d v + d n + d , , v and d 2 = 0 is equivalent to seven conditions:

dnv o dnv= O,

dnod,,v+d,,zodn=O,

dvod,,v + d,,vodv + d 2 = O,

d,,zod,,n + d,,nod,,v+ dvodu + dnodv=O,

dn o d,,n + d~n o d n + d2v = O,

dvod~n+ d.nodv=O,

d~., o d n . = O,

bigrade = ( - 2 , +4)

bigrade = ( - 1, + 3)

bigrade = (0, +2)

bigrade = ( + 1, + 1 )

bigrade = ( + 2, 0)

bigrade = ( + 3, - 1 )

bigrade = ( +4, - 2 )


If a differential form e is vertical, then dve is vertical, dno~ is 1-vertical, and d,,vO~ is 2-vertical. From Definition 1 it follows for the phenomenological field theory {E, Ver, g2} (13) that

~2 -vertical if Ver is integrable

is [4-vertical otherwise

3.3. Dirac Operator and Codifferential

Let g: ~ r - - * H o r , and let y: ~er ^ --*lin(~er ̂ ) be the left Clifford multiplication, for e e ~fkr and 0t e ('fbr) ^,

y ~ - e ^ ct + igEo~ (27)

Let ~Uermspan{e "} and Hor-=span{h ,} , where e"hv-6~. Denote ?,u - y(e') and V u =- Vh,.

Definition 11. An operator D - y u o V u : A ~ A is said to be the splitting-dependent Dirac operator. When Ver = 0, D collapse to the usual Dirac operator.

We have

- e " ^ V~ s skew der A,

~ = d v + Tv

grade ~7 = + 1 (28)

Because of (27) we have D = ~ + 6, where 6 is a splitting-dependent codifferential,

6 ~_/~,o V,, 161 = -1

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

~So,~= + ,~o~7 or + i,~o(V,**)

,~.o~or _~o,~. +e~o(v,,,~.)

Therefore

where

fi = (sign deto g)(*r ~ ~ ~ *tg*,O,)o (-@)lol + "V*")

terT , , m �9 V = l g e q ~ ~ 1 7 6 1 7 6 1 + [ 6 1

1398 Oziewicz

If the Hodge map is parallel, V . - -0 , and T v = 0 (28), then q? = dv and the codifferential has the form

6 = (sign deto g) *~g,o~) ~ dvo *~g*.O,) ~ ( - t p ) IOl

62 = det g* [det g[ *g ~ d v ~ *g ~ ~ / I + 10]

(29)

If Hor is integrable, then d 2 , = 6 2 = 0 . The square of the Dirac operator A g - ( d v + 6 ) E = 6 o d v + d v o 6 is said to be the splitting-dependent Laplace-Beltrami-Hodge operator, which commutes with the Hodge map if g is symmetric,

ZlgO,=.oAg

C o n f o r m a l Map, g ~ fg. For f ~ ~ , gSg = f6g and Dsg = d + f . 6. Because fi o f = fo 6 + ig dr, then

DfgoDg=do6+ f . 6od

Plebafiski's Product. Plebaflski invented in 1972 an associative product, c t ~ f l ~ , ( , ~ ^ , f l ) , such that 6Eskewder /~. An algebra generated by {1, d, ,} has been studied by Plebafiski (1979) who shows that this algebra is 8-dimensional over the ring R[A] and possesses 4-dimensional faithful representation. The basis is

{ 1, *, D • * o D • ~- +D • o *, D_g o Dg, * o D_g o Dg}

3.4. Legendre's T r a n s f o r m s

Let {qA, qA .... } be the collection of the vertical forms, [qAI = IqAI, and let

q2=-- I g ~ qA | l-- l(q 2) (30)

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

N(IE qA A ,qA):(q2l)t dE (qA A *qA)--q21t Z g(qA(~q A) dO (31)

If f , g ~ ~,~, then

d ( f ^ * g ) = d f ^ *g+dg ^ *f +fgdO


This is no more valid for arbitrary grade. Formula (v) of Lemma 4 suggest the simplifying assumption

d(~ ^ *gfl)~-d~ ^ *gfl+ dfl ^ *g, OC (assumption) (32)

This could be the case if 0 is a cocycle and ~ and fl are the vertical differential forms on which d=dv. Let 0 be a cocycle. With the help of (31)-(32) we get

d(l~.q '4 A ,qA)~-2(qZl)'~dqA A ,qA (33)

Every vertical form of highest grade can be expressed in terms of the independent vertical forms. In particular, the Hamiltonian, H e Al~ can be expressed in terms of {PA, qA}. Consider the example

H - ~k(p 2) ~'. PA A ,pA + �89 ~ qA A *qA (34)

More general expressions are possible. Looking at the table after formulas (12) we see, for example, that magnetostatics offers more possibilities than electrostatics for which the above expression (34) for the Hamiltonian is unique up to the scalar factors.

From (33)-(34) it follows that

OH OH ~--(p2k)',pA and - -~--(qZl) ' ,qA Op A Oq A

With the help of the Legendre transforms among different Poincar6- Cartan subbundles of E and with the help of identities (16), we can calculate a vertical differential forms L, S, and T defined by (20), on other Poincar6-Cartan subbundles. For example,

L=L~'* PA ^ 0--~A - H , where Z~a=~/Tlo.~

We will assume that the Legendre transforms commute with the Hodge map

s o . = . o Z,a*

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

k(p'-) -pEk- and l(q2)=_cq 21, where m, c e R m

(35)

825/24/10-6

1400 Oziewicz

Denote

m~.-(s ign d e t g ) I * 0 1 - ' k + " / ' z k + " 1 + 2 k ( m ,~l/,Zk+,, 1 + k \ 1 - ~ 1

ct - (sign det g) I*0l I1 + i v~2/+ 1 ) 1 + /

1 + 21 [ (1 + l) c] I/(2/+ l J

For the given Hamiltonian (34)-(35) we get

H = p2k S" cq2t 2m ~ PA A , p A "-b ~ - - ~ qA ^ *qA

c q 21 197k --(2k)/(2k + 1 ) L = T v E vA ^ * v A - - - - 2 - Z q A ^ *qA

p2k

S = _j_~m ~. p. 4 ^ , p A

m k -(2k) / (2k + I ) T = -~- v Z vA ^ *vA

1 ____f-,2,~/,zt+ 17 ~ f A ^ ,fA

2c/

1 _ _ f -,z,v,2,+,, ~.UA ^ , f a

- 2cl

Assuming that mk and ct are constant, we have

O H k + 1 O H ~_ pZk , pA, ap A m Oq A ~- ( l + l ) cq21* q A

OL k + 1 OL mkv-Zk/~2k + I) , VA ' "~ - - ( l+ 1) cq 21 * qA

OV A -- 2k + 1 Oq A -

OS k + 1 OS 1 1+ 1 _ _ _ p 2 k *PA, _ _ _ f-2t/(21+1~.,fA

Op.4 m O f A ct 21 + 1

OT k + l OT 1 1+1 ... f-21/~2t+ I I , f a O v A " m k 2 - - f f - ~ O-2k/(2k+l~*v'l' Of^ cl21+ 1

Examples:

2 /= 0, harmonic oscillator

- 3, Kepler problem

3.5. Second-Order Field Equations

In this section, for simplicity we will drop the indices. Let a and b be scalar functions responsible for nonlinear theories. The results of the


previous section suggest to consider the closed ideals generated by the relations

{(q, p)-subbundleldq - .ehp, dp - .e~q, fi(e"q), ~(ehp)}

{(q, v)-subbundle l dq - v, 6(e-by) -e~q, dr, 6(e"q)} (36)

{(f, p )-subbundle [ dp - f, 6( e -~f ) - ehp, df, O(ehp)}

{ ( f v)-subbundle Id( *e-"f) - v, d( . e - % ) - f dv, df}

In the notations of the electromagnetic field theory, gauge potential q--* A, field strength v--* F, induction p ~ G, and current f--* j, the Maxwell equations on four Poincar6-Cartan subbundles have the following forms

d A = F {dA =*ehG fi(e-hF) = e~A dG *eUA

{ ~(e-'7) = ehG ~ ( e oj) = *F dG = j [fi(e -hF) = . j

If I f [ =0 , then 6 ( f ~ ) = f f o c + i g a / o c Marcel Riesz in 1946 noticed that the Maxwell equations can be equivalently presented in the Dirac form by means of the Dirac operator D - d + 6. For example,

~ DA = F - i ~ a ~ A , ~ i [ D F = e A + xahF

The first-order relations lead to the second-order relations (field equations) of two types,

f idq=ea+bq+igabdq and 8q= -iga~ q

6dp=e"+hp+i.~a,,dp and 3p= - i , ah p

d 3 f =e"+hf + d ( a + b ) ^ (3- i .~a , , ) f +digauf and df =O

d f i v = e ~ + % + d ( a + b ) /x ( f i - ixal , )v+digahv and dv=O

Therefore

Aq = e~+~q + i~ah dq -diga~q (37)

and the same for Ap if we interchange q .--, p and a*--, b in (37). For the "harmonic oscillator" case, a and b are constants, and the second-order equations for q and p are the London equations. In case of mechanics, strings, and electrostatics, [q] = 0, the "gauge fixing" condition, 3q = 0, is an identity, and Eq. (37) is the Newton equation.

1402 Oziewicz

A scalar function a is responsible for the nonlinear relation among current and gauge potential, j - * e a A . The conservation of current, dj = O, is equivalent to the a-dependent gauge condition 6 ( e " A ) = O ~ O A = --i,d,,A. The Lorentz gauge, 6A = 0 , is possible only if a = c o n s t .

REFERENCES

De Donder, Th6odore, Thkorie hwariantive du Caleul des Variations (Nouv. l~d. Gauthier- Villars, Paris, 1935).

Gaw~:dzki, Krzysztof (1972), Rep. Math. Phys. 3(4), 307-326. Gusiew-Czud~ak, Magdalena (1993), "Formalizm multisymplektyczny w mechanice i

elektrodynamice klasycznej," Diploma Thesis, University of Wroclaw, Institute of Theoreti- cal Physics.

Jadczyk. Arkadiusz, and Modugno, Marco (1992), "New geometrical approach to Galilei general relativistic quantum mechanics," XXI Conference on Differential Geometry Methods in Theoretical Physics, China.

Kijowski, Jerzy (1973), "A finite-dimensional canonical formalism in the classical field theory," Comm. Math. Phys. 30, 99-128.

Kijowski, Jerzy, and Tulczyjew, Wlodzimierz (1979), A Symplectie Framework for FieM Theories (Lectures Notes in Physics 107) (Springer, Berlin).

Kondracki, Witold (1978), "Geometrization of the classical field theory and its application in Yang-Mills field theory," Rep. Math. Phys. 16(1), 9-47.

Krupkova, Olga (1992), "Higher-order mechanics," Habilitation Thesis, University of Opava. Newton, Isaac, Principes Math~matiques de la Philosophie Naturelle, 1686 (Reprinted from

French edition of 1759, by Editions Jacques Gabay, Paris, 1990). Oziewicz, Zbigniew, and Gruhn, Wojciech (1983), "On Jacobi's theorem from the year 1838,"

Hadronic J. 6(6), 1579-1605. Oziewicz, Zbigniew (1992), "Calculus of variations for multiple-valued functionals," Rep.

Math. Phys. 31(1), 85-90. Piron, Constantin (1983), "New quantum mechanics," in OM and New Questions h~ Physics,

Cosmology, Philosophy and Theoretical Biology: Essays #~ Honour of Wolf gang Yourgrau, Alwyn van der Merwe, ed. (Plenum, New York).

Piton, Constantin (1989), Lectures on Electrodynamics, l'Universit6 de Gen~ve. Plebafiski, Jerzy F. (1972), "'Forms and Riemannian geometry," Summer School on

Cosmological Models at the Ettore Majorana Centre for Scientific Culture, Erice. Plebafiski, Jerzy F. (1979), "The algebra generated by Hodge's star and external differential,"

J. Math. Phys. 20(7), 1415-1422. Thirring, Walter (1979), Classical FieM Theory (Springer, New York). Tulczyjew, Wlodzimierz (1979), Rep. Math. Phys. 16, 233. Weyl, Hermann, Am1. Math. 36, 607 (1935).

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Classical field theory and analogy between Newton's and Maxwell's equations