Global characterization of variational first-order quasi-linear equations

Vol. 56 (2005)

REPORTS ON MATHEMATICAL PHYSICS

No. 1

G L O B A L CHARACTERIZATION OF VARIATIONAL FIRST-ORDER QUASI-LINEAR EQUATIONS J. M u I q o z MASQUt~ Instituto de Ffsica Aplicada, CSIC C/Serrano 144, 28006-Madrid, Spain (e-mall: [email protected])

and L. M. POZO CORONADO Departamento de Geometria y Topologfa Universidad Complutense de Madrid Ciudad Universitaria s/n, 28040-Madrid, Spain (e-mail: [email protected]) (Received September 6, 2004)

The global inverse problem of the calculus of variations for the particular case of first-order quasi-linear PDEs is solved. Some examples in the field theory are discussed. Keywords: affine Lagrangian, Euler-Lagrange equations, first-order quasi-linear equations, jet bundles, Poincar6-Cartan form. Mathematics Subject Classification 2000: Primary 58E30; Secondary 35F05, 37K05, 58A20. PACS numbers: 02.30.Jr, 02.30.Xx, 02.40.Ma, 02.40.Vh, 11.10.Ef, 11.10.Kk.

1.

Introduction

Let p : M - - > N be a fibred manifold, i.e., p is a surjective submersion. We set dim N = n, dim M = m + n. Throughout the paper we assume that the base manifold N is connected and oriented by a volume form v. Greek indices run from 1 to m, and Latin indices run from 1 to n. Let Pl j I M --~ N be the 1-jet bundle of local sections of p, and let Pl0 : J1M --~ J °M = M be the canonical projection: "

Plo(jls)

= S(X).

We denote by (x i, y"; y~) the coordinate system induced on j 1 U = p ~ l ( u ) by a fibred coordinate system (x i, y~) for the submersion p, defined on an open subset U ___ M (see e.g., [9]): y~(jlxS) = O(y ~ o s)/Oxi(x). We confine ourselves to coordinate systems (x i) on N adapted to the volume v; that is, v = d x 1 A . . . A d x n. Let L v be a Lagrangian density on JXM. As is known (see [4]), the P o i n c a r r -Cartan form ® of L v is Pa0-projectable (this means that L v admits a Hamiltonian [23]

24

J. MUlqOZ MASQUt~ and L. M. POZO CORONADO

formalism of order zero) if and only if L is an a n n e function over the a n n e bundle Pl0 : J1M --+ M, or, in other words, locally we have

L = A ° + aiy~,

a °, a i E C ~ ( M ) .

(1)

This property is an easy consequence of the local expression of the Poincar6-Cartan form [3]: 0L (2) tO = (--1)i-1--~.~0= A vi-t-Lv,

oYi A

where vi = dx 1 A . . . A d x i A . . . A d x n, and 0 ~ are the standard 1-contact forms on J1M, i.e. 0 ~ = dy ~ - y~dx i. In this case, the Euler-Lagrange equations for L v are the following system of m first-order quasi-linear equations on J1M:

OZ ° Oy"

OA_j~ _ (OA i Oxi \ Oy~

OAi~] OY----~ Oy~ / OXi "

(3)

This result poses the problem of determining which systems of first-order quasi-linear equations on j IM come from an a n n e Lagrangian as above. In [4] the following local characterization has been obtained. THEOREM 1.1. With the previous hypotheses and assumptions, the system of first-order quasi-linear equations

OYB F~ = FiB Oxi,

i i F~B + F}~ = O,

F~, FiB E C ~ ( M ) ,

(4)

is locally variational with respect to an affine Lagrangian if and only if the differential form ~ ~ f2n+l(M) defined by = F~dy ~ A v + (-1)iFS~dY ~ A d y ~ AVi

(5)

is closed. In this case ~ is the exterior differential of the Poincard-Cartan form associated with Lv. REMARK 1.1. The result can also be formulated by saying that an ( n - 1)horizontal differential (n + 1)-form on M is locally variational if and only if it is closed. Also see [5] for a different approach to the topic. The g0al of the present paper is to solve the inverse problem introduced above from a global point of view. As Theorem 4.1 below shows, the basic obstruction for an (n - 1)-horizontal exact (n + 1)-form on a fibred manifold p : M ~ N to be globally variational lies in the cohomology of the fibres of the submersion p. In particular, we obtain a global version of Theorem 1.1 aforementioned. To this end, we assume the system (4) is given as the kernel of an affine epimorphism (7). This imposes a slight global restriction to the system; it suffices however to cover the applications to classical field theory as shown in the Section 5.

GLOBAL CHARACTERIZATIONOF VARIATIONALEQUATIONS

2.

25

Preliminaries and notation

We denote by V M = Given two integers 0 over N with respect to Xo . . . . . X r - s ~ V M . In fibred coordinates,

OAr=

{X ~ T M ' p . X = 0} the vertical subbundle of p. < s < r, an r-form mr on M is said to be s-horizontal p if i x o . . , iXr-smr = 0, for all vertical tangent vectors

an s-horizontal r-form mr can be written as

E

f a , t d y a A d x I,

fa,I C C~(U),

(6)

Ill>_s IAl+lll=r

where I = (il . . . . . i~), A = (ai . . . . . ar-k) are two sequences of indices such that 1
d y A = d y cq A . . . A d y c~r-k.

For s----r we obtain the usual notion of a horizontal form or even a completely p-horizontal form in view of its local expression (6). We denote by ~2r(u) the space of s-horizontal r-forms defined on an open subset U ___ M. Hence f2~ is a sheaf of C~t-modules. We denote by A r T * M C A r T * M the vector subbundle corresponding to the sheaf f2r, i.e. f 2 r ( u ) = F(U, As~T*M) for every open subset U~M. We recall that the canonical projection Pl0 : j 1 M ~ M is endowed with an affine bundle structure modelled over the vector bundle p * T * N @ V M --+ M . In what follows, we consider an epimorphism of affine bundles F" JiM

/x~+IT*M

~

(7)

with associated vector-bundle homomorphism ~" " p * T * N ® V M -+ A~+~T*M.

(8)

If Z denotes the zero section in A~+IT*M, then k e r F = F - a ( z ) C J1M is an a n n e subbundle of corank m, since the corank is preserved under inverse image by a submersion. First of all, let us describe the equations of F. As above, let (x ', y~) be a fibred coordinate system on an open domain U _ M for the projection p adapted to v. Let ~o " U ~ J 1 M be the zero section associated to this system, i.e. o'0 is defined by the equations y~ o o0 = 0. Every j l s c j I u can uniquely be written as jlxs = w + ~ro(s(x)),

w C T * N ® Vs(x)M.

Hence, from the very definition of or0, we deduce

/

w =- Yi (JxS)( d x )x ®

(+)

s(x)"

26

J. MUI'7,1OZMASQUt~ and L. M. POZO CORONADO

Accordingly, if for every y ~ U, with x = p(y), we set

F(ao(y)) -= FE(y)(dy~)y A Vx, 0 F((dxi)x®(~ya)y)=-fi~(Y)(dYfl)yAVx, for certain functions Fu, Fi E ~ Ca(U), then we obtain

F(j~s) = F(w) + F(ao(y)) 0 : y.~(jlxS)['((dxi)x @ ( ~ y ~ ) y ) + F(ao(y)) = -y'~(jls)Fi~,~(y)(dyZ)y A v:, + F E ( y ) ( d y ~ ) y A Vx = (Fz(y) - F ~ ( y ) y ~ ( 2 s )

) (dyE), A Vx

= [ ( F E - Fi~,~y~)dy ~ A v]jjs. Therefore, the local equations for ker F are Eft

i iol = O. -- F}ozy

(9)

Moreover, every local section s of p, defined on an open neighbourhood of x, induces a retract pjlxs " Ts(x)M --+ Vs(x)m,

pjlxs(X ) = X - s,p,X,

of the inclusion V~(~)M C Ts(~)M, whose local expression is

Hence

Pj~s (dy [VM)s(x) = (Oa)s(x)• Let us denote by I t, 12, I~, and 1~ the identity maps of the vector spaces T~*N, Ts(x)M, Ts*(x)M, and A~+ITs*(x)M, respectively. For every y 6 M let

Uy: ry*M ® (A."+~rs*M) -+ A."+2TyM be the homomorphism defined as follows: /Zy(t01 @ Wn+l) ~- W1 A Wn+l,

and let Cy : T y M * (An+2ry*M) -+

An+ITyM

be the homomorphism induced by the interior product, Cy(X @ lBn+2) = iXWn+2 .

GLOBAL CHARACTERIZATION OF VARIATIONAL EQUATIONS

3.

T h e fornl

~r

27

defined

Let EF be the differential (n + 1)-form on j I M

defined as

=

where

Xj~s : TxN ® Vs*(x)M ® An+lTs*(x)M -+ An~+'lTs*(x)M

is the composition of the following maps:

TxN ® Vs*(x)M ® A~+ITs*(x)M $ I~®p*j]s @ If Tx(U) ® T~*(x)M ® A~+IT~x)M + s,,x o I; ® If * * Ts(x)M ® T~(x)M @ A n+l n Ts(x)M

$ I2 ® ~t~(x)

Ts(x)M@ An.+2 Ts(x)M • $ c~(x)

Ann+llTs*(x)M. From the very definition above it follows that F,F is an ( n with respect to Pl.

1)-horizontal form

LEMMA 3.1. We have

~F = F;c~y:dy" A V - F;c~y~dy' A V + (--1) i F ~i d y

A dy ~

A

vi.

Proof: When we apply I~ ® pj], ® If to F~(x), we obtain (~V)j~, = --F'~,(s(x)) ~x i ~ ® ( ) j ) , *

,

Similarly, when. we apply S.,x @ 12 @ 13 ,,,~,t\

(~F)jlxs

i ( O = -F'~=(s(x)) ~X i

Also, /2 ® A~(x) acting ,,~,,,,

I~F)jl s

,~:

(~F)jlxs , we obtain

O(yr os) ~ ) Jr-

o n i.~F)jlxs

i (~ ~--- -F}~(s(x))

to

+

OX i

® (dy~),(:O A Vx.

a

s(x)

05 ® ( )j~ ® (dy~)~(x)/x vx.

yields

O(y y o s) 0 ) ax i Oy× ~(x) ®

(O°t)jlxS

A

(dy~)~(x) A Vx,

(10)

28

J. MUlqOZ MASQUI~ and L. M. POZO CORONADO

and finally, C~(x) acting

on

{ ~/it "-,

k~F)jl s yields

O(YOXi

=

(x)(O°t)jlsAVx"[-(--l)i-1 (0 a)j}~ A (dY~)s(x~ A ( V i ) x ) .

Hence

SF = --Fi~ot( \ - y ~ d y ~ A

V -Jw(-- 1 ) i - 1 0 ce A

dy~ A

Vi)

= -Fi~o~ ( - y ~ d y °~ A + v ( - 1 ) i - l ( d y °' - y ] d x j) A d y '8 A vi) --

--

~,~y ~i d - Y '~ A v - F ~ c ~"y . ~ d y ~ A v + ( - 1 )

i F ~i d y

F i

o~A d y ~ A v i .

[]

The homomorphism /~ • p * T * N ® V M --~ A~+IT*(M) can be considered as a section of the bundle p * T N ® V * M ® A nn + l T * M. The volume form induces an epimorphism .

* n + l T * /I.4 Ts(x)M ---> ^,,n ~s(x) ....

ce(COs(x))= Ws(x) A Vx,

whose kernel is p * T * N . Hence, by passing to the quotient modulo ker~, we obtain an isomorphism .

* "-' * n + l T * .,1A" Vs(x)M = Ts(;c)M/p * Tx* N -+ ^"'n "s(x) ....

By composing A = I1 ® I~ ® ~-1, TxN ® Vs(x)M @ An+lTs*(x)M a > TxN ® V~(x)M , • ® V~x)M, and /3 we obtain a section /3 = A o F of the bundle p * T N ® V * M ® V * M . Let sym • p * T N @ V * M @ V * M -+ p * T N

®

alt " p * T N ® V * M @ V * M --+ p * T N ®

S2V*M,

A2V*M,

be the projections onto the first and second summands, respectively of the decomposition p*TN @ V*M @ V*M = p*TN @ szV*M

G p*TN ®

AzV*M.

DEFINITION 3.1. An affine morphism F • J 1 M ~ A nn + l T * M is said to be symmetric (resp. skew-symmetric) if alto/~ = 0 (resp. sym o P = 0). REMARK 3.1. According to Definition 3.1, the symmetric or skew-symmetric character of F depends only on /~. REMARK 3.2. The form ~ defined in formula (5) depends only on the skewsymmetric part of F, as we have F ~i d y ,~ A dy ~ + F~o~dy~ A dy ~ = ( F ~ - F~o~)dy '~ A dy ~.

GLOBAL CHARACTERIZATION OF VARIATIONAL EQUATIONS

29

Moreover, if the quasi-linear system (4) is variational, then F is skew-symmetric, i.e. F/~ + F ~ = 0, as in this case, according to (3), we have

oa ° F~-- OY~

oa i Oxi ,

oa i Fi~-- ~

oai~ OY~.

LEMMA 3.2. If F is skew-symmetric, then the form EF is given by

EF --- - 2 ( F ; ~ y : d y ~ A v + ~ ( - 1 ) i

F~otdy fi A dy a AVi). /

k

fl
Proof: From the formula (10), the result follows directly.

[]

PROPOSITION 3.1. Let iF : k e r F --+ J1M be the canonical injection. Then, the pull-back (iF)*EF is PlO-projectable; namely, there exists a unique differential (n + 1)-form ~F on M such that 1~. ,*,~ - - ~ U F ) ~ F = (PlO]kerF)*~F.

Proof: In fact, taking Eqs. (9) of F into account, we have ½(iF)*gF = Fedye A v + ~ ( - - 1 ) i F~e,dy ~ A dy ~ A vi, fl
thus concluding the proof. 4. 4.1.

[]

Global formulation Fibre derivative

For every f ~ Coo(M) we denote by dM/Nf : V M --+ ]R the restriction to V M of the differential d f : T M -+ R, which is usually called (cf. [3, 9]) the fibre derivative. More generally, let ~2rM/U, f~rN be the sheaf of sections of the vector bundles ArV*M, ArT*N, over M, N, respectively. Then, there exists a unique antiderivation r * i o r + l ® p*~2%

diM/U " ~2M/N ® P g2U '-> ~"M/N

of bidegree (1, 0) such that:

1. diM/N o d~/N = O, 2. dO/Nf = dM/sf for every f C C°°(M), and i 3. dM/sw = 0 for every local section w in f2~. Hence

diM/N (fA,Hdy A @ dx H) = dM/Nfa,H A dY A ® dx H,

(11)

with dy a = (oq < ... < oer), H = (hi < ... < hi) (cf. formula (6)). We recall that the algebra structure of the tensor product of the algebras f2"M/m and p*fa~v is given by (Ok ® Wl) A (Ok' ® Wl') = (--1)k'l(o)k A O)kO ® (Wl A W l , ) , k kt

for every Wk ~ ~M/N, COk,~ g2M/N, Wl C p*~21U, Wt' C p*~2%.

30

J. MUIqOZ MASQUt~ and L. M. POZO CORONADO

As the parametric version of Poincar6's lemma holds (see e.g. [10, Corollary 4.2]) we have an exact sequence of sheaves over M,

0--> p-l~2iN

* i diM/N 1 * i d~M/N ) P ~"~N ) ~~M/N ~ P ~-~U ) "'"

diM/N

diM~N>

) ~m-IM/N@ P*~"2iN

~-~mM/N@ P*~'2iN "+ 0,

(12)

i = C ~ @ c ~ P-I~'2N • where we have set p * ~"~U

4.2.

Splittings and connections

Assume that p • M --+ N is endowed with a nonlinear connection Y " T M -+ V M (cf. [7]), i.e. Y is a homomorphism of vector bundles such that F ( X ) = X for every X ~ V M . On a fibred coordinate system for p, F is locally described by Y = Y ~ ® O/OY~, where y ~ = dy ~ - Fiadx i, Fi°t C C a ( M ) . The tangent vectors in H z = ker Y are called F-horizontal. The map p , • H× --+ p * T N is a vector-bundle isomorphism. We denote by X v and X h the vertical component and the horizontal component respectively of X c T M in the decomposition T M = V M @ H×. In particular, we denote by X h ~ Y2(M) the horizontal lift of a vector field X 6 •(N), and by Oi the horizontal lift of O/Ox i. Hence 0 0 - - , Di = Ox---7 + F~ Oy,~ (13) Let A z : T * M ~ V * M (9 H~, be the isomorphism A×(w) = (win×, W l v i ) , for all w c T*M. In a fibred coordinate system for p, the equations of A× are az(dxi)=p

i,

(14)

A z (dy ~) = dy ~ + Fi~p i ,

(15)

where (pl,..., pn) is the dual basis of (D1 . . . . . Dn) in H×. The equations of the inverse isomorphism A~ 1 • V * M (9 H~, --+ T * M are a;l(pi)=dx

i,

(16)

a ~ 1 (dy~lvM) = y ~.

(17)

By passing to the exterior algebra, A× induces an isomorphism of graded algebras A'A,/ • A ' T * M --+ A ' V * M ® A ' H ~ . For every r > 0, we thus have a linear isomorphism

r r-k V , M ® A k H ; ) . ArAy " A r T * M -+ (~k=0(A For every couple of integers r > k > 0, let W r'k be the sheaf on M of sections of the vector bundle A r A ~ I ( A r - k V * M ® AkH~), i.e. for every open subset U _c M,

"}/~r'k(g) = F ( g , ArAy 1 (Ar-kg*M ® Akg;)).

31

GLOBAL CHARACTERIZATIONOF VARIATIONALEQUATIONS

A local basis for W r'~ is ( y A A ~"~rM :

(dx)I)lAl=r_k,

lll=k .

r We have f2~

:

mr 1 A ) r ' k , and "~k=s''

m r )A)r,k ~SJk= 0 , , .

By using the isomorphism A°A×, we define an anti-derivation dv~ • ~2~ -+ ~2~t of degree +1 on the algebra of germs of differential forms on M, by setting d~lw~.k = ArAy 10dM/M o A~A×, which is uniquely determined by (d~)2 = 0 and d ~ f = A~I(dM/Nf), for all f E C~(M). We thus have d ~ f fy T , Y f c C~(M). Let d~4 • ~ t --~ a i r be the operator d~: = d - d~. Then, d~/ is an anti-derivation of degree +1 completely determined by the following formulae: -

k

d~ f = Di ( f ) d x i ,

(18)

d~(dx i) -- 0,

(19) O Y ic~

i

d~ y ~ = - d y ~ o R ~"+ ~y~ dX A T e ,

(20)

where R y denotes the curvature of V, i.e. the 2-form on M with values in V M defined by RY(X, Y) = IX h, Yh] v, X, Y ~ 5E(M). Hence

n ×=~R~jadx

iAdx j®

,

with

Oy~'

0×;

0×i_2

R~,o -- Oxi

°y;

0yf

OxJ + V[ Oy-~ - ~'7 0y~"

Moreover, (d~) 2 is a derivation of degree +2 completely determined by

(d;,) 2 ( S ) = dfIvM o (d~/) 2 (dx i) = 0 ,

(21) (22) (23)

h
4.3.

3y~

A sufficient condition for ~V to be globally variational

PROPOSITION 4.1. A globally defined affine Lagrangian L ~ C ~ ( j 1 M )

exists

such that ~F =

dOLv,

(24)

if and only if an (n - 1)-horizontal n-form On exists on f2n(M) such that ~F = dl]n.

(25)

Proof: If a globally defined affine Lagrangian L ~ C~(JaM) exists satisfying (24), then we can take tln = ®Cv, as the Poincar6-Cartan form--which is also globally defined on M in this case--is ( n - 1)-horizontal, as follows from its local expression ®Lv = ( _ 1) i-1 A Si y /xvi + A°v, L being given by (1).

32

J. MUNOZ MASQUI~ and L. M. POZO CORONADO

Conversely, if an ( n - 1)-horizontal globally defined n-form on M exists, say O, = B i d y ~ A vi + B°v, B °, B i ~ C a ( M ) , satisfying Eq. (25), then we can define a global affine Lagrangian by the formula Lv = h (On) i c~ = ( ( - 1 ) i--1 B~y i + B 0) V,

where h ' f 2 n ( M ) --+ S2n(J1M) denotes the horizontalization operator: h (O,)j~s = (Pl)* s* (On)s(x),

Vjlx s ~ JaM,

and it is readily checked that 0n = ®Lv, thus finishing the proof.

[]

COROLLARY 4.1. If n = d i m e = 1, then a globally defned affine Lagrangian L ~ C ~ ( J 1 M ) exists satisfying Eq. (24) if and only if ~V is exact. THEOREM 4.1. Let p • M ~ N be a fibred manifold over a connected and oriented manifold N. Assume that Hn-k(d~/N) = 0 for k = 0 . . . . . n - 2. Then every ( n - 1)-horizontal exact (n ÷ 1)-form on M is globally variational. Proof: Let us choose a nonlinear connection g " T M --+ V M . First of all, we remark that the hypothesis is equivalent to saying that im(W~_k_ 1 d(,> W~_~ ) = ker(W~_k d~ w~_k+l),

0 < k < n - 2.

(26)

Let ~ be an ( n - 1)-horizontal exact (n + 1)-form on M; hence ~ = don for a globally defined form 0n 6 fan(M). According to the graduation produced by y, the form 0n can be uniquely decomposed as 0n = 0~ + " " + 0In, where 7; 6 W~, for i = 0 . . . . . n. Let k be the least nonnegative integer for which the form 0~ does not vanish identically. If k > n - 1, we can conclude the proof by virtue of Proposition 4.1. Otherwise (i.e., if k < n - 2 ) , we can proceed by recurrence on k. As d~W~ c- - W~ +1 and dH'FV × ~r c- - )A2r+l @ ~A;~+I by virtue of the formulae ' "i+1 • vi+2' g / (18)-(20), we have dvO ~ = 0, and, according to (26), there exists (k c 14;~-1 such that 0~ = d~fk. Then, the form 0 n - dfk is a (k + 1)-horizontal primitive for ~, as 0n

-

=

0n

-

-

Y

d

t

¢k

t

= --dn(k + 0~+1 + "'" + On, and we can conclude the proof by virtue of the recurrence hypothesis.

[]

REMARK 4.1. If all the fibres of the submersion p " M ~ N are acyclic in dimensions 2 < d < n ; i.e., H n - k ( p - l ( x ) ; N ) = 0 for k = 0 . . . . . n - 2 , then the r i H r assumption in Theorem 4.1 holds, as H (dM/N) = (dM/N)®I'(M, p*f2iN) since d~t/N acts trivially on f~r, and there is an injection Hr(dM/N) ~ l~xsN H r ( p - ~ ( x ) ; 1R). In particular, this applies to the case of vector or affine bundles. Nevertheless, if the fibres of p are not contractible, the conclusion of Theorem 4.1 may fail, as shown in the next example.

GLOBAL CHARACTERIZATION OF VARIATIONAL EQUATIONS

33

EXAMPLE 4.1. Let p : M --+ N be as follows: N = S 1 x S 1 , parametrizing its points as x = (exp(ixl), exp(ix2)), x 1, x 2 6 R; M = N x (S 1 x S1), parametrizing its points as y = ( x ; e x p ( i y l ) , e x p ( i y 2 ) ) , x 6 N, yl, y2 E R, and p being the canonical projection onto the first factor. Hence we have n = 2, m = 2 and the smooth functions on M can be identified to the smooth functions on I~4 which are periodic with period 2zr with respect to each variable x h, yh, for h = 1, 2, i.e. f (x I + 27r, x 2, yl y2) = f (x 1, x 2, yl y2), and similarly for x 2, yl, y2. Let us choose f ~ C~(N), and let ~ be the (n - 1)-horizontal (n + 1)-form on M given by = d ( / ( x 1, x2)dy 1/x dy 2)

= fxldx 1 A dy 1/x dy 2 + fx2dx 2 A dy I/x dy 2. This form is globally defined as the differentials dx 1, dx2, dy 1, dy 2 are a global basis for the module of the linear differentials on M. Moreover, ~ admits a globally defined primitive, which is the n-form f d y l A dy 2. Assume a ( n - 1)-horizontal n-form on M primitive of ~ exists, say

= gdx 1 A dx 2 Jr-g11dx 1/X dy I + g12dx 1/x dy 2 Jr-g21dx 2 A dy 1 +g22dx 2 A dy 2, g, gab c= C~(M),

a, b = 1, 2.

Then, by comparing the corresponding coefficients of the forms d x l / x dx 2/x dy 1, dx 1/x dx 2/x dy 2, dx 1 /x dy 1 /x dy 2, dx 2/x dy 1/x dy 2, in the equation = d~, we respectively have

gyl = ( g l l ) x 2 -- (g21)x 1 , gy2=(gl2)x2 -- (g22)x 1 , fx 1 = ( g l l ) y 2 -- ( g l Z ) y l , fx 2 = (gsl)y2 -- (g22)yl • Let us fix an arbitrary point x0 E N and let us integrate the third and fourth equations above along the fibre p - l ( x 0 ) , which means to integrate the variables yl, y2 from 0 to 2zr and letting x = x0. For the third equation we have

~a0(y 1, y2) = gab(XO,ya, y2). The left-hand side of this equation equals 4zr2fxl(Xo), whereas the right-hand side (using Fubini's theorem to invert the integration order) is equal to

where

f2Jr ( fo2~r @ll)y2 dy2) dyl - fo2Jr ( foo2~r(~12)yl dyl) dy2 = O,

34

J. MUNOZ MASQUE andL. M. POZO CORONADO

since

fo rr (gab)yl dy 1 = gab (X0, 27r, y2) _ gab (Xo, O, y2) = 0, and similarly for the variable y2. Hence, fxl - 0 . Proceeding along the same way for the fourth equation we also obtain fx 2 - 0 . In summary: If f is not constant, then f cannot exist.

5.

Some examples

EXAMPLE 5.1. Assume N = IR, M = IR x Q and let p ' M - - + N be the projection onto the first factor. Let (ql . . . . . q m ) be a coordinate system on Q, m = dim Q, and let us denote by t the standard coordinate on I~. Then, using a more familiar notation, the system (4) can be written as follows: F~ = F,#~0/~. As above, we assume that F is skew-symmetric. Moreover, we can identify T N to the trivial line bundle by means of the vector field O/Ot, and we have (see (6)) a natural identification A2T*M = dt/~ (R x T'Q). Hence, from (8), we deduce that can be viewed as a vector bundle homomorphism F " IR x T Q --+ ]R x T* Q, i.e. /? is a section of the vector bundle lR x A2T*Q; or even, /" can be viewed as a 2-form on M, horizontal over Q. With this identification, we have ~F = co/X dt + [;, where co = F~dq ~. If the system (4) is assumed to be variational and the associated Lagrangian does not depend on time, then co = dH, where H is the Hamiltonian of the system. We also remark that if M is an arbitrary m-dimensional manifold endowed with a nondegenerate 2-form o)2 = F~dq ~/x dq ~, then the system (4) determines the integral curves of the vector field X defined by ixco2 = co, with co = F~dq ~. Hence, in mechanics, the systems studied above constitute a natural class. EXAMPLE 5.2. In this example we consider Lagrangian densities invariant under diffeomorphisms defined on the linear frame bundle 7r • F(N) --+ N of a manifold N. Hence, here we have M = F(N), m = n 2, where n = d i m N , and a Greek index is a pair of Latin indices, say oe = (i, j ) , i, j = 1 . . . . . n. First of all, we introduce some notation. Each coordinate system (x i) on an open domain U __ N induces a coordinate s y s t e m (xi,x}) on 7 r - l ( U ) by setting u = ((O/OXl)x . . . . . (O/Oxn)x) • (x}(u)), x = :r(u), and a coordinate system

(xi,x},x},k) on J1F(U).

General relativity can be formulated in the bundle of metrics as well as in the linear f r a m e ' b u n d l e of a manifold (tetrad or vierbein formalism, Einstein-Cartan theory, etc.). For the important role that such bundles play in the field theory we refer the reader to [11]. Let Z;)k : J I ( M ) - + ]~, j < k, be the Lagrangian

£~k(jlx s) = coi([Xj, Xk])(x), where s = (X1 . . . . . X , ) and (o) 1. . . . . co") is the dual coframe. The definition makes sense as coi([xj, Xk])(X) only .depends on jlxS. From the very definition we have [Xj, Xk]x = £.~k(jlxs)(Xi)x . Such Lagrangians are a basis for the ring of DiffN-invariant Lagrangians e.g. see [8].

GLOBAL CHARACTERIZATION OF VARIATIONAL EQUATIONS

35

Let 0 = (0 a. . . . . 0 n) be the soldering form on M = F(N). There are only two classes of variational problems defined by the densities g2~ = £ ~ k O a A . . . A on; precisely, those equivalent to S213 and those equivalent to ~'212 (see [8, Proposition 2.2]). The Euler-Lagrange equations for the extremals to such problems are a system of first-order quasi-linear equations, to which the theory developed in the present paper can be applied. The vertical bundle V(M) is trivial; as a matter of fact, there is an isomorphism M x g[(n, N) ---> V(M) given by (u, A) ~ A*, where A* is the fundamental vector field attached to A ~ g[(n, R) (see e.g. [6]). Furthermore, Ann+IT*M can be identified to Anrr*T*N ® V*(M) -~ Anrr*T*N x t~[(n, N)*. Accordingly, from the formula (8) we deduce that, in the present case, the homomorphism /? is defined on the bundle of g[(n, IR)-valued 1-forms on M horizontal over N, and takes values into the vector bundle of 9[(n, R)*-valued n-forms on M horizontal over N; that is,

if" : rc*T*N @ ~t[(n, IR) ~ 7r* An T*N × l~[(n, JR)*. Making use of the identification Horn (V, W) ~- V*® W, we can see /3 as a section of the vector bundle n-*TNNn-*A n T*N®I~[(n, R)*®9[(n, R)*. By taking the interior product in the two first components, and the exterior product in the two last ones, we transform /T into a A2g[(n, ®)*-valued ( n - 1)-form on M, horizontal over N. Finally, again taking into consideration the identification M x g[(n, R) ~ V(M), we arrive to a section of zr* An-1 T*N ® A2V(M), which in turn may be seen as an ( n - 1)-horizontal (n + 1)-form. This form is precisely the form ~. We can check this by taking the local expression of /T (given by (8)) into account,

[; (dx i ® (O/Ox~))

i --F;q,kldX qp A *¢,

which, considered as an element of rc*T*N @ Aner*T*N @ ~[(n, R)* @ ~[(n, R)*, reads = - - f p q , k ( ~ ® V ® (Ek) * ® ( E P ) * , i 1J , k, l = 1. . . . . n, are the standard basis where the matrices E~ = (aij), aij = Sk(~ in g[(n, IR), and {(E~)*} is its dual basis. The transformation with the interior and exterior products gives the ( n + 1)-form ( - - 1 ) i r;q,kl vi A d x ~ A d x p, which is precisely the local expression for the form $, since the Euler-Lagrange equations for this variational problem are homogeneous (see [8]), that is, F~ = 0.

EXAMPLE 5.3. Finally, we consider the free Dirac electron fields. Let us consider the linear representation p : SL(2, C) --+ GL(4, C) given by

p(A) =

(AO) O

A t-1

'

A c SL(2, C),

and let N be a connected 4-manifold endowed with a metric g of signature (l, 3), which is assumed to be space- and time-orientable. Hence its bundle of orthonormal

36

J. MUIqOZ MASQUI~ and L. M. POZO CORONADO

frames rc : F ( N ) --+ N has four connected components. Let 7r : Fo(N) --+ N be one of such components, so that Fo(N) is a principal bundle with structure group O+t(1, 3) = {B 6 O(1, 3) • d e t B = 1, boo > 1}. We also assume that N admits a spin structure, that is, a principal bundle rCs : S(N) -+ N with structure group SL(2, C) and a map )~ : S(N) ~ Fo(N) such that, (i) 7r(X(u)) = 7rs(u), for every u ~ S(N), and (ii) X(u • A) = X(u)A(A), for every u ~ S(N) and every A ~ SL(2, C), where A " SL(2, C ) ~ O+~(1, 3) is the two-sheet covering A(A)(x) = a x . A t,

x = (xo, xl, x2, xa) c N4,

X, = ( Xo + X3 Xl -- ix2 ) xl+ix2 Let (e0 . . . . .

xo-x3

e3) be the standard basis in ]~4 and let y : ~;~4 ~

~,(x)=

O x*

x.) O

~1[(4, C) be the map

'

with

x, _. ( -xO + x3 Xl + ix2

xl -- ix2 ). -Xo -- x3

The Dirac matrices are Vi = v(ei), 0 < i < 3. We have

y ( x ) y ( y ) + y ( y ) y ( x ) = 2 (x, y) I,

Vx, y 6 •4,

where I is the identity map in g[(4, C) and Ix, y) is the standard Lorentzian scalar product in ~4, i.e. (x, Y / = - x o Y o + x l y l +x2Y2+x3y3. If a~g is the connection form on Fo(N) defined by the Levi-Civita connection of g, then &g -- A , 1 o 3.*COg is a connection form on S(N), called the spin connection. We remark that the Lie-algebra homomorphism A. • ~[(2, C) --> 0(1, 3) induced by A • SL(2, C) --~ O+t(1, 3), is an isomorphism. Let 0 = )~*0, where 0 is the canonical 1-form in Fo(N). Let p : M = Ts(N) --+ N be the associated bundle to S(N) with respect to the representation p of SL(2, C) onto C 4. The spin connection induces a derivation law on Ts(N) and the Dirac operator is the differential operator on the sections of 1 J h Ts(N) given by (cf. [1, 2]) ~ - - - - ~ / J o V t / O x j + ~I~hkV VjV k ~, where yh = ~hiyi, and r / = 07hi) = diag ( - 1 , +1, +1, +1). Let us denote by (., .) : C 4 × C 4 ~ C the twisted Hermitian product, which is defined by (z, w) = -z-lw3 - z-2w4 + #3wl + z-4tb2. It induces a Hermitian bilinear map by ((., .)) • C 4 × C 4 ~ ]~ by setting ((z, w)) = ½ ((z, w) + (w, z)), and ((., .)) admits p(A) as an isometry, for every A ~ SL(2, C) (see [2, §20.2a]). Hence ((., .)) defines a metric on Ts(N), denoted by ((Vtl, ~2)), Then, the Dirac Lagrangian is defined by L(j~Tt) = ((~9~, ~ ) ) - m((~, ~)). Hence L is an affine function on

37

GLOBAL CHARACTERIZATION OF VARIATIONAL EQUATIONS

Pl0 : J l ( M ) ~ M and, accordingly, its Euler-Lagrange equations are of first order. As is well known (e.g., see [1, 2]), this is the Dirac equation: g ~ = m ~ . For the sake of simplicity, let us assume that m = 0, this is to say, we consider a massless electron. If we write ja~p with respect to the local coordinate system (xi, ~ , aPe,i) (i, 15 = 0 . . . . . 3), then the Euler-Lagrange equations are obtained upon variation of the ~ and of the ~ . The former yield

E0 "0 = ( ~ ) 2 ,

El "0 = (g~r)3,

E3 ' 0 = - ( ~ / r ) l ,

E2 ' 0 = - - ( ~ / r ) 0 ,

and the latter E4 " 0 :

-(~3~.r)2,

E5 " 0 :

-(~.r)3,

which are, in fact, equivalent to ~ equation as F~ :

E6 " 0 :

(~91/r)o,

E 7 "0 :

(~)~t)l,

----0. If we write the ot-th Euler-Lagrange

i i F~x[t15,i "~- F~,/3+4~r/3,i,

0<~<7,

0
1 J h k then, setting A = ~I'hkY y j y ~ , w e have

Fo = - A 2 ,

F1 = -~z~3,

F2 = ~z~0,

F4 :

F5 :

F6 =

A2,

A3,

-Ao,

F3 = ~Z~l, F7 = --Am,

whereas the F/g are given by o --rliiyOg i

0

'

i:0

. . . . . 3.

This allows us to construct ~ explicitly. Acknowledgement Supported by Ministerio de Educaci6n y Ciencia of Spain, under grant BFM200200141. REFERENCES [1] D. Bleecker: Gauge Theory and Variational Principles, Addison-Wesley Publishing Company, Inc., Reading, MA 1981. [2] Th. Frankel: The Geometry of Physics, Cambridge University Press, Cambridge, UK 1997. [3] H. Goldschmidt and S. Sternberg: The Hamilton-Cartan formalism in the Calculus of Variations, Ann. Inst. Fourier, Grenoble 23 (1973), 203-267. [4] J. Grifone, J. Mufioz Masqu6 and L. M. Pozo Coronado: Variational first-order quasi-linear equations, L. Kozma, P. T. Nagy and L. Tam~isy (Eds.), Proc. of the Colloquium on Differential Geometry, Steps in Differential Geometry, July 25-30, 2000, Debrecen (Hungary), Institute of Mathematics and Informatics, University of Debrecen, Debrecen, 2001, pp. 131-138. [5] A. Hakov~i and O. Krupkov~i: Variational first-order partial differential equations, J. Diff. Equations 191 (2003), 67-89. [6] S. Kobayashi and K. Nomizu: Foundations of Differential Geometry, Volume I, Wiley, New York 1963.

38

J. MUIqOZ MASQUI~ and L. M. POZO CORONADO

[7] J. E. Marsden and S. Shkoller: Multisymplectic geometry, covariant Hamiltonians, and water waves, Math. Proc. Cambridge Philos. Soc. 125 (1999), no. 3, 553-575. [8] J. Mufioz Masqu6 and E. Rosado Mar/a: Invariant variational problems on linear frame bundles, Z Phys. A: Math. Gen. 35 (2002), 2013-2036. [9] D. J. Saunders: The Geometry of Jet Bundles, Cambridge University Press, Cambridge, UK 1989. [10] S. Sternberg: Lectures on Differential Geometry, Second edition, with an appendix by Sternberg and Victor W. Guillemin, Chelsea Publishing Co., New York 1983. [11] J. J. Stawianowski: Field of linear frames as a fundamental self-interacting system, Rep. Math. Phys. 22 (1985), 323-371.