D-modules, contact valued calculus and Poincare-Cartan form

Czechoslovak Mathematical Journal, 49 (124) (1909), Praha

^-MODULES, CONTACT VALUED CALCULUS AND POINCARE-CARTAN FORM RICARDO J. ALONSO BLANCO, Salamanca (Received October 30, 1996)

1. INTRODUCTION AND NOTATION The geometric formulation of the calculus of variations has been carried out in the last decades in different fashion (see, for instance, [1], [2], [3], [4], [6], [7], [9 [12], [13], [16]). These methods, used in the process of generalizing from mechanics to the higher order case and several variables, may seem somewhat artificial. For instance, in the construction of a general Poincare-Cartan form, one utilizes a method involving determination of coefficients under certain prescribed conditions, and other adhoc techniques. It is not always clear what the real significance is of the role played by new objects (e.g. a linear connection on the base manifold) that are extrinsic t the problem. We offer a new approach that unifies the classical methods and their generalizations to field theories of higher order and the study of the variational bicomplex in a natural and rather simple way. We make use of two basic tools; first, a decomposition of operators that is trivial in the first order case. This is where the above mentioned connection comes in. Second, we introduce a formal covariant derivation law which is the geometric analogue of the derivative of variations with respect to time in mechanics. For a differentiable manifold M, we make the key remark that the algebra nn of differential forms of maximal degree is a right module over the ring @ of differential operators. With each linear connection on M we associate a morphism whose effect is to decompose the higher order tangent fields &k, k > I (^k C @) as a composition of ordinary tangent fields with differential operators of order k — 1. With this we achieve a factorization, via the exterior differential, of the action of &k on the *&module Hn. Mutatis mutandis, it is possible to generalize the method to the valued 585

case Qn®^/, when sd is a left ^-module through the action of the covariant derivative associated with a derivation law. Let ^ be the contact module on the jet spaces of a fibred manifold with base manifold M. By making use of the formal or total derivative, we define a covariant derivation law on M with values in <&. This law yields a differential ^-valued calculus on M that allows us to recover in a natural way the main objects and manipulations of the classical variational calculus and then to generalize it to the higher order and several variables case; in particular, the Euler-Lagrange form appears as a result of a certain action on the Lagrangian density. The integration by parts formula (called 'decomposition formula' in [6]) relates the differential of the Lagrangian density, the Euler-Lagrange form and an exact differential form. As shown in [6], to find a Poincare-Cartan form is equivalent to finding an 'integral' of this exact form. We do this by means of a constructive process which is based on the factorization of the action of yk on fin <8> <^. Obviously, the ^-valued calculus admits extension to a calculus with values in the exterior algebra of ^'. This extension allows us to define the so-called variational bicomplex 4>r's. We give a canonical construction (without reference to the coordinate expressions) of the Euler-Lagrange resolution. Finally, with the same technique with which we have built the Poincare-Cartan form (relative to a specific linear connection on M), we give the decomposition of the spaces $r'n (dirnM = n) explicitly finding an 'integral' in $r'n-1 of the exact term in this decomposition. The paper is structured as follows. In §2 we see how a linear connection on M produces a notion of degree in ^. By making use of the homogeneous components of §2 we find in §3 the decomposition morphism of STk. In §4 we recall some properties of the ^-modules. In §5 we will see how to use the decomposition morphism (§3) to factorize the action of ?7k on Ctn. In §6 we make some remarks on Weil's definition of the jet spaces and the vertical lift of tangent vectors. In §7 we recall the definition of the formal or total derivatives. With the results of §7 we give in §8 the definitions of the structure form, the contact module ^ and the 'if-valued differential calculus. In §9 we recover the fundamental elements of the calculus of variations, generalizing them to higher order and several variables, and we give our construction of the Poincare-Cartan form. Finally, in §10, we apply the above techniques to the EulerLagrange resolution. We now fix notation and a few conventions that we will use later. For a differentiable manifold M we will write ^°°(M} for the ring of infinitely differentiable functions, TM for the tangent bundle, T*M for the cotangent bundle and APM for the bundle of exterior forms of degree p. If IT: N —> M is a projection between differentiable manifolds, we will write TV(N/M) for the bundle of vertical tangent vectors of TT; if F —» M and G —> M are vector bundles, we will write FN for the 586

pullback TT*F and F <S>M G for the tensor product ; when there is no risk of confusion we will understand that the tensor product is taken before a suitable pullback. Calligraphic letters will be reserved for the modules of sections of the corresponding vector bundles; for instance, &M will mean the ^°°(M)-module of sections of TM, i.e. the module of tangent vector fields. As an exception np(M) will mean the •^(M^-module of sections of A^Af.

2. GRADUATION OF DIFFERENTIAL OPERATORS Let M be a differentiate manifold, dimM = n and mx the maximal ideal of functions vanishing at a point a; of M. A differential operator of order k on M is, by definition, an D?-linear morphism P: ^°°(M) that sends m£+1 to mx for each re e M. We will denote the set of differential operators of order k on M by ®k. Let p: M x IR —> M be the projection onto the first factor, Jkp the fiber bundle of A-jets of sections of p (i.e. the functions on M) and ^kp the module of the corresponding sections. The map jk: t^°°(M) —» ^kP, associating with every function its k-ih Taylor expansion, represents @k in the following sense: for every P € @>k there is a unique morphism P: ^kp —> ^°°(M) such that P = P o jk In other words, @k is the dual ^"(MJ-module of a?kp. Therefore @k is locally free, with rank (n*k), and, if (xi,... ,xn) is a local chart on M, @k is generated by da = (gf^)"1 o • • • o (gf^) Qn as a = (ai,... ,a n ) runs through the multi-indexes |a| ^ k (d° means the constant function 1). If k < r, then there is a natural immersion @k C ®T (observe that 0° = ^f°°(M)). Besides the structure mentioned, every 3>k has a second canonical ^^(MJ-module structure. If g 6 ^>00(M) and P € Sk, we define the operator g * P € @k according to the following rule: (g * P)f = P(g/)for every / e <^>00(M). This new structure for 3>k will be denoted by "W*. The ^°° (M)-module Hfi is also locally free with the same local generators. The subbundle T*>kM of <j?kp comprised of those fc-jets whose projections over M x IR are zero is, by definition, the fe-th cotangent fibre bundle of M. The fibre of each a; e M is T*'kM = m!C/m£+1 C ^°°(M)lmkx+l = Jkp. The module of sections of T*'fc will be denoted by ^*'fc. The dual of T*'k is called the fc-th tangent fiber bundle of M. We will denote it by TkM. Equivalently, TkM is the incident of R in Jkp. The module of sections of TkM will be denoted by £TkM. The elements of &kM can be characterized by Leibniz's rule of derivations of products. On the other hand &kM is, at the same 587

time, a quotient and a submodule of @k; in fact we have the decomposition

where we put P = P(l) + (P - P(l)) for each P 6 3>k. For the sake of simplicity, we will write &k instead of &kM and & instead of &1M. For k ^ 0, the quotient of the <^°° (M)-modules S>k/^k~l has the fibre (m^/m*"1"1)* c± Sk(mx/ml)* at each point x 6 M, where ( )* means taking the dual and 5fc means the fc-th tensor symmetric product. From the above, making use of the isomorphism (tn x /m^)* ~ Tx, we infer the exact sequence The image under the projection @k —>• Sh & of an element P € 3>k is known as the symbol of P. Remark 2.1. The same exact sequence holds for the tangent bundles (taking into account the inclusions 3?k C @h):

We can consider C/°°(M) and Sk& as the homogeneous minimal and maximal degree components of ^ fc , respectively. We will show how to define the remaining components for a specific linear connection. Lemma 2.2.

A linear connection V on M gives the module @k a graduation k

via an isomorphism © Sk& ~ 2>k (where S°& = ^°°(M)). r=0

Proof.

It suffices to find sections sr for the exact sequences

For r = 1, the sequence is

and we take the section si: ^ —> @l as the natural inclusion. For r > 1 we define the section sr: Sr' & —>• 3>r by the rule

588

where £>i • • • Dr is the symmetric product of r tangent vector fields Di and D? is the covariant derivative that V induces over Sr~1?7. It is easy to check that each sr is well defined, ^""'(MJ-linear and it is also a section of $r —>• Sr&. See [5] for another construction of this graduation of 3>k. We obtain the expression of this in coordinates by letting (zi,..., xn) be a local chart for M, dt = gfr, d*a = d"1 • • -9° 6 Sr&, da = df1 o • • • o d£ e 0r with a = (ai,..., a n ), |a| = r and V the local linear connection denned by (xi,..., xn)Then sr(dta) = da.

3. DECOMPOSITION MORPHISM OF HIGHER ORDER TANGENT BUNDLES If P is a differential operator of order k - I and D is a tangent vector field, the composition P o D belongs to &k. This type of compositions defines a Sf°°(M)module morphism that we will explain in what follows. Define a map

where G(P ® D) = P o D and extending by linearity. The map G is well defined by the choice of the structure for f^*"1. However, G is not ^>00(M)-linear. Now let jYk be the ^°°(M)-module obtained from Sik~lM O & by defining the product by functions as follows: for each / 6 <^'°°(M), we put f • (P ® D) = (fP) ® D; i.e., only multiplying / by the first factor and throughout the first structure for ^k~1. Over ^, the map G is <*?°°(M)-linear. Definition 3.1. morphism.

Henceforth G: jV^ —> 3?k shall be known as the composition

We are interested in determining a section of the morphism G for subsequent use. The following is the precise statement of this. Theorem 3.2. Each linear connection V on M produces a section Hy : 2fk —>• J/k of G, i.e. G o HV = Id- We will call Hy the decomposition morphism. P r o o f . It suffices to define the images by Hy for each of the homogeneous components of the graduation produced by V (Lemma 2.2). Let DI, ..., Dr be vector fields on M. For r = 1, we define #y(si(£>i)) = DI.

589

For r > 1, we put

where the symbol V means composing D{ with the first factor of the tensor product. As with sr, it is easy to see that #v is well denned and ^°° (M)-linear. We will verify that #v is a section of G:

By the induction hypothesis, GH^sr-i = s r _i, hence

Remark 3.3. For orders k = I and k = 2 the morphism Hy is independent of the connection V. Indeed, in order 1 it is clearly independent, and in order 2 we have where [Di,Di] is the Lie bracket of DI and D2. In general it can be deduced from the construction of Hy that Hy depends only on the (k — 2)-jet of the symmetric connection associated with V. In local coordinates, if V is the connection that the local chart (xi,... ,xn) induces on M, we have that Hv(da) = £ &\da~Si <8> &i where E* is the multi-ind

(Q,...,o,(t^--.,Q). 590

4. ^-MODULES

We have taken certain definitions and properties of ^-modules from Schneiders [18]. These properties will be used in the subsequent sections. Definition 4.1. The injective limit of the system (@k)k&* > where the inclusion gk c @h for k ^ h are considered, will be called the ring of the differential operators (of the manifold M) and denoted by @. The ring 0 inherits the ^^(MJ-module structure of @k. On the other hand, ^ is a non-commutative ring under composition of operators. Therefore, we need to distinguish between left and right D-modules. The proof of the next results can be found in [18]. Proposition 4.2. Let stf be a ^°°(M}-module and let a: ^°°(M) —>• EndR sf be the associated multiplication morphism. Let us assume that there exists a map X '• & —> EndR jtf defining an R-linear action of ^ on f^ such that: 1) [X(D), a(f)] = a(Df) (resp. -a(Df)), 2) (x(Di),x(D2)} = x((Di,D2}) (resp. -x([I>2,Z?i]);, 3) «(/) o X(D) = x(fD) (resp. X(D) ° a(f) = x(fD)) for every £>i, D2, D3 6 ff and every f 6 ^°°(M). Then there exists a unique structure of the left (resp. right) ^-module enlarging the actions a and xCorollary 4,3. The exterior algebra $ln(M) comprised of the forms of maximal degree possesses a unique structure of the right ^-module that extends its structure, a, of the tf00 (M)-module, and also extends the action of ^ given by LJ • D = —Lou for any a; € fln(M),D 6 & (Lo being the Lie derivative). Corollary 4.4. Let £^\, stfi be two left ^-modules and & a right ^-module. Then: 1) The action of t? on the ft00(M)-module $4\ ®M ^i given by D(a\ <8> 02) = Dai ® ^2 + «i ® Da,2 for any D € &, a\ € J&i, 0,2 6 J&2, extends to a unique structure of the left ^-module on stfi <8>M -#22) The action of & on the ^°° (M)-module SB ®M £& given by (b ® o) • D = b • D <8> a — b®Da for any D £ & ,b £ &, a € ^ 2 , extends to a unique structure of the right ^-module on & ®M ^2-

591

5. FACTORIZATION OF THE ACTION OF HIGHER ORDER TANGENT FIELDS ON fin Let D be a (tangent) vector field on M and u> a form of maximal degree. Then for the action denned in 4.3 we have w • D = —LouJ = d(-irju) (irju being the inner contraction of u> with D), in other words, we can build an explicit integral for the exact form u> • D. We want now to extend this result for the action of higher order vector fields P e &k> k > 1. The key to this resides in the decomposition morphism (Theorem 3.2). Theorem 5.1. Given a linear connection V on M, for each P € ^k, there is an K-h'near map $p: ftn(M) —» H n ~ 1 (M) that makes the next diagram commutative:

where -P is the action of P and d is the exterior differential. In other words, for any u> £ n n (M), $pw is an integral ofui • P. P r o o f . If Hv(P) = Y,Q®D £ ^k then we can define $ pw = - £ J D ( W - < ? ) which gives

where we have made use of the fact that LD = dio on fP(M). Remark 5.2. When P is of order 1 or 2, the decomposition morphism is independent of the linear connection, as a result of which $p is also independent. On the other hand, provided that dim M = 1, V is another connection and $p is the associated operator, then $p can only differ from $p in a constant term. Both $P and $p depend linearly on P, thus we deduce that this constant term is zero. Therefore, if dim M = 1 for any order fc, $p is also independent of the connection. Remark 5.3. The result obtained in the theorem is automatically generalizable to the following ('valued') case: let st be a (M)-module with a covariant derivation law such that the Lie covariant derivative converts £/ into a left ^-module (Lemma 4.2). The <^°°(M)-module nn(M) ®M ^ is now a right ^-module (Corollaries 4.3 and 4.4). Explicitly: if D 6 &, u 6 fln(M) and a € ^ we put

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where LD means the covariant derivative of a by D and LD(U®O) is the Lie covariant derivative. Hence Theorem 5.1 holds, with the same proof, when one replaces fT(M) by fln(M) jtf, and d by the correspondent exterior covariant differential. Corollary 5.4. Let L: !?k ®M fin(M) —>• fin(M) be the morphism associated with the action of &k C @ on fin(M). Given a fixed linear connection there exists a map $ that makes the next diagram commutative:

P r o o f . Define $(P®w) = $pw with P e &k, u 6 fin(M) and $P the operator denned in 5.1. Extending $ by linearity the proof is complete. Remark 5.5. In the same way as 5.1, Corollary 5.4 is valid in the 'valued' case.

6. JET SPACES AND VERTICAL LIFT Let TT: E —> M be a fibred manifold and Jk = Jk(E/M) the space of fc-jets of sections of TT. It is known that there is an isomorphism

i.e., if p1 is a 1-jet of J1 that projects over p 6 E and x G M, with each pair (w x , Dp), comprised by ujx £ T*(M) and Dp 6 T£(E/M), we can associate canonically a tangent vector in T£(Jl/E). It will be shown in this section how to generalize this operation in higher orders. The result obtained is equivalent to [7] ((14) of §1). Nevertheless, our calculations make use of an alternative construction of the jet spaces [14], following Weil's suggestions (see [20] and [8]). In agreement with [14], a fc-jet pk € Jk over a point x 6 M is a morphism of algebras tf°°(E) —> <^0°(M)/m£+1 such that, restricted to C°°(M), this is the quotient by m£+1. In categorical terminology we would say that pk is 'a point of E with values in
The

principal

advantages

of

this

construction

stem

from

always

working over the same ring C£°°(E}. 593

In accordance with this, the tangent space to Jk at pk vertical to M, T£k(Jk/M), is identified canonically with the derivations over ^°°(M) of the ring x on ^>00(M)/mk+1 by -ux. We have a composition

Further, if pfe+1 is a (k + l)-jet over p fc , (•(*>„) o Dpfc can be identified as above with a tangent vector at pk+1 which is vertical to Jk+l —* E (and not only to jk+i __). M). Thereby this defines a morphism:

(recall that the calligraphic letters mean modules of sections). Definition 6.1.

Henceforth the morphism

obtained from (*) by transposing ,^'*>fc+1 shall be called the vertical lift. To express this in coordinates, let (xi,...,xn) be a local chart on M and let (j/ii • • • 12/m) be local coordinates on the fibres of TT : E —> M. These charts produce the usual fibred coordinates (EJ, j/j a ), i = 1,... ,n, j = 1,... m, |a| < k on Jk. Then

where 9" = (jf^-i o ... o (^-)«-, a = (a,,...,an). 7. THE FORMAL DERIVATIVE

In this section the concept of 'formal' (also known as 'total') derivative will be defined (refer to [10] or [15]). By means of the immersion J fc+1 c J1 J fe , we associate with each pk+l e Jk+l an element of J1 Jk that will be denoted by pk>1. Following [14], pk'1 is a morphism tf°°(Jk) —>• «700(M)/m^. Thus for each function / e ^°°(Jfe) we have pM/ € if 00 (M)/m2. 594

Remark 7.1. With the usual construction of jet spaces, pk+l is the (k + l)-jet of some local section s in x of TT: E —»• M, i.e. pk+l = jk+1s. Let s = j fe s be the local fc-jet prolongation of s. Then pk>l = j'^s and pk'lf is the class of the function / o s modulo m£. Definition 7.2. Each tangent vector field D 6 ^ produces a derivative

denned by the rule (Df)pk+l = Dx(pk>1 /) 6 ^°°(M)/mx ~ R. We will call D the holonomic lift of D to the jets. Similarly, we can consider D as a section of (TJk)jb+i or as a map Jk+1 —t TJk over Jk. Later on we will use the notation D = bk+i(D). Remark 7.3. The holonomic lifts to different orders are compatible with each other. This fact allows us to use the notation D without reference to k. Let ( x i , y j a ) be fibred coordinates as above. Then ^ = ^- + 53 Vja+eiQ^— • |a|
Definition 7.4. Let D be a vector field on M. We will call the map defined below the formal inner contraction with respect to D:

defined by the rule (iQ(r)pk+i = ig apk , for any a 6 $lp(Jk) and pk+l £ Jk+1 over pk; the second member of the equality being an ordinary contraction (observe that £>pfc+i isinT p fcJ f e ). Definition 7.5. We will denote the formal Lie derivative with respect to the vector field D on M as the derivative

defined by Cartan's formula Lg = igd + dig . The formal Lie derivative verifies the usual properties of the ordinary Lie derivative (see [10], [15]).

595

8. DIFFERENTIAL CALCULUS WITH VALUES IN THE CONTACT MODULE In this section we will recall the definition of the contact module and then see how the restriction of the formal Lie derivative (Definition 7.5) produces a covariant derivation law. The holonomic lift, &&: T(M)jk —> T(Jk~l)jk, is a canonical splitting of the exact sequence

Definition 8.1. The retract 6k associated with && is known as the structure form of Jk. Through pull-back to Jk, Ok defines a section of the bundle T* Jk ®jt.-. T v (J k ~ l /M). With the above notation, in a fibred local chart ( x i , y j a ) we have

As is known, a section s of the projection Jk —>• M is holonomic (i.e., is the fc-jet prolongation of some section of ?r: E —> M) if and only if 6k vanishes on the image of s. Definition 8.2. We will call the (^'00(Jk)-module comprised by those 1-forms of Jk that become zero over the holonomic sections as the contact module of J*, denoted by %. The module % is generated by the components of Ok (the above I0ja-s'). The fibre bundle associated with ^ will be denoted by Ck. It is useful to observe that the dual map of 6k produces an immersion

which identifies the dual of Tv(Jk 1/M)Jk with Ck. In particular, the holonomic lifts of tangent vectors on M are incident with Ck • Denote the natural projections Jk —> Jr, 0 ^ r < k, by irk. Each ^ with r < k induces a submodule of % via the pull back by ifk. Definition 8.3. We will call the injective limit of the system (^,71-*) the contact module *&:

596

The restriction of the formal Lie derivative in Definition 7.5 to ^fc C ft1(Jk) takes values in ^t+i. Indeed, if s is a local section of •n: E —> M, jfc+1s is the (k + l)-prolongation and a G %, then because i^a = 0 we have (j fc+1 s)*Lgcr = (jk+1s)*igda = iDd(jks)*a = iDdQ = 0, hence Lga € tfk+iIn a local chart (xi,yja) as above, we have L~g~9ja = Oja+a • Qa-.i

Because of the compatibility of the holonomic lifts with the projections TT£ it is possible to define, for any tangent vector field D on M, the formal Lie derivative

Recall that ^ is, in particular, a ^°° (M)-module. Proposition 8.4. The assignment D —> Lg produces a covariant derivation law on the
This law allows us to define a "^-valued differential calculus on M. In particular, we will have a formal covariant exterior differential

and a formal covariant Lie derivative

(by putting LD — iod + dirj). Remark 8.5. d and LD satisfy the relations

and

Moreover, in local coordinates we have d&ja = $Z d&i ® ^jQ+ei • i

Proposition 8.6. Tie module ^ has the structure of a left ^-module, precisely that denned by the LD action. 597

Proof. Let a: ^°°(M) —> Endj? ^ be the multiplication corresponding to the structure of the <^"X) (M)-module of tf and let us define x- & —>• EndR ^ by X(D) = LD for any D 6 &. We must check properties 1), 2) and 3) of Proposition 4.2 as follows: For D,Di,Da 6 &, f € V°°(M) and yeV, 1) [x(D),a(f)]a = LD(fa)- fLDa = Df • a +JLDa - fLDa = a(Df)a. 2) [x(Di),x(D2)}cr = LDlLD^a - LD2LDlff = L[DltDz]a = x([Di,D2])a. 3) &(f) ° x(D)a = fLov = fioda = ifDda = L5t>a = x ( f D ) a . Corollary 8.7. fin(M) ®M *& has a right ^-module structure (n = dim M). P r o o f . According to Corollaries 4.3 and 4.4, the former structure is obtained by extending the following action of &\ for any w € f l n ( M ) , a € *& and D € ST we put (u<S>a)-D = ui-D®cr — u<S> D • a = -Lpt^ ® a - u> ® LDCT = — LD(W ®
and analogously

where (-L 9 )^ = (-£ ai ) ft o • • • o (-Ldn)^. Remark 8.8. The ^-valued differential calculus is extensible in an obvious way to a calculus with values in A fc( if, Sfc#, etc. Remark 8.9. If r/ 6 fT(M) ®M % and P 6 j^1 or 6 &h then TJ • P € fln(M) ®M %+h-

Remark 8.10. Corollary 8.7 and Remark 5.3 say that the factorization Theorem 5.1 is available for the fin(M) <8>M ^ case.

598

The passage of the "^-valued calculus to the ordinary one is carried out via the morphism This process consists of a combination of introducing Q P (M) and ^ into £l'(Jk) and then alternating. This is equivalent to taking the exterior product, multiplying by the structure form 0k relative to the bilinear product that the duality between % and ^v(Jk~1/M)Jk produces (see Definition 8.2). Definition 8.11. We call 77 the antisymmetry operator. The morphism T] relates dto & certain differential on Jk. Definition 8.12. The unique antiderivation of degree 1,

such that 1) for p = 0, dH = H o d, where H: n^J*) —> n^MJ^+i C fll(Jk+1) is the transposed operator of the holonomic lift bk (Definition 7.2); 2) dh o d — —do dn

will be called the horizontal differential. In local coordinates ( x i t y j a ) we have

It is a simple process to check the following: Proposition 8.13. On H P (M) ®M %, d>n and d satisfy rj o d — du ° rj.

9. CONSTRUCTION OF THE POINCAR£-CARTAN FORM In the classical variational calculus the equations for the critical sections are found by using integration by parts. Briefly, if the problem is defined by a lagrangian function Jf = _£f(t,<7i,<7i), where t, qi and <& are the time, position and velocit coordinates of a mechanical system, respectively, the variation of J£ dt is written as

599

by making use of -gj% = Sqi (this '£' does not relate to the vertical lift). The construction that we give below is based on an adequate interpretation of the elements in the above formula. The variation is viewed as a type of vertical differential in such a way that with the obvious notation, Sqi = dqi - qt dt, Sqi = dqt — q\ dt, 8^f = i/jrbqi + §T"&?t are contact forms; the second term of the second member of (*) is the formal derivative with -^ of ^Sqidt, and the last is simply the result of applying the vertical lift to the structure form (dqi — qi dt) ® •/- and then to acting on & dt. From the above we deduce that the ^-valued differential calculus of §8 appears as the best framework for the generalization of the calculus of variations to several independent variables and the higher order case. In general, let A be an element of £ln(M)jk, i.e. a Lagrangian density of order k over TT: E —> M, and let 0fe e % ®jk &v(Jk~l/M) be the structure form on Jfe. Definition 9.1. We will call the 1-form 6k = SoBk 6 % ®Jk &k®Jh &v(Jk/E) the vertical lift of the structure form (6 is the vertical lift morphism defined in 6.1). We will consider also 0 fe (A) e &k ®M tfk ®M ftn(M), the result of contracting d\ with the index of Sk in &v(Jk/E) (i.e., to 'derive A by <9fel). Now let us denote by L the morphism given by the action of 2Fk C !& onffk ®M fln(M) (Corollary 8.7 and Remark 8.9):

Definition 9.2. Let Ax = L(6k(X)) 6 ^k ®M fi"(M) and let us define the vertical differential of A as cf A = 6k+1(\) e %+i ®M fln(M) (in the same sense as the definition of 0 fc (A)). Then we will call the ^-valued form S = A\ + dv\ the Euler-Lagrange form associated with the Lagrangian density A. Remark 9.3. Definition 9.2, written in the form dv\ = S — A^, is the identity corresponding to integration by parts (*) in the classical variational calculus. Let (xi,yja) with 0 ^ |a| ^ k — 1 be fibred local coordinates in J*0"1 and analogously, in Jk with 0 < |a| < k, and let A = Jfw, where 3? = 5?(xi,yja) € ^°°(Jk) 600

and w = dxi A ... A dxn. Then

and

Taking into account the coordinate expression for § we obtain Proposition 9.4. The Euler-Lagrange form & belongs to (^i)j2fc ®M H n (M). The utility of the integration by parts (Remark 9.3) depends critically on the fact that A* is an exact differential. Theorem 9.5. (n — I)-form

By fixing a linear connection V on M, there is a ^-valued

from which it can be deduced that dv\ = § — d&i. P r o o f . The operator (depending on V)

is constructed as in Corollary 5.4 in such a way that L = do$. Hence A\ = L9k(X) = d($0k(\)). By putting 0j = $$fe(A) the proof is complete. 601

Via the antisymmetry operator 77 (Definition 8.11), the objects &, ©i and dv\ can be understood as forms in £lp(Jr) (for suitable values of p and r). We will use the following notation:

~§ - r) ° §,

0i = r) o ©j

and

d"\ = ijodv\ = d\.

Definition 9.6. We will call the n-form 0 e n n (J 2fc " 2 )ja fc -i denned by 0 = ©i + A the Poincare-Cartan form associated with the Lagrangian density A (and with the linear connection V). Remark 9.7. In fact © 6 Cln(Jk~1)J2k-i with |a| ^ k.

because J£ depends only on the yja

In a fibred local chart (xi,yja), if V is the local connection produced by (xi) on M and if #v is the associated decomposition morphism (3.2), we have

where u>i = z'a;<j. Finally, we make use of the combinatorial identity

to obtain

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The above expression coincides with that obtained in [12] and [16]. Theorem 9.8. dX = H - dH®. P r o o f . This is a direct consequence of Theorem 9.5 and the relations (rjod]Q1 = (dHorj)el=dHQi, da\ = 0. Remark 9.9. The property enunciated in the last theorem or, equivalently, the correspondent one in Theorem 9.5 is known as 'decomposition formula' in [7]. This formula makes it possible to find the equations for the critical sections in a variational problem defined for a given Lagrangian density (see [7]). As demonstrated in [6], any other form verifying 'decomposition formula' as 61 is of the form &i + G, where G <E (^j^-' ®M n n ~ 2 (M). Remark 9.10. There is no total agreement on this topic. One can consult [4] and [13] for a discussion concerning the construction of the Poincare"-Cartan form. See [1], [2], [3], [9], [11], [12] and [16] for other references of the geometric formula the calculus of variations (naturally, the above list is not exhaustive). 10. APPLICATION TO THE EULER-LAGRANGE RESOLUTION The ^-valued differential calculus of §8 allows us to define intrinsically the elements of the Euler-Lagrange resolution. We will follow Tulczyjew's paper [19] (it can also be consulted in [17]). We fix the projection TT: E —> M and work on the infinity jet space J°° = proj lim Jk. Let us consider the tensor products $r's = Ar
In the same way as in Proposition 8.6, one checks that /\^®M tln(M) (dimM = • n) is a right ^-module. Let us denote by L the action of @ over /\ # ®M O"(M), i.e.

where • means 'action'. 603

Remark 10.1. d corresponds, via the antisymmetry operator, to the horizontal differential dn of [19]. Moreover, this notation is coherent with Definition 8.12. On the other hand, the vertical lift morphism (Definition 6.1) can be dualized giving us a morphism 5*k : %+i —>• % ®j**+i ^k • With notation as above, in local coordinates we have

The morphisms b*k are compatible with the inclusions % C^r, 3^k C &*, k < r. In this way it is possible to define 8% on the injective limit, thus obtaining a map

where &°° = projlim^ fc = {P € 0/P(l) = 0}. The local expression of 5* is the same as that of 5%. Now we extend S* to a derivation over the exterior algebra of ff. We get the map (keeping the notation)

The composition L o 5* is denoted by T:

We will make use of the notation

In this way the elements of f\rc& ®M tln(M) have the form

Therefore

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and

Since for any ju € $r's, r is obtained by applying L, we deduce (Corollary 5.4, Remark 5.5) that for each linear connection V on M, there exists an element F^ € $r'n-1 such that Tfj, = dFp. Now let us modify the definition of r over each $ r>n by putting TT = f + r • Id, where Id is the identity map. In local coordinates, if n is as above, we have

Moreover, r r ju = dF^ +r • /J,. Finally, if we put rr = ^rr and F^ = ^F^, then

One can deduce as in [19] that TT o d = 0. By applying this equality we deduce that rr o Tr\i, — rr o dFp + Tr/J, = TT/J,. In other words, rr is a projector in $ r > n . Theorem 10.2. The subspace A.r = Tr$Tn;

P r o o f . With each fj, € $r'™ we assign the decomposition /x = Trfj, - dF^. The rest follows since rr is a projector and rr o d = 0. Remark 10.3. Even though F^ depends on the choice of the connection, the element dF'^ is canonically denned. Remark 10.4. The above theorem can be viewed as a generalization of the procedure in Theorem 9.5 (by taking there p, = d"j£?w). As in Theorem 9.5, it is easy to find the explicit expression of F^. Spreading the vertical differential d" to every <&r'n as in Definition 9.2 and following [19] we can define the variational operators vr = rr+1 o d". Ultimately, the EulerLagrange resolution is the sequence (see [19])

The operator v° is known as the Euler-Lagrange operator and i>1 is known as the Helmholtz-Sonin operator (inverse problem of the calculus of variations). Acknowledgements. The author wishes to thank Prof. Munoz Diaz and Prof. Rodriguez Lombardero for stimulating discussions and continuous encouragement during all stages of this work. 605

References [1] P.L. Garcia: The Poincare-Cartan invariant in the calculus of variations. Symp. Math. XIV (1974), 219-246. [2] P.L. Garcia and J. Munoz: On the geometrical structure of higher order variational calculus. Proceedings of the IUTAM-ISIMM. Symposium on Modern Developments in Analytical Mechanics (Bologna). Tecnoprint, 1983, pp. 127-147. [3] H. Goldschmidt and S. Sternberg: The Hamilton-Cartan formalism in the calculus of variations. Ann. Inst. Fourier (Grenoble) 23 (1973), 203-267. [4] M. Gotay: An exterior differential system approach to the Cartan form. Geometric Symplectique et Physique Mathematique (Boston). P. Donato et al., Boston, 1991, pp. 160-188. [5] H. Hess: Symplectic connections in geometric quantization and factor orderings. Ph.D. thesis, Berlin, 1981. [6] M. Horde and I. Kolar: On the higher order Poincare-Cartan forms. Czechoslovak Math. J. 33 (1983), 467-475. [7] /. Koldr: A geometric version of the higher order Hamilton formalism in fibered manifolds. J. Geom. Phys. 1 (1984), 127-137. [8] /. Koldr, P. Michor and J. Slovak: Natural Operations in Differential Geometry. Springer-Verlag, Berlin, 1993. [9] D. Krupka: A geometric theory of ordinary first order variational problems in fibered manifolds I. Critical sections. J. Math. Anal. Appl. 49 (1975), 180-206. [10] A. Kumpera: Invariants differentiels d'un pseudogroupe de Lie, I. J. Differential Geom. 10 (1975), 289-345. [11] B. Kupershmidt: Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalism. Lect. Notes in Math. 775. 1980, pp. 162-217. [12] J. Munoz: Poincare-Cartan forms in higher order variational calculus on fibred manifolds. Rev. Mat. Iberoamericana 1 (1985), no. 4, 85-126. [13] P.J. Olver: Equivalence and the Cartan form. Acta Appl. Math. 31 (1993), 99-136. [14] J. Rodriguez: Sobre los espacios de jets y los fundamentos de la teoria de los sistemas de ecuaciones en derivadas parciales. Ph.D. thesis, Salamanca, 1990. [15] C. Ruiz: Prolongament formel des systemes differentiels exterieurs d'ordre superieur. C. R. Acad. Sci. Paris Ser. I Math. 285 (1977), 1077-1080. [16] D. Saunders: An alternative approach to the Cartan form in Lagrangian field theories. J. Phys. A 20 (1987), 339-349. [17] D. Saunders: The Geometry of Jet Bundles. Lecture Notes Series, vol. 142. London Mathematical Society, Cambridge University Press, New York, 1989. [18] J.P. Schneiders: An introduction to the D-Modules. Bull. Soc. Roy. Sci. Liege 63 (1994), 223-295. [19] W.M. Tulczyjew: The Euler-Lagrange resolution. Lect. Notes in Math. 836. 1980, pp. 22-48. [20] A. Weil: Theorie des points proches sur les varietes differentiables. Colloque de Geometric Differentielle, C.N.R.S. (1953), 111-117. Author's address: Departamento de Matematica Pura y Aplicada, Universidad de Salamanca, Plaza de la Merced, 1-4. E-37008 Salamanca, Spain, e-mail: ricardoQgugu.usal.es.

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