Generalized Bianchi identities in gauge-natural field theories and the curvature of variational principles

Vol. 56 (2005)

REPORTS ON MATHEMATICAL PHYSICS

No. 1

G E N E R A L I Z E D BIANCHI IDENTITIES IN GAUGE-NATURAL FIELD THEORIES AND THE CURVATURE OF VARIATIONAL PRINCIPLES* M. FRANCAVIGLIA, M. PALESE and E. WINTERROTH Department of Mathematics, University of Torino via C. Alberto 10, 10123 Torino, Italy (e-malls: [email protected], [email protected], [email protected]) (Received July 26, 2004)

By resorting to Noether's Second Theorem, we relate the generalized Bianchi identities for Lagrangian field theories on gauge-natural bundles with the kernel of the associated gauge-natural Jacobi morphism. A suitable definition of the curvature of gauge-natural variational principles can be consequently formulated in terms of the Hamiltonian connection canonically associated with a generalized Lagrangian obtained by contracting field equations. 2000 MSC: 58A20, 58A32, 58E30, 58E40, 58J10, 58J70. Keywords: jets, gauge-natural bundles, generalized Bianchi identities, generalized Jacobi morphisms, curvature.

1.

Introduction

To investigate conservation laws for covariant field theories, see e.g. [1] and references quoted therein, the general problem has been tackled of coherently defining the lifting of infinitesimal transformations of the basis manifolds to the bundle of fields (namely bundles of tensor fields or tensor densities which could be obtained as suitable representations of the action of infinitesimal space-time transformations on frame bundles of a given order [23]). Such theories were also called geometric or natural [28]. An important generalization of natural theories to gauge fields theories passed through the concept of jet prolongation of a principal bundle and the introduction of a very important geometric construction, namely the gauge-natural bundle functor [4, 17]. Within the mentioned above general program, generalized Bianchi identities for geometric field theories were introduced to get (after an integration by parts procedure) a consistent equation involving divergences within the first variation formula. It was also stressed that in the general theory of relativity these identifies coincide with the contracted Bianchi identities for the curvature tensor of the pseudo-Riemannian metric. *Partially supported by GNFM of INdAM, MIUR (PRIN 2003) and University of Torino. Lecture given at the 36th Symposium on Mathematical Physics, Torufi, June 9-12, 2004. [11]

12

M. FRANCAVIGLIA, M. PALESE and E. WINTERROTH

Our general framework is the calculus of variations on finite-order fibered bundles. Fibered bundles will be assumed to be gauge-natural bundles (i.e. jet prolongations of fiber bundles associated to some gauge-natural prolongation of a principal bundle P [4, 17]). Such geometric structures have been widely recognized to suitably describe so-called gauge-natural field theories, i.e. physical theories in which right-invariant infinitesimal automorphisms of the structure bundle P uniquely define the transformation laws of the fields themselves (see e.g. [5] and references quoted therein). We shall in particular consider finite order variational sequences on gauge-natural bundles. The gauge-natural lift enables one to define the generalized gauge-natural Jacobi morphism where the variation vector fields are Lie derivatives of sections of the gauge-natural bundle with respect to gauge-natural lifts of infinitesimal automorphisms of the underlying principal bundle P. Within such a picture, as a consequence of the Second Noether Theorem, it is possible to relate the generalized Bianchi morphism to the second variation of the Lagrangian and then to the associated Jacobi morphism [8, 9, 25, 26]. In the case of geodesics in a Riemannian manifold vector fields which make the second variation to vanish identically modulo boundary terms are called Jacobi fields and they are solutions of a second-order differential equation known as Jacobi equation (of geodesics). The notion of Jacobi equation as an outcome of the second variation is in fact fairly more general: formulae for the second variation of a Lagrangian functional and generalized Jacobi equations along critical sections have been already considered (see e.g. the results of [2] and classical references quoted therein). In this paper we show that, as a consequence of the gauge-natural invariance of the Lagrangian, there exists a covariantly conserved current associated with the contraction of the Euler-Lagrange morphism with a gauge-natural Jacobi vector field. Such a conserved current can be considered as a Hamiltonian for the Lagrangian corresponding to such a contraction. The curvature of the variational principle can be then defined as the curvature of the corresponding Hamiltonian connection.

2.

Finite order jets of gauge-natural bundles and variational sequences

In this section we recall some basic facts about jet spaces. We introduce jet spaces of a fibered manifold and the sheaves of forms on the s-th order jet space. Moreover, we recall the notion of horizontal and vertical differential [17, 27]. Our framework is a fibered manifold Jr : Y --+ X, with dim X = n and dimY = n + m. For s > q >_ 0 integers we are concerned with the s-jet space JsY of s-jet prolongations of (local) sections of Jr; in particular, we set JoY =- Y. We recall the natural fiberings Jrs : J~g -+ JqY, s > q, Jr~ • J~g ---> X , and, among these, the affine fiberings Jr~-l" ~v~e denote by V Y the vector subbundle of the tangent bundle T Y of vectors on g which are vertical with respect to the fibering Jr. Greek indices or,/x . . . . run from 1 to n and they label basis coordinates, while Latin indices i, j . . . . run from 1 to m and label fibre coordinates, unless otherwise specified. We denote multi-indices of dimension n by boldface Greek letters such as

13

CURVATURE OF VARIATIONAL PRINCIPLES

ot = (oq . . . . . otn), with 0 < oe,, # = 1 . . . . . n; by an abuse of notation, we denote by a the multi-index such that or, = 0, if # # a , otu = 1, if /z = o-. We also set lot[ : = o q + . . . +otn and ot! : = c q ! . . . a n ! . The charts induced on JsY are denoted by (x °, yi), with 0 < Iotl < s; in particular, we set y~-= yi. The local vector fields and forms of JsY induced by the above coordinates are denoted by (0~) and (d~), respectively. For s > 1, we consider the natural complementary fibered morphisms over JsY---> Js-aY (see [19, 20, 29]):

D" JsY × T X ~ T J s - I Y ,

0 " JsY

X

x

T J s - I Y -+ V J s - I Y ,

Js_lY

with coordinate expressions, for 0 _< Iotl -< s - 1, given by

J o~), 79 = dXD), = d )' (O~,+ yot+)Oj

19-=

~=(d Ootj ,Oj

j

--

j ~.)Oj. y,~+)d

The morphisms above induce the following natural splitting (and its dual),

JsY

x

T*Js-IY = (JsY

Js-lY

where C*_l[Y]::imO* and O * ' J ~ Y

x

T * X ) GC*_I[Y],

(1)

Js_l Y

× V*Js-IY-+ JsY "Is 1 r

x

T*J~-IY.

Js-1 Y

If f " J ~ Y - + I ~ is a function, then we set D o f : = D ~ f , D~+~f : = D ~ D ~ f , where D~ is the standard formal derivative. Given a vector field E • J~Y --~ T J y , the splitting (1) yields E o 7rf+1 EH + Ev where, if ~ = EY0× + ~aOi, then we have ~H = ,~YD× and Ev = ( E / _ y. ai + y ~ ~ y a)0' ~i . We shall call ~H and Ev the horizontal and the vertical part of E, respectively. The splitting (1) induces also a decomposition of the exterior differential on Y, (zr~_a)*od = dH + d r , where dH and dv are defined to be the horizontal and vertical differential. The action of dH and dv on functions and 1-forms on JsY uniquely characterizes dH and dv (see, e.g., [27, 29] for more details). A projectable vector field on Y is defined to be a pair (u,~), where u • Y -+ T Y and ~ • X -+ T X are vector fields and u is a fibered morphism over ~. If there is no danger of confusion, we will denote simply by u a projectable vector field (u, ~). A projectable vector field (u, ~) can be conveniently prolonged to a projectable vector field (jsU, ~); coordinate expression can be found e.g. in [19, 27, .29].

2.1.

Gauge-natural bundles

Let P ---> X be a principal bundle with structure group G. Let r < k be integers and w(r'k)P := JrP x Lk(X), where Lk(X) is the bundle of k-frames x in X [4, 17], w(r'k)G := JrG 63 GLk(n) the semidirect product with respect to the action of GLk(n) on JrG given by the jet composition and GLk(n) is the group of k-frames in IRn. Here we denote by Jr G the space of (r, n)-velocities on G. The bundle w(r'k)P is a principal bundle over X with structure group w(r'k)G. Let F be any manifold and ff : w(r'k)G × F----> F be a left action of w(r'k)G

14

M. FRANCAVIGLIA, M. PALESE and E. WINTERROTH

on F. There is a naturally defined right action of w(r'k)G on w(r'k)P × F so that we can associate in a standard way to w(r'k)P the bundle, on the given basis X, Y¢ := w(r'k)P ×~ F. DEFINITION 1. We say (Y¢, X, :re; F, G) to be the gauge-natural bundle of order (r, k) associated to the principal bundle W(r'k)P by means of the left action of the group w
.0 t

cb¢ • Y¢ -+ Y+ • [(Jr )/, Jk ), f ] ( ~

.x .0 [ q b ( J r Y, Jkt), f ] ¢ ,

(2)

where ] c F and [., .]~ is the equivalence class induced by the action ~. DEFINITION 2. We define the vector bundle over X of right-invariant infinitesimal automorphisms of P by setting Jt = T P / G . We also define the vector bundle over X of right invariant infinitesimal automorphisms of w(r'k) P by setting ~(r,k) := TW(r,k) p/w(r,k)G (r <_k). Denote by 7"x and .A(r'k) the sheaf of vector fields on X and the sheaf of fight-invariant vector fields on W(r'k)P, respectively. A functorial mapping @ is defined which lifts any fight-invafiant local automorphism (qb, q~) of the principal bundle w(r'k)P into a unique local automorphism (qbC, ~b) of the associated bundle Y~. Its infinitesimal version associates to any F, E A (r'k), projectable over ~ 6 Tx, a unique projectable vector field a,: :=@(~,) (the gauge-natural lift) on Y~ in the following way, ^

@ • Y~

x

~4(r'k) ~ TYc

•

,~ (y, E) ~+ ~(y),

(3)

X

where, for any y c YC, one sets: ~(y) = d[(qbCt)(y)]t= 0, and qSct denotes the (local) flow corresponding to the gauge-natural lift of qst. This mapping fulfils the following properties (see [17]): 1. ~b is linear over idrc. 2. We have Trr¢ o@ = idTx oft (r,k), where z~(r,k)is the natural projection YC x .A(r'k) --+ X

TX. 3. For any pair (.~, F,) 2.2.

6

.A(r'k), we have @([A, F,]) = [@(.~), ~5(~)].

Lie derivative

DEFINITION 3. Let F be a (local) section of Y~, ~ ~ A (r'k) and ~ its gauge-natural lift• Following [17] we define the generalized Lie derivative of F along the vector field F, to be the (local) section £~F " X -+ VY¢, given by £gF : TF o ~ - - ~ o y . REMARK 2• The Lie derivative operator acting on sections of gauge-natural bundles satisfies the following properties:

CURVATURE OF VARIATIONAL PRINCIPLES

15

1. For any vector field 2= 6 . 4 (r'~), the mapping ~, ~-+ £~y is a first-order quasi-linear differential operator. 2. For any local section V of Y¢, the mapping 2= ~ £~V is a linear differential operator. 3. We can regard £~ : J1Y¢ "-+ VY¢ as a morphism over the basis X. By using the canonical isomorphisms VJ~Y¢ ~ J~VY¢ for all s, we have £~[AV] = fi[£~V], for any (local) section ~, of Yc and for any (local) vector field 2= E A (~'k). 2.3.

Variational sequences

For the sake of simplifying notation, sometimes we will omit the subscript ~, so that all our considerations shall refer to Y as a gange-natural bundle as defined above. REMARK 3. According to [19, 20, 29], the fibered splitting (1) yields the p p-t t sheaf splitting H(s+l,s ) = ~)P-0 C(s+l,s) A Hs+l, which restricts to the inclusion Asp C ~)P=0 CP-ts A 7-[t,hs+a, where A, H, C are respectively sheaves of differential, horizontal and contact forms in the sense of [19], H p,hs+l := h(AsP) for 0 < p < n and the surjective map h is defined to be the restriction to Asp of the projection of the above splitting onto the nontrivial summand with the highest value of t (see e.g. [25, 26] for notation). [] By an abuse of notation, let us denote by d k e r h the sheaf generated by the presh'eaf d k e r h in the standard way. We set 0* := kerh + dkerh. In [19] it was proved that the following sequence is an exact resolution of the constant sheaf N r over Y: 0~

~r-+

1 1 El> A s2 / O ~ 2 £2 > ... £t-1> A s1/ O s1 Ez> As~+ 1 A ° ~0> A~/t3s

d > 0.

DEFINITION 4. The above sequence, where the highest integer I depends on the dimension of the fibers of JsY -+ X (see, in particular, [19]), is said to be the s-th order variational sequence associated with the fibered manifold Y -+ X. For practical purposes we shall limit ourself to consider the truncated variational sequence: 0

> by

> ]~O E0), •1

E1~ . . .

En> ~,~n+l En+l En+l ("1~,+2 ) E,+2 0,

where, following [29], the sheaves V p :=Cff-nAHn'h+l/h(dkerh ) with 0 _ p _< n + 2 are suitable representations of the corresponding quotient sheaves in the variational sequence by means of sheaves of sections of tensor bundles. Let ot 6 Cs1 A H n,hs+l c wn+l "s+l. Then there is a unique pair of sheaf morphisms [15, 18, 29] Eot E C (2s,0) 1 A H n'h2s+l,

Fa E C1(2s,s) A H n,h2s+l'

(4)

16

M. FRANCAVIGLIA, M. PALESE and E. WINTERROTH

2 s +1l ,), o / = E~ - F~ and F~ is locally of the form F~ = drips, with such that -~zr~+ 1 p e C ~ ( 2 s - l , s - 1 ) /~ 7-{n-12s"

Let 0 ~ C1 A C~s,O) A ~ n,hs+l C "l)n+2" "s+a, then there is a unique morphism 1

1

K~ ~ C(2s,s) ® C(2s,0) A ~

n,h

es+l

such that, for all E • g ~ V Y , Ejszj~ = C~(ja~EK~), where C~ stands for tensor contraction on the first factor and J denotes inner product (see [18, 29]). Furthermore, there is a unique pair of sheaf morphisms 1 1 H, 6 C(2s,s) m C(2s,0) m

~ , h2s+l,

G~ E

2

n,h

C(2s,s) /~ 7~ 2s+l,

(5)

such that ~_2s+1,*._ ~ s + l ) q = H ~ - G , and Ho = 1 A(K~), where A stands for antisymmetrisa2 A~_~n-1 2s, tion. Moreover, G~ is locally of the type G~ = dnq~, where qo 6 C(2s_l,s_a) hence [~] = [H~] [18, 29]. REMARK 4. A section )v ~ V2 is just a Lagrangian of order (s + 1) of the standard literature. Furthermore g~()0 6 V2+1 coincides with the standard higher order Euler-Lagrange morphism associated with )v.

3.

Variations and generalized Jacobi morphisms

We recall some previous results concerning the representation of generalized gauge-natural Jacobi morphisms in variational sequences and their relation with the second variation of a generalized gauge-natural invariant Lagrangian. We consider formal variations of a morphism as multiparameter deformations and relate the second variation of the Lagrangian )~ to the Lie derivative of the associated Euler-Lagrange morphism and to the generalized Bianchi morphism; see [25] for details. DEFINITION 5. Let zr • Y ~ X be any bundle and Ljs~k be the Lie derivative operator acting on Let Ek, 1 < k < i, be (vertical) variation vector [8, 9, 12, 25]. We define the i-th formal variation operator: ~io/ : ZJsZl .. Ljs~iOt.

P

,

and let oe • J~Y ~ A T JsY differential fibered morphism. fields on Y in the sense of of the morphism c~ to be the

REMARK 5. Let oe 6 (V2)v. We have ~i[ol]:m-[~/O~] £zi ... £s~ [od.

:

[Lzi...L~lOt]

=

DEFINITION 6. We call the operator 8i the i-th vertical variational derivative.

As a consequence of the Second Noether Theorem (see e.g. [10, 25]), the following characterization of the second variational vertical derivative of a generalized Lagrangian in the variational sequence holds true. PROPOSITION 1. Let )~ ~ (V'sn )y and let ~ be a variation vector field; then we have ~2)~ =

[gn(j2s,~Jh3k) +

1

'

C 1 (J2s ~Khd~X)].

(6)

CURVATURE OF VARIATIONAL PRINCIPLES

17

3.1. Generalized gauge-natural Jacobi morphisms Let now specify Y to be a gange-natural bundle and let ~ -= ~ ( E ) be a variation vector field associated to some F~ c .4 r'k. Let us consider Js,~V, i.e. the vertical part according to the splitting (1). We shall denote by jsEv the induced section of the vector bundle .4 (r+s'k+~). The set of all sections of this kind defines a vector subbundle of J~A (~'k), which by a slight abuse of notation (since we are speaking about vertical parts with respect to the splitting (1)), we shall denote by V J s A (r'k). By applying an abstract result due to Kolaf, see [15], concerning a global decomposition formula for vertical morphisms, and by using Proposition 1 we can prove the following lemma. ^

LEMMA 1. Let X be a Lagrangian and E v a variation vector field. Let us set

X()~, ~)(~)V)::C~(j2s~Khd£.jZs~V~

) ~ Ejs~jhd~j2s+l~V x. Let DH be the horizontal

differential on J4sY¢ x VJ4sA (r'k). Then we have X

4s+l •

(Zr2s+l) X(;v, ~5(~)v) = Ez(z,e~(~)v) + Fx(z,cs(g)v ),

(7)

where Ex(z,~(~)v " J4~Y¢

x WJ4s.A (r'k) ---> C~[¢4(r'k)]c~[.A(r'k)] A (~T*X),

x

and locally Fx(z,~(~)v)= DHMx(~,~(Z)v) with

Mx(z,~(~)v) • J 4 s Y (

n--1 x V J4s~4 (r'k) ---> C~s_l[j4(r'k)]c~[J4 (r'k)] A ( A T'X). X

Proof: As a consequence of linearity properties of both X0~, E) and the Lie derivative operator £ we have X( )~, O(~)v) " J2sY¢ x VJ2sA (r'k) --+ C~s[~A(r'k)]c~[~z~(r'k)] A (ffxT*X) X and

DHX(X, qS(~)v) = O.

Thus KolAf's decomposition formula can be applied.

[]

DEFINITION 7. We call the morphism ,.70~, ~5(~)v) := Ez(x,~(~)v ) the gaugenatural generalized Jacobi morphism associated with the Lagrangian )~ and the variation vector field E (it depends on the gauge-natural lift O(E)v). Coordinate expressions can be found e.g. in [25]. The morphism ,7()~, O ( ~ ) v ) is a linear morphism with respect to the projection J4sY¢ x VJ4s A(r'k) --+ J4sY~. We have then (see [25]) the following theorem. x THEOREM 1. Let ~2 )~ be the variation of )~ with respect to vertical parts of gauge-natural lifts of infinitesimal principal automorphisms. We have

¢5(~)vJgn(ff3(~)VJgn()O) = ~2 )~ = £n(~5(~)vJh(dS)O).

(8)

18

M. FRANCAVIGLIA, M. PALESE and E. WINTERROTH

^ PROPOSITION 2. Let )~ c ?~ be a gauge-natural Lagrangian and (F,, ~) a gaugenatural symmetry of )~. Then we have 0 =-£~.AE~()O + d~l(--js£~JPdvz + ~J~.).

(See e.g. [28] for geometric field theories and, in particular, [10, 26]). 3.2.

Bianchi morphism and curvature

Generalized Bianchi identities for field theories are necessary and (locally) sufficient conditions for the Noether conserved current ~ =--Js£~JPdv~ +~J)~ to be not only closed but also the divergence of a skew-symmetric (tensor) density along solutions of the Euler-Lagrange equations (see e.g. [28]). Let now )~ be a gauge-natural Lagrangian. We set n

~o()~, ~ ( ~ ) v ) -= £~J£n(~) " J2sY~ -~ d~s[A(r'k)]C~[A (r'k)] A (AT*X).

(9)

The morphism w()~,CS(E)v) so defined is a generalized Lagrangian associated with the field equations of the original Lagrangian )~. It has been considered in applications, e.g. in general relativity (see [7] and references quoted therein)._ We have DnoJ()~, ¢5(F,)v) = 0 and by the linearity of £ we can regard w()~, ¢$(E)v) as the extended morphism CO()~, ~ ( ~ ) V )

" J2sY¢ × V J2s.A (r'k)) ~ C~s[A(r'k)]C~s[A(r'k)]c~[~z~ (r'k)] A (~T*X).

x Thus we can state the following [25] lemma.

LEMMA 2. Let co()~, ¢5(F,)v) be as above. Then we have globally 4s+l * ~, (Tgs+ 1 ) O)()~, Q~(~)V) = fl(~', ~ ( ~ ) V )

-'[-

Fo,(z,e~(g)v),

where fi(X, O(E)v) -- E
(10)

J4sY¢ x V J4sA (r'k) --+ C~s[A("k)]C~[A(r'k)]C~[A (r'k)] A (~T*X) x

(11)

and, locally, F~o(z,e~(~)v)= Dt~Mo~(z,~(~)v), with Mo4z,e~(~,)v) : J4s_lY¢ x V J 4 s - l . A (r'k) ~ X

n-1 C~s[~Z~(r'k)]C~s_l[~z~(r'k)]c~[A(r'k) ] A ( A

T'X).

DEFINITION 8. W e call the global morphism fiO~,C3(E)v):=E~o(z,e(~)v ) the generalized Bianchi morphism associated with the Lagrangian ~ and the variation vector field E. REMARK 6. For any (~, ~) ~ .,4(r'k), as a consequence of the gauge-natural invariance of the Lagrangian, the morphism fi0~, ¢5(F,)v) --gn(co(X, O(F~)v)) is locally identically vanishing. We stress that these are just local generalized Bianchi identities [1].

CURVATURE OF VARIATIONAL PRINCIPLES

19

Let E be the kernel of J()~, O(~)v). As a consequence of the considerations above we have the following result [25]. THEOREM 2. The generalized Bianchi morphism is globally vanishing for the = 0, i.e. if and only if variation vector field ~ if and only if 8~)~ =- J()~, O(,~)v) ;"

OS(~,)v c ~. From now on we shall write co()~,J~) to denote co()~, ~b(2~)v) when ~5(~)v belongs to J~. Analogously for fi and other morphisms. As we already recalled, the classical Jacobi equation for geodesics of a Riemannian manifold defines in fact the Riemann curvature tensor of the metric g; because of this it was suggested in [2] that the second variation and the generalized Jacobi equations define the 'curvature' of any given variational principle. This general concept of 'curvature' takes, for example, a particularly significant form in the case of generalized harmonic Lagrangians, giving rise to suitable 'curvature tensors' which satisfy suitable 'generalized Bianchi identities' [2]. Following this guideline, we shall provide a definition of the curvature of a given gauge-natural invariant variational principle. First of all let us make the following important consideration. PROPOSITION 3. For each ~

fJJs ~H O)()~' j~)

E

=

A (r'k) such that ~v E ~ we have

--DH (-js£avJ PDvw(~.,J~)).

(12)

Proof: The horizontal splitting gives us

f--.js~,O)()~, .f~) = £js~,H O)()~, a¢~)--~ ~-.js~,v (.O()~, .f~). Furthermore, co()~, J~) =-- -£:zJgn(k) = £.js~)~ - d14(--js£~JPdvZ + ~J)~); so that

~-~js~,v(-O()~, ~) = £js~.V~-~js~)~ = ~-~js[~,V,~H])~. On the other hand we have

Recall now that from the theorem above we have ~,v E R if and only if fl (X, J~) = 0. Since

f-.js~,vO)(~, J~) = --£~,v JEn(O)(~., J~)) -Jr-DH(--jS£~v J PDvw(X,~ ) ----fl(~., ~) + D~(--js£~.v J PDvo)(Z,Y~)), we get the assertion.

[]

It is easy to realize that, because of the gauge-natural invariance of the generalized Lagrangian )~, the new generalized Lagrangian o~0~, R) is gauge-natural invariant too, i.e. £Js ~ co()~, ~) = 0.

20

M. FRANCAVIGLIA, M. PALESE and E. WINTERROTH

Even more, we can state the following, PROPOSITION 4. Let Ev E ~. We have (13)

£ j ~ i ~ w ( )~, YO = O. Proof: In fact, when E v 6 J~, since Z ; j ~ v o ) ( X , ~ ) = [ E- v , ~ v ] J C . ( z )

+ ¢" ~.vJ£Js~vgn(z) = O,

we have

0 = Ejs~CO()~, Y~) = Ejs~vCO(~., Y~) + Ejs~Ho)()~, Y~) = Ej~.~io)()~, ~).

(14) []

As a quite relevant byproduct we get also the following (this result can be compared with [3]). COROLLARY 1. Let ~ v c SL We have the covariant conservation law

DH (--L£~v J PDv
(15)

Proof: 0 = £.js~HO2(X, ~) = --fi(X, ~) -- DH(--jS£~vJPDveo(;~,y~) ) = --Dlf(--js£~vJPDvco(Z,~)).

[]

DEFINITION 9. We define the covariantly conserved current

~-L()~, ~) = - js£ J PDv~o(X,~),

(16)

to be a Hamiltonian form for w on the Legendre bundle n-1

I-[ I V * ( J 2 s Y ¢ x V J 2 s A (r'k)) A ( A T ' X ) X

(in the sense of [21]).

[]

Let f2 be the multisymplectic form on 1-I. It is well known that every Hamiltonian form ~()~,J~) admits a canonical Hamiltonian connection YT-t(~,N) such that y~(X, N)Jf2 = d~0~, N). Let then y~(X, N) be the corresponding Hamiltonian connection form (see [21]). DEFINITION 10. We define the curvature of the given gauge-natural invariant variational principle to be the curvature of the Hamiltonian connection form y~ (~., N). We claim that the Euler-Lagrange and the generalized Jacobi equations for )~ can be related with the zero curvature equations for the Hamiltonian connection form yT-t(X, ~). According to [2, 8, 9], they must be, in fact, the Lagrangian counterpart of Hamilton equations for the Lagrangian o9 obtained by contracting Euler-Lagrange equations with Jacobi variation vector fields. Investigations are in progress and results will appear elsewhere [13].

CURVATURE OF VARIATIONAL PRINCIPLES

21

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