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arXiv:1601.07193v1

[math-ph

]26Jan2016

Variational derivatives in locally Lagrangian fieldtheories and NoetherBessel-Hagen currents

F. Cattafi

Department of Mathematics, Universiteit Utrecht

3508 TA Utrecht, The Netherlands

email: [email protected]

M. Palese and E. WinterrothDepartment of Mathematics, University of Torino

via C. Alberto 10, 10123 Torino, Italy

email: [email protected], [email protected]

January 28, 2016

Abstract

The variational Lie derivative of classes of forms in the Krupkas

variational sequence is defined as a variational Cartan formula at anydegree, in particular for degrees lesser than the dimension of the basis

manifold. As an example of application we determine the condition for

a NoetherBessel-Hagen current, associated with a generalized sym-

metry, to be variationally equivalent to a Noether current for an in-

variant Lagrangian. We show that, if it exists, this Noether current is

exact on-shell and generates a canonical conserved quantity.

Key words: fibered manifold, jet space, Lagrangian formalism, variationalsequence, variational derivative, cohomology, symmetry, conservation law.2000 MSC: 58A20,58E30,46M18.

1 Introduction

The study of calculus of variations for field theories (multiple integrals) asa theory of differential forms and their exterior differential modulo contact

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F. Cattafi, M. Palese and E. Winterroth 2

forms (congruences) was initiated [22]by Lepage in 1936; see e.g.[23] for a

brief review. One of the most important fact within such a geometric formu-lation of the calculus of variations is the fact that considering the ambientmanifold to be a fibered manifold and the configuration space a jet prolonga-tion of it,variations can be described by Lie derivatives of forms with respectto projectable vector fields; see, e.g.[8,13].

In[14,15] Krupka described the contact structure at a given finite pro-longation order and initiated the project of framing the calculus of variationswithin a differential sequence obtained as a quotient sequence of the de Rhamsequence. Krupkas variational sequence is a sequence of differential formsmodulo a contact structure inspired by the Lepage idea of a congruence.Krupka also showed that the Lie derivative of forms with respect to pro-

jectable vector fields preserves the contact structure naturally induced bythe affine bundle structure of jet projections order-by-order. This fact sug-gests that a Lie derivative of classes of forms, i.e.a variational Lie derivative,can be correctly defined as the equivalence class of the standard Lie derivativeof forms and represented by forms.

By a representation of the quotient sheaves of the variational sequence assheaves of sections of tensor bundles, in [3] explicit formulae were providedfor the quotient Lie derivative operators, as well as corresponding versions ofNoether Theorems interpreted in terms of conserved currents for Lagrangiansand EulerLagrange morphisms (only classes of forms up to degreen + 2, the

latter assumed to be exact, were considered). Such a representation madeuse of intrinsic decomposition formulae for vertical morphisms due to Kolar[9], expressing geometrically the integration by part procedure (a geomet-ric decomposition was proposed earlier, e.g. by Goldschmidt and Sternberg[8]). Such decomposition formulae, corresponding to the first and secondvariational formulae, in particular introduce local objects such as momentawhich could be globalized by means of connections. In particular, besides theusual momentum associated with a Lagrangian, a generalized momentumis associated with an Euler-Lagrange type morphism. Its interpretation inthe calculus of variations has not been exaustively exploited; in [26] it wassuggested that it could play a role within the multisymplectic framework for

field theories.We recall that, by using the so-called interior Euler operator adapted to

the finite order case, a complete representation of the variational sequence bydifferential forms was given in [10, 11, 12] and independently in [19, 20,31].This is an operator involved with the integration by parts procedure. We

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shall exploit the relation between the interior Euler operator and the Cartan

formula for the Lie derivative of differential forms. The representation ofthe variational Lie derivative provides, in a quite simple and immediate way,the Noether Theorems as quotient Cartan formulae [25]. In this paper,inspired by (and extending) a formalism developed by [10,12], we shall derivethe variational counterpart of the Cartan formula for the Lie derivative offorms of degree q n 1, where n is the dimension of the basis manifold,thus explicating and making more precise previous results stated in [3,25].In particular formulae for the Lie derivative of classes of q-forms will beobtained, not only for the representations. To this aim we define an interiorEuler operator associated with a contact component of degree k of a formof degreeq n 1. We shall also define a momentum associated with the

differential of such a q-form and provide an example of application.The above general results concerning variational derivatives of forms of

degree q n 1 were motivated by the need to have at hand suitable tech-niques in order to investigate variational problems for (conserved) currentsassociated to symmetries and invariant variational problems in locally La-grangian field theories. As it is well known, invariance properties of fielddynamics are an effective tool to understand a physical system without solv-ing the equations themselves: the existence of conservation laws associatedwith symmetries of equations strongly simplifies their study and correspond-ing conserved currents along solutions (on-shell) appear to be significant for

the description of the system.It turns out to be fundamental to understand whether conserved currentsassociated with invariance of equations could be identified with Noether con-served currents for a certain Lagrangian; in fact, a symmetry of a Lagrangianis also a symmetry of its Euler-Lagrange form, but the converse in generalis not true. We are interested to this converse problem which belongs toaspects of inverse problems in the calculus of variations.

2 Representation of the variational sequence

We assume the r-th order prolongation of a fibered manifold :Y

X

,with dimX = n and dimY = n+ m, to be the configuration space; thismeans thatfields are assumed to be (local) sectionsofr :JrY X. We re-fer to the geometric formulation of the calculus of variations as a subsequenceof the de Rham sequence of differential forms on finite order prologations of

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fibered manifolds.

Due to the affine bundle structure ofr+1

r : Jr+1

Y J

rY

, we have anatural splitting JrY Jr1Y TJr1Y = JrY Jr1Y (T

XVJr1Y),

which induces natural spittings in horizontal and vertical parts of vectorfields, forms and of the exterior differential on JrY (see the Appendix forsome more technical details and properties which will be used here).

Letbe aq-form onJrY; in particular we obtain a natural decompositionof the pull-back by the affine projections of, as

(r+1r )=

q

i=0

pi ,

where pi is the i-contact component of (by definition a contact form iszero along any holonomic section ofJrY).

Starting from this splitting one can define sheaves of contact forms r,suitably characterized by the kernel ofpi [14]; the sheaves

r form an exact

subsequence of the de Rham sequence on JrY and one can define the quotientsequence

0 IRY En1 nr/

nr

En n+1r /n+1r

En+1 n+2r /n+2r

En+2 0

i.e. the rth order variational sequence on Y X which is an acyclicresolution of the constant sheaf IRY; see [14]. In the following, if

kr ,

then [] Vkr.

= kr/kr denotes the equivalence class of modulo contact

forms as defined by Krupka (by an abuse of notation we therefore denote inthis way a local or a global section of the sheaf, when there is no possibilityof misunderstanding).

The quotient sheaves in the variational sequence can be represented assheaves ofq-forms on jet spaces of higher order. For 1 q n, the represen-tation mapping is just given by the horizontalization p0= h. For q > n,say, q = n+k, it is clear that any form is contact; therefore, in this case,

pk denotes the component of with the lowest degree of contactness. Forq n+ 1, a representation can be given by the interior Euler operatorIwhich is uniquely intrinsically defined by the decomposition

pk= I() +pkdpkR() ,(where R() is a local (q 1)-form) together with the properties

(2r+1r ) I() n+k2r+1 I(pkdpkR()) = 0 ;

I2() = (4r+32r+1)I() kerI

.= n+kr .

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It is defined a representation mapping Rq :Vqr

qs, : []Rq([]), with

Rq([]) .=p0 h for 0 q n, s= r+ 1;

Rq([]) .=I() for n + 1 q P, s= 2r+ 1;

Rq([]) .= for P+ 1 q N, s= r;

where N = dimJrY and P is the maximal degree of non trivial contactforms on JrY (see e.g. [10, 12, 14, 19, 31], whereby also local coordinateexpressions can be found).

The representation sequence {0 R(Vr ) , E}, is also exact and we

have Eq Rq([]) = Rq+1 Eq([]) = Rq+1([d]). Currents are sheaf sectionsofVn1r and En1 =dHis the total divergence; Lagrangians are sections

ofVnr , while En is called the Euler-Lagrange morphism; sections ofVn+1rare called source forms or also dynamical forms, while En+1 is called theHelmholtz morphism.

It is well known that in order to obtain a representation of Euler-Lagrangetype forms the following integration formula is used [14]

i1irj 0= di1ir

j.

and the corresponding representation is obtained by taking the p1componentobtained by iterated integrations by parts; here

I =dy

I y

Ij dxj , with I

a multindex of lenght r, are local contact 1-forms on JrY, while we denoteby0 the volume density on Xand byi=

xi 0,ij =

xj i and so on.

In order to integrate by parts k-contact components of (p+ k)-forms withp < n, we notice that

i1ir1[irj] i1ir1[ir j]=

i1ir1[irj] d

i1ir1

irj.

This enabled one of us to generalize results given in[12] as follows (see[25]).

Proposition 1 Let p+kr , 1 p n, k 1. Letpk=r

|J|=0 J

J ,

withJ (k 1)-contact(p+k 1)-forms. Then we have the decomposition

pk= I() +pkdpkR() , (1)

whereR() is a localk-contact(p+k 1)-form such that

I() +pkdpkR() =

r

|J|=0

(1)|J|dJJ

+r

|I|=1

dI( I) ,

withI =r|I|

|J|=0(1)J|I|+|J|

|J|

dJ

JI

.

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Note that dJJ

are also (k 1)-contactp-horizontal (p+k 1)-forms. Of

course,I

= I andR

= R in the case p= n.

Remark 1 In the case p= n 1, R() is defined by

pkdpkR() =r

|I|=1

dI( I) =dH[

r1

|I|=0

(1)kdII[lj] lj] .

2.1 Variational Lie derivatives of classes

Avariational Lie derivativeoperator Ljracting on the sections of sheaves inthe variational sequence is well defined: the basic idea is to factorize modulo

contact structures [16,17, 19, 21]. This enables us to define symmetries ofclasses of forms of any degree in the variational sequence and correspondingconservation theorems; see also [25].

We define the interior product of a projectable vector field with the equiv-alence class of as the equivalence class of the interior product of the vectorfield with the representation of the equivalence class of, that is

jr[]jr[]

.= [jsRq[]] .

This definition is well given. In fact, we need only to check that the image ofjrdoes not change while changing the representative inside the equivalence

class. If [] = [], thenRq([] []) = 0 (by linearity), thereforej r([

]) = [jsRq[ ]] = [0] and j r[] =jr[], as we wanted.

Accordingly, in the following we will sometimes skip to specify the jetprolongation of a projectable vector field when it appears within formulaefor the variational classes (the order in the variational sequence is fixed andso is for the jet order of prolongations).

Therefore we can define the variational Lie derivative with respect toa projectable vector field (, ) of a class of forms in Vqr simply by takingthe class of the Lie derivative of its representative with respect to the s-prolongation of , i.e.

L([]) .= [LjsRq[]] .

As before we can easily check that this definition is well given.

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Let be a projectable vector field on Y, a q-form defined (locally) on

JrY

. We define an operator Rq : V

q

r q

s by the following commutativityrequirement

Rq L= Ljs Rq,

i.e. Rq L [] = Rq[Ljs Rq[]]. This operator is uniquely defined and isequal, respectively, to the following expressions:

Ljsh 0 q n, s= r+ 1;

LjsI() n + 1 q P, s= 2r+ 1;

Ljs P+ 1 q N, s= r.

This means that R together with the (variational) operator jrreturn the (differential) operator ijs,i.e.

Rq1 [] =ijsRq[] = Rq1[ijsRq[]] ;

in the same way, we have that Rq+1 Eq =d Rq.This enables us to deal with ordinary Lie derivatives of forms on qs,

then apply the Cartan formula for differential forms, therefore return backto the classes of forms to obtain a sort ofvariational Cartan formulae; see in

particular also [25] where the case q n + 1 has been worked in detail andpartial results concerning the case case q n have been obtained.

We shall also need the following naturality property.

Proposition 2 We haveEqL= LEq.

Proof. For every [] Vqr we have

Eq(L[]) =Eq[LjrRq[]] = [d(LjrRq[])] = [LjrdRq[]] ,

on the other hand,L(Eq[]) =L([d]) = [LjrRq+1[d]]. The commutatorofd and Rq is contact, hence it vanishes in the quotient.

In the following we shall make use thoroughly of a technical result due toKrbek[10] (Theorem III.11), which we recall here for the convenience of thereader; see also [12].

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Lemma 1 Let be a-vertical vector field on Y and a differential q-

form onJrY

. Then the following holds true for i= 1, . . . , q

jr+2pidpi= pi1d(jr+1pi) ,

and

Ljr+2(r+2r+1)

pi= jr+2pi+1dpi +pid(j

r+1pi) ,

2.2 The case q n 1

Definition 1 The momentum associated with the density = [] Vqr and

the projectable vector field is defined as a section pdV V

q

r+1, of whichthe representation Rk(pdV) = pdVRk = pdVh is a local 1-contact q-formsatisfying the identity

dH(js1VpdVh) = dH(j

s1Vp1R(d)) ,

where Ris defined by the splitting of Proposition1.

We have the following.

Theorem 1 Let Vqr , 0 q n 1, and let be a-projectable vectorfield onY; the following holds locally

L= HEq() + Eq1(VpdV+ H) .

Proof. Recalling the decomposition of the vector fields and of the ex-terior differential we have

(r+3r+1)RqL[] = (

r+3r+1)

Ljr+1Rq[] = (r+3r+1)

Ljr+1(h) =

= (r+3r+1)jr+1d(h) + (r+3r+1)

d(jr+1h) =

= (jr+1H+ jr+1V)(dH+ dV)h+ (dH+ dV)(jr+1H+ jr+1V)h .

By applying lemmas7,6 and 5, we easily see that:

jr+1V(dHh) =jr+1Vh(dh

2) = 0 ,

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it is also easy to check thatdH(jr+1Vh) anddV(j

r+1Vh) vanish, while

jr+1H(dVh) =jr+1H(p1dh

2) , dV(jr+1Hh) =p1dh(j

r+1Hh) ,

are contact pieces. On the other hand, by using Lemma8

jr+1HdH(h) =jr+1HdHRq[] =

=jr+1H(r+3r+1)

Rq+1Eq[] = (r+3r+1)

Rq(HEq[]) ,

and

dH(jr+1Hh) =dH(jr+1HRq[]) = (dH+dV)(jr+1HRq[]) =

= (r+3r+1)d(jr+1HRq[]) = (

r+3r+1)

Rq[d(jr+1HRq[])] =

= (r+3r+1)RqEq[j

r+1HRq[]] = (r+3r+1)

RqEq(H[]) .

Analogously, one can see that

jr+1VdVh= (r+3r+1)

Rq[jr+1VdV] ,

so that up to contact terms

(r+3r+1)RqL[] =j

r+1VdVh +jr+1HdH(h) +dH(j

r+1Hh) =

= (r+3r+1)Rq([j

r+1VdV] + HEq[] + Eq1(H[])) .

By taking the class, which makes the remaining contact pieces vanish,

L[] = [jr+1VdV] + HEq([]) + Eq1(H[]) .

By splittingdVaccording with Proposition1, again by Lemma1, and since[j2r+1VI(d)] = 0, the result is obtained by denoting = [].

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2.3 The case q=n

Definition 2 The momentum associated with the Lagrangian = [] Vnrand the projectable vector field is defined as a section pdV V

nr+1, of

which the representation Rk(pdV) = pdVRk = pdVh is a local 1-contactn-form satisfying the identity

dH(VpdVh) =dH(Vp1R(d)) ,

where Ris defined by the splitting given by the interior Euler operator.

Theorem 2 (Noethers Theorem I)Let Vnr and be a-projectable vector field onY; the following holds

(locally):

L= VEn() + En1(VpdV+ H) .

Proof. As before, by the representation Rn and the pullback (r+3r+1)

(r+3r+1)RnL[] =j

r+1VdVh+jr+1HdHh+dH(j

r+1Hh) ,

however (unlike the casek n 1) the termj r+1HdHhvanishes becausedis contact anddHh= (

r+3r+1)

h(d) = 0. On the other hand

dVh= p1dh2

= (r+3r+2)

p1dh=

= (r+3r+2)[(r+2r+1)

(p1d) p1dp1]=

= (r+3r+1)I(d) + (r+3r+1)

p1dp1R(d) (r+3r+2)

p1dp1 .

Thus, by Krbeks Lemma we get

(r+3r+1)RnL[] =j

r+1V(r+3r+1)

I(d) hdH(jr+1Vp1R(d))+

+(r+3r+2)hd(jr+1Vp1) +dH(j

r+1Hh) =

=jr+1V(r+3r+1)

I(d) +hdH(jr+1VpdVh)+

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+(r+3r+2)hd(jr+1Vp1) +dH(j

r+1Hh) .

However, by Lemma 8, (r+3r+2)hd(jr+1Vp1)() = dHhp1(V, ) = 0 .Lastly, we use again the representations in each remaining term:

jr+1V(r+3r+1)

I(d) =jr+1V(r+3r+1)

Rn+1[d] =

=jr+1V(r+3r+1)

Rn+1En[] = (r+3r+1)

Rn(VEn[]) ;

hdH(jr+1VpdVh) = (

r+3r+1)

hd(jr+1VpdVh) =

= (r+3r+1)Rn[d(j

r+1VpdVRn[])] = (r+3r+1)

RnEn1[jr+1VRn(pdV[])] =

= (r+3r+1)RnEn1(VpdV[]) = (

r+3r+1)

RnEn1(VpdV[]) ;

dH(jr+1Hh) = (dV +dH)(j

r+1Hh) = (r+3r+1)

d(jr+1Hh) =

= (r+3r+1)d(jr+1Hh) = (

r+3r+1)

Rn[d(jr+1Hh)]

= (r+3r+1)RnEn1[j

r+1HRn[]] = (r+3r+1)

Rn En1(H[]) .

As before, by calling = [], we get the conclusion.

2.4 The case q n + 1

In [25] it was proved the following variational Cartan formula for classes offorms of degree q n+ 1 (the case q = n + 1 for locally variational dy-namical forms encompasses Noethers Theorem II, or so-called Bessel-Hagensymmetries).

Theorem 3 Letq= n + k, withk 1 and Vqr . Let be a-projectablevector field onY; we have

L

= V

Eq() + E

q1(

V) .

These variational Cartan formulae will be the underlying mathematicalcore of the next Section, which deals with currents associated with invari-ance of (locally) variational dynamical forms, invariance of currents and cor-responding generalized momenta.

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3 NoetherBessel-Hagen currents

Consider now conserved currents associated with invariance properties of(locally) variational global field equations, i.e. with so-called generalized orBessel-Hagen symmetries[1]. Noether currents for different local Lagrangianpresentations and correspondingconserved currents associated with each localpresentationhave been characterized in[4,5, 6,28]. There exist cohomolog-ical obstructions for such local currents be globalized and such obstructionsare also related with the existence of global solutionsfor a given global fieldequation[7].

We will denote by a subscript i the fact that in general a sheaf sec-tion is defined only locally, i.e. that it is a 0-cochain in Cech cohomology;

analogously by two subscripts ij we shall denote that a sheaf section is a1-cochain. In the following we shall also denote simply by since we aredealing with classes and there is no danger of confusion. Let for simplic-ity i denote a global EulerLagrange class of forms for a (local) varia-tional problem represented by (local) sheaf sections i. Notice that in thiscase Theorem 2 (Noether Theorem I) reads Li = Vi +dHi; wherei = VpdVi + Hi is the Noether current associated with it.

Definition 3 A generalized symmetry of a (locally variational) dynamicalformi is a projectable vector field j

r onJrY such thatLi = 0.

Since we assume i to be closed, Theorem 3 reduces (case q = n+ 1) toLi =En(Vi), and ifjr is such thatLi = 0, thenEn(Vi) = 0;

therefore, locally we have Vi = dHi. Notice that, although Vi isglobal, in general it defines a non trivial cohomology class [4]; it is clear thati is a (local) current which is conserved on-shell (i.e.along critical sections).On the other hand,and independently(see [24]), we get locallyLi= dHithus we can write V+ dH(i i) = 0, where i is the usual canonicalNoether current.

Definition 4 We call the (local) current i i a NoetherBessel-Hagencurrent.

A NoetherBessel-Hagen current i i is a current (conserved along crit-ical sections) associated with a generalized symmetry; in [6, 7] we provedthat a NoetherBessel-Hagen current is variationally equivalent to a global(conserved) current if and only if 0 = [[VEn(i)]] H

ndR(Y).

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According with our general considerations, in view of the precise state-

ments which the Noether Teorems provide concerning the existence and thenature of conservation laws for invariant variational problems, it is of impor-tance to determine whether a NoetherBessel-Hagen current is variationallyequivalent to a Noether conserved current for a suitable invariant Lagrangian.It is known that this is involved with the existence of a variationally triviallocal LagrangiandHi, and with a condition on the current associated withit[29]. In the following we will relax some of the conditions and investigatethe outcome.

Proposition 3 A NoetherBessel-Hagen current i i associated with ageneralized symmetry ofi is a Noether conserved current (for that symme-

try) if and only if it is of the form i Li, withi a current satisfyingL(i dHi) = 0.

Proof. FromLi = 0, we get Li = dHi. It is easy to see that thecurrent i i is a Noether conserved current if and only if there exists isuch that i Li is closed,i.e. if and only if

dHi = dH(VpdVdHi+ HdHi) .

This means of course, that, locally, i = Li+ dHij. On the other handdHi= dHLi= dH(HdHi). Notice that, comparing the two expressionwe getd

H(

Vp

dVdHi) 0; thus, in particular, this identity is a consequence

of the fact that L commutes with dH.

Proposition 4 Leti= Li(i.e. dHij = 0). The Noether currentidHiis exact on-shell and it is equal to dH(VpdVi + Hi).

Proof. As it is well known, along any section pulling back to zero Viwe get the on-shell conservation law dH(i i) = 0. If there exists acurrentisuch thati = Li HdHi+dH(VpdV +Hi), thereforeVpdVi + H(idHi)dH(VpdVi + Hi) is closed on-shell. Bya uniqueness argument, we see that the latter expression must be equal to

VpdVdHi ; therefore dH(VpdVi+ Hi) = idHi on-shell.

Remark 2 It turns out that, on-shell, a canonical potential of the NoethercurrentidHi , then a corresponding conserved quantity, is defined. An off-shellexact Noether current associated with the invariance ofidHiwould

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be generated by a generalized symmetry jr such that Vi = 0; the cor-

responding cohomology class would be, therefore, trivial (see the discussionin[6,7]).

Our next goal is to relax the results above by recasting the problem byusing directly the definition of canonical Noether current rather than thesplittings of the Lie derivative given by the Cartan identities. So, it could beuseful to considerinot just as a single current but as a morphism of the typei : i i , from the sheaf of the Lagrangians to the one of the currents.It is obviously linear since the interior product, the vertical differential andthe momentum are so.

It is now instructive to obtain the result of Proposition 3as a consequence

of Krbeks Lemma.First we state some preliminary technical results.

Lemma 2 We have

dH(VdVi) =dH(VI(di)) .

Proof. Since, up to pullback, hd= dHh= hdH, we have

VdVi= VI(di) + Vp1dp1R(di) = VI(di) hd(Vp1R(di)) =

= VI(di) hdH(Vp1R(di)) = VI(di) dH(VpdVi) .

The statement follows immediately.

Lemma 3 We have

VpdV(dHi)= VI(di) +dH(VpdVi+ Hi) .

Proof. It is a consequence of the naturality of the variational Lie deriva-tive: L(dHi) =dH(Li). In fact, from one side

L(dHi) =dH(VpdV(dH)+ HdHi) =dH(VpdV(dHi)) +dHi .

while on the other side

dH(Li) =dH(VdVi+ HdHi+dH(Hi)) =dH(VdVi) +dHi;hence dH(VdVi) = dH(VpdV(dHi)). Therefore, by the formula provedabove,dH(VI(di)) =dH(VpdV(dHi)) as well, and VI(di) = VpdV(dHi)+dHij. We therefore get the result.

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Lemma 4 Giveni Vn1r , we have

dH(VpdVdHi) = 0 .

Proof. From one side we have:

(r+5r+3)dV(RndHi) = dV((

r+3r+1)

RndHi) =dV(dHRn1i) = dV(dHhi) =

=dV(hdHi) =dV((hdH+ hdV)i) = dVh(r+2r )

di= (r+5r+3)

dVhdi .

On the other hand, since the pullback (r+5r+3) is injective, we have by defi-

nition

RndH(VpdVdHi) =dHRn1(VpdVdHi) = dH(VRn(pdVdHi)) =

=dH(VpdVRn(dHi)) = dH(VpdVh(di)) =dH(Vp1R(d(di)) = 0 ,

which gives us the assertion.

Let us suppose now that i is a Noether current associated with theLagrangian i i, withi = dHi,i.e. i= idHi .

Proposition 5 The Noether-Bessel-Hagen currenti i is the canonical

Noether current associated with the Lagrangiani dHi if and only ifi =HdHi modulo a locally exact current.

Proof. First of all, by linearity, we see that i i is a Noether currentassociated withi dHi if and only ifi i = idHi =i dHi ,i.e.if and only ifi= dHi .

On the other hand, the previous Lemma implies that VpdVdHi is closed,hence locally exact (VpdVdHi = dHij). Thus, by definition of Noethercurrent associated to dHi,

dHi = VpdVdHi+ HdHi = dHij+ HdHi .

This means that i is the Noether current associated with i dHi ifand only ifi = HdHi+dHij.

As we saw, the if implication could be weakened further since

L(dHi) =En1(VpdVdHi+ HdHi) =En(HdHi) ,

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and, by the uniqueness of the decomposition of the Lie derivative, the con-

dition i = HdHi is sufficient in order to have i = dHi , i.e. we cantake dHij = 0. Conversely, the indetermination remains because, wheni = dHi , only the differential ofi and HdHi are equal. However, we

can still state the following relaxed result.

Proposition 6 Under the hypothesis i = HdHi+ dHij, the Noether-Bessel-Hagen currenti i = idHi is exact on-shell, and its potentialVpdVi+ Hi is defined up to a cohomology class.

Proof. The on shell conservation law dH(i i) = 0 implies

HdHi+dH(VpdVi+ Hi) + VI(di) = VpdVi + Hi+dHij;

by simple manipulations, thanks to the Lemmas above, we obtain

idHi =dH(VpdVi+ Hi+dHij) .

Note that (unlike the case dHij = 0), generally speaking, the potential of

the NoetherBessel-Hagen current is not a canonical one.

Acknowledgement

Research supported by Department of Mathematics-University of Torino re-search project Geometric methods in mathematical physics and applications2013 14 (M.P.) and 2014 15 (E.W.); F.C. was partially supported by theNWO VIDI project Poisson Geometry Inside Out639.033.312.

4 Appendix

For the convenience of the reader, we recall some useful technical tools neededin Section2; details can be found e.g.in [2,10,30].

Lemma 5 Given the vector fieldXand the differential form, the contrac-tion betweenXVand the horizontal componenth is zero; the same holds forthe contraction betweenXH and then-contact componentpn.

Lemma 6 For every k(JrY), p2i = (r+2r+1)

(pi) =pi(r+1r )

i.

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Proof. Since for every k(JrY), pjpi= 0 i = j, it is enough to

apply the decomposition formula twice, first on pi, then on :

(r+2r+1)(pi) =

k

j=1

pj(pi) =p2i = pi(pi) =pi

k

j=1

pj= pi(r+1r )

.

In particular, the operators pi behave almost like projectors: their composi-tion is not the Kronecker delta, but we have the following formula

pipj =ij(r+2r+1)

pj =ijpj(r+1r )

.

Lemma 7 We have the following decomposition of the exterior differential:

(r+2r )d= dH+ dV.

Proof. Thanks to the contravariance of the pullback,

(r+2r )(d) = (r+1r

r+2r+1)

(d) = (r+2r+1)((r+1r )

(d)) =

= (r+2r+1)

k

i=0

pi(d) =k

i=0

(r+2r+1)pi(d) =

k

i=0

(pidpi1+pidpi) =

=k

i=0

(pidpi1) +k

i=0

(pidpi) =k1

i=0

(pi+1dpi) +k

i=0

(pidpi) =dV+dH .

We have several fundamental properties, which relates the operators pi,h, d, dH and dV

Lemma 8 For everyi 1, supposing the operators are applied to k-forms,

1. pidH=dHpi

2. pidV =dVpi1

3. (r+2r+1)(pid) =pid(pi+pi1)

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4. hdH=dHh

5. hdV = 0

6. (r+3r+1)(hd) = dHh= hdH

7. d2H= 0

8. d2V = 0

9. dHdV =dVdH.

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