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Olga Krupkov´

a and Radka Mal´ıkov´

a

Dedicated to the 70-th anniversary of Professor Constantin Udriste

Abstract. In the paper, properties of morphisms in the variational se-quence are investigated. The Euler–Lagrange morphism E1 is well-understood. It is also known that the kernel of the Helmholtz mor-phismE2consists of locally variational dynamical forms, and is character-ized by Helmholtz conditions. We study the image of E2 and the kernel of the next morphism E3, and solve the corresponding local and global inverse problem when a three-form comes (via a variational map) from a dynamical form, i.e., corresponds to a system of differential equations. We find identities, that are a generalization of the Helmholtz conditions to this situation, and show that the problem is closely related to the question on existence of a closed three-form. The obtained results extend known results on Lagrangians and locally variational dynamical forms to gen-eral dynamical forms, and open a new possibility to study non-variational equations by means of closed three-forms, as a parallel to extremal prob-lems (variational equations) that are studied by means of closed two-forms (Cartan forms, symplectic geometry).

M.S.C. 2000: 49N45, 49J27, 58E99, 58A20.

Key words: Euler–Lagrange equations; Helmholtz conditions; the variational se-quence; dynamical form; source forms; variational morphisms; Helmholtz form; Helm-holtz mapping; closed three-form.

1

Introduction: Helmholtz conditions and the inverse

problem of the calculus of variations

Consider a system of second order ordinary differential equations

(1.1) Ei(t, xk,x˙k,x¨k) = 0, 1im,

for curvesc:I→Rm,c(t) = (xk(t)), where the functions Ei, 1im, are defined

on an open subset ofR×R3m. The problem to decide when there exists a function

L(t, xk,x˙k), such that for alli,

(1.2) Ei= ∂L

∂xi − d dt

∂L ∂x˙i,

Balkan Journal of Geometry and Its Applications, Vol.15, No.1, 2010, pp. 80-89. c

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is called the(covariant) inverse problem of the calculus of variations. Above,d/dt is the total derivative operator defined by

(1.3) df

dt = ∂f

∂t + ∂f ∂xix˙

i+ ∂f ∂x˙ix¨

i,

where summation over repeated indices is understood.

Equations (1.1) satisfying (1.2) are called variational, and a corresponding func-tionL is called a Lagrangian. Solutions of variational equations are extremals of a variational functional defined by the LagrangianL.

It is well-known that equations (1.1) are variational if and only if the “left-hand-sides”Ei satisfy the following identities, calledHelmholtz conditions[4, 15]:

(1.4)

∂Ei ∂x¨k −

∂Ek ∂x¨i = 0,

∂Ei ∂x˙k +

∂Ek ∂x˙i −

d dt

³∂Ei ∂x¨k +

∂Ek ∂x¨i ´

= 0,

∂Ei ∂xk −

∂Ek ∂xi −

1 2

d dt

³∂Ei ∂x˙k −

∂Ek ∂x˙i ´

= 0.

Then, a Lagrangian for equations (1.1) can be constructed by the following formula

(1.5) L=xi

Z 1

0

Ei(t, uxk, ux˙k, ux¨k)du

due to Tonti [16] and Vainberg [17] (see [13]).

Within the modern calculus of variations, variational objects and their properties can be effectively studied by methods of differential and algebraic geometry. In this paper we shall use the framework of the theory of variational sequences on fibred man-ifolds, introduced by Krupka [9, 8]. The variational sequence is a quotient sequence of the De Rham sequence, such that one of the morphisms is theEuler–Lagrange mapping E1 :λ→Eλ, assigning to a Lagrangian (one-form λ=Ldt) its Euler-Lagrange form (two-formEλ=Ei(L)dxi∧dt, whereEi(L) are the Euler–Lagrange expressions (1.2)).

The next morphism,E2:E→HE, assigns to a two-formE=Eidxidta three-form HE, calledHelmholtz form. Erepresents a system of differential equations (1.1) that, of course, need not be variational. Due to exactness of the sequence, dynamical forms with identically zero Helmholtz forms are locally variational, i.e. representvariational

equations. Components of a Helmholtz formHE are “left-hand sides” of Helmholtz conditions, and nonzero Helmholtz forms correspond tonon-variational equations. In other words, Helmholtz conditions describe the kernel of the morphismE2.

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Our results extend known results on Lagrangians and locally variational dynamical forms to general dynamical forms, and open a new possibility to studynon-variational equations by means of closed three-forms, as a parallel to extremal problems (vari-ational equations) that are studied by means of closed two-forms (Cartan forms, symplectic and pre-symplectic geometry).

2

The variational sequence in fibred manifolds

Let π : Y → X be a smooth fibred manifold, dimX = 1, dimY = m+ 1, and

πr:JrY X,r1, its jet prolongations. Denote byπr,s:JrY JsY,r > s0,

canonical jet projections. A differentialq-form (q >1) η on JrY is called contactif Jrγη= 0 for every sectionγofπ,horizontalor 0-contactifiξη= 0 for every vertical vector fieldξ on JrY, and k-contact, 1 kq, if for every vertical vector fieldξ, iξη is (k−1)-contact. If lifted toJr+1Y, everyq-formη onJrY can be canonically

decomposed into a sum ofk-contact components,ηk, wherek= 0,1, . . . , q, We write

ηk=pkη, andp0=h, then

(2.1) π∗r+1,rη=hη+p1η+· · ·+pqη.

A contactq-form is calledstrongly contactifπ∗

r+1,rη=pqη.

In what follows, we denote Ωr

q the sheaf of q-forms on J rY, Ωr

0,c ={0}, Ωrq,c the

sheaf of strongly contact q-forms onJrY, dr

q−1,c the image sheaf of Ωrq−1,c by the

exterior derivatived, and we put

(2.2) Θr

q = Ω r q,c+dΩ

r q−1,c.

The De Rham sequence

(2.3) 0→R→Ωr

0→Ωr1→Ωr2→Ωr3→ · · ·

(where morphisms are the exterior derivativesd) has a subsequence

(2.4) 0→Θr

1→Θr2→Θr3→ · · ·

which is an exact sequence of soft sheaves. The quotient sequence

(2.5) 0→R→Ωr

0→Ωr1/Θ1r→Ωr2/Θr2→Ωr3/Θr3→Ωr4/Θr4→ · · ·

is also exact, and is calledthe variational sequenceof order r. As proved in [9], the variational sequence is an acyclic resolution of the constant sheaf Rover Y. Hence, due to the abstract De Rham theorem, the cohomology groups of the cochain com-plex of global sections of the variational sequence are identified with the De Rham cohomology groupsHqY of the manifold Y.

By construction,morphismsin the variational sequence are quotients of the exte-rior derivative operatord. In turns out that

(2.6) E1: Ωr1/Θ

r 1→Ω

r 2/Θ

r 2

is the well-known Euler–Lagrange mapping of the calculus of variations. The next morphism

(2.7) E2: Ωr2/Θ

r 2→Ω

r 3/Θ

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is called Helmholtz mapping. It should be stressed that this mapping, discovered within the variational sequence theory, has not been known earlier in the calculus of variations.

Objectsin the variational sequence are elements of the quotient sheaves Ωr

q/Θrq,q≥

1, i.e., they areequivalence classes of localrth-order differentialq-forms. We denote by [ρ]∈Ωr

q/Θrq theclassofρ∈Ωrq. By definition, the kernel of the Euler–Lagrange

mappingE1 consists ofnull Lagrangians, and the image arelocally variational forms

(Euler–Lagrange forms of locally defined Lagrangians). Due to the exactness of the variational sequence, if [α]∈Ωr

2/Θr2 is such that

(2.8) E2([α]) = [dα] = 0

then there exists [ρ]∈Ωr

1/Θr1 such that [α] = [dρ] =E1([ρ]), i.e. [α] is the image by

the Euler–Lagrange mapping of a class [ρ]. In other words, the class [α] is locally variational—comes from a class [ρ] that has the meaning of a local Lagrangian. If moreover H2Y = {0} then a global Lagrangian exists. Condition (2.8) for “local variationality” then provides Helmholtz conditions (of orderr).

Classes in the variational sequence can be represented by differential forms. We shall explore the representation by so-called source forms, (q−1)-contact q-forms belonging to the ideal generated by contact forms ωi = dxi x˙idt, 1 i m.

A canonical source forms representationis obtained by means of the interior Euler operator, I, introduced to the variational bicomplex theory by Anderson [1, 2], and adapted to the finite order situation of the variational sequence theory in [6, 10]. This operator reflects in an intrinsic way the procedure of getting a distinguished representative of a class [ρ] ∈ Ωr

q/Θrq by applying to ρ the operator pq−1 and the

factorization by Θr

q. I is anR-linear mapping Ωrq→Ω2r+1q such that

(1) Iρbelongs to the same class asπ∗2r+1,rρ,

(2) I2=I (up to a canonical projection),

(3) the kernel of I: Ωr

q →Ω2r+1q is Θrq.

Source forms for classes [ρ]∈Ωr

1/Θr1arehorizontal formsλ=Ldt, calledLagrangians.

Source forms for classes [α]∈Ωr

2/Θr2 are two-formsE =Eiωi∧dt, calleddynamical forms(corresponding to differential equations). Note that in this representation, if [ρ] is represented byλ then [dρ] =E1([ρ]) is represented by the dynamical formEλ, the Euler–Lagrange form ofλ. If [α]∈Ωr

2/Θr2 is represented by a dynamical formE

then [dα] =E2([α]) is represented by a source three-form HE, the Helmholtz-form of

E. According to [10], for a general class [ρ]∈Ωr

3/Θr3, where

(2.9) p2ρ=

r X

k,l=0 Hijklω

i k∧ω

j l ∧dt,

we get the canonical source form

(2.10) Iρ= 1

2

r X

k,l=0 k X

p=0

(−1)k

µ k p ¶

dk−p dtk−p(H

lk ji −H

lk ij)ω

j p+l∧ω

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Above,

(2.11) ωi

k =dx i k−x

i

k+1dt, 1≤i≤m, 0≤k≤r−1

are basic contact forms of orderr,ωi 0=ωi.

3

The image of the Helmholtz mapping: generalization of

Helmholtz conditions to

3

-forms

The aim of this paper is to study theimage of the Helmholtz mapping

(3.1) E2: Ωr

2/Θr2→Ωr3/Θr3,

and thekernelof the next variational morphism

(3.2) E3: Ωr

3/Θr3→Ωr4/Θr4.

First, we note that ImE2= KerE3. Indeed, if [β]∈KerE3, i.e., [β]∈Ωr

3/Θr3is such

thatE3([β]) = [dβ] = 0 then due to exactness of the variational sequence there exists [α] ∈ Ωr

2/Θr2 such that [β] = [dα] = E2([α]), i.e. [β] is the image by the Helmholtz

mapping of a class [α] (that has the meaning of a differential equation, and can be represented by a localdynamical form). Conversely, if [β]∈ImE2, i.e. [β] =E2([α]) for a class [α] ∈ Ωr

2/Θr2 then [β] = [dα], hence E3([β]) = E3([dα]) = [ddα] = 0 (the

zero class in Ωr

4/Θr4), so that [β]∈KerE3.

In the representation by source forms, classes [dα] ∈ Ωr

3/Θr3 are represented by

3-forms which arise from local dynamical forms as their Helmholtz forms. Let us introduce the following definitions:

Definition 3.1. We shall call source forms representing elements in Ωr

3/Θr3 Helmholtz-like forms.

LetU ⊂JsY be an open set. We say that a Helmholtz-like formH isHelmholtz overU if there exists a dynamical formE onU such thatH|U =HE. If there is an open covering{Wι}ofJsY such thatH is Helmholtz overfor everyι, we say that H islocally Helmholtz. We callH globally Helmholtzif there exists a dynamical form

EonJsY such thatH =HE.

Note that relation between locally and globally Helmholtz forms is similar to that between locally and globally variational dynamical forms. A three-form H which isglobally Helmholtz is a Helmholtz form of a (globally defined) dynamical formE. A form H which is locally Helmholtz comes from a family of local dynamical forms whose Helmholtz forms glue together to a global differential form; in this case a global dynamical formE such thatH=HE need not exist.

Theorem 3.2. LetH be a Helmholtz-like form. H is locally Helmholtz if and only if there exists a (possibly local) three-formβ such that p2β=H, anddβ= 0.

Proof. Let H be locally Helmholtz. Then (locally) H = HE for a dynamical form

E. E is a source form for a class [α]∈Ωr

2/Θr2 such thatH is a source form for the

class [dα]∈Ωr

3/Θr3. By assumption, there is a representative ¯αof the class [α] such

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p2β¯=p2dα¯ ∼HE. This means thatHE =p2dα¯+p2dηwhereηis 2-contact. Putting

α0 = ¯α+η andβ =dα0 we get a two-form equivalent with ¯αsuch thatp1α0 =E, and a 3-form such thatdβ= 0 andp2β=HE, as desired.

Conversely, let H be a source form such thatH =p2β for a closed three-formβ. Putα=Aβ whereA is the contact homotopy operator [7]. Indeed, by definition of

A, β =dAβ+Adβ =dAβ =dα. Moreover, A is adapted to the decomposition to contact components (meaning that ifηisk-contact thenAηis (k−1)-contact). Then

(3.3) E˜ =p1α=p1Aβ=p1(Ap2β) =p1AH=AH

is a (local) 1-contact two-form, representing the class [α]. We shall show that ˜E is equivalent with a dynamical formE (that is, with a source form for the class [α]).

In fibred coordinates,

Theorem 3.3. If H is locally Helmholtz and the cohomology group H3Y is trivial thenH is equivalent with a globally Helmholtz form.

Proof. We have seen above that if H is locally Helmholtz then H is a source form for a class [dα]. If, moreover, the group H3Y ={0} then the class [α] has a global

representative ¯α. Putting E =Iα¯ where I is the interior Euler operator, we get a global dynamical form. The global formHE=Idα¯ is then the Helmholtz form ofE,

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With help of Theorem 3.2 we can compute explicit conditions for a three-form be a locally Helmholtz form (a generalization of Helmholtz conditions to three-forms), and obtain a corresponding dynamical formE, and a closed three-form (closed counterpart ofHE), which, as we shall show, isunique. In what follows, we shall be interested in second-order Helmholtz-like forms; a generalization to the 3rd order can be found in [14].

Theorem 3.4. A second-order source three-form (Helmholtz-like form) H is locally Helmholtz if and only if components ofH, given by

(3.5) H =Hij0ω

satisfy the following identities:

(3.6)

where [ ]and ( ) denotes skew-symmetrization and symmetrization in the indicated indices, respectively.

Proof. By Theorem 3.2 we have to search for a 3-contact three-form

(3.7) G= skew-symmetric in the upper indices whenever at least two of the indices take the same value. Denoteβ =H+G. Conditiondβ = 0 means that p2dβ = 0 and p3dβ = 0. Computing the former we get thatp2dβ= 0 if and only if (3.6) are satisfied, proving that (3.6) are necessary forH be locally Helmholtz. However, (3.6) are also sufficient, since, by a straightforward computation,p3dβ= 0 is a consequence of p2dβ= 0. ¤

Theorem 3.5. Let H be a second-order source three-form (Helmholtz-like form) (3.5).

(1)Assume that there exists a second-order3-contact formGsuch thatβ=H+G is closed. ThenH is locally Helmholtz, Gis unique and takes the coordinate form

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A corresponding dynamical formE forH is given by the formula

(2) Conversely, if H is locally Helmholtz then there exists a unique (global) 3 -contact formGonJ2Y such thatβ =H+G is closed. Gis given by formula (3.8). Proof. (1) Denote H as above, then G should take the form (3.8), where G000

ijk is

completely skew-symmetric inijk,G001

ijk =−G001jik,G011ijk =−G011ikj, etc. Fromp2dβ= 0

we get the following identities:

(3.12)

From (3.12) we get formulas for the components ofGand conditions (3.6), henceH

is locally Helmholtz. Since dβ = 0 we have locally β = dα, where α= Aβ, and a corresponding local dynamical formE forH is a source form for α, i.e., E =Iα=

Ip1α=IAH.

(2) By Theorem 3.2, ifH is locally Helmholtz thenGexists locally. However, G

is unique, hence global (defined onJ2Y). ¤

Note that we have proved also the following result which can be viewed as a geometric version of the “generalized Helmholtz conditions” (3.6) above, and is an extension to three-forms of a well-known result in the calculus of variations on mani-folds (see [3, 5, 11, 12]).

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(1) His locally Helmholtz (comes from a possibly local dynamical form as its Helmholtz form).

(2) p2dβ= 0.

(3) dβ= 0.

Acknowledgments. Research supported by grant GA ˇCR 201/09/0981 of the Czech Science Foundation.

References

[1] I. Anderson,Aspects of the inverse problem to the calculus of variations, Arch. Math. (Brno) 24 (1988), 181–202.

[2] I. Anderson, The Variational Bicomplex, Utah State University, Technical Re-port, 1989.

[3] M. Crampin, G.E. Prince and G. Thompson, A geometric version of the Helmholtz conditions in time dependent Lagrangian dynamics, J. Phys. A: Math. Gen. 17 (1984), 1437–1447.

[4] H. Helmholtz, Ueber die physikalische Bedeutung des Prinzips der kleinsten Wirkung, J. f¨ur die reine u. angewandte Math. 100 (1887), 137–166.

[5] L. Klapka, Euler–Lagrange expressions and closed two–forms in higher order mechanics, in: Geometrical Methods in Physics, Proc. Conf. on Diff. Geom. and Appl. Vol. 2, Nov´e Mˇesto na Moravˇe, Sept. 1983, D. Krupka, ed., J. E. Purkynˇe Univ. Brno, Czechoslovakia, (1984), 149–153.

[6] M. Krbek and J. Musilov´a, Representation of the variational sequence, Rep. Math. Phys. 51 (2003), 251–258.

[7] D. Krupka, Lepagean forms in higher order variational theory, in: Modern De-velopments in Analytical Mechanics I: Geometrical Dynamics, Proc. IUTAM-ISIMM Symposium, Torino, Italy 1982, S. Benenti, M. Francaviglia and A. Lich-nerowicz, eds., Accad. Sci. Torino, Torino, (1983), 197–238.

[8] D. Krupka,Variational sequences in mechanics, Calc. Var. 5 (1997), 557–583. [9] D. Krupka,Variational sequences on finite order jet spaces, in: Differential

Ge-ometry and Its Applications, Proc. Conf., Brno, Czechoslovakia, 1989, J. Janyˇska and D. Krupka eds., World Scientific, Singapore, (1990), 236–254.

[10] D. Krupka and J. ˇSedˇenkov´a, Variational sequences and Lepagean forms, in: Differential Geometry and its Applications, Proc. Conf., Prague, 2004, J. Bureˇs, O. Kowalski, D. Krupka, J. Slov´ak, Eds., Charles Univ., Prague, Czech Republic, (2005), 605–615.

[11] O. Krupkov´a,Lepagean 2-forms in higher order Hamiltonian mechanics, I. Reg-ularity, Arch. Math. (Brno) 22 (1986), 97–120.

[12] O. Krupkov´a,The Geometry of Ordinary Variational Equations, Lecture Notes in Mathematics 1678, Springer, Berlin, 1997.

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[14] R. Mal´ıkov´a, On a generalization of Helmholtz conditions, Acta Math. Univ. Ostraviensis, 17 (2009), 11–21.

[15] A. Mayer,Die Existenzbedingungen eines kinetischen Potentiales, Berich. Verh. Konig. Sach. Gesell. wissen. Leipzig, Math. Phys. Kl. 84 (1896), 519–529. [16] E. Tonti,Variational formulation of nonlinear differential equations I, II, Bull.

Acad. Roy. Belg. Cl. Sci. 55 (1969), 137–165, 262–278.

[17] M.M. Vainberg, Variational methods in the theory of nonlinear operators (in Russian), GITL, Moscow, 1959.

Authors’ addresses:

O. Krupkov´a

Department of Mathematics, Faculty of Science, The University of Ostrava, 30. dubna 22, Ostrava, Czech Republic;

Department of Mathematics, La Trobe University, Bundoora, Victoria 3086, Australia.

E-mail: olga.krupkova@osu.cz, O.Krupkova@latrobe.edu.au

R. Mal´ıkov´a

Department of Mathematics, Faculty of Science, The University of Ostrava, 30. dubna 22, Ostrava, Czech Republic.

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