Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations

We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-Kähler theorem. We consider a linear partial differential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-f...

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Datum:	2012
1. Verfasser:	Constantinescu, O.
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spelling	irk-123456789-1483852019-02-19T01:25:44Z Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations Constantinescu, O. We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-Kähler theorem. We consider a linear partial differential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that P is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor R of the induced nonlinear connection. We recover some of the classes of Lagrangian semisprays: flat semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional manifolds. 2012 Article Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations / O. Constantinescu // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 40 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 49N45; 58E30; 34A26; 37J30 DOI: http://dx.doi.org/10.3842/SIGMA.2012.059 http://dspace.nbuv.gov.ua/handle/123456789/148385 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-Kähler theorem. We consider a linear partial differential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that P is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor R of the induced nonlinear connection. We recover some of the classes of Lagrangian semisprays: flat semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional manifolds.
format	Article
author	Constantinescu, O.
spellingShingle	Constantinescu, O. Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Constantinescu, O.
author_sort	Constantinescu, O.
title	Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations
title_short	Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations
title_full	Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations
title_fullStr	Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations
title_full_unstemmed	Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations
title_sort	formal integrability for the nonautonomous case of the inverse problem of the calculus of variations
publisher	Інститут математики НАН України
publishDate	2012
url	http://dspace.nbuv.gov.ua/handle/123456789/148385
citation_txt	Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations / O. Constantinescu // Symmetry, Integrability and Geometry: Methods and Applications. — 2012. — Т. 8. — Бібліогр.: 40 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT constantinescuo formalintegrabilityforthenonautonomouscaseoftheinverseproblemofthecalculusofvariations
first_indexed	2025-07-12T19:17:04Z
last_indexed	2025-07-12T19:17:04Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 059, 17 pages Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations Oana CONSTANTINESCU Faculty of Mathematics, Alexandru Ioan Cuza University, Bd. Carol no. 11, 700506, Iasi, Romania E-mail: oanacon@uaic.ro URL: http://www.math.uaic.ro/~oanacon/ Received March 16, 2012, in final form September 03, 2012; Published online September 06, 2012 http://dx.doi.org/10.3842/SIGMA.2012.059 Abstract. We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan–Kähler theorem. We consider a linear partial differential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that P is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor R of the induced nonlinear connection. We recover some of the classes of Lagrangian semisprays: flat semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional manifolds. Key words: formal integrability; partial differential operators; Lagrangian semisprays; Helmholtz conditions 2010 Mathematics Subject Classification: 49N45; 58E30; 34A26; 37J30 1 Introduction One of the most interesting problems of geometric mechanics is related to the integrability conditions of the inverse problem of the calculus of variations for time-dependent second-order ordinary differential equations (SODE). The inverse problem can be formulated as follows. Given a time-dependent system of SODE d2xi dt2 + 2Gi ( t, x, dx dt ) = 0, i ∈ {1, . . . , n}, under what conditions this system can be made equivalent, using a multiplier matrix gij , with the system of Euler–Lagrange equations of a regular Lagrangian gij ( t, x, dx dt )( d2xi dt2 + 2Gi ( t, x, dx dt )) = d dt ( ∂L ∂yi ) − ∂L ∂xi ? In this case such a system is called variational. The necessary and sufficient conditions under which such a system is variational are known as the Helmholtz conditions. This inverse problem was solved for the case n = 1 by Darboux [17], and for n = 2 by Douglas [20]. Douglas’s approach consists in an application of the Riquier theory of systems of partial differential equations [32], to a certain associated linear differential system. The generalization of its results in the higher dimensional case is a very difficult problem because the system provided by the Helmholtz conditions is extremely over-determined. Some of the first studies of the inverse problem in spaces of arbitrary dimension are those of Davis [18] and Kosambi [26]. mailto:oanacon@uaic.ro http://www.math.uaic.ro/~oanacon/ http://dx.doi.org/10.3842/SIGMA.2012.059 2 O. Constantinescu There are different attempts to solve this problem. First, there are some reformulations of the Helmholtz conditions in better geometric forms, which are close enough to the first analytical formulations [20, 33, 34, 35], but undercover more of the geometry behind them [11, 13, 15, 16, 19, 28, 29]. The system of SODE is identified with a semispray on the first jet bundle of a fibred manifold over R. The most important geometric tools induced by a semispray are nonlinear connection, Jacobi endomorphism, dynamical covariant derivative, linear connections and their curvatures. Some reformulations of the Helmholtz conditions are using either the special derivations along the tangent bundle projection introduced in [38], or the semi-basic 1-forms [5] and the Frölicher–Nijenhuis theory of derivations on the algebra of vector-valued forms [21]. Anderson and Thompson [2] analyzed the inverse problem based on the exterior differential system approach [4]. Using the variational bicomplex associated to a system of arbitrary order ordinary differential equations, they derived the fundamental system of equations for the va- riational multiplier and proved their sufficiency. They made a detailed study of two dimensional sprays and they proved, for general degrees of freedom, that all isotropic semisprays are varia- tional. It means semisprays that have the associated Jacobi endomorphism a multiple of the identity. This correspond to the Case I of Douglas’s classification. This approach is continued in [1], where the case of Φ diagonalizable, with distinct eigenfunctions, is exposed in detail. The same Case I was proved to be variational also in [36]. This paper uses Riquier theory, but in a more geometric way. The process of repeated differentiations of equations and searching for new nontrivial relations is realized by intrinsic operations. Another subcase of Douglas’s case II is discussed in [14]: separable systems of SODE. Any systems of SODE from this subcase is variational. They showed that any system of SODE in Case II1 with n degrees of freedom can be separated into n separate systems of two first- order equations. They also proved that there are systems separable in the above sense but not separable into single independent second-order equations. This case was treated in [9]. In [37] the authors reinvestigated the case n = 2 with their more intrinsic version of the Riquier algorithm. Their approach is based on the same underlying methodology as the analytical work of Douglas. Another method of studying the integrability conditions of the inverse problem of the calculus of variation is the Spencer–Goldsmchmidt theory of formal integrability of partial differential operators, using two sufficient conditions provided by Cartan–Kähler theorem [12, 22, 40]. This method was applied for autonomous SODE in [23], using the Frölicher–Nijenhuis theory of derivations of vector-valued differential forms. Grifone and Muzsnay gave the first obstructions so that a spray (homogeneous semispray) is variational, for general degrees of freedom. In order to obtain a complete classification of variational sprays, they restricted their work to some particular cases. The Spencer theory is fully applied to the two dimensional case, corresponding to Douglas’s paper. For the general n-dimensional case, it is proved only that isotropic sprays are variational. It is important to notice that Grifone and Muzsnay’s analysis starts from the Euler–Lagrange partial differential operator, and not from the Helmholtz conditions. For time independent, homogeneous SODE, the inverse problem is known as the projective metrizability problem. This problem and its formal integrability is studied in [7] using Spencer theory. It was shown that there exists only one first obstruction for the formal integrability of the projective metrizability operator, expressed in terms of the curvature tensor of the nonlinear connection induced by the spray. This obstruction correspond to second obstruction for the formal integrability of the Euler–Lagrange operator. An interesting and new approach regarding variational PDE’s is the one of A. Prásta- ro [30, 31]. Using suitable cohomologies and integral bordism groups, the author characterizes variational systems constrained by means of PDE’s of submanifolds of fiber bundles. He presents a new algebraic topological characterization of global solutions of variational problems. Formal Integrability for the Inverse Problem of the Calculus of Variations 3 In this paper we address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using also the Spencer version of the Cartan–Kähler theorem. The proper setting is the first jet bundle J1π of an (n+ 1) manifold M fibred over R. In [5] it is proved that a time-dependent semispray is Lagrangian if and only if there exists a semi-basic 1-form θ on J1π, that satisfies a differential system. This gives rise to a linear partial differential operator P . We study the formal integrability of P using two sufficient conditions provided by Cartan–Kähler theorem. We prove that the symbol σ1(P ) is involutive (Theorem 3) and hence there is only one obstruction for the formal integrability of the opera- tor P , which is due to curvature tensor R (Theorem 4). Based on this result, we recover some of the classes of Lagrangian semisprays: flat semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional jet spaces (n = 1). The motivation for this article is double-folded. So far all the results about the inverse problem of the calculus of variations were obtained separately, in the autonomous and nonauto- nomous settings. This is due to the different frameworks involved: the tangent bundle TM (a vector bundle) and respectively the first jet bundle J1π (an affine bundle). The geometric tools are usually constructed in different ways, and special attention was given to the time- depending situation. This paper follows the line of [7] but naturally the proofs of the main theorems have some particularities due to the different setting. Secondly, there are similarities between the formulation of the Helmholtz conditions for sprays in the autonomous setting and respectively for semisprays in the nonautonomous one [5, 6]. This is natural because J1π can be embedded in T̃M (the tangent bundle with the zero section removed). Due to this embedding one can associate to any regular Lagrangian on the velocity- phase space J1π a homogeneous degenerate Lagrangian on the extended phase space T̃M , such that the action defined by a curve in the jet formalism coincides with the action defined by the corresponding curve in the extended formalism. There are correspondences between the main geometric objects associated to these Lagrangians: Poincaré 1- and 2-forms, energies, canonical semisprays-sprays [3, 8, 10, 24]. Therefore, due to this homogeneous formalism, it is natural to expect such kind of similarities between the results corresponding to homogeneous structures on T̃M and nonhomogeneous one on J1π. The paper is organized as follows. In Section 2 we introduce the principal geometric tools induced by a time-dependent semispray on J1π and characterize Lagrangian vector fields with respect to semi-basic 1-forms. Section 3 is dedicated to the application of the Spencer theory to the study of formal integrability of the partial differential operator (PDO) P = (dJ , dh). The most important results are Theorems 3 and 4. Section 3.3 presents classes of semisprays for which the obstruction in Theorem 4 is automatically satisfied. For these classes, the PDO P is formally integrable, and hence these semisprays will be Lagrangian SODE. 2 Preliminaries 2.1 The first-order jet bundle J1π The appropriate geometric setting for the study of time-dependent SODE is the affine jet bundle (J1π, π10,M) [39]. We consider an (n + 1)-dimensional, real, smooth manifold M , which is fibred over R, π : M → R, and represents the space-time. The first jet bundle of π is denoted by π10 : J1π → M , π10(j1 t φ) = φ(t), for φ a local section of π and j1 t φ the first-order jet of φ at t. A local coordinate system (t, xi)i∈{1,...,n} on M induces a local coordinate system on J1π, denoted by (t, xi, yi). Submersion π10 induces a natural foliation on J1π such that (t, xi) are transverse coordinates for this foliation, while (yi) are coordinates for the leaves of the foliation. Throughout the paper we consider Latin indices i ∈ {1, . . . , n} and Greek indices α ∈ {0, . . . , n}, using the notation (xα) = (t = x0, xi). 4 O. Constantinescu In this article we use the Frölicher–Nijenhuis theory [21, 23, 25] of derivations of vector-valued differential forms on the first jet bundle J1π. We adopt the following notations: C∞(J1π) for the ring of smooth functions on J1π, X(J1π) for the C∞ module of vector fields on J1π and Λk(J1π) for the C∞ module of k-forms on J1π. The C∞ module of (r, s)-type vector fields on J1π is denoted by T rs (J1π) and the tensor algebra on J1π is denoted by T (J1π). The graded algebra of differential forms on J1π is written as Λ(J1π) = ⊕ k∈{1,...,2n+1} Λk(J1π). We denote by Sk(J1π) the space of symmetric (0, k) tensors on J1π and by Ψ(J1π) = ⊕ k∈{1,...,2n+1}Ψk(J1π) the graded algebra of vector-valued differential forms on J1π. Throughout the paper we assume that all objects are C∞-smooth where defined. A parametrized curve on M is a section of π: γ : R → M , γ(t) = (t, xi(t)). Its first-order jet prolongation J1γ : t ∈ R → J1γ(t) = ( t, xi(t), dxi/dt ) ∈ J1π is a section of the fibration π1 := π ◦ π10 : J1π → R. Let V J1π be the vertical subbundle of TJ1π, V J1π = {ξ ∈ TJ1π, Dπ10(ξ) = 0} ⊂ TJ1π. The fibers VuJ 1π = KerDuπ10, u ∈ J1π determine a regular, n-dimensional, integrable vertical distribution. Remark that VuJ 1π = spann{∂/∂yi} and its annihilators are the contact 1-forms δxi = dxi−yidt, i ∈ {1, . . . , n} and basic 1-forms λdt, λ ∈ C∞(J1π). The vertical endomorphism J = ∂ ∂yi ⊗δxi is a vector-valued 1-form on J1π, with Im J = V (J1π), V (J1π) ⊂ Ker J and J2 = 0. Its Frölicher–Nijenhuis tensor is given by NJ = 1 2 [J, J ] = − ∂ ∂yi ⊗ δxi ∧ dt = −J ∧ dt. Consequently, d2 J = dNJ = −dJ∧dt 6= 0 and therefore dJ -exact forms on J1π may not be dJ - closed. Here dJ is the exterior derivative with respect to the vertical endomorphisms. Remark 1. For A ∈ Ψ1(J1π) a vector-valued 1-form, the exterior derivative with respect to A is a derivation of degree 1 given by dA = iA ◦ d− d ◦ iA. A k-form ω on J1π, k ≥ 1, is called semi-basic if it vanishes whenever one of the arguments is vertical. A vector-valued k-form A on J1π is called semi-basic if it takes values in the vertical bundle and it vanishes whenever one of the arguments is vertical. A semi-basic k-form satisfies the relation iJθ = 0 and locally can be expressed as θ = θ0dt+ θiδx i. For example, contact 1-forms δxi are semi-basic 1-forms. If a vector-valued k-form A is semi-basic, then J ◦ A = 0 and iJA = 0. The vertical endomorphism J is a vector-valued, semi-basic 1 -form. Locally, a semi-basic k-form θ has the next form θ = 1 k! θi1...ik(xα, yj)δxi1 ∧ · · · ∧ δxik + 1 (k − 1)! θ̃i1...ik−1 (xα, yj)δxi1 ∧ · · · ∧ δxik−1 ∧ dt. For simplicity, we denote by T ∗ the vector bundle of 1-forms on J1π, by T ∗v the vector bundle of semi-basic 1-forms on J1π and by ΛkT ∗v the vector bundle of semi-basic k-forms on J1π. We also denote by Λkv = Sec ( ΛkT ∗v ) the C∞(J1π)-module of sections of ΛkT ∗v and by SkT ∗ the vector bundle of symmetric tensors of (0, k)-type on J1π. S1T ∗ will be identified with T ∗. A semispray is a globally defined vector field S on J1π such that J(S) = 0 and dt(S) = 1. The integral curves of a semispray are first-order jet prolongations of sections of π◦π10 :J1π→R. Locally, a semispray has the form S = ∂ ∂t + yi ∂ ∂xi − 2Gi(xα, yj) ∂ ∂yi , (1) where functions Gi, called the semispray coefficients, are locally defined on J1π. Formal Integrability for the Inverse Problem of the Calculus of Variations 5 A parametrized curve γ : I →M is a geodesic of S if S ◦ J1γ = d dt(J 1γ). In local coordinates, γ(t) = (t, xi(t)) is a geodesic of the semispray S given by (1) if and only if it satisfies the system of SODE d2xi dt2 + 2Gi ( t, x, dx dt ) = 0. (2) Therefore such a system of time-dependent SODE can be identified with a semispray on J1π. Canonical nonlinear connection. A nonlinear connection on J1π is an (n+1)-dimensional distribution H : u ∈ J1π 7→ Hu ⊂ TuJ1π, supplementary to V J1π: ∀u ∈ J1π, TuJ 1π = Hu⊕Vu. A semispray S induces a nonlinear connection on J1π, given by the almost product structure Γ = −LSJ + S ⊗ dt, Γ2 = Id. The horizontal projector that corresponds to this almost product structure is h = 1 2 (Id− LSJ + S ⊗ dt) and the vertical projector is v = Id− h. The horizontal subspace is spanned by S and by δ δxi := ∂ ∂xi − N j i ∂ ∂yj , where N i j = ∂Gi ∂yj . In this paper we prefer to work with the following adapted basis and cobasis:{ S, δ δxi , ∂ ∂yi } , {dt, δxi, δyi}, (3) with δxi the contact 1-forms and δyi = dyi +N i αdx α, N i 0 = 2Gi −N i jy j . Functions N i j and N i 0 are the coefficients of the nonlinear connection induced by the semispray S. With respect to basis and cobasis (3), the horizontal and vertical projectors are locally ex- pressed as h = S ⊗ dt + δ δxi ⊗ δxi, v = ∂ ∂yi ⊗ δyi. We consider the (1, 1)-type tensor field F = h ◦ LSh − J , which corresponds to the almost complex structure in the autonomous case. It satisfies F3 + F = 0, which means that it is an f(3, 1) structure. It can be expressed locally as F = δ δxi ⊗ δyi − ∂ ∂yj ⊗ δxi. Curvature. The following properties for the torsion and curvature of the nonlinear connec- tion induced by the semispray are proved in [5]. The weak torsion tensor field of the nonlinear connection Γ vanishes: [J, h] = 0, which is equivalent also with [J,Γ] = 0. The curvature tensor R = Nh of the nonlinear connection Γ is a vector-valued semi-basic 2-form, locally given by R = 1 2 [h, h] = 1 2 Rkij ∂ ∂yk ⊗ δxi ∧ δxj +Rji ∂ ∂yj ⊗ dt ∧ δxi, (4) where Rijk = δN i j δxk − δN i k δxj and Rij = 2 ∂Gi ∂xj − ∂Gi ∂yk ∂Gk ∂yj − S ( ∂Gi ∂yj ) . (5) The Jacobi endomorphism is defined as Φ = v ◦ LSh = LSh− F− J. (6) Jacobi endomorphism Φ is a semi-basic, vector-valued 1-form and satisfies Φ2 = 0. Locally, can be expressed as Φ = Rji ∂ ∂yj ⊗ δxi, where Rij are given by (5). The Jacobi endomorphism and the curvature of the nonlinear connection are related by the following formulae: Φ = iSR, (7) [J,Φ] = 3R+ Φ ∧ dt. (8) Remark that R = 0 if and only if Φ = 0. 6 O. Constantinescu Definition 1. A semispray S is called isotropic if its Jacobi endomorphism has the form Φ = λJ, (9) where λ ∈ C∞(J1π). Next we express the isotropy condition (9) for a semispray in terms of the curvature tensor R. Proposition 1. A semispray S is isotropic if and only if its curvature tensor R has the form R = α ∧ J, where α is a semi-basic 1-form on J1π. Proof. Suppose that S is an isotropic SODE. Then there exists λ ∈ C∞(J1π) such that Φ = λJ . From (8) it results 3R = [J, λJ ]− Φ ∧ dt, [J, λJ ] = (dJλ) ∧ J − dλ ∧ J2 + λ[J, J ] ⇒ R = 1 3 (dJλ) ∧ J − λJ ∧ dt = α ∧ J, with α = 1 3dJλ+ λdt ∈ T ∗v . In the above calculus we used the formula [23] [K, gL] = (dKg) ∧ L− dg ∧KL+ g[K,L], for K, L vector-valued one-forms on J1π and g ∈ C∞(J1π). For the converse, suppose that R = α∧J , with α ∈ T ∗v . Formula (7) implies Φ = iS(α∧J) = (iSα)J − α ∧ iSJ = (iSα)J . � 2.2 Lagrangian semisprays In this subsection we recall some basic notions about Lagrangian semisprays. Definition 2. 1) A smooth function L ∈ C∞(J1π) is called a Lagrangian function. 2) The Lagrangian L is regular if the (0, 2) type tensor with local components gij ( xα, yk ) = ∂2L ∂yi∂yj has rank n on J1π. The tensor g = gijδx i⊗ δxj is called the metric tensor of the Lagrangian L. Remark 2. More exactly, [39], a function L ∈ C∞(J1π) is called a Lagrangian density on π. If Ω is a volume form on R, the corresponding Lagrangian is the semi-basic 1-form Lπ∗1Ω on J1π. Using a fixed volume form on R, for example dt, it is natural to consider the function L as a (first-order) Lagrangian. For the particular choice of dt as volume form on R, the Poincaré–Cartan 1-form of the Lagrangian L is θL := Ldt + dJL. The Lagrangian L is regular if and only if the Poincaré– Cartan 2-form dθL has maximal rank 2n on J1π. For a detailed exposition on the regularity conditions for Lagrangians see [27]. Formal Integrability for the Inverse Problem of the Calculus of Variations 7 The geodesics of a semispray S, given by the system of SODE (2), coincide with the solutions of the Euler–Lagrange equations d dt ( ∂L ∂yi ) − ∂L ∂xi = 0 if and only if gij ( t, x, dx dt )( d2xi dt2 + 2Gi ( t, x, dx dt )) = d dt ( ∂L ∂yi ) − ∂L ∂xi . (10) Therefore, for a semispray S, there exists a Lagrangian function L such that (10) holds true if and only if S ( ∂L ∂yi ) − ∂L ∂xi = 0, which can be further expressed as LSθL = dL ⇔ iSdθL = 0. (11) Definition 3. A semispray S is called a Lagrangian semispray (or a Lagrangian vector field) if there exists a Lagrangian function L, locally defined on J1π, that satisfies (11). In [5] it has been shown that a semispray S is a Lagrangian semispray if and only if there exists a semi-basic 1-form θ ∈ Λ1 v with rank(dθ) = 2n on J1π, such that LSθ is closed. This represents a reformulation, in terms of semi basic 1-forms, of the result in terms of 2-forms obtained by Crampin et al. in [15]. The characterization of Lagrangian higher order semisprays in terms of a closed 2-form appears also in [2]. Based on this result we can obtain the following reformulation in terms of semi-basic 1-forms of the known Helmholtz conditions [5, Lemma 4.2, Lemma 4.3, Theorem 4.5, Theorem 5.1]. Theorem 1. A semispray S is a Lagrangian vector field if and only if there exists a semi-basic 1-form θ ∈ Λ1 v, with rank(dθ) = 2n on J1π, such that dJθ = 0, dhθ = 0. (12) Proof. In order to make this paper self contained, we give a direct proof of this theorem. Suppose that S is a Lagrangian semispray. It results that there exists a regular Lagrangian L on J1π with LSθL = dL, or equivalently iSdθL = 0, where θL = Ldt + dJL is its Poincaré 1-form. Evidently θL is a semi-basic 1-form with rank(dθL) = 2n on J1π. We will prove that dJθL = dhθL = 0. Indeed, dJθL = dJL ∧ dt+ LdJdt+ d2 JL = iJdL ∧ dt− dJ∧dtL = iJdL ∧ dt− iJ∧dtdL = 0. From the formula iJLS − LSiJ = iΓ−S⊗dt and iJLSdθL = 0 we obtain LSdJθL + iΓdθL − iS⊗dtdθL = 0. We also compute iΓdθL = i2h−IddθL = 2ihdθL − 2dθL = 2dhθL. Therefore 2dhθL = iS⊗dtdθL = −iSdθL ∧ dt = 0. Conversely, suppose that there exists a semi-basic 1-form θ ∈ Λ1 v, with rank(dθ) = 2n on J1π, such that dJθ = 0, dhθ = 0. In order to prove that S is a Lagrangian vector field, we will first show that LSθ = d(iSθ). The hypothesis dJθ = 0 implies θ = (iSθ)dt+ dJ(iSθ). Indeed, dJ iS + iSdJ = LJS − i[S,J ] = ih−S⊗dt−v ⇒ dJ(iSθ) = ihθ − (iSθ)dt− ivθ = θ − (iSθ)dt. Next, from dhiS + iSdh = LhS − i[S,h] and dhθ = 0 it results that dhiSθ = LSθ − iF+J+Φθ = LSθ − iFθ. From iFθ = iF ((iSθ)dt+ dJ(iSθ)) = iF (dJ(iSθ)) and iFdJ − dJ iF = dJ◦F − i[F,J ] ⇒ iF (dJ(iSθ)) = dv(iSθ). It results that LSθ = dh(iSθ) + dv(iSθ) = d(iSθ). Consider L = iSθ. Then θ is the Poincaré–Cartan 1-form of L and LSθL = dL. From rank(dθ) = 2n on J1π it results that L is a regular Lagrangian and S is a Lagrangian vector field. � In the next section we discuss the formal integrability of these Helmholtz conditions using two sufficient conditions provided by Cartan–Kähler theorem. 8 O. Constantinescu 3 Formal integrability for the nonautonomus inverse problem of the calculus of variations In order to study the integrability conditions of the set of differential equations (12), we associate to it a linear partial differential operator and study its formal integrability, using Spencer’s technique. The approach in this work follows the one developed in [7] for studying the projective metrizability problem for autonomous sprays. For the basic notions of formal integrability theory of linear partial differential operators see [7, 23]. Consider T ∗v the vector bundle of semi-basic 1-forms on J1π and Λ1 v the module of sections of T ∗v . For θ ∈ Λ1 v and k ≥ 1 we denote by jkuθ the kth order jet of θ at the base point u in J1π. The bundle of kth order jets of sections of T ∗v is denoted by JkT ∗v . The projection π0 : JkT ∗v → J1π is defined by π0(jkuθ) = u. If l > k, one defines the projections πk as follows: πk(j l uθ) = jkuθ and J lT ∗v is also a fibred manifold over JkT ∗v . If f1, . . . , fk ∈ C∞(J1π) are functions vanishing at u ∈ J1π and θ ∈ Λ1 v, we define ε : SkT ∗ ⊗ T ∗v −→ JkT ∗v by ε(df1 � · · · � dfk ⊗ θ)u = jku(f1 · · · fkθ), where � is the symmetric product. Then the sequence 0 −→ SkT ∗ ⊗ T ∗v ε−→ JkT ∗v πk−1−→ Jk−1T ∗v −→ 0 is exact. Consider the linear partial differential operator of order one P : Λ1 v → Λ2 v ⊕ Λ2 v, P = (dJ , dh) . (13) Remark that P (θ) can be expressed in terms of first-order jets of θ, for any θ ∈ Λ1 v, and therefore it induces a morphism between vector bundles: p0(P ) : J1T ∗v → Λ2T ∗v ⊕ Λ2T ∗v , p0(P )(j1 uθ) = P (θ)u, ∀ θ ∈ Λ1 v. We also consider the lth order jet prolongations of the differential operator P , l ≥ 1, which will be identified with the morphisms of vector bundles over M , pl(P ) : J l+1T ∗v → J l ( Λ2T ∗v ⊕ Λ2T ∗v ) , pl(P ) ( jl+1 u θ ) = jlu (P (θ)) , ∀ θ ∈ Λ1 v. Remark that for a semi-basic 1-form θ = θαδx α, its first-order jet j1θ = δθα δxβ δxβ ⊗ δxα + ∂θα ∂yi δyi ⊗ δxα determines the local coordinates ( xα, yi, θα, θαβ, θαi ) on J1T ∗v . In this work all contravariant or covariant indices, related to vertical components of tensor fields will be under- lined. Consider θ = θαδx α, a semi-basic 1-form on J1π. Then dθ = ( ∂θi ∂t − θjN j i − δθ0 δxi ) δx0 ∧ δxi + ( θi − ∂θ0 ∂yi ) δx0 ∧ δyi + 1 2 ( δθj δxi − δθi δxj ) δxi ∧ δxj + ( ∂θj ∂yi ) δyi ∧ δxj , dJθ = ( θi − ∂θ0 ∂yi ) δx0 ∧ δyi + 1 2 ( ∂θi ∂yj − ∂θj ∂yi ) δxj ∧ δxi, dhθ = ( ∂θi ∂t − θjN j i − δθ0 δxi ) δx0 ∧ δxi + 1 2 ( δθi δxj − δθj δxi ) δxj ∧ δxi. Using these formulae we obtain p0(P ) ( j1θ ) = (( θi − ∂θ0 ∂yi ) δx0 ∧ δyi + 1 2 ( ∂θi ∂yj − ∂θj ∂yi ) δxj ∧ δxi,( ∂θi ∂t − θjN j i − δθ0 δxi ) δx0 ∧ δxi + 1 2 ( δθi δxj − δθj δxi ) δxj ∧ δxi ) . Formal Integrability for the Inverse Problem of the Calculus of Variations 9 The symbol of P is the vector bundle morphism σ1(P ) : T ∗ ⊗ T ∗v → Λ2T ∗v ⊕Λ2T ∗v defined by the first-order terms of p0(P ). More exactly, σ1(P ) = p0(P ) ◦ ε. For A ∈ T ∗ ⊗ T ∗v , A = Aαβδx α ⊗ δxβ +Aiβδy i ⊗ δxβ, we compute σ1(dJ)A = −Ai0δx0 ∧ δxi + 1 2 ( Aji −Aij ) δxj ∧ δxi, σ1(dh)A = (A0i −Ai0) δx0 ∧ δxi + 1 2 (Aji −Aij) δxj ∧ δxi and hence σ1(P )A = (τJA, τhA) , (τJA) (X,Y ) = A(JX, Y )−A(JY,X), (τhA) (X,Y ) = A(hX, Y )−A(hY,X), for X,Y ∈ X(J1π). In the above formulae τJ , τL are alternating operators [7]. Remark 3. The alternating operators are defined in general as follows. For K ∈ Ψk(J1π), a vector-valued k-form, we consider τK : Ψ1(J1π)⊗Ψl(J1π)→ Ψl+k(J1π), (τKB)(X1, . . . , Xl+k) = 1 l!k! ∑ σ∈Sl+k ε(σ)B(K(Xσ(1), . . . , Xσ(k)), Xσ(k+1), . . . , Xσ(k+l)), (14) where X1, . . . , Xl+k ∈ X(J1π) and Sl+k is the permutation group of {1, . . . , l+k}. The restriction of τK to Ψl+1(J1π), is a derivation of degree (k− 1) and it coincides with the inner product iK . The first-order prolongation of the symbol of P is the vector bundle morphism σ2(P ) : S2T ∗⊗ T ∗v → T ∗ ⊗ ( Λ2T ∗v ⊕ Λ2T ∗v ) that verifies iX ( σ2(P )B ) = σ1(P ) (iXB) , ∀B ∈ S2T ∗ ⊗ T ∗v , ∀X ∈ X(J1π). Therefore σ2(P )B = ( σ2(dJ)B, σ2(dh)B ) ,( σ2(dJ)B ) (X,Y, Z) = B(X, JY, Z)−B(X, JZ, Y ),( σ2(dh)B ) (X,Y, Z) = B(X,hY, Z)−B(X,hZ, Y ). In local coordinates we obtain the following formulae. If B ∈ S2T ∗ ⊗ T ∗v , then it has the local decomposition B = Bαβγδx α ⊗ δxβ ⊗ δxγ +Biαβδy i ⊗ δxα ⊗ δxβ +Bαiβδx α ⊗ δyi ⊗ δxβ +Bijαδy i ⊗ δyj ⊗ δxα, (15) with Bαβγ = Bβαγ , Bijα = Bijα, Bi0α = B0iα, Bijα = Bjiα. (16) The first-order prolongation of the symbol of P is given by σ2(dJ)B = Bαi0δx α ⊗ δxi ∧ δx0 +Bij0δy i ⊗ δxj ∧ δx0 + 1 2 ( Bαij −Bαji ) δxα ⊗ δxi ∧ δxj + 1 2 ( Bijk −Bikj ) δyi ⊗ δxj ∧ δxk, σ2(dh)B = 1 2 (Bαβγ −Bαγβ) δxα ⊗ δxβ ∧ δxγ + 1 2 ( Biαβ −Biβα ) δyi ⊗ δxα ∧ δxβ. 10 O. Constantinescu For each u ∈ J1π, we consider gku(P ) = Kerσku(P ), k ∈ {1, 2}, g1 u(P )e1...ej = {A ∈ g1 u(P )\|ie1A = · · · = iejA = 0}, j ∈ {1, . . . , n}, where {e1, . . . , en} is a basis of Tu(J1π). Such a basis is called quasi-regular if it satisfies dim g2 u(P ) = dim g1 u(P ) + n∑ j=1 dim g1 u(P )e1...ej . Definition 4. The symbol σ1(P) is called involutive at u in J1π if there exists a quasi-regular basis of TuJ 1π. A first-order jet j1 uθ ∈ J1T ∗v is a first-order formal solution of P at u in J1π if p0(P )(θ)u = 0. For l ≥ 1, a (1 + l)th order jet j1+l u θ ∈ J1+l u T ∗v is a (1 + l)th order formal solution of P at u in J1π if pl(P )(θ)u = 0. For any l ≥ 0, consider R1+l u (P ) = ker plu(P ) the space of (1 + l)th order formal solutions of P at u. We denote also π̄l,u : R1+l u (P ) → Rlu(P ) the restriction of πl,u : J1+l u (T ∗v ) → J lu(T ∗v ) to R1+l u (P ). Definition 5. The partial differential operator P is called formally integrable at u in J1π if R1+l(P ) = ⋃ u∈J1π R 1+l u (P ) is a vector bundle over J1π, for all l ≥ 0, and the map π̄l,u : R1+l u (P )→ Rlu(P ) is onto for all l ≥ 1. The fibred submanifold R1(P ) of π0 : J1 u(T ∗v )→ J1π is called the partial differential equation corresponding to the first-order PDO P . A solution of the operator P on an open set U ⊂ J1π is a section θ ∈ Λ1 v defined on U such that Pθ = 0⇔ p0(P )(j1 uθ) = 0, ∀u ∈ U . The Cartan–Kähler theorem [23] takes the following form for the particular case of first-order PDO. Theorem 2. Let P be a first-order linear partial differential operator with g2(P ) a vector bundle over R1(P ). If π1 : R2(P ) → R1(P ) is onto and the symbol σ1(P ) is involutive, then P is formally integrable. 3.1 The involutivity of the symbol of P In this subsection we prove that the operator P satisfies one of the two sufficient conditions for formal integrability, provided by Cartan–Kähler theorem: the involutivity of the symbol σ1(P ). Theorem 3. The symbol σ1(P ) of the PDO P = (dJ , dh) is involutive. Proof. First we determine g1(P ) = { A ∈ T ∗ ⊗ T ∗v \|σ1(P )A = 0 } , and compute the dimension of its fibers. We obtain g1 u(P ) = { A = Aαβδx α ⊗ δxβ +Aiβδy i ⊗ δxβ \| Ai0 = 0, Aji = Aij , Aαβ = Aβα } . From Ai0 = 0 and Aji = Aij it results that Aij contribute with n(n + 1)/2 components to the dimension of g1 u(P ), and from Aαβ = Aβα it follows that Aαβ contribute with (n+ 1)(n+ 2)/2 components to the dimension of g1 u(P ). So dim g1 u(P ) = n(n+ 1) 2 + (n+ 1)(n+ 2) 2 = (n+ 1)2. Next we determine g2(P ) = { B ∈ S2T ∗ ⊗ T ∗v \|σ2(P )B = 0 } . Formal Integrability for the Inverse Problem of the Calculus of Variations 11 If B ∈ S2T ∗ ⊗ T ∗v has the local components (15), then B ∈ g2(P ) if and only if the following relations are satisfied: Bαi0 = 0, Bij0 = 0, Bαij = Bαji, Bijk = Bikj , Bαβγ = Bαγβ, Biαβ = Biβα. (17) From the relations (16) and (17) it results that B ∈ g2(P ) if and only if its local components Bαβγ , Bijk, Bijk, Bijk are totally symmetric and the rest are vanishing. Therefore Bαβγ con- tribute with (n+ 1)(n+ 2)(n+ 3)/6 components to the dimension of g2 u(P ), and Bijk, Bijk with n(n+ 1)(n+ 2)/6 components each of them. It results dim g2 u(P ) = (n+ 1)(n+ 2)(n+ 3) 6 + 2 n(n+ 1)(n+ 2) 6 = (n+ 1)2(n+ 2) 2 . Consider B = {h0 = S, h1, . . . , hn, v1 = Jh1, . . . , vn = Jhn} a basis in TuJ 1π with h0 = S, h1, . . . , hn horizontal vector fields. For any A ∈ g1(P ), we denote A(hα, hβ) = aαβ, A(vi, hα) = biα. Because A is semi-basic in the second argument it follows that these are the only components of A. Since A ∈ Kerσ1(dJ) it follows that Ai0 = 0, Aij = Aji ⇒ bi0 = 0, bij = bji. Because A ∈ Kerσ1(dh) it results that A0i = Ai0, Aij = Aji and hence a0i = ai0, aij = aji. Consider j ∈ {1, . . . , n} arbitrarily fixed and B̃ = { e0 = S + hj + vn, e1 = h1, e2 = h2 + v1, . . . , ei = hi + vi−1, . . . , en = hn + vn−1, v1, . . . , vn } a new basis in TuJ 1π. If we denote A(eα, eβ) = ãαβ, A(vi, eα) = b̃iα, a simple computation and the fact that A is semi-basic in the second argument determine ã00 = a00 + 2a0j + ajj + bnj , ãik = aik + bi−1,k 6= ãki = aki + bk−1,i, ãi0 = ai0 + aij + bi−1,j 6= ã0i = a0i + aji + bni, b̃ik = bik = b̃ki, b̃i0 = bij . It can be seen that all the independent components of A in the basis B can be obtained from the components of A in the basis B̃, and hence we can use the later for determining the dimensions of (g1 u)e0e1...ek . If A ∈ (g1 u)e0 it results ã0α = 0, so using this new basis we impose n + 1 supplementary independent restrictions. It follows that dim(g1 u)e0 = (n+ 1)2 − (n+ 1) = (n+ 1)n. If A ∈ (g1 u)e0e1 it results that together with the previous restrictions we impose also ã1α = 0, so another independent n+1 restrictions. Hence dim(g1 u)e0e1 = (n+1)n−(n+1) = (n+1)(n−1). In general dim(g1 u)e0e1...ek = (n + 1)(n − k), ∀ k ∈ {1, . . . , n}. Hence dim(g1 u)e0e1...en = 0 ⇒ dim(g1 u)e0...env1...vk = 0, ∀ k ∈ {1, . . . , n}, dim(g1 u) + n∑ k=0 dim(g1 u)e0e1...ek = (n+ 1)2 + (n+ 1)[n+ n− 1 + · · ·+ 1] = (n+ 1)2(n+ 2) 2 = dim g2 u(P ). We proved that B̃ is a a quasi-regular basis, hence the symbol σ1(P ) is involutive. � 12 O. Constantinescu 3.2 First obstruction to the inverse problem In this subsection we determine necessary and sufficient conditions for π̄1 to be onto. We will obtain only one obstruction for the integrability of the operator P . The obstruction is due to the curvature tensor of the nonlinear connection induced by the semispray. Theorem 4. A first-order formal solution θ ∈ Λ1 v of the system dJθ = 0, dhθ = 0 can be lifted into a second-order solution, which means that π̄1 : R2(P )→ R1(P ) is onto, if and only if dRθ = 0, where R is the curvature tensor (4). Proof. We use a known result from [23, Proposition 1.1]. If K is the cokernel of σ2(P ), K = T ∗ ⊗ ( Λ2T ∗v ⊕ Λ2T ∗v ) Imσ2(P ) , there exists a morphism ϕ : R1(P )→ K such that the sequence R2(P ) π̄1−→ R1(P ) ϕ−→ K is exact. In particular π̄1 is onto if and only if ϕ = 0. After defining ϕ, we will prove that for θ ∈ Λ1 v, with j1 uθ ∈ R1 u(P ), a first-order formal solution of P at u ∈ J1π, we have that ϕuθ = 0 if and only if (dRθ)u = 0. The construction of the morphism ϕ is represented in the next diagram by dashed arrows. We denote F = Λ2T ∗v ⊕ Λ2T ∗v . Remark that dimT ∗ = 2n+ 1, dimT ∗v = n+ 1, dim Λ2T ∗v = (n+1)n 2 , dimS2T ∗ = (2n+1)(2n+2) 2 , dimT ∗ ⊗ ( Λ2T ∗v ⊕ Λ2T ∗v ) = (2n+ 1)n(n+ 1). Therefore dimK = dim [ T ∗ ⊗ ( Λ2T ∗v ⊕ Λ2T ∗v )] − dim ( Imσ2(P ) ) = dim [ T ∗ ⊗ ( Λ2T ∗v ⊕ Λ2T ∗v )] − [ dim ( S2T ∗ ⊗ T ∗v ) − dim ( kerσ2(P ) )] = (n− 1)n(n+ 1) 2 = 3 ( n+ 1 3 ) . It results from this that K ' ⊕(3)Λ3T ∗v . Formal Integrability for the Inverse Problem of the Calculus of Variations 13 Next we define τ : T ∗ ⊗ ( Λ2T ∗v ⊕ Λ2T ∗v ) → ⊕(3)Λ3T ∗v such as the next sequence is exact: 0→ g2(P ) i−→ S2T ∗ ⊗ T ∗v σ2(P )−→ T ∗ ⊗ ( Λ2T ∗v ⊕ Λ2T ∗v ) τ−→ ⊕(3)Λ3T ∗v → 0. (18) For B1, B2 ∈ T ∗ ⊗ Λ2T ∗v , we define τ(B1, B2) = (τ1(B1, B2), τ2(B1, B2), τ3(B1, B2)), where τi : T ∗ ⊗ ( Λ2T ∗v ⊕ Λ2T ∗v ) → Λ3T ∗v , i ∈ {1, 2, 3}, are given by τ1(B1, B2) = τJB1, τ2(B1, B2) = τhB2, τ3(B1, B2) = τhB1 + τJB2. Using the definition (14) of the alternating operators τJ , τh, we prove that τ ◦ σ2(P ) = 0. Indeed, using that any B ∈ S2T ∗⊗T ∗v is symmetric in the first two arguments, it follows that( τ ◦ σ2(P ) ) (B) = τ ( σ2(dJ)B, σ2(dh)B ) = ( τJσ 2(dJ)B, τhσ 2(dh)B, τhσ 2(dJ)B + τJσ 2(dh)B ) = 0, ∀B ∈ S2T ∗ ⊗ T ∗v . For example, τJ ( σ2(dJ)B ) (X,Y, Z) = [ σ2(dJ)B (JX, Y, Z)− σ2(dJ)B (JY,X,Z) + σ2(dJ)B (JZ,X, Y ) ] = [B(JX, JY, Z)−B(JX, JZ, Y )−B(JY, JX,Z) +B(JY, JZ,X) +B(JZ, JX, Y )−B(JZ, JY,X)] = 0, ∀X,Y, Z ∈ X(J1π). The relation τ ◦ σ2(P ) = 0 implies that Im(σ2(P )) ⊆ Ker τ . Using that τ is onto (τJ , τh are both onto) it results that dim [ Im(σ2(P )) ] = dim (Ker τ) and hence Im(σ2(P )) = Ker τ and the sequence (18) is exact. The last step before defining ϕ : R1(P ) → K is to consider a linear connection ∇ on J1π such that ∇J = 0. It means that ∇ preserve semi-basic forms and ∇ can be considered as a connection in the fiber bundle Λ2T ∗v ⊕ Λ2T ∗v → J1π. As a first-order PDO we can identify ∇ with the bundle morphism p0(∇) : J1 ( Λ2T ∗v ⊕ Λ2T ∗v ) → T ∗ ⊗ ( Λ2T ∗v ⊕ Λ2T ∗v ) . We will also use two derivations of degree 1 introduced in [7], defined byDJ = τJ∇, Dh = τh∇. Both derivations DJ , Dh preserve semi-basic forms and dJ−DJ , dh−Dh are algebraic derivations. It means that if ω ∈ Λk ( J1π ) vanishes at some point u ∈ J1π, then (DJω)u = (dJω)u and (Dhω)u = (dhω)u [7, Lemma 2.1]. Now we are able to define ϕ : R1(P )→ K such that the sequence R2(P ) π̄1−→ R1(P ) ϕ−→ K is exact. Let θ ∈ Λ1 v such that j1 uθ ∈ R1 u(P ) ⊂ J1 uT ∗ v is a first-order formal solution of P at u ∈ J1π, which means that (dJθ)u = (dhθ)u = 0. Consider ϕuθ = τu∇Pθ = τu (∇dJθ,∇dhθ) . Using the fact that dJ − τJ∇ and dh− τh∇ are algebraic derivations and (dJθ)u = (dhθ)u = 0 it results dJ (dJθ)u = τJ∇ (dJθ)u and dh (dhθ)u = τh∇ (dhθ)u. We will compute the three components of the map ϕ. It follows that τ1 (∇Pθ)u = τ1 (∇dJθ,∇dhθ)u = τJ (∇dJθ)u = ( d2 Jθ ) u = 1 2 ( d[J,J ]θ ) u = − (dJ∧dtθ)u = − (dJθ)u ∧ dt = 0, τ2 (∇Pθ)u = τ2 (∇dJθ,∇dhθ)u = τh (∇dhθ)u = ( d2 hθ ) u = 1 2 ( d[h,h]θ ) u = (dRθ)u , 14 O. Constantinescu where R is given by (4), τ3 (∇Pθ)u = τ3 (∇dJθ,∇dhθ)u = τh (∇dJθ)u + τJ (∇dhθ)u = (dhdJθ)u + (dJdhθ)u = ( d[h,J ]θ ) u = 0. Hence ϕ = 0 if and only if dRθ = 0. � Remark 4. Locally, dRθ has the following form: R = 1 2 Rkij ∂ ∂yk ⊗ δxi ∧ δxj︸︷︷︸ R̃= 1 3 [J,Φ] −Φ ∧ dt ⇒ dRθ = d R̃ θ − dΦθ ∧ dt, dΦθ = Rji ( θj − ∂θ0 ∂yj ) dt ∧ δxi + 1 2! ( ∂θj ∂yk Rki − ∂θi ∂yk Rkj ) δxj ∧ δxi ⇒ dRθ = 1 3! ( ailR l jk + ajlR l ki + aklR l ij ) δxi ∧ δxj ∧ δxk + 1 2! ( ajkR k i − aikRkj ) dt ∧ δxi ∧ δxj , where we denoted aij = ∂θi ∂yj . Hence dRθ = 0 if and only if ailR l jk + ajlR l ki + aklR l ij = 0 and ajkR k i − aikRkj = 0. The first identity represents the algebraic Bianchi identity for the curvatures of the nonlinear connection. The second identity is one of the classical Helmholtz condition for the multiplier matrix aij . These obstructions appear also in [2]. It can be seen that for n = 2 the formula of dRθ becomes dRθ = 1 2! ( ajkR k i − aikRkj ) dt ∧ δxi ∧ δxj = −dΦθ ∧ dt. Therefore, for n = 2, the obstruction is equivalent with dΦθ ∧ dt = 0. 3.3 Classes of Lagrangian time-dependent SODE We present now some classes of semisprays for which the obstruction in Theorem 4 is automa- tically satisfied. Therefore the PDO P is formally integrable, and hence these semisprays will be Lagrangians SODEs. These classes of semisprays are: • flat semisprays, R = 0⇔ Φ = 0; • arbitrary semisprays on 2-dimensional manifolds; • isotropic semisprays, Φ = λJ , for λ a smooth function on J1π. All these classes of semisprays were already studied in the articles cited in the introduction. In the flat case, the obstruction is automatically satisfied. If dimM = 1 then for a semi-basic 1-form θ on J1π, dRθ is a semi-basic 3-form on J1π. Because dim Λ3 (T ∗v ) = (n+ 1)n(n− 1)/6 and it is zero if n = 1, dRθ will necessarily vanish. We consider now the last case, of isotropic semisprays. Proposition 2. Any isotropic semispray is a Lagrangian second-order vector field. Proof. Assume now that S is an isotropic SODE and θ a semi-basic 1-form on J1π such that (dJθ)u = (dhθ)u = 0, for some u ∈ J1π, (dRθ)u = dα∧Jθ = α ∧ dJθ + (−1)2dα ∧ iJθ = 0. We used that iJθ = 0, (dJθ)u = 0 and the formula [23] dω∧Kπ = ω ∧ dKπ + (−1)q+kdω ∧ iKπ, for ω a q-form on J1π and K a vector-valued k-form on J1π. Since (dRθ)u vanishes, S is a Lagrangian semispray. � Formal Integrability for the Inverse Problem of the Calculus of Variations 15 Next we give some simple examples of Lagrangian semisprays, corresponding to the above general classes. We start with the semispray expressed by the SODE d2x1 dt2 + f ( t, dx2 dt ) = 0, d2x2 dt2 + g(t) = 0, with f an arbitrary smooth function depending only on t and y2 = dx2 dt , and g an arbitrary smooth function depending only on t. The only possible nonvanishing local component of the Jacobi endomorphism is R1 2 = −S ( N1 2 ) = −1 2 ∂2f ∂t∂y2 + g(t) ∂2f ∂(y2)2 . Hence, if ∂2f ∂t∂y2 = 2g(t) ∂2f ∂(y2)2 the semispray is flat (Φ = 0). This example is a generalization of the one given by Douglas [20, (8.14)]. If ∂2f ∂t∂y2 6= 2g(t) ∂2f ∂(y2)2 , then the semispray is isotropic Φ = 0 −1 2 ∂2f ∂t∂y2 + g(t) ∂2f ∂(y2)2 0 0  = ( −1 2 ∂2f ∂t∂y2 + g(t) ∂2f ∂(y2)2 ) J. Another example of isotropic (or flat) semispray is the one given by the SODE d2x1 dt2 + f ( t, x2 ) = 0, d2x2 dt2 + g(t) = 0, with f an arbitrary smooth function depending only on t and x2, and g an arbitrary smooth function depending only on t. All the local coefficients of the associated nonlinear connection are vanishing. Evidently Φ = 0 ∂f ∂x2 0 0  = ( ∂f ∂x2 ) J. This example was treated in [2, (6.1)] and [20, (15.4)] for f(x2) = −x2 and g = 0. The first paper also presents all the Lagrangians corresponding to the given SODE. Consider also the semispray given by the SODE d2x1 dt2 + 2 dx2 dt = 0, d2x2 dt2 − ( dx2 dt )2 = 0. Evidently Φ = ( 0 y2 0 0 ) , hence the semispray is isotropic. 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[40] Spencer D.C., Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 179–239. http://dx.doi.org/10.1016/B978-044452833-9.50017-6 http://dx.doi.org/10.1023/A:1006346916360 http://dx.doi.org/10.1016/j.jmaa.2005.08.037 http://dx.doi.org/10.1088/0305-4470/14/9/018 http://dx.doi.org/10.1088/0305-4470/14/9/018 http://dx.doi.org/10.1088/0305-4470/15/5/013 http://dx.doi.org/10.1023/A:1006102121371 http://dx.doi.org/10.1090/S0002-9947-02-02994-X http://dx.doi.org/10.1016/0926-2245(95)00013-T http://dx.doi.org/10.1017/CBO9780511526411 1 Introduction 2 Preliminaries 2.1 The first-order jet bundle J1 2.2 Lagrangian semisprays 3 Formal integrability for the nonautonomus inverse problem of the calculus of variations 3.1 The involutivity of the symbol of P 3.2 First obstruction to the inverse problem 3.3 Classes of Lagrangian time-dependent SODE References

Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations

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