The Helmholtz conditions revisited. A new approach to the inverse problem of Lagrangian dynamics

Sarlet, Willy

doi:10.1088/0305-4470/15/5/013

Cited by 130 publications

(110 citation statements)

References 11 publications

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“…and the Helmholz condition (14) with the Tonti integral (15) resolve such "incomplete" inverse problem.…”

Section: Theorem 15 For Any Intermediate Term the Identitymentioning

confidence: 99%

“…Indeed, the Helmholz conditions were already proved by many methods [6], [12], [13], [10], [14]: the potential operators, the variational bicomplex, the Helmholz-Sonin mapping, analysis of Poincaré-Cartan form, and so on. It is however only our naive approach that can be still developed as follows.…”

Section: Proposition 14 (Tonti) Let Us Introduce the Lagrange Functioñmentioning

confidence: 99%

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On the Exact Inverse Problem of the Calculus of Variations

Chrastinová¹,

Tryhuk²

2017

AAN

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The article is devoted to the problem of whether or not a given system of differential equations is identical with the Euler-Lagrange system of an appropriate variational integral. The actual theories which rest on the Helmholz solvability condition and the local Tonti formula are revised. Quite elementary approach is applied. Then the Helmholz condition turns into an easy matter together with unexpected consequence, the solution of incomplete inverse problem. Since the Tonti formula does not give the economical solution, new direct and even global approach is proposed for the determination of all first-order variational integrals related to the second-order Euler-Lagrange system. It employs the fibered de Rham theory where the multiple-valued (ramified) solutions are included as well. The article is of a certain interest also for nonspecialists.Keywords: Euler-Lagrange expression; divergence; Helmholz condition; exact inverse problem; de Rham theory.Informally, the inverse problem of the calculus of variations concerns the indication of hidden extremality principles which undoubtedly belongs to the most important topics both in theoretical and in applied sciences. Various setings are possible. Roughly, the absolute inverse problem is without any restrictions: to decide whether a given system of differential equations is equivalent in the broadest possible sense to an appropriate Euler-Lagrange system. Alas, only some rudiments of the absolute calculus of variations exist [1, Section 7] and partly [2], [3]. Then the general inverse problem deals with equivalences preserving the dependent and the independent variables. Only the particular subcase where the equivalences are linear combinations of equations was systematically investigated after the famed initiating article [4], however, the general case with one independent variable was treated in [5]. Finally, the exact inverse problem appears as a very strict topic: to determine if a given system of differential equations is identical with the Euler-Lagrange system of an appropriate variational integral. This problem looks as the easiest one, however, though it was investigated for a long time in a huge number of articles, the results still cannot be regarded as satisfactory.Our aim is twofold. First, to demonstrate the simplicity of the well-known fundamental achievements on the exact inverse problem. Second, to propose a direct method of determining the "most economical" solutions of the exact inverse problem for the case of the second-order Euler-Lagrange systems of partial differential equations on rather general domains. This is already a new global approach. It should be noted that very advanced tools were applied to the global inverse problem [6], [7], [8], [9], [10], however, only the abstract existence of smooth and single-valued solutions on the total jet space of all cross-sections of a fibered manifold was ensured if some cohomology of the underlying manifold vanishes. It is nevertheless well-known that just the multiply-valued (ramified) obj...

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“…and the Helmholz condition (14) with the Tonti integral (15) resolve such "incomplete" inverse problem.…”

Section: Theorem 15 For Any Intermediate Term the Identitymentioning

confidence: 99%

Section: Proposition 14 (Tonti) Let Us Introduce the Lagrange Functioñmentioning

confidence: 99%

On the Exact Inverse Problem of the Calculus of Variations

Chrastinová¹,

Tryhuk²

2017

AAN

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“…In this case we will say that the semispray S is Lagrangian. Locally it means that the solutions of the system (1.1) are among the solutions of the Euler-Lagrange equations of some Lagrangian L. Necessary and sufficient conditions for the existence of such Lagrangian are called Helmholtz conditions and were expressed in terms of a multiplier matrix [10,12,17,24], a semi-basic 1-form [3,10], or a 2-form [1,9,16,19].…”

Section: 2mentioning

confidence: 99%

“…The problem has a long history and the literature about the subject is vast. There are various approaches to this problem, using different techniques and mathematical tools, [1,3,7,9,12,17,19,24].…”

Section: Introductionmentioning

confidence: 99%

Generalized Helmholtz Conditions for Non-Conservative Lagrangian Systems

Bucataru

Constantinescu

2015

Math Phys Anal Geom

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Abstract. In this paper we provide generalized Helmholtz conditions, in terms of a semi-basic 1-form, which characterize when a given system of second order ordinary differential equations is equivalent to the Lagrange equations, for some given arbitrary non-conservative forces. For the particular cases of dissipative or gyroscopic forces, these conditions, when expressed in terms of a multiplier matrix, reduce to those obtained in [18]. When the involved geometric structures are homogeneous with respect to the fibre coordinates, we show how one can further simplify the generalized Helmholtz conditions. We provide examples where the proposed generalized Helmholtz conditions, expressed in terms of a semi-basic 1-form, can be integrated and the corresponding Lagrangian and Lagrange equations can be found.

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“…For specific aspects of time-dependence in this context, see for example [4,9,8,13]. The very concrete problem we are addressing here is much more computational in nature, so that it will be sufficient to rely on analytical expressions of the conditions for the existence of a Lagrangian, such as the ones developed in [10].…”

Section: Introductionmentioning

confidence: 99%