Towards a geometrical understanding of Douglas's solution of the inverse problem of the calculus of variations

Crampin, M.; Sarlet, Willy; Martínez, Eduardo; Byrnes, G B; Prince, G. E.

doi:10.1088/0266-5611/10/2/005

Cited by 56 publications

(81 citation statements)

References 8 publications

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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%

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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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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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“…Some final comments: Berwald-type connections have been successfully applied in a number of applications concerning qualitative feature of SODEs. A few examples of such problems are mentioned below (see also [2,4,5]):…”

Section: Proposition 6 a Linearρ-connection Onπ Is Associated With Anmentioning

confidence: 99%

“…Two important tools are extensively used in the geometrical analysis of (time-dependent) SODEs (see [2,4,5]):…”

Section: Introductionmentioning

confidence: 99%

Generalized connections on affine bundles

Mestdag¹

2004

AIP Conference Proceedings

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“…Nowadays, geometric and global rather than analytical aspects of the inverse variational problem are of main interest, and relations between variationality and geometry of differential equations are intensively studied and explored (see e.g. [4,21,22]). …”

Section: Introductionmentioning

confidence: 99%

Lepage Equivalents of Second-Order Euler–Lagrange Forms and the Inverse Problem of the Calculus of Variations

Krupková¹,

Smetanová²

2021

JNMP

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In the calculus of variations, Lepage (n + 1)-forms are closed differential forms, representing Euler-Lagrange equations. They are fundamental for investigation of variational equations by means of exterior differential systems methods, with important applications in Hamilton and Hamilton-Jacobi theory and theory of integration of variational equations. In this paper, Lepage equivalents of second-order Euler-Lagrange quasilinear PDE's are characterised explicitly. A closed (n + 1)-form uniquely determined by the Euler-Lagrange form is constructed, and used to find a geometric solution of the inverse problem of the calculus of variations.Keywords: Second-order Euler-Lagrange equations; Euler-Lagrange form; Lepage form; Lepage equivalent of a Lagrangian; Lepage equivalent of an Euler-Lagrange form; inverse problem of the calculus of variations.

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Towards a geometrical understanding of Douglas's solution of the inverse problem of the calculus of variations

Abstract: We describe a novel approach to the study of the inverse problem of the calculus of variations, which gives new insights into Douglas's solution of the two degree of freedom case.

Cited by 56 publications

References 8 publications

Generalized Helmholtz Conditions for Non-Conservative Lagrangian Systems

Generalized Helmholtz Conditions for Non-Conservative Lagrangian Systems

Generalized connections on affine bundles

Lepage Equivalents of Second-Order Euler–Lagrange Forms and the Inverse Problem of the Calculus of Variations

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