Variational Principles For Second-Order Differential Equations

Grifone, Joseph; Muzsnay, Zoltán

doi:10.1142/3996

Cited by 44 publications

(79 citation statements)

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“…When σ = 0, depending on the values of p, we obtain two important particular cases of the inverse problem of Lagrangian mechanics. The case σ = 0, p = 1, and L = F , for a Finsler function F , is known as the projective metrizability problem [5,6,15,16]. The case σ = 0, p = 2, and L = F 2 , for a Finsler function F , is known as the Finsler metrizability problem [20].…”

Section: The Homogeneous Casementioning

confidence: 99%

“…In Section 2 we use the Frölicher-Nijenhuis formalism [13,15] to provide a geometric framework associated to a given system (1.1). This framework includes: a nonlinear connection, dynamical covariant derivative and curvature type tensors.…”

Section: Introductionmentioning

confidence: 99%

“…A geometric setting for semisprays. In this section, we use the Frölicher-Nijenhuis formalism [13] to associate a geometric setting to a given system of second order ordinary differential equations, [2,3,14,15].…”

mentioning

confidence: 99%

“…For various commutation formulae, within the Frölicher-Nijenhuis formalism, we will use the Appendix A of the book [15]. We will use some vector valued forms on T M to associate a differential calculus on T M .…”

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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: The Homogeneous Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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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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“…With an explicit computation based on the theory developed in [9], we examine a 3-web whose linearizability was claimed in [9]. We show that, contrary to the statement of [6] and [7], this particular web is linearizable.…”

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confidence: 99%

On the problem of linearizability of a 3-web

Muzsnay

2008

Nonlinear Analysis: Theory, Methods & Applications

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In this paper we study the linearizability problem for 3-webs on a 2-dimensional manifold. With an explicit computation based on the theory developed in [9], we examine a 3-web whose linearizability was claimed in [9]. We show that, contrary to the statement of [6] and [7], this particular web is linearizable. We compute explicitly the affine deformation tensor and the corresponding flat linear connection adapted to the web which linearizes the 3-web.

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Invariant Metrizability and Projective Metrizability on Lie Groups and Homogeneous Spaces

2016

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In this paper we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left-invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given.2000 Mathematics Subject Classification: 53B05, 53B40, 70H03, 70H30, 70F17.

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Variational Principles For Second-Order Differential Equations

Cited by 44 publications

References 0 publications

Generalized Helmholtz Conditions for Non-Conservative Lagrangian Systems

Generalized Helmholtz Conditions for Non-Conservative Lagrangian Systems

On the problem of linearizability of a 3-web

Invariant Metrizability and Projective Metrizability on Lie Groups and Homogeneous Spaces

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