Generalized Helmholtz conditions for non-conservative Lagrangian systems

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.

final version, for more details of some proofs please see the old version

References in corpus (6)

• Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations
• On the generalized Helmholtz conditions for Lagrangian systems with dissipative forces
• Projective Metrizability and Formal Integrability
• Projective and Finsler metrizability: parameterization-rigidity of the geodesics
• Geometric properties of Lagrangian mechanical systems
• Formal Integrability for the Nonautonomous Case of the Inverse Problem of the Calculus of Variations

Cited by in corpus (5)

• The inverse problem of the calculus of variations and the stabilization of controlled Lagrangian systems
• A class of Finsler metrics admitting first integrals
• Finsler metrizabilities and geodesic invariance
• First integrals for Finsler metrics with vanishing $\chi$-curvature
• Alternative Lagrangians obtained by scalar deformations