Generalized Helmholtz conditions for non-conservative Lagrangian systems

TL;DR

This paper introduces generalized Helmholtz conditions using semi-basic 1-forms to determine when second order ODE systems with non-conservative forces are equivalent to Lagrangian systems, extending previous results.

Contribution

It provides a new geometric characterization of non-conservative Lagrangian systems through generalized Helmholtz conditions involving semi-basic 1-forms, applicable to dissipative and gyroscopic forces.

Findings

Conditions reduce to known cases for dissipative and gyroscopic forces.

Simplifications possible for homogeneous geometric structures.

Examples demonstrate integrability and construction of Lagrangians.

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.