On the Jacobi metric for a general Lagrangian system

TL;DR

This paper derives an explicit series expansion for the Jacobi metric in general Lagrangian systems, linking geodesics to energy-dependent Finsler metrics and force fields, with implications for understanding system trajectories.

Contribution

It provides a novel explicit series expansion for the Jacobi metric in general Lagrangian systems and describes geodesics using non-linear connections and curvature.

Findings

Trajectories are approximated by Randers metrics at low kinetic energies.

Higher kinetic energies lead to geodesics of energy-dependent Finsler metrics.

Force fields generalize electromagnetic and gravitational interactions.

Abstract

An explicit expression for the Jacobi metric for a general Lagrangian system is obtained as a series expansion in the square root of the kinetic energy of the system and the corresponding geodesics are described in terms of an appropriate non-linear connection and the associated curvature. In the limit of low kinetic energies the trajectories of motion of any Lagrangian system are very well approximated by the geodesics of an energy dependent Randers metric or, equivalently, by the paths in configuration space of a representative point subject to electromagnetic- and gravitational-like force fields. For higher kinetic energy values the trajectories of motion are instead the geodesics of a general energy dependent Finsler metric, corresponding also to the paths in configuration space of a representative point subject to a hierarchy of a potentially infinite number of covariant force fields that generalise the electromagnetic and gravitational ones. Some general implications of these findings are discussed.