Jacobi-Maupertuis-Eisenhart metric and geodesic flows

TL;DR

This paper explores the Jacobi-Maupertuis-Eisenhart metric, demonstrating its reduction to standard forms in non-relativistic limits, deriving it for stationary metrics, and extending it to time-dependent metrics via the Eisenhart-Duval lift.

Contribution

It introduces a unified formulation of the Jacobi-Maupertuis metric for stationary and time-dependent metrics using the Eisenhart-Duval lift, expanding its applicability.

Findings

Reduction of Jacobi metric to standard form in non-relativistic limit

Derivation of Jacobi metric for stationary metrics

Formulation of Jacobi-Eisenhart metric for time-dependent metrics

Abstract

The Jacobi metric derived from the line element by one of the authors is shown to reduce to the standard formulation in the non-relativistic approximation. We obtain the Jacobi metric for various stationary metrics. Finally, the Jacobi-Maupertuis metric is formulated for time-dependent metrics by including the Eisenhart-Duval lift, known as the Jacobi-Eisenhart metric.