On the Jacobi-Metric Stability Criterion

TL;DR

This paper explores the relationship between the stability equations of mechanical systems and geodesic deviation in the Jacobi metric, revealing that the dynamical and geometrical methods are not equivalent.

Contribution

It clarifies the exact relation between stability equations and geodesic deviation, highlighting differences between dynamical and geometrical approaches.

Findings

Dynamical and geometrical stability approaches are not equivalent.

The stability equation for mechanical solutions relates differently to geodesic deviation.

The Jacobi metric approach has limitations in stability analysis.

Abstract

We investigate the exact relation existing between the stability equation for the solutions of a mechanical system and the geodesic deviation equation of the associated geodesic problem in the Jacobi metric constructed via the Maupertuis-Jacobi Principle. We conclude that the dynamical and geometrical approaches to the stability/instability problem are not equivalent.