On The Geometrical Description of Dynamical Stability II

TL;DR

This paper examines the geometrization of dynamical stability using Eisenhart and Jacobi metrics, revealing that the Jacobi metric's instability measures are non-physical unless kinetic energy is constant, especially in large ergodic systems.

Contribution

It clarifies the conditions under which the Jacobi metric provides meaningful stability measures, highlighting its limitations and equivalence to Eisenhart metric in large ergodic systems.

Findings

Jacobi metric yields positive Lyapunov exponents due to non-affine parametrization.

Non-physical instabilities are linked to fluctuations in kinetic energy.

Jacobi and Eisenhart metrics are equivalent for large ergodic systems at equilibrium.

Abstract

Geometrization of dynamics using (non)-affine parametization of arc length with time is investigated. The two archetypes of such parametrizations, the Eisenhart and the Jacobi metrics, are applied to a system of linear harmonic oscillators. Application of the Jacobi metric results in positive values of geometrical lyapunov exponent. The non-physical instabilities are shown to be due to a non-affine parametrization. In addition the degree of instability is a monotonically increasing function of the fluctuations in the kinetic energy. We argue that the Jacobi metric gives equivalent results as Eisenhart metric for ergodic systems at equilibrium, where number of degrees of freedom $N\to\infty$. We conclude that, in addition to being computationally more expensive, geometrization using the Jacobi metric is meaningful only when the kinetic energy of the system is a positive constant.