Non-affine geometrization can lead to nonphysical instabilities

TL;DR

This paper critically examines geometrization methods for dynamical systems, demonstrating that the Eisenhart metric accurately reflects stability while the Jacobi metric can produce misleading chaos predictions.

Contribution

It shows that the Eisenhart metric correctly represents dynamical stability, whereas the Jacobi metric can lead to nonphysical chaos predictions, clarifying the appropriate geometrization approach.

Findings

Jacobi metric predicts chaos in harmonic oscillators

Eisenhart metric accurately reflects true dynamics

Jacobi metric's correspondence with actual dynamics is ill-defined

Abstract

Geometrization of dynamics consists of representing trajectories by geodesics on a configuration space with a suitably defined metric. Previously, efforts were made to show that the analysis of dynamical stability can also be carried out within geometrical frameworks, by measuring the broadening rate of a bundle of geodesics. Two known formalisms are via Jacobi and Eisenhart metrics. We find that this geometrical analysis measures the actual stability when the length of any geodesic is proportional to the corresponding time interval. We prove that the Jacobi metric is not always an appropriate parametrization by showing that it predicts chaotic behavior for a system of harmonic oscillators. Furthermore, we show, by explicit calculation, that the correspondence between dynamical- and geometrical-spread is ill-defined for the Jacobi metric. We find that the Eisenhart dynamics corresponds to the actual tangent dynamics and is therefore an appropriate geometrization scheme.