# Coherent Riemannian-geometric description of Hamiltonian order and chaos with Jacobi metric

## TL;DR

This paper defends the use of the Jacobi metric in geometrizing Hamiltonian dynamics, showing it can reliably describe chaos and stability without inconsistencies, countering recent criticisms.

## Contribution

It demonstrates that the Jacobi metric provides a consistent geometric framework for analyzing Hamiltonian chaos, clarifying misconceptions about its limitations.

## Key findings

- Jacobi metric accurately describes chaos in Hamiltonian systems.
- Observed instabilities with Jacobi metric are artefacts, not fundamental flaws.
- Supports geometric approach as a valid method for stability analysis.

## Abstract

By identifying Hamiltonian flows with geodesic flows of suitably chosen Riemannian manifolds, it is possible to explain the origin of chaos in classical Newtonian dynamics and to quantify its strength. There are several possibilities to geometrize Newtonian dynamics under the action of conservative potentials and the hitherto investigated ones provide consistent results. However, it has been recently argued that endowing configuration space with the Jacobi metric is inappropriate to consistently describe the stability/instability properties of Newtonian dynamics because of the non-affine parametrization of the arc length with physical time. To the contrary, in the present paper, it is shown that there is no such inconsistency and that the observed instabilities in the case of integrable systems using the Jacobi metric are artefacts.