Hamiltonian chaos and differential geometry of configuration space-time

TL;DR

This paper explores Hamiltonian chaos through Riemannian geometry, using the Eisenhart metric to relate chaotic dynamics to curvature properties of configuration space-time, with applications to specific models like Hénon-Heiles and Heisenberg XY.

Contribution

It introduces a geometric framework for analyzing Hamiltonian chaos via the Eisenhart metric, filling a gap in the application of differential geometry to chaotic dynamics.

Findings

Chaos is primarily driven by curvature-induced parametric instability.

Comparison of Jacobi and Eisenhart metrics reveals insights into geometric effects on chaos.

Applications to Hénon-Heiles and Heisenberg XY models demonstrate the framework's effectiveness.

Abstract

This paper tackles Hamiltonian chaos by means of elementary tools of Riemannian geometry. More precisely, a Hamiltonian flow is identified with a geodesic flow on configuration space-time endowed with a suitable metric due to Eisenhart. Until now, this framework has never been given attention to describe chaotic dynamics. A gap that is filled in the present work. In a Riemannian-geometric context, the stability/instability of the dynamics depends on the curvature properties of the ambient manifold and is investigated by means of the Jacobi--Levi-Civita (JLC) equation for geodesic spread. It is confirmed that the dominant mechanism at the ground of chaotic dynamics is parametric instability due to curvature variations along the geodesics. A comparison is reported of the outcomes of the JLC equation written also for the Jacobi metric on configuration space and for another metric due to Eisenhart on an extended configuration space-time. This has been applied to the H\'enon-Heiles model, a two-degrees of freedom system. Then the study has been extended to the 1D classical Heisenberg XY model at a large number of degrees of freedom. Both the advantages and drawbacks of this geometrization of Hamiltonian dynamics are discussed. Finally, a quick hint is put forward concerning the possible extension of the differential-geometric investigation of chaos in generic dynamical systems, including dissipative ones, by resorting to Finsler manifolds.