Relativistic chaos in the anisotropic harmonic oscillator

TL;DR

This paper demonstrates that relativistic two-dimensional harmonic oscillators, especially anisotropic ones, exhibit chaotic behavior due to momentum coupling, revealing chaos as a common feature in relativistic corrections to classical systems.

Contribution

It provides the first analysis of chaos in relativistic, conservative, two-dimensional harmonic oscillators, highlighting the role of momentum coupling in non-integrability.

Findings

Relativistic anisotropic harmonic oscillator is chaotic.

Chaos arises from momentum coupling in the Hamiltonian.

Relativistic corrections induce chaos in classical integrable systems.

Abstract

The harmonic oscillator is an essential tool, widely used in all branches of Physics in order to understand more realistic systems, from classical to quantum and relativistic regimes. We know that the harmonic oscillator is integrable in Newtonian mechanics, whether forced, damped or multidimensional. On the other hand, it is known that relativistic, one-dimensional driven oscillators present chaotic behavior. However, there is no analogous result in the literature concerning relativistic conservative, two-dimensional oscillators. We consider in this paper different separable potentials for two-dimensional oscillators in the context of special relativistic dynamics. We show, by means of different chaos indicators, that all these systems present chaotic behavior under specific initial conditions. In particular, the relativistic anisotropic, two-dimensional harmonic oscillator is chaotic. The non-integrability of the system is shown to come from the momentum coupling in the kinetic part of the Hamiltonian, even if there are no coupling terms in the potential. It follows that chaos must appear in most integrable classical systems once we introduce relativistic corrections to the dynamics. The chaotic nature of the relativistic, anisotropic harmonic oscillator may be detected in the laboratory using Bose condensates in a two-dimensional optical lattice, extending the recent experiments on the one-dimensional case.