Destructive relativity

TL;DR

This paper formulates necessary conditions for the integrability of relativistic Hamiltonian systems with homogeneous potentials, revealing that relativity generally destroys integrability in such systems.

Contribution

It introduces the first necessary integrability conditions for relativistic Hamiltonian systems using differential Galois theory, showing relativity's impact on system integrability.

Findings

Relativity almost completely destroys integrability in systems with two degrees of freedom.

The integrability conditions are simple and effective for practical applications.

Most relativistic versions of integrable systems are non-integrable.

Abstract

Relativistic Hamiltonian equations describing a motion of a point mass in an arbitrary homogeneous potential are considered. For the first time, the necessary integrability conditions for integrability in the Liouville sense for this class of systems are formulated. These conditions are obtained by means of an analysis of the differential Galois groups of variational equations. They are simple and effective in applications. For instance, an application of the necessary integrability conditions for systems with two degrees of freedom shows that relativity almost completely destroys integrability, that is, in almost all cases relativistic versions of integrable systems are not integrable. The paper has been already published in ,,Chaos: An Interdisciplinary Journal of Nonlinear Science'', and the final journal version is available under the link: https://doi.org/10.1063/5.0140633