Non-integrability of a system with the Dyson Potential

TL;DR

This paper proves that a Hamiltonian system with Dyson potential is analytically and formally non-integrable, using advanced mathematical theories to analyze the properties of its solutions and period functions.

Contribution

It demonstrates the non-integrability of the Dyson potential system through formal and analytical methods, extending the understanding of its dynamical complexity.

Findings

The system has no additional analytic first integrals.

The period function is infinitely branched around equilibrium.

Formal non-integrability is established via Ziglin-Moralez-Ruiz-Ramis theory.

Abstract

In this paper it is shown that the Hamiltonian system with Dyson potential is analytical non-integrable and formal non-integrable. The approach is based on the following: when a system has a family of periodic solutions around an equilibrium and if the period function is infinitely branched then the system has non additional analytic first integral. We prove formal non-integrability using Ziglin-Moralez-Ruiz-Ramis theory.