Symmetric Liapunov center theorem for orbit with nontrivial isotropy group

TL;DR

This paper extends the Liapunov center theorem to symmetric systems with Lie group actions, establishing conditions for non-stationary periodic solutions near symmetric critical points.

Contribution

It introduces two versions of the Liapunov center theorem tailored for systems with symmetry groups, broadening the scope of periodic solution existence results.

Findings

Proves existence of non-stationary periodic solutions near symmetric critical orbits.

Develops symmetric Liapunov center theorems applicable to Lie group actions.

Provides mathematical conditions for periodic solutions in symmetric dynamical systems.

Abstract

In this article we prove two versions of the Liapunov center theorem for symmetric potentials. We consider a~second order autonomous system $\ddot q(t)=-\nabla U(q(t))$ in the presence of symmetries of a compact Lie group $\Gamma$ acting linearly on $\mathbb{R}^n.$ We look for non-stationary periodic solutions of this system in a~neighborhood of an orbit of critical points of the potential $U.$