Symmetric Liapunov center theorem for minimal orbit

TL;DR

This paper proves the existence of non-stationary periodic solutions in symmetric systems near minimal orbits, demonstrating new periodic orbits in classical physics problems like Lennard-Jones and Schwarzschild three-body systems.

Contribution

It introduces a symmetric Liapunov center theorem for minimal orbits, expanding bifurcation theory to symmetric dynamical systems.

Findings

Existence of new periodic orbits near minimal orbits

Application to Lennard-Jones two- and three-body problems

Application to Schwarzschild three-body problem

Abstract

Using the techniques of equivariant bifurcation theory we prove the existence of non-stationary periodic solutions of $\Gamma$-symmetric systems $\ddot q(t)=-\nabla U(q(t))$ in any neighborhood of an isolated orbit of minima $\Gamma(q_0)$ of the potential $U$. We show the strength of our result by proving the existence of new families of periodic orbits in the Lennard-Jones two- and three-body problems and in the Schwarzschild three-body problem.