Symmetric Liapunov center theorem for orbit with nontrivial isotropy group

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 $U.$

17 pages