Periodic Solutions of Symmetric Hamiltonian Systems

This paper is devoted to the study of periodic solutions of a Hamiltonian system z˙(t)=J∇H(z(t))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{z}(t)=J \nabla H(z(t))$$\end{document}, where H is symmetric under an action of a compact Lie group. We are looking for periodic solutions in a neighborhood of non-isolated critical points of H which form orbits of the group action. We prove a Lyapunov-type theorem for symmetric Hamiltonian systems.