# Periodic solutions of symmetric Hamiltonian systems

## TL;DR

This paper investigates the existence of periodic solutions in symmetric Hamiltonian systems, especially near non-isolated critical points, by establishing a Lyapunov-type theorem for systems with symmetry.

## Contribution

It introduces a Lyapunov-type theorem specifically for symmetric Hamiltonian systems, addressing solutions near non-isolated critical points formed by group orbits.

## Key findings

- Proves a Lyapunov-type theorem for symmetric Hamiltonian systems.
- Establishes conditions for the existence of periodic solutions near critical orbits.
- Analyzes the role of symmetry and group actions in the dynamics of Hamiltonian systems.

## Abstract

This paper is devoted to the study of periodic solutions of Hamiltonian system $\dot z(t)=J \nabla H(z(t))$, where $H$ is symmetric under an action of a compact Lie group. We are looking for periodic solutions in a nearby of non-isolated critical points of $H$ which form orbits of the group action. We prove Lyapunov-type theorem for symmetric Hamiltonian systems.