# Homework Help: Periodic orbit of Hamiltonian dynamical system in R^2 is stable

1. Jun 11, 2010

### MatthijsV

1. The problem statement, all variables and given/known data
We are given a Hamiltonian dynamical system with a smooth Hamiltonian $$H:\mathbb{R}^2\to\mathbb{R}$$ on $$\mathbb{R}^2$$, with canonical symplectic structure. Suppose this Hamiltonian has a periodic orbit $$H^{-1}(h_0)$$. Prove that there exists an $$\epsilon>0$$ such that for all $$h\in ]h_0-\epsilon,h_0+\epsilon[$$: $$H^{-1}(h)$$ is also a periodic orbit.

2. Relevant equations
Liouville's theorem tells us that the flow of a Hamiltonian vector field preserves the Liouville volume form in $$\mathbb{R}^2$$.

3. The attempt at a solution
It is enough to prove that $$H^{-1}(h)$$ is bounded. It then follows from Liouville's theorem that it is also a periodic orbit. However, I have no idea how to do this.