# Hamiltonian system - qual question

1. Aug 1, 2011

### triamine

This was on a previous qualifying exam. Let H(x,y) be C^2. Assume that $$\lim_{x^2+y^2}{|H(x,y)|} = \infty$$ and that the system $$\dot{x} = H_y, \quad \dot{y} = -H_x$$ has only finitely many critical points. Prove that it has at least one Lyapunov stable critical point.

Now, what I know is that for a planar Hamiltonian system, all of the critical points must be saddles or centers (look at the trace of the linearization of the system). Thus, the problem is equivalent to saying that a planar Hamiltonian system must have at least one center point (as saddles are not Lyapunov stable). I'm pretty much stuck after that though. Any ideas? Thanks in advance.