Hamiltonian system - qual question

[triamine]

This was on a previous qualifying exam. Let H(x,y) be C^2. Assume that [tex]\lim_{x^2+y^2}{|H(x,y)|} = \infty[/tex] and that the system [tex]\dot{x} = H_y, \quad \dot{y} = -H_x[/tex] 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.

 

[lippyka]

Here is one way to approach the problem: First, note that the given limit implies that H(x,y) must be unbounded as x^2+y^2 tends to infinity. Thus, there must be at least one critical point on the boundary of the region for which the limit is taken. Now, consider a Lyapunov function V(x,y) for the system, such that V(x,y) is bounded and tends to infinity as x^2+y^2 tends to infinity. This implies that all critical points of the system must be in the region for which V(x,y) is bounded. Next, note that since there are only finitely many critical points, there must be at least one critical point in the interior of the region for which V(x,y) is bounded. Since this point lies in the interior of the region, it follows that it must be a center point (as any saddle point would lie on the boundary of the region). Thus, the system must have at least one Lyapunov stable critical point.