Ostrogradsky's instability theorem says that under some conditions, a system governed by a Lagrangian which depends on time derivatives beyond the first is "unstable". In the proof, one computes the Hamiltonian and shows that the dependence on one of the canonical momenta is linear.
I don't understand either the statement or the proof. I think this is because I don't remember my Hamiltonian mechanics very well.
What is meant by saying that the system is unstable? There are several notions of stability out there. Is the theorem saying the for any choice of initial conditions, the system is, say, Lyapunov unstable?
What does the linearity of the Hamiltonian have to do with stability?
My sense is that Ostrogradsky's theorem is sometimes taken to justify the idea that Lagrangians in physics shouldn't depend on higher time derivatives. Why is this? What's wrong with studying systems which are unstable? If I understand what "unstable" means, isn't it just another word for "chaotic"? And certainly there are chaotic systems in the real world...