I'm starting an undergraduate research on Mathematical-Physics and the topic I chose is Hamiltonian mechanics (and its formalism, Symplectic Geometry). I'm looking for some interesting open problems so that I can pick one to study now and then maybe take it to a graduate level later on. I was interested in integral invariants, topology of manifolds and Hamilton-Jacobi theory when I studied Analytical Mechanics.
I would like some direction on where to look for these open problems in Hamiltonian Mechanics, that is, articles, pages or even keywords that could be used as useful material.
The books I'm currently using are:
- "Mathematical Methods of Classical Mechanics", V.I. Arnold
- "Global Formulations of Lagrangian and Hamiltonian Dynamics on Manifolds", Lee, Leok and McClamroch
- "Semi-Riemannian geometry with applications to relativity", Barrett O'Neill - "Gravitation", (you all know the authors of this one)