Georgia Institute of Technology College of Sciences

One of the major tools in the study of periodic solutions of Hamiltonian systems is the Maslov-type index theory for symplectic matrix paths. In this lecture, I shall give first a brief introduction on the Maslov-type index theory for symplectic matrix paths as well as the iteration theory of this index. As an application of these theories I shall give a brief survey about the existence, multiplicity and stability problems on periodic solution orbits of Hamiltonian systems with prescribed energy, especially those obtained in recent years. I shall also briefly explain some ideas in these studies, and propose some open problems.