Research In Hamiltonian Dynamics

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Professor Xia proposes to continue his investigation of dynamics ofHamiltonian systems. One of the most important problems in Hamiltoniandynamical systems is whether orbits in typical systems are stable. Theproposed research will address these problems, with research topics such asAubry-Mather theory, Arnold diffusions, chaotic behaviors and Newtonian$n$-body problem. The ultimate goal is to show that typical near integrableHamiltonians in higher dimensions are topologically unstable.The proposed research concerns the stability problems in Hamiltoniandyanmics. Hamiltonian dynamical systems model many systems arisingfrom classical mechanics, celestial mechanics and physics. Theseproblems has a long history, going back to Poincare and Birkhoff. Atypical question one often asks is the following: Is our solar systemstable? One of our goals is to find an answer to this and a largeclass of related questions. With recent progresses in the theory ofmodern dynamical systems, we understand much better the chaotic natureof typical systems. However, the stability problem remains open exceptin some special cases and it is one of the most important areas ofstudy, from both theoretical point of view and wide applications itfinds in various physical systems.