Research In Hamiltonian Dynamics

Northwestern University, Evanston IL

Investigators

Abstract

Professor Xia proposes to continue his investigation of dynamics of Hamiltonian systems. One of the most important problems in Hamiltonian dynamical systems is whether orbits in typical systems are stable. The proposed research will address these problems, with research topics such as Aubry-Mather theory, Arnold diffusions, chaotic behaviors and Newtonian $n$-body problem. The ultimate goal is to show that typical near integrable Hamiltonians in higher dimensions are topologically unstable. The proposed research concerns the stability problems in Hamiltonian dyanmics. Hamiltonian dynamical systems model many systems arising from classical mechanics, celestial mechanics and physics. These problems has a long history, going back to Poincare and Birkhoff. A typical question one often asks is the following: Is our solar system stable? One of our goals is to find an answer to this and a large class of related questions. With recent progresses in the theory of modern dynamical systems, we understand much better the chaotic nature of typical systems. However, the stability problem remains open except in some special cases and it is one of the most important areas of study, from both theoretical point of view and wide applications it finds in various physical systems.