Conservative mechanical systems, such as our planetary system, are examples of Hamiltonian systems. Geodesic flows are other examples of Hamiltonian systems. One of the most important problems in Hamiltonian dynamical systems is whether orbits in typical systems are stable. In fact, the stability of our planetary system has long been an important subject of mathematical research. Hamiltonian systems usually exhibit some extremely complicated and chaotic behavior. The proposed research will address various problems related to stability and chaos, with research topics such as the Aubry-Mather theory, Arnold diffusions, chaotic behaviors and the Newtonian n-body problem. The ultimate goal is to show that typical near integrable Hamiltonians in higher dimensions are topologically unstable. The proposed research addresses problems with a long history and wide applications to classical mechanics, celestial mechanics, as well as areas of engineering. Professor Xia's research in Hamiltonian systems and celestial mechanics has been featured in many popular science magazines, books and even in some elementary undergraduate textbooks. Another aspect of this work, is the study of escaping and capture trajectories in the three-body problem, which may have potential applications in NASA orbit design. Professor Xia will continue to be actively involved in student training at both graduate and undergraduate level.
