Research in Hamiltonian Systems

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Professor Xia proposes to continue his investigation of dynamics of Hamiltonian systems, Newtonian N-body problems and partially hyperbolic dynamical systems. Conservative mechanical systems, such as our planetary system, are examples of Hamiltonian systems. Hamiltonian systems typically exhibit some of the most complicated and chaotic behavior. These chaotic behaviors not only are important in applications, but also have interesting mathematical structures. The proposed research focuses on these chaotic structures, including distribution of periodic trajectories, homoclinic phenomenon and instabilities or Arnold diffusion in higher dimensional system.Professor Xia will also continue his study on partially hyperbolic systems by establishing connections between the geometric, topological and analytic properties of the invariant foliations.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 some elementary undergraduate textbooks. As he has learned, Professor Xia's work on escaping and capture trajectories in the three-body problem 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. In the last three years, Professor Xia has graduated five Ph.D students.