Research in Hamiltonian Systems

夏教授打算继续研究哈密顿系统、牛顿n体问题和部分双曲动力系统的动力学。保守的机械系统，比如我们的行星系统，就是哈密顿系统的例子。哈密顿系统通常表现出一些最复杂和混沌的行为。这些混沌行为不仅具有重要的应用价值，而且具有有趣的数学结构。本文提出的研究重点是这些混沌结构，包括周期轨迹分布、同斜现象和高维系统中的不稳定性或阿诺德扩散。夏教授还将继续研究部分双曲系统，建立不变叶化的几何、拓扑和解析性质之间的联系。本课题研究的问题在经典力学、天体力学和工程领域具有悠久的历史和广泛的应用。夏教授在哈密顿体系和天体力学方面的研究被许多科普杂志、书籍甚至一些初级本科教科书所介绍。据他了解，夏教授在三体问题中关于逃逸和捕获轨迹的工作可能在NASA轨道设计中有潜在的应用。夏教授将继续积极参与研究生和本科生的培训工作。在过去的三年中，夏教授培养了五名博士研究生。

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.