Research in Hamiltonian Dynamics

摘要DMS-0401334PI：ZJ夏保守的力学系统，如我们的行星系统，都是哈密顿系统的例子。测地线流是哈密顿系统的另一个例子。哈密顿动力系统中最重要的问题之一是典型系统的轨道是否稳定。事实上，我们行星系统的稳定性一直是数学研究的一个重要课题。哈密顿系统通常表现出一些极其复杂和混沌的行为。拟议的研究将解决与稳定性和混沌有关的各种问题，研究主题包括Aubry-Mather理论、Arnold扩散、混沌行为和牛顿n体问题。最终目的是证明高维典型的近可积哈密顿量在拓扑上是不稳定的。所提出的研究解决了历史悠久的问题，并广泛应用于经典力学、天体力学以及工程领域。夏教授在哈密顿系统和天体力学方面的研究出现在许多科普杂志、书籍上，甚至出现在一些小学本科生教科书中。这项工作的另一个方面是研究三体问题中的逃逸和捕获轨迹，这可能在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。

Abstract DMS-0401334PI: ZJ XiaConservative mechanical systems, such as our planetary system, are examples of Hamiltonian systems. Geodesic flows are other examplesof Hamiltonian systems. One of the most important problems inHamiltonian dynamical systems is whether orbits in typical systems are stable.In fact, the stability of our planetary system has long been an importantsubject of mathematical research. Hamiltonian systems usually exhibit someextremely complicated and chaotic behavior. The proposed research willaddress various problems related to stability and chaos, with research topicssuch as the Aubry-Mather theory, Arnold diffusions, chaotic behaviors and theNewtonian n-body problem. The ultimate goal is to show that typical nearintegrable Hamiltonians in higher dimensions are topologically unstable.The proposed research addresses problems with a long history and wideapplications to classical mechanics, celestial mechanics, as well as areas ofengineering. Professor Xia's research in Hamiltonian systems and celestialmechanics has been featured in many popular science magazines, books andeven in some elementary undergraduate textbooks. Another aspect of this work, is the studyof 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.