Research in Hamiltonian Dynamics

保守力学系统，如我们的行星系统，是哈密顿系统的例子。测地线流是哈密尔顿系统的另一个例子。Hamilton动力系统中最重要的问题之一是典型系统中的轨道是否稳定，事实上，行星系统的稳定性一直是数学研究的重要课题。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相关的各种问题，研究主题包括Aubry-Mather理论，Arnold扩散，混沌行为和牛顿n体问题。本文的最终目标是证明高维近似可积Hamilton算子是拓扑不稳定的，所提出的研究内容涉及经典力学、天体力学以及工程领域中具有悠久历史和广泛应用的问题.夏教授在哈密顿系统和天体力学方面的研究成果已被许多科普杂志、书籍甚至一些初级本科教材收录。本文工作的另一个方面是研究三体问题中的逃逸和捕获轨道，这在NASA轨道设计中可能有潜在的应用。夏教授将继续积极参与研究生和本科生的培养工作。

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.