Research in Hamiltonian Systems

哈密​​顿系统研究

• 批准号: 0701004
• 负责人: Zhihong Xia
• 金额: $ 17.39万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701004&HistoricalAwards=false
• 关键词: Research; Hamiltonian; Systems

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.

夏教授提议继续研究汉密尔顿系统动力学、牛顿N体问题和部分双曲动力学系统。保守力学系统，例如我们的行星系统，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乱的行为。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。夏教授将继续研究部分双曲系统的混沌结构，包括高维系统中的周期轨迹分布、同宿现象、不稳定性或Arnold扩散等。拓扑和分析性质的不变叶理。拟议的研究解决的问题与历史悠久，广泛应用于经典力学、天体力学以及工程学领域。夏教授在哈密尔顿系统和天体力学方面的研究成果被许多科普杂志、书籍甚至一些初级本科教材收录。据了解，夏教授在三体问题中逃逸和捕获轨道方面的工作可能在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养工作。在过去的三年里，夏教授已经毕业了五名博士生。