Collaborative Research: Exterior Differential System Approach to Periodic Orbits in Hamiltonian Systems

合作研究：哈密顿系统中周期轨道的外微分系统方法

• 批准号: 0505468
• 负责人: Joseph Landsberg
• 金额: $ 6.2万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-15 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505468&HistoricalAwards=false
• 关键词: Collaborative; Research; Exterior; Differential; System

The study of periodic orbits in Hamiltonian systems is of great mathematical and physical interest and has been the subject of much effort over the past century. In particular, an important basic problem is the determination of how many periodic orbits a dynamical system has, or, equivalently, the "probability" that a given initial condition leads to a periodic orbit. In many classical systems, such as billiards, it is conjectured that the probability that an orbit is periodic is zero. Surprisingly, this classical mechanics problem is also related to the asymptotic distribution of eigenvalues in the corresponding Helmholtz problem. The structure of the set of periodic orbits also plays important role in the study of quantum chaos and in physics of microcavities. For example, some special sets of periodic orbits support quasimodes, which have been used in microlasers.We expect the results of this research to help elucidate the general understanding of the structure of periodic orbits in Hamiltonian systems. The proposed research will help physicists working on spectral problems and on the physics of microlasers, which have important applications in materials science and medicine. Graduate students will participate in the research project and broaden their education through interactions with industrial researchers.

对哈密顿系统中周期轨道的研究具有重大的数学和物理意义，并且是过去一个世纪中人们付出巨大努力的课题。 特别是，一个重要的基本问题是确定一个动力系统有多少个周期轨道，或者等效地，确定给定初始条件导致周期轨道的“概率”。 在许多经典系统中，例如台球，人们推测轨道具有周期性的概率为零。 令人惊讶的是，这个经典力学问题也与相应的亥姆霍兹问题中特征值的渐近分布有关。 周期轨道组的结构在量子混沌和微腔物理学的研究中也发挥着重要作用。 例如，一些特殊的周期轨道集支持准模，这些准模已在微型激光器中使用。我们期望这项研究的结果有助于阐明哈密顿系统中周期轨道结构的一般理解。 拟议的研究将帮助物理学家研究光谱问题和微型激光器物理学，这些问题在材料科学和医学中具有重要应用。 研究生将参与研究项目，并通过与工业研究人员的互动来扩大他们的教育。