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

在哈密顿系统中对周期轨道的研究具有极大的数学和物理意义，并且在过去的一个世纪里一直是人们努力的主题。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，等价地，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，人们推测轨道是周期的概率为零。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的特征值的渐近分布有关。周期轨道集的结构在量子混沌和微腔物理的研究中也起着重要的作用。例如，一些特殊的周期轨道支持准模式，这些准模式已用于微激光器。我们期望这项研究的结果有助于阐明对哈密顿系统周期轨道结构的一般理解。这项提议的研究将帮助物理学家研究光谱问题和微激光的物理学，这在材料科学和医学中有重要的应用。研究生将参与研究项目，并通过与工业研究人员的互动扩大他们的教育。

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.