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

哈密顿系统中周期轨道的研究在数学和物理上都有很大的兴趣，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，确定一个给定的初始条件导致一个周期轨道的“概率”。在许多经典系统中，例如台球，人们猜想轨道是周期的概率为零。令人惊讶的是，这个经典力学问题还与相应的Helmholtz问题中本征值的渐近分布有关。周期轨道集的结构在量子混沌的研究和微腔物理中也起着重要作用。例如，一些特殊的周期轨道集支持已用于微激光的准模。我们期望这项研究的结果有助于阐明对哈密顿系统中周期轨道结构的一般理解。这项拟议的研究将有助于物理学家研究光谱问题和微激光的物理学，这些问题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.