Nonlinear Dynamics with Applications to Physical Systems

该项目涉及应用于物理系统或由这些应用驱动的数学的三个领域。这些领域中的第一个是对（一维）薛定谔方程和“自行车轨道”几何之间新发现的联系的探索。薛定谔方程是数学、物理和工程许多领域的基本模型。许多物理现象，如氢原子的光谱，粒子加速器的工作，等等，都可以用这个方程的性质来解释。一个完全不同的几何研究对象——所谓的“自行车轨迹”——已经被研究了一个多世纪，与希尔方程没有明显的联系。PI最近注意到，这两个主题密切相关。拟议研究的一部分是利用这种关系来获得对这两个主题的新见解。拟议研究的第二个领域涉及具有快速施加振动的系统。这种振动被用于粒子加速器、低温实验的粒子阱和激光“镊子”。这种现象背后的几何原理直到最近才被理解。PI建议将他早期的工作扩展到更广泛的物理背景下，并在这个问题的背景下进一步探索微分几何、力学和平均理论之间的联系。这项工作有望展示微分几何（曲率、法族）的概念如何在力学中得到体现。在项目的第三个领域，研究人员的目标是双重的：首先，开发时间依赖的哈密顿系统的变分技术，其次，在由物理或几何驱动的特定示例中，为阿诺德扩散问题（哈密顿动力学中的不稳定性问题）提供新的线索。这个项目涉及物理应用中出现的三个数学领域。该项目的第一部分是探索两个领域之间新发现的联系，这两个领域直到最近看起来都不相关：一方面是薛定谔方程，另一方面是“自行车轨道”的几何。前者描述了氢原子的光谱、粒子加速器的工作、机械振动等等。这位研究者最近注意到，这两个科目是密切相关的。这种联系打开了每个领域令人兴奋的前景，给对方带来了新的见解。其中一种可能性是，水波的运动与轨道的变形之间存在着惊人的联系。提出的第二个研究方向是研究受快速振动影响的机械系统。快速振动在粒子加速器、用于低温实验的粒子陷阱和激光“镊子”中发现了意想不到的实际应用。激光“镊子”使生物学家能够以一种非侵入性的方式通过光操纵细胞的某些部分。这个领域的一个关键现象的要点是在研究者早期的工作中发现的。目前的项目旨在将这项工作扩展到更广泛的应用，包括经受快速振动的流体和气体，应用包括冷却、去除气体中的杂质、影响湍流等等。本项目将运用微分几何和微分方程的工具来更好地理解上述物理现象。在项目的第三个领域，研究人员的目标是双重的：首先，开发用于分析时间相关哈密顿系统的变分技术，其次，在由物理和几何驱动的特定示例中，为阿诺德扩散问题（动力学中的不稳定问题，例如卫星运动）提供新的线索。

This project deals with three areas of mathematics applied to physical systems or motivated by such applications. The first of these areas is a proposed exploration of a newly discovered connection between the (one-dimensional) Schroedinger's equation and the geometry of "bicycle tracks." Schroedinger's equation is a fundamental model in many areas of mathematics, physics and engineering. Many physical phenomena, such as the spectrum of the hydrogen atom, the working of particle accelerators, and many more are explained by the properties of this equation. A completely different object of geometrical study--the so-called "bicycle tracks"--has been studied for over a century, with no apparent connection to Hill's equations. It was recently noticed by the PI that the two subjects are closely related. Part of proposed research is to exploit this relationship to gain new insights into both subjects. The second area of proposed research deals with systems with rapid imposed vibrations. Such vibrations are used in particle accelerators, in particle traps for low-temperature experiments and in laser "tweezers." Underlying geometry of the phenomenon was understood only recently. The PI proposes to extend his earlier work to broader physical contexts, and to further explore the connection between differential geometry, mechanics and averaging theory in the context of this problem. This work is expected to show how concepts from differential geometry (curvature, normal family) find their manifestations in mechanics. In the third area of project, the researcher's goal is two-fold: first, to develop variational techniques for time--dependent Hamiltonian systems, and second, to shed new light on the problem of Arnold diffusion--an instability problem in Hamiltonian dynamics--in specific examples motivated by physics or geometry. This project deals with three areas of mathematics arising in physical applications. The first part of the project is an exploration of a newly discovered connection between two fields which until recently seemed unrelated: Schroedinger's equation on the one hand, and the geometry of "bicycle tracks" on the other. The former describes the spectrum of the hydrogen atom, the working of particle accelerators, mechanical vibrations, and more. It was recently noticed by this investigator that the two subjects are closely related. This connection opens exciting prospects of each area giving new insights into the other. One such possibility is a striking connection between the motion of water waves on the one hand, and the deformation of tracks on the other. The second direction of proposed research addresses study of mechanical systems subjected to rapid vibrations. Rapid vibrations have found unexpected practical use in particle accelerators, in particle traps used for low-temperature experiments and in laser "tweezers," the latter enabling biologists to manipulate parts of a cell by light, in an non--invasive way. The gist of a key phenomenon in this area was discovered in the investigator's earlier work. The present project aims to extend this work to wider applications, including fluids and gases subjected to rapid vibration, with applications including cooling, removing impurities from gases, affecting turbulence, and more. This project will apply tools of differential geometry and differential equations to better understand physical phenomena mentioned above. In the third area of project, the researcher's goal is two-fold: first, to develop variational techniques for analyzing time-dependent Hamiltonian systems, and second, to shed new light on the problem of Arnold diffusion--an instability problem in dynamics (e.g., of motion of satellites)--in specific examples motivated by physics and geometry.