Nonlinear Dynamics with Applications to Physical Systems

非线性动力学及其在物理系统中的应用

• 批准号: 1009130
• 负责人: Mark Levi
• 金额: $ 29.99万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1009130&HistoricalAwards=false
• 关键词: Nonlinear; Dynamics; Applications; Physical; Systems

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.

该项目涉及应用于物理系统或由此类应用激发的三个数学领域。 其中第一个领域是一个新发现的提议探索 （一维）薛定谔方程和“自行车轨道”几何之间的联系。“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，以及更多的解释这个方程的性质。一个完全不同的几何学研究对象--所谓的“自行车道”--已经被研究了世纪，与希尔方程没有明显的联系。PI最近注意到这两个受试者密切相关。 拟议的研究的一部分是利用这种关系， 对这两个主题都有新的见解。 建议研究的第二个领域涉及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”。“这一现象的基本几何学直到最近才被理解。PI建议将他早期的工作扩展到更广泛的物理环境，并进一步探索微分几何，力学和平均理论之间的联系。 这项工作预计将显示如何从微分几何（曲率，正常家庭）的概念找到他们的表现在力学。 在项目的第三个领域，研究人员的目标是双重的：第一，开发变分技术的时间依赖的哈密顿系统，第二，揭示了新的光的问题阿诺德扩散-不稳定的问题，在哈密顿动力学-在具体的例子，由物理或几何。该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的新发现的联系，直到最近似乎无关：一方面是薛定谔方程，另一方面是“自行车轨道”的几何形状。 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。研究者最近注意到这两个受试者密切相关。这种联系为每个领域开辟了令人兴奋的前景，为其他领域提供了新的见解。其中一种可能性是水波运动与轨道变形之间的惊人联系。 第二个研究方向是研究承受快速振动的机械系统。快速振动在粒子加速器、用于低温实验的粒子阱和激光“镊子”中发现了意想不到的实际用途，后者使生物学家能够以非侵入性的方式用光操纵细胞的一部分。这一领域的一个关键现象的要点是在调查人员的早期工作中发现的。本项目旨在将这项工作扩展到更广泛的应用，包括受到快速振动的流体和气体，其应用包括冷却，从气体中去除杂质，影响湍流等。 本计画将应用微分几何与微分方程式的工具，以更好的了解上述的物理现象。在项目的第三个领域，研究人员的目标是双重的：首先，开发分析时间相关哈密顿系统的变分技术，其次，揭示阿诺德扩散问题的新观点-动力学中的不稳定问题（例如，卫星的运动）--在具体的例子中是由物理学和几何学驱动的。