Nonlinear dynamics with applications to physical systems

该项目将开发数学工具，目的是在三个不同但相关的领域理解一些重要的物理现象。第一个领域是研究受高频、高强度振动影响的机械系统。用于操纵生物细胞内物体的激光镊子就属于这类系统，保罗陷阱-？离子阱的发明使保罗获得了1989年的诺贝尔物理学奖。该项目旨在对这些现象和相关现象建立一种新的、更直观的几何理解。除了以一种更简单的新方式解释物理之外，这项研究还将在不同的数学领域之间建立新的联系。项目的第二部分涉及希尔方程，这是一个在数学、物理和工程许多领域具有重要基础意义的系统。这部分项目的主要目的是解释这个系统的一个基本和优雅的特征，这个特征已经被数值观察到，但其原因仍然不清楚。该项目的第三部分涉及一个基本问题，即理解潮汐耗散对天体运动的长期影响。提出的研究由理解物理环境中产生的动力系统的共同目标统一的三个领域组成。第一个领域是研究受高频、高强度振动影响的机械系统。这项研究将发展对相关现象的新的几何理解（可能对应用有益），并将平均理论与微分几何（作为两个数学领域之间的桥梁）联系起来。该项目的第二部分将探索与希尔方程相关的分岔图的几何形状，以定性地了解系统如何依赖于参数。这部分项目的主要目的是解释斯塔克效应中共振间隙的崩溃，这是通过数值观察到的，但其原因仍不清楚。该项目的第三部分是分析潮汐耗散对天体运动的长期影响的模型。

This project will develop mathematical tools with the goal of understanding some important physical phenomena, in three different but related areas. The first of these areas deals with the study of mechanical systems subjected to high frequency, high intensity vibrations. Laser tweezers used to manipulate objects inside biological cells belong to this class of systems, as does the Paul trap -? an ion trap for the invention of which W. Paul received the 1989 Nobel Prize in physics. This project aims to develop a new, more intuitive and geometric understanding of these and related phenomena. In addition to explaining physics in a simpler new way, this research would develop new connections between different mathematical areas. The second part of the project deals with Hill's equation, a system of fundamental importance in many areas of mathematics, physics, and engineering. The main aim of this part of the project is to explain a fundamental and elegant feature of this system which has been observed numerically but the reason for which is still not understood. The third part of the project deals with a basic problem of understanding the long-time effect of tidal dissipation on the motion of celestial bodies. The proposed research consists of three areas unified by the common goal of understanding dynamical systems arising in physical settings. The first of these areas deals with the study of mechanical systems subjected to high frequency, high intensity vibrations. The research will develop a new geometric understanding of related phenomena (as a possible benefit to applications), and will relate averaging theory with differential geometry (as a bridge between two areas of mathematics). The second part of the project will explore the geometry of the bifurcation diagram associated with Hill's equation, to understand qualitatively how the system depends on parameters. The main aim of this part of the project is to explain the collapse of resonance gaps in the Stark effect, which is observed numerically but the reason for which is still not understood. The third part of the project deals with analysis of models for the long-time effect of tidal dissipation on the motion of celestial bodies.