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。这些领域中的第一个涉及研究受到高频，高强度振动的机械系统。用于操纵生物细胞内物体的激光镊子属于这类系统，保罗陷阱也是如此。用于本发明的离子阱，W.保罗获得了1989年诺贝尔物理学奖。该项目旨在对这些现象和相关现象进行新的、更直观的几何理解。除了以一种更简单的新方式解释物理学之外，这项研究还将在不同的数学领域之间建立新的联系。该项目的第二部分涉及希尔方程，这是一个在数学，物理学和工程学的许多领域具有根本重要性的系统。本项目这一部分的主要目的是解释该系统的一个基本而优雅的特征，该特征已被数值观察到，但其原因仍不清楚。该项目的第三部分涉及了解潮汐消散对天体运动的长期影响的基本问题。拟议的研究包括三个领域，统一的共同目标是理解物理环境中产生的动力系统。这些领域中的第一个涉及研究受到高频，高强度振动的机械系统。该研究将发展相关现象的新几何理解（作为应用的可能益处），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叉图的几何形状，以定性地了解系统如何依赖于参数。该项目这一部分的主要目的是解释斯塔克效应中共振间隙的崩溃，这是数值观察到的，但其原因仍然不清楚。该项目的第三部分涉及潮汐耗散对天体运动的长期影响的模型分析。

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.