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Nonlinear Dynamics with Applications to Physical Systems

非线性动力学及其在物理系统中的应用

基本信息

批准号：
2206500
负责人：
Mark Levi
金额：
$ 20万
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：
2022
资助国家：
美国
起止时间：
2022-09-01 至 2025-08-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206500&HistoricalAwards=false
关键词：
Nonlinear Dynamics Applications Physical Systems

项目摘要

The proposed research aims at discovering new phenomena of fundamental mathematical interest as well as physical relevance, and at explaining them in the simplest possible way. The three main proposed projects are (i) study of the robustness or fragility of resonances arising in many physical applications, e.g. in solid state physics, (ii) explaining the geometrical mechanism of the recently discovered ponderomotive magnetism (observed in many physical experiments) and developing a connection between averaging theory and differential geometry – two seemingly unrelated fields, and (iii) exploiting the recently discovered connection between Hill's equation on the one hand and the tire track problem on the other, thus unifying two seemingly unrelated areas. Some results of this project are likely to give a new and simpler understanding of some phenomena of fundamental interest, and may make their way into textbooks. The project consists of three parts, unified by the desire to discover and understand new phenomena in dynamics. Some years ago J. B. Keller and V. Arnold discovered and analyzed a fundamental feature of resonances in two different classes of problems: circle maps and Mathieu-type equations. The goal of the first part of the project is to add a yet one more class to the two studied by Arnold and Keller, namely the area-preserving cylinder maps. Besides of its basic interest, the latter example comes up in solid-state physics among many other settings. The second part of the project aims at understanding the somewhat mysterious mechanism of ponderomotive magnetism, and also aims to explore a connection between two fundamental objects, one from mechanics (the gyroscopic effect) and the other from differential geometry (Jacobi fields). The third project proposes to exploit the recently found connection between two seemingly unrelated objects: (i) Hill's equation, studied extensively over a couple of centuries, and (ii) tire tracks, a more recently studied object. It is hoped that the results from one area will give new insights into the other.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

拟议的研究旨在发现新的现象的基本数学利益以及物理相关性，并在解释他们在最简单的方式。三个主要的建议项目是（i）研究在许多物理应用中，例如在固态物理中产生的共振的鲁棒性或脆弱性，（ii）解释最近发现的有质动力磁性的几何机制（在许多物理实验中观察到）并发展了平均理论和微分几何之间的联系-两个看似无关的领域，和（iii）利用最近发现的希尔方程之间的联系，一方面和轮胎的轨道问题，从而统一两个看似无关的领域。这个项目的一些结果可能会给一些基本利益的现象一个新的和更简单的理解，并可能使他们的方式进入教科书。该项目由三个部分组成，统一的愿望，发现和理解新的动力学现象。几年前，J. B。Keller和V. Arnold发现并分析了两类不同问题中共振的基本特征：圆映射和Mathieu型方程。该项目的第一部分的目标是增加一个又一类的阿诺德和凯勒研究的两个，即面积保持圆柱映射。除了它的基本兴趣之外，后一个例子出现在固态物理学和许多其他环境中。该项目的第二部分旨在了解有质动力磁性的神秘机制，并旨在探索两个基本对象之间的联系，一个来自力学（陀螺效应），另一个来自微分几何（雅可比场）。第三个项目建议利用最近发现的两个看似无关的物体之间的联系：（一）希尔方程，几个世纪以来被广泛研究，（二）轮胎痕迹，最近研究的对象。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。