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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：
1909200
负责人：
Mark Levi
金额：
$ 29.12万
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：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1909200&HistoricalAwards=false
关键词：
Nonlinear Dynamics Applications Physical Systems

项目摘要

The project involves three directions. The first one deals with so-called twist maps -- a basic building block in studying almost every kind of moving or stationary physical system where friction is negligible. As an illustrating example, a twist map can arise from the equation governing the distribution of electrons in models of crystal lattices. Prior work by the investigator suggests an unexpected mathematical effect; interpreted for the case of a crystal lattice, this general mathematical result amounts to an unexpectedly low electric resistance. The mechanism of this effect is not understood, and the investigator works to develop a mathematical theory explaining it. The second part of the project deals in particular with the recently discovered "ponderomotive magnetism." Recall that any change in an electric field gives rise to a magnetic field, as has been known since Faraday. Surprisingly, a superficially similar effect takes place in mechanics (without electric fields present), as was discovered only recently by the investigator and collaborators. For example, particles in the gravitational field of a spinning (non-spherical) asteroid behave as if they were charged and in the presence of a magnetic field -- with neither charge nor the field actually present. This is a mysterious and fundamental mathematical phenomenon. A goal of this project is to develop a geometrical explanation of this effect using tools of differential geometry. The third part of the project explores the recently discovered connection between Hill's equation (ubiquitous in innumerable problems in physics and engineering) on the one hand, and the "tire track" problem on the other. The main value of this topic lies in unifying and simplifying two seemingly distinct areas of mathematics, resulting in richer understanding of both, and perhaps in new discoveries.The project involves three directions unified by use of geometry and analysis with physical motivation. The first direction involves area-preserving twist maps of the cylinder; such maps arise in many settings, e.g. the Frenkel-Kontorova (F-K) model of a crystal lattice. The investigator studies whether the robustness of periodic orbits with respect to certain perturbations (e.g., addition of a tilt to the periodic potential in the F-K example) increases with the number of harmonics in the map. Two entirely different (and slightly simpler) objects are known to show an analogous effect: circle mappings and Mathieu-type Hill's equations. If successful, the project would add a new class of systems to the latter two classes. The second direction of research includes the goal to explain geometrically the recently discovered "ponderomotive magnetism," in which rapid time-periodic oscillation of a conservative force field gives rise to a Faraday-like effect discovered by the investigator and collaborators through a normal-form computation: particles in such a field behave as if they possessed electric charge in the presence of a magnetic field, though neither an electric charge nor a magnetic field is present. One of the goals is to explain the geometry underlying this effect. The third direction explores questions raised by the connection between Hill's equation (arising in many areas of mathematics and applications) on the one hand and the geometrical "tire track" problem on 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.

该项目涉及三个方向。第一种是所谓的扭曲图--这是研究几乎所有摩擦可以忽略的运动或静止物理系统的基本构件。作为一个例证，扭曲图可以从控制晶格模型中电子分布的方程中产生。研究人员先前的工作表明了一种意想不到的数学效应；对于晶格的情况，这个一般的数学结果相当于一个出乎意料的低电阻。这种效应的机制尚不清楚，研究人员致力于开发一种数学理论来解释它。该项目的第二部分特别涉及最近发现的“有重量运动的磁性”。回想一下，电场的任何变化都会产生磁场，这是从法拉第开始就知道的。令人惊讶的是，研究人员和合作者最近才发现，表面上类似的效应在力学中也发生了(没有电场)。例如，在自转(非球形)小行星的引力场中，粒子的行为就像它们带电一样，在磁场存在的情况下--实际上既没有电荷，也没有磁场。这是一个神秘而基本的数学现象。这个项目的一个目标是利用微分几何的工具对这种效应进行几何解释。该项目的第三部分探索了最近发现的希尔方程(在无数物理和工程问题中普遍存在)与“轮胎印”问题之间的联系。这个主题的主要价值在于统一和简化两个看似不同的数学领域，导致对这两个领域的更丰富的理解，也许还会有新的发现。这个项目涉及三个方向，通过使用几何和物理动机的分析来统一。第一个方向涉及圆柱体的面积保持扭曲映射；这种映射出现在许多环境中，例如晶格的Frenkel-Kontorova(F-K)模型。研究人员研究周期轨道相对于某些扰动(例如，在F-K例子中对周期势增加倾斜)的稳健性是否随着映射中的谐波的数量而增加。已知有两个完全不同的(稍微更简单的)物体显示出类似的效应：圆映射和马蒂厄类型的希尔方程。如果成功，该项目将在后两类系统的基础上增加一类新的系统。第二个研究方向包括从几何上解释最近发现的“有质动力磁性”，即保守力场的快速时间周期振荡会产生一种类似法拉第的效应，这是由研究人员和合作者通过正态计算发现的：这种磁场中的粒子在磁场存在的情况下表现得就像拥有电荷一样，尽管既不存在电荷，也不存在磁场。其中一个目标是解释这种效应背后的几何原理。第三个方向探讨了希尔方程(在许多数学和应用领域中出现)和几何“轮胎印”问题之间的联系所引起的问题。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。