Nonlinear Dynamics with Applications to Physical Systems

该项目涉及三个方向。 第一个涉及所谓的扭曲图，这是研究几乎所有摩擦力可以忽略不计的移动或静止物理系统的基本构件。 作为说明性示例，扭曲图可以由控制晶格模型中电子分布的方程产生。 研究人员之前的工作表明了意想不到的数学效应；对于晶格的情况进行解释，这个一般数学结果相当于意想不到的低电阻。 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“有质动力磁力”。 回想一下，电场的任何变化都会产生磁场，正如法拉第以来人们所知道的那样。 令人惊讶的是，正如研究人员和合作者最近才发现的那样，表面上相似的效应也发生在力学中（不存在电场）。 例如，旋转（非球形）小行星的引力场中的粒子的行为就好像它们带电并且存在磁场一样——实际上既不存在电荷也不存在磁场。 这是一个神秘而基本的数学现象。 该项目的目标是使用微分几何工具对这种效应进行几何解释。 该项目的第三部分一方面探讨了最近发现的希尔方程（在物理和工程中无数问题中普遍存在）与“轮胎痕迹”问题之间的联系。 该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，从而使人们对这两个领域有更丰富的理解，或许还会带来新的发现。该项目涉及三个方向，通过使用几何和具有物理动机的分析来统一。 第一个方向涉及圆柱体的面积保留扭曲图；此类地图出现在许多环境中，例如晶格的 Frenkel-Kontorova (F-K) 模型。 研究人员研究周期性轨道相对于某些扰动（例如，在 F-K 示例中向周期性势添加倾斜）的鲁棒性是否随着图中谐波数量的增加而增加。 已知两个完全不同（且稍微简单）的对象显示出类似的效果：圆形映射和马蒂厄型希尔方程。 如果成功，该项目将在后两类系统的基础上添加一类新的系统。 第二个研究方向包括从几何角度解释最近发现的“有质动力磁学”，其中保守力场的快速时间周期振荡会产生研究人员和合作者通过正常形式计算发现的法拉第效应：这种场中的粒子的行为就好像它们在磁场存在下具有电荷一样，尽管既不存在电荷也不存在磁场。 目标之一是解释这种效应背后的几何结构。 第三个方向一方面探讨希尔方程（出现在数学和应用的许多领域）与几何“轮胎痕迹”问题之间的联系所提出的问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.