Nonlinear Dynamics with Applications to Physical Systems

非线性动力学及其在物理系统中的应用

• 批准号: 0605878
• 负责人: Mark Levi
• 金额: --
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0605878&HistoricalAwards=false
• 关键词: Nonlinear; Dynamics; Applications; Physical; Systems

Proposal ID: 0605878PI: Levi, MarkInsitution: Pennsylvania State Univ University ParkTitle: Nonlinear Dynamics with Applications to Physical SystemsProposed research consists of three different parts, united by the common theme of understanding dynamical systems arising in physical applications. In part I, a geometrical approach to the problem of parametric resonance aims at gaining new insight into the model that arises in numerous applications in mechanics, physics, engineering, and plays an important role inmathematics. The problem of parametric resonance has been extensively studied by analysts, but recent numerical observations suggest that a topological approach may give a principally new insight. This problem is treated in many texts in mathematics, physics and engineering; basic new insight into the problem will be of considerable interest. It is hoped that this research will shed new light on Stark effect (splitting of atomic spectral lines). Part II of proposed research addresses study of systems with imposed rapid vibrations. Stabilization by vibration is used in particle acclerators, in particle traps and in laser ``tweezers". Underlying geometry of the phenomenon was understood only recently. The author proposes to extend his earlier work to broader physical contexts, and to further explore the fruitful connection between differential geometry, mechanics and averaging theory. This work will show how concepts from differential geometry (curvature, normal family) find their manifestations in mechanics. Part III of proposed research deals with Arnold diffusion -- a fundamental aspect of stability of Hamiltonian systems. The researcher's goal is two-fold: first, to develop variational techniques for time--dependent Hamiltonian systems, and second, to shed new light on the problem of Arnold diffusion in specific examples motivated by physics or geometry. The unifying theme of proposed research is to establish new connections between abstract mathematical concepts on the one hand and their physical manifestations (e.g., in mechanics) on the other. Such connections enrich mathematics and benefit applications by providing the latter with tools for better understanding physical phenomena. A recent example of such mutually beneficial interaction was the author's use of differential geometry to provide new insight into the functioning of the Paul trap -- a device used to suspend charged particles by electric field. In 1989 W. Paul was awarded Nobel prize for his invention. Proposed research should give new insights into some resonance phenomena of basic importance in mechanics, quantum mechanics and engineering. It is hoped that some results of proposed research will make their way into upper undergraduate and graduate texts in differential equations, engineering and mechanics. Both graduate and undergraduate students will be closely involved with this research.

提案ID：0605878 PI：Levi，MarkInsitution：Pennsylvania State University Park题目：Nonlinear Dynamics with Applications to Physical SystemsProposed Research由三个不同的部分组成，由理解物理应用中出现的动力系统的共同主题统一起来。在第一部分，几何方法的参数共振问题的目的是获得新的洞察模型，出现在众多的应用在力学，物理，工程，并发挥了重要作用inmathematics。参数共振的问题已经被分析家广泛研究，但最近的数值观测表明，拓扑方法可能会给一个主要的新见解。这个问题在数学、物理和工程学的许多教科书中都有涉及;对这个问题的基本新见解将引起相当大的兴趣。希望这项研究能对斯塔克效应（原子光谱线的分裂）有新的认识。拟议研究的第二部分涉及对具有施加快速振动的系统的研究。振动稳定用于粒子加速器、粒子阱和激光"镊子”。这种现象的基本几何学直到最近才被理解。作者建议将他早期的工作扩展到更广泛的物理背景，并进一步探索微分几何，力学和平均理论之间富有成效的联系。这项工作将显示如何从微分几何（曲率，正常家庭）的概念找到他们的表现在力学。第三部分提出的研究涉及阿诺德扩散-一个基本方面的稳定性的哈密顿系统。研究人员的目标是双重的：第一，开发变分技术的时间依赖的哈密顿系统，第二，揭示了新的光的问题阿诺德扩散在具体的例子，物理或几何的动机。拟议研究的统一主题是一方面在抽象数学概念与其物理表现形式之间建立新的联系（例如，在力学上，另一方面。这种联系丰富了数学，并通过为后者提供更好地理解物理现象的工具而使应用受益。这种互利互动的一个最近的例子是作者使用微分几何来提供对保罗陷阱功能的新见解-一种用于通过电场悬浮带电粒子的设备。1989年W.保罗因他的发明而获得诺贝尔奖。拟议中的研究应给予新的见解，在力学，量子力学和工程的一些基本重要的共振现象。人们希望，拟议的研究的一些结果将进入高年级本科生和研究生的文本微分方程，工程和力学的方式。研究生和本科生都将密切参与这项研究。