Nonlinear Dynamics with Applications to Physical Systems

提案ID：0605878PI：Levi，MarkInsition：宾夕法尼亚州立大学公园标题：非线性动力学与应用于物理系统拟议的研究由三个不同的部分组成，由理解物理应用中产生的动力系统这一共同主题联系在一起。在第一部分中，参数共振问题的几何方法旨在获得对该模型的新见解，该模型在力学、物理、工程中的许多应用中出现，并在数学中扮演重要角色。参数共振问题已被分析人员广泛研究，但最近的数值观测表明，拓扑方法可能会给出一种主要的新见解。这个问题在数学、物理和工程学的许多课本中都有讨论；对这个问题的基本新见解将会引起相当大的兴趣。人们希望这项研究将为斯塔克效应(原子谱线分裂)提供新的线索。拟议研究的第二部分涉及对施加快速振动的系统的研究。振动稳定被用于粒子加速器、粒子捕捉器和激光镊子中。直到最近，人们才了解了这一现象的潜在几何学。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卓有成效的联系。这项工作将展示来自微分几何(曲率、正规族)的概念如何在力学中找到它们的表现形式。拟议研究的第三部分涉及阿诺德扩散--哈密顿系统稳定性的一个基本方面。研究人员的目标有两个：第一，发展依赖时间的哈密顿系统的变分技术；第二，在物理或几何驱动的特定例子中，阐明阿诺德扩散问题的新线索。拟议研究的统一主题是一方面在抽象的数学概念和它们的物理表现(例如，在力学中)之间建立新的联系。这种联系丰富了数学，并为应用程序提供了更好地理解物理现象的工具。最近这种互惠互动的一个例子是作者使用微分几何为保罗陷阱的功能提供了新的见解--保罗陷阱是一种通过电场使带电粒子悬浮的装置。1989年，W·保罗因他的发明被授予诺贝尔奖。提出的研究应该对一些在力学、量子力学和工程学中具有基本重要性的共振现象提供新的见解。希望所提出的一些研究成果能进入高年级本科生和研究生的微分方程式、工程学和力学课本。研究生和本科生都将密切参与这项研究。

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.