RUI: Hamiltonian Instability

RUI：哈密顿不稳定性

• 批准号: 0601016
• 负责人: Marian Gidea
• 金额: $ 9.73万
• 依托单位: Northeastern Illinois University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0601016&HistoricalAwards=false
• 关键词: RUI; Hamiltonian; Instability

This research is devoted to dynamical systems, with emphasis on Arnold instability and chaotic phenomena in Hamiltonian systems. A quintessential example of instability in Hamiltonian systems was described by Arnold in 1964: it represents a simple mechanical system, consisting in a pendulum and two rotators with a weak coupling, whose trajectories wander `wildly' and `arbitrarily far' from their points of departure in the phase space. This example gave rise to a conjecture, referred as Arnold instability, stating that such a behavior occurs in rather general systems. The first objective of this project is to investigate this conjecture on some general models and identify various geometric and topological mechanisms of instability. Perturbations of integrable Hamiltonian systems will be considered. Under certain non-degeneracy assumptions, there exist families of KAM tori that posses invariant manifolds; the invariant manifolds give rise to connecting orbits between nearby tori, but separating gaps also appear. The main goal is to find diffusing trajectories that move along the connections that link these tori and also move across the gaps, traveling a uniform distance as the size of the perturbation tends to zero. Some of these trajectories should be able to make chaotic excursions. The methodology of this research will be based on a topological technique of correctly aligned windows, reinforced with geometric approximation theory and variational methods. The main goals of this research are to overcome the large gap problem, obtain optimal estimates on the diffusion time, relax the transversality and non-degeneracy assumptions on the system, and study the existence of diffusion for large perturbations. The second objective of this project is to apply the knowledge gained from the study of Arnold instability to the three-body problem. The goal is to understand the geometric mechanisms that produce chaotic motions and find optimal trajectories with prescribed itineraries. The objectives of this project will be addressed both theoretically and numerically.The general context of this research is the study of the cumulative effect of small perturbations applied periodically to a stable system. This study originated from a question that goes back to Netwon, Lagrange and Laplace, whether the Solar System in the distant future will keep the same form as it is now, or will undergo a catastrophic change. If in some perturbed systems, like the Solar System, stability is predominant, in some other systems instability is typical. The Arnold instability can be viewed as a recipe on how to increase the energy of physical systems with small periodic forcing. Arnold instability ideas applied to the three-body problem have potential applications to spacecraft dynamics, in designing fuel efficient space missions exploring various regions of the Solar System. The techniques outlined in this project can also be used in studying the dynamics of planets orbiting systems of binary stars, and of mass transfers in tight binary star systems (e.g. Algol in Betta Persei). Other possible applications include heart pacemaking, plasma confinement, and accelerator physics. Additionally, this project will enhance the knowledge and professional development of students and K-12 educators. It will directly engage students, including members of underrepresented groups, in research activities and mathematics education.

这项研究致力于动力系统，重点是哈密顿系统中的阿诺德不稳定性和混沌现象。阿诺德在 1964 年描述了哈密顿系统不稳定性的一个典型例子：它代表了一个简单的机械系统，由一个摆和两个弱耦合的旋转器组成，其轨迹“疯狂地”和“任意远离”其出发点在相空间中。这个例子引起了一个猜想，称为阿诺德不稳定性，指出这种行为发生在相当一般的系统中。该项目的首要目标是在一些通用模型上研究这一猜想，并确定各种不稳定的几何和拓扑机制。将考虑可积哈密顿系统的扰动。在某些非简并假设下，存在具有不变流形的KAM tori族；不变流形在附近环面之间产生连接轨道，但也会出现分离间隙。主要目标是找到沿着连接这些圆环的连接移动并穿过间隙的扩散轨迹，当扰动的大小趋于零时，移动均匀的距离。其中一些轨迹应该能够进行混乱的偏移。这项研究的方法将基于正确对齐窗口的拓扑技术，并通过几何近似理论和变分方法得到加强。本研究的主要目标是克服大间隙问题，获得扩散时间的最优估计，放宽系统的横向性和非简并性假设，研究大扰动下扩散的存在性。该项目的第二个目标是将阿诺德不稳定性研究中获得的知识应用于三体问题。目标是了解产生混沌运动的几何机制，并找到具有规定行程的最佳轨迹。该项目的目标将从理论上和数值上得到解决。这项研究的总体背景是研究周期性应用于稳定系统的小扰动的累积效应。这项研究源于一个可以追溯到Netwon、拉格朗日和拉普拉斯的问题：遥远的未来太阳系是否会保持现在的形态，还是会发生灾难性的变化。如果说在某些受扰动的系统中，例如太阳系，稳定性占主导地位，那么在其他一些系统中，不稳定则是典型的。 阿诺德不稳定性可以被视为如何通过小周期性强迫来增加物理系统能量的秘诀。应用于三体问题的阿诺德不稳定性思想在航天器动力学、设计探索太阳系各个区域的燃料高效太空任务方面具有潜在的应用。该项目中概述的技术还可用于研究双星轨道系统的行星动力学，以及紧密双星系统中的质量传递（例如英仙座斗鱼中的大陵五）。其他可能的应用包括心脏起搏、等离子体限制和加速器物理。此外，该项目还将增强学生和 K-12 教育工作者的知识和专业发展。它将直接让学生（包括代表性不足群体的成员）参与研究活动和数学教育。