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.

本研究致力于动力学系统的研究，重点研究哈密顿系统中的Arnold不稳定性和混沌现象。1964年，Arnold描述了哈密顿系统中不稳定性的一个典型例子：它代表了一个简单的机械系统，它由一个摆和两个弱耦合的旋转体组成，它们的轨迹在相空间中从它们的出发点“疯狂地”和“任意地”地漂移。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。这个项目的第一个目标是在一些一般模型上研究这一猜想，并确定不稳定的各种几何和拓扑机制。我们将考虑可积哈密顿系统的摄动。在一定的非退化假设下，存在具有不变流形的Kam环面族；不变流形导致相邻环面之间的轨道连通，但也出现了分离的间隙。主要目标是找到沿着连接这些环的连接移动的扩散轨迹，并在缝隙中移动，随着扰动的大小趋于零，移动一段均匀的距离。其中一些轨迹应该能够进行混乱的漂移。这项研究的方法论将基于正确对齐窗口的拓扑技术，并辅之以几何近似理论和变分方法。本研究的主要目的是克服大间隙问题，获得扩散时间的最优估计，放宽对系统的横截性和非退化假设，并研究大扰动下扩散的存在性。这个项目的第二个目标是将从阿诺德不稳定性研究中获得的知识应用于三体问题。目标是了解产生混乱运动的几何机制，并找到具有指定路线的最佳轨迹。这个项目的目标将从理论和数值两个方面来阐述。这项研究的总体背景是研究周期性施加小扰动对稳定系统的累积影响。这项研究源于一个可以追溯到奈特旺、拉格朗日和拉普拉斯的问题，即太阳系在遥远的未来是会保持现在的形式，还是会经历灾难性的变化。如果说在一些扰动系统中，如太阳系，稳定性占主导地位，那么在其他一些系统中，不稳定是典型的。阿诺德不稳定性可以看作是关于如何增加具有小周期作用力的物理系统的能量的配方。应用于三体问题的阿诺德不稳定性思想在航天器动力学方面具有潜在的应用，在设计探索太阳系不同区域的燃料效率高的空间任务方面。该项目中概述的技术也可用于研究行星围绕双星系统运行的动力学，以及致密双星系统(例如，英仙座木卫二)中的质量转移的动力学。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学。此外，该项目还将加强学生和K-12教育工作者的知识和专业发展。它将直接让学生，包括代表人数不足的群体的成员，参与研究活动和数学教育。