Hamiltonian and Celestial Mechanics

哈密​​顿量和天体力学

• 批准号: 0500443
• 负责人: Richard Moeckel
• 金额: $ 8.5万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500443&HistoricalAwards=false
• 关键词: Hamiltonian; Celestial; Mechanics

Project Abstract DMS-0500443This is a research project in the general area of dynamical systems theory with emphasis on Hamiltonian and celestial mechanics. The gravitational n-body problem remains an active topic for research three centuries after Newton proposed it. Over the years it has served as a stimulus for the development of new mathematics. Three aspects of the problem will be pursued. In brief, they are i) the development of computational algebra techniques for systems of polynomial equations and their application to the problem of relative equilibria of the n-body problem, ii) the study of continuation of families of periodic solutions by a combination of the topological method of isolating blocks and classical variational methods, and iii) the analysis of a simplified model problem which may lead to better understanding of the phenomenon of Arnold diffusion in Hamiltonian systems.The first part of the project concerns the simplest periodic motions of the n-body problem, the so-called relative equilibria. These are special arrangements of the masses such that the gravitational forces can be exactly balanced by centifugal forces when the configuration rotates. For example, three masses arranged in the shape of an equilateral triangle can be rigidly rotated about the center of mass to produce a solution of the three-body problem. For four or more masses, it is a difficult algebraic problem to find or even count these special solutions. New computational algebra methods have met with some success recently the plan is to continue to develop and apply these ideas. Numerical studies show that nearby some of the relative equilibrium solutions there are other interesting families of periodic orbits. The second part of the project is about developing methods to mathematically prove the existence of such solutions. Two different types of proofs will be attempted, one based on qualitative geometrical arguments and another based on a venerable analytical technique. Both of these have been used before in celestial mechanics but the problem to be studied here is essentially different and new ideas are needed. Finally, the third part of the project moves from the study of orderly periodic motions to the realm of chaotic dynamics. It has been known for many years that certain kinds of chaotic motions can lead to instability in mechanical systems. Instability in celestial mechanics can give rise to such phenomena as the slow drifting of the orbital parameters of planets or asteroids. A specific mechanism for such chaotic drifting, known as Arnold diffusion, is currently a very active area of dynamical systems research. However, it is difficult to understand in a system as complex as the gravitational n-body problem. The approach taken in this project will be to study a simplified model problem which is designed to capture the essential features of the real situation but is more amenable to mathematical analysis. Broader impacts of the project include development of scientific software, mentoring of young mathematicians and possible applications to spacecraft mission design.

项目摘要 DMS-0500443这是动力系统理论一般领域的一个研究项目，重点是哈密顿量和天体力学。 在牛顿提出引力多体问题三个世纪后，它仍然是一个活跃的研究课题。 多年来，它一直刺激着新数学的发展。 我们将从三个方面来解决这个问题。 简而言之，它们是i）多项式方程组计算代数技术的发展及其在n体问题相对平衡问题中的应用，ii）通过隔离块拓扑方法和经典变分方法相结合来研究周期解族的连续性，以及iii）对简化模型问题的分析，这可能会导致更好地理解阿诺德扩散现象 哈密顿系统。该项目的第一部分涉及 n 体问题的最简单的周期运动，即所谓的相对平衡。 这些是质量的特殊排列，使得当结构旋转时重力可以通过离心力精确平衡。 例如，以等边三角形排列的三个质量可以绕质心刚性旋转以产生三体问题的解决方案。 对于四个或更多的质量，找到甚至计算这些特殊的解决方案是一个困难的代数问题。 新的计算代数方法最近取得了一些成功，计划是继续开发和应用这些想法。 数值研究表明，在一些相对平衡解附近还有其他有趣的周期轨道族。 该项目的第二部分是关于开发方法以数学方式证明此类解决方案的存在。 将尝试两种不同类型的证明，一种基于定性几何论证，另一种基于古老的分析技术。 这两种方法以前都在天体力学中使用过，但这里要研究的问题本质上是不同的，需要新的想法。 最后，该项目的第三部分从有序周期运动的研究转向混沌动力学领域。 多年来，人们已经知道某些类型的混沌运动会导致机械系统不稳定。 天体力学的不稳定性会引起行星或小行星轨道参数缓慢漂移等现象。 这种混沌漂移的一种特定机制被称为阿诺德扩散，目前是动力系统研究的一个非常活跃的领域。 然而，在像引力n体问题这样复杂的系统中，这是很难理解的。 该项目采用的方法将是研究一个简化的模型问题，该模型旨在捕获真实情况的基本特征，但更适合数学分析。 该项目更广泛的影响包括开发科学软件、指导年轻数学家以及在航天器任务设计中的可能应用。