Hamiltonian and Celestial Mechanics

项目摘要DMS-0500443这是一个动力系统理论的一般领域的研究项目，重点是哈密顿和天体力学。自牛顿提出引力n体问题三个世纪以来，它一直是一个活跃的研究课题。多年来，它一直是新数学发展的推动力。将从三个方面探讨这一问题。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)用隔离块拓扑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)一个简化的模型问题的分析，这可能导致对哈密顿系统中的Arnold扩散现象的更好的理解。这是质量的特殊排列，当构型旋转时，引力可以被离心力精确地平衡。例如，以等边三角形的形状排列的三个质量可以绕质心刚性地旋转，以产生三体问题的解。对于四个或四个以上的质量，寻找甚至计算这些特解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从数学上证明此类解的存在的方法。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，项目的第三部分从有序周期运动的研究转移到混沌动力学领域。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的不稳定。天体力学中的不稳定性会导致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殊机制，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。本项目采用的方法将是研究一个简化的模型问题，该问题旨在捕捉真实情况的基本特征，但更容易进行数学分析。该项目的更广泛影响包括开发科学软件、指导年轻数学家以及可能将其应用于航天器飞行任务设计。

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.