Hamiltonian and Celestial Mechanics

项目摘要dms -0500443这是一个动力系统理论一般领域的研究项目，重点是哈密顿和天体力学。在牛顿提出引力n体问题三个世纪后，它仍然是一个活跃的研究课题。多年来，它一直在促进新数学的发展。将从三个方面探讨这个问题。简而言之，它们是i)多项式方程系统的计算代数技术的发展及其在n体问题的相对平衡问题中的应用；ii)结合隔离块的拓扑方法和经典变分方法对周期解族延拓的研究；iii)分析一个简化的模型问题，这可能有助于更好地理解哈密顿系统中的阿诺德扩散现象。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。这些是质量的特殊安排，当结构旋转时，重力可以通过离心力精确地平衡。例如，排列成等边三角形的三个质量可以围绕质心进行刚性旋转，从而得到三体问题的解。对于四个或更多的质量，找到甚至计数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，计划是继续发展和应用这些想法。数值研究表明，在一些相对平衡解附近存在其他有趣的周期轨道族。该项目的第二部分是关于开发数学方法来证明这些解的存在。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，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可以导致机械系统的不稳定。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。这种混沌漂移的具体机制，被称为阿诺德扩散，是目前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很难理解。本项目采用的方法将是研究一个简化的模型问题，该问题旨在捕捉实际情况的基本特征，但更适合于数学分析。该项目的更广泛影响包括科学软件的开发，年轻数学家的指导以及航天器任务设计的可能应用。

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.