Hamiltonian and Celestial Mechanics

这是一个研究项目，在一般领域的动力系统理论与应用中出现的问题，力学，特别是天体力学。 引力n体问题在牛顿提出三个世纪后仍然是一个活跃的研究课题。 除了与天文学和空间使命设计直接相关外，它还促进了新数学技术的发展。 研究集中在三个主要领域：奇异性的正则化，周期轨道的拓扑和变分存在性证明，以及相对平衡解的研究。 天体力学的特征之一是存在碰撞奇点。 人们早就知道，简单的二元碰撞可以通过巧妙地选择坐标的变化而“规则化”。 更复杂的奇点，如三体问题中的三重碰撞，可以被“爆破”，以揭示近碰撞解的相当复杂的行为。 部分建议是给一个完整的正规化和爆破的三体问题相结合的老工作的问题与现代发展辛约化理论。 一个类似的四体问题项目将涉及许多新的想法，并与代数曲面理论有着惊人的联系。 周期运动在动力系统理论中一直扮演着重要的角色。 它们通常是运动方程的最简单的解，并为理解其他更复杂的解提供了一个框架。 最近，变分方法被用来构造一些显着的对称周期解的n体问题。 这个项目将开发拓扑技术来寻找这样的解决方案，并将探讨变分和拓扑方法之间的关系。最简单的周期轨道来自中心构型--质量的特殊排列，使得当构型旋转时，重力可以被离心力精确地平衡。 寻找甚至计算中心构型是一个困难的代数问题。 为这个问题开发的技术可以应用于其他复杂的代数方程组。

This is a research project in the general area of dynamical systems theory with applications to problems arising in mechanics, especially celestial mechanics. The gravitational n-body problem remains an active topic for research three centuries after Newton proposed it. In addition to its direct relevance for astronomy and space mission design, it has served as a stimulus for the development of new mathematical techniques. The research is focussed on three main areas: regularization of singularities, topological and variational existence proofs for periodic orbits, and studies of relative equilibrium solutions. One of the characteristic features of celestial mechanics is the presence of collision singularities. It has long been known that simple binary collisions can be ``regularized'' by cleverly chosen changes of coordinates. More complicated singularities like the triple collision in the three-body problem can be ``blown-up'' so as to reveal the quite complicated behavior of near-collision solutions. Part of the proposal is to give a complete regularization and blow-up for the three-body problem which combines the older work on the subject with modern developments in symplectic reduction theory. A similar project for the four-body problem will involve many new ideas and has surprising connections to algebraic surface theory. Periodic motions have always played an important role in dynamical systems theory. They are often the simplest solutions of the equations of motion and provide a framework for understanding other, more complicated solutions. Recently, variational methods have been used to construct some remarkable symmetric periodic solutions of the n-body problem. This project will develop topological techniques for finding such solutions and will explore the relationship between the variational and topological approaches. The simplest periodic orbits arise from central configurations -- special arrangements of the masses such that the gravitational forces can be exactly balanced by centrifugal forces when the configuration rotates. It is a difficult algebraic problem to find or even count the central configurations. Techniques developed for this problem could be applied to other complicated systems of algebraic equations.