Hamiltonian and Celestial Mechanics

这是一个在动力系统理论的一般领域的研究项目，应用于力学，特别是天体力学中出现的问题。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。除了与天文学和空间任务设计直接相关外，它还促进了新数学技术的发展。研究主要集中在三个方面：奇点正则化，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碰撞奇点的存在。人们早就知道，简单的二元碰撞可以通过巧妙地选择坐标变化来“正则化”。更复杂的奇点，如三体问题中的三重碰撞，可以被“放大”，从而揭示出相当复杂的近碰撞解的行为。该建议的一部分是给出三体问题的完全正则化和扩展，该问题将该主题的旧工作与辛约简理论的现代发展结合起来。四体问题的类似项目将涉及许多新思想，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起着重要的作用。它们通常是运动方程的最简单解，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n体问题的一些重要的对称周期解。这个项目将开发拓扑技术来寻找这样的解决方案，并将探索变分和拓扑方法之间的关系。最简单的周期轨道产生于中心结构——质量的特殊安排，当结构旋转时，引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.