Hamiltonian and Celestial Mechanics

这项研究项目涉及动力系统理论，重点是经典力学和天体力学中的问题。动力系统理论致力于对随时间演化的系统进行数学研究，包括引力n体问题。该项目的研究对于理解行星、卫星和小行星等天体可能的运动具有直接的应用。但它的首要重要性在于开发新的数学方法，这些方法可以在其他情况下应用。从数学的角度来看，所研究的问题涉及寻找和理解一个复杂的微分方程组的解。这样的方程太复杂了，无法显式求解，取而代之的是使用计算机模拟和理论、数学推理相结合的方法进行研究。平面三体问题是一个有着悠久历史的经典动力学系统，至今仍面临着严峻的数学挑战。已经发现了许多简单而美丽的周期运动，但在数学证明的层面上理解的很少。该系统可表示为五维常微分方程组。利用庞加莱截面，周期轨道可以作为四维映射的不动点来求解。这个项目的一部分致力于开发新的拓扑方法来寻找周期轨道。天体力学的特征之一就是奇点的存在。碰撞附近轨道的行为可能是混乱的。作为研究的一部分，这一领域有几个悬而未决的问题正在调查中。N体问题中最简单的周期轨道是平面中心构型产生的相对平衡运动。了解平面中的中心构型及其向更高维度的推广是本项目的另一个目标。寻找甚至计算中心构型是一个困难的代数问题。将研究中心构型数目的有限性问题和由此产生的周期解的稳定性问题。

This research project concerns dynamical systems theory, with emphasis on questions in classical and celestial mechanics. Dynamical systems theory is devoted to the mathematical study of systems that evolve in time, including the gravitational n-body problem. The research in this project has direct applications to understanding the possible motions of celestial bodies such as planets, moons, and asteroids. But its primary importance lies in the development of new mathematical methods that can be applied in other contexts. From the mathematical point of view, the questions under study involve finding and understanding the solutions of a complex system of differential equations. Such equations are too complicated to solve explicitly, and instead are studied using a combination of computer simulations and theoretical, mathematical reasoning. The planar three-body problem is a classical dynamical system with a long history that still presents formidable mathematical challenges. Many simple and beautiful periodic motions have been discovered, but only a few have been understood at the level of mathematical proof. The system is formulated as ordinary differential equations in five dimensions. Using Poincare sections, the periodic orbits can be found as fixed points of four-dimensional mappings. One part of this project is devoted to developing new topological methods for finding periodic orbits. One of the characteristic features of celestial mechanics is the presence of singularities. The behavior of orbits near collisions can be chaotic. There are several open questions in this area under investigation as part of the research. The simplest periodic orbits in the n-body problem are relative equilibrium motions that arise from planar central configurations. Understanding central configurations in the plane and their generalization to higher dimensions is another goal of this project. It is a difficult algebraic problem to find or even count the central configurations. The problems of finiteness of the number of central configurations and the question of stability of the resulting periodic solutions will be investigated.