Mathematical and numerical methods in celestial mechanics

天体力学历来是数学发现和新问题的源泉。今天仍然如此，当代空间科学的需求带来了对天体力学理论和实践方面的新一波兴趣，使其与数学和应用的联系比以往任何时候都更加紧密。天体力学的一个主要方面是对牛顿N体问题的研究，即在动力学由牛顿运动定律决定的情况下分析N点粒子的运动。现在，对n体问题的研究包括了几乎任何与牛顿n体问题相类似的动力系统，而且除了它在天文学和空间科学中提供的见解之外，它在科学和工程中有许多应用。我的研究项目主要有两个方向。第一个问题涉及理论性质的研究。第二部分强调应用方面。从理论的角度来看，我感兴趣的是研究n体问题的一些特殊解的性质（即同形运动和相对平衡）以及在这些运动中允许的特殊构型（即中心构型）。中心构型的研究很重要，因为它们在理解n体行为中起着“核心作用”：它们表征碰撞和膨胀的动力学行为，并在n体问题的积分流形的拓扑结构中起关键作用。从更面向应用的角度来看，我的研究计划是针对开发数值方法，提高计算效率，并允许以更高的精度模拟新系统。此外，我对分析全身问题的几个方面很感兴趣。这是一类引人入胜的问题，有许多悬而未决的问题，涉及科学和工程中的许多重要问题，例如双星小行星，地月系统的动力学，分子的反应和电离，水下航行器的稳定性和控制以及空间飞行器的交会和对接。

Celestial mechanics has historically served as a source of mathematical discovery and new problems.  This is still true today where the contemporary needs of space science bring a new wave of interest in the theoretical and practical aspects of celestial mechanics making its connections with mathematics and applications stronger than ever before. One of the main aspects of celestial mechanics is the study of the Newtonian N-body problem, namely the analysis of the motion of N point particles in a setting where the dynamics are dictated by Newton's law of motion. The study of the N-body problem now includes just about any dynamical system that remotely resembles the Newtonian N-body problem, and beyond the insight it provides in astronomy and space science it  has many applications in science and engineering.My research program has two main directions. The first concerns investigations of a theoretical nature. The second emphasizes applied aspects. From the theoretical point of view I am interested in studying properties of some special solutions of the N-body problem (i.e. homographic motions and relative equilibria) and the special configurations that are allowed in such motions (i.e. central configurations). The study of central configurations is important because they play a ``central role" in understanding N-body behavior: they characterize the dynamical behavior of collisions and expansions and play a key role in the topology of the integral manifolds of the N-body problem. From a more application oriented  point of view my research program is directed towards developing numerical methods that improve computational efficiency and permit new systems to be simulated with greater accuracy. Additionally, I am interested in analyzing several aspects of the full-body problems.  This is a fascinating class of problems that has many open questions and touches on numerous important issues in science and engineering,  as for example binary asteroids, the dynamics of the Earth-Moon system, reaction and ionization of molecules, stability and control of underwater vehicles and  rendezvous  and docking of space vehicles.