Mathematical and numerical methods in celestial mechanics

天体力学在历史上一直是数学发现和新问题的来源。 这仍然是真实的今天，当代空间科学的需求带来了新的兴趣波的理论和实践方面的天体力学，使其与数学和应用的联系比以往任何时候都更强。天体力学的一个主要方面是研究牛顿的N体问题，即分析N个点粒子在牛顿运动定律支配下的运动。现在，对N体问题的研究几乎包括了任何与牛顿N体问题有一点相似的动力学系统，而且除了在天文学和空间科学中提供的洞察力之外，它在科学和工程学中也有许多应用。第一个问题涉及理论性质的调查。第二，注重应用方面。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（即单应运动和相对平衡）以及在这种运动中允许的特殊构型（即中心构型）。中心构型的研究是重要的，因为它们在理解N体行为中起着“中心作用”：它们表征了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着关键作用。从一个更面向应用的角度来看，我的研究计划是针对发展的数值方法，提高计算效率，并允许新的系统进行模拟，具有更高的精度。此外，我对分析全身问题的几个方面感兴趣。 这是一个迷人的问题，有许多开放的问题，并涉及许多重要的科学和工程问题，例如双小行星，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.