Mathematical Methods in Classical and Celestial Mechanics

经典力学和天体力学的一个重要方面是研究N体问题。最经典的N体问题是牛顿的N体问题，即分析N个点粒子在牛顿引力定律支配下的运动。现在，对N体问题的研究几乎包括了任何与牛顿N体问题稍有相似之处的动力学系统。除了牛顿的N体问题外，与这个建议最相关的还包括全身问题、常曲率空间上的N体问题和N涡问题。旋涡问题可以被看作是N体问题，因为在理想流体中，人们可以研究旋涡（即流体中流体旋转的点）如何相互作用，而不参考背景流体。特别感兴趣的是在平面牛顿N体问题和平面N涡问题的解决方案，似乎固定时，在一个均匀旋转的框架。这样的解称为相对平衡，在这样的运动中允许的特殊构型称为相对平衡构型。我对研究相对平衡的性质和它们所定义的构型很感兴趣。对这种构型的研究很重要，因为它们在理解N体行为中起着重要作用：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中起着关键作用。 我想研究的另一类问题是全身问题。整体问题涉及两个或多个分布式物体的动力学相互作用。 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，以及水下航行器的稳定性和控制。此外，我有兴趣在分析几个方面的N体问题的空间常曲率。在这一领域发现的一些数学现象是如此新颖和令人惊讶，以至于需要更好地理解它们。

One important aspect of classical and celestial mechanics is the study of N-body problems. The most classical of the N-body problems is 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 gravitational law. The study of N-body problems now includes just about any dynamical system that remotely resembles the Newtonian N-body problem. The most relevant for this proposal, besides the Newtonian N-body problem, include the full-body problems, the N-body problem on constant curvature spaces and the N-vortex problem. The vortex problem can be viewed as a N-body problem since, in an ideal fluid, one can study how vortices (i.e. points in a fluid where the fluid is spinning) interact without reference to the background fluid. Of particular interest in the planar Newtonian N-body problem and in the planar N-vortex problem are solutions that appear fixed when viewed in a uniformly rotating frame. Such solutions are called relative equilibria and the special configurations that are allowed in such motions are called relative equilibria configurations. I am interested in studying properties of relative equilibria and the configurations they define. The study of such configurations is important because they play a major 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. Another class of problems that I would like to study are the Full-body problems. Full body problems are concerned with the dynamical interaction of two or more distributed bodies. 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, and stability and control of underwater vehicles. Additionally, I am interested in analyzing several aspects of the N-body problem on spaces of constant curvature. Some of the mathematical phenomena discovered in this area are so new and surprising that they need to be better understood.