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N体问题，即分析在动力学受牛顿引力定律支配的情况下N点粒子的运动。现在对N-体问题的研究包括几乎所有与牛顿N-体问题极其相似的动力系统。除了牛顿N体问题外，与这一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涡旋问题。涡旋问题可以看作是一个N体问题，因为在理想流体中，人们可以研究涡旋(即流体中的点在流体旋转的地方)如何相互作用，而不需要参考背景流体。在平面牛顿N体问题和平面N涡旋问题中，特别感兴趣的是在均匀旋转的坐标系中看起来是固定的解。这样的解被称为相对平衡，而在这种运动中允许的特殊构型称为相对平衡构形。我对研究相对平衡的性质及其定义的构型很感兴趣。对这些构型的研究很重要，因为它们在理解N体行为方面发挥了重要作用：它们描述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发挥了关键作用。我想研究的另一类问题是全身问题。全身问题涉及两个或多个分布的物体之间的动力学相互作用。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、分子的反应和电离以及水下机器人的稳定性和控制。此外，我还感兴趣地分析了常曲率空间上N体问题的几个方面。在这一领域发现的一些数学现象是如此新鲜和令人惊讶，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.