Mathematical Methods in Classical and Celestial Mechanics

点击翻译按钮获取中文摘要

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.