Hamiltonian and Celestial Mechanics

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This research project concerns dynamical systems theory, with emphasis on questions in classical and celestial mechanics. Dynamical systems theory is devoted to the mathematical study of systems that evolve in time, including the gravitational n-body problem. The research in this project has direct applications to understanding the possible motions of celestial bodies such as planets, moons, and asteroids. But its primary importance lies in the development of new mathematical methods that can be applied in other contexts. From the mathematical point of view, the questions under study involve finding and understanding the solutions of a complex system of differential equations. Such equations are too complicated to solve explicitly, and instead are studied using a combination of computer simulations and theoretical, mathematical reasoning. The planar three-body problem is a classical dynamical system with a long history that still presents formidable mathematical challenges. Many simple and beautiful periodic motions have been discovered, but only a few have been understood at the level of mathematical proof. The system is formulated as ordinary differential equations in five dimensions. Using Poincare sections, the periodic orbits can be found as fixed points of four-dimensional mappings. One part of this project is devoted to developing new topological methods for finding periodic orbits. One of the characteristic features of celestial mechanics is the presence of singularities. The behavior of orbits near collisions can be chaotic. There are several open questions in this area under investigation as part of the research. The simplest periodic orbits in the n-body problem are relative equilibrium motions that arise from planar central configurations. Understanding central configurations in the plane and their generalization to higher dimensions is another goal of this project. It is a difficult algebraic problem to find or even count the central configurations. The problems of finiteness of the number of central configurations and the question of stability of the resulting periodic solutions will be investigated.