Hamiltonian and Celestial Mechanics

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Project Abstract DMS-0500443This is a research project in the general area of dynamical systems theory with emphasis on Hamiltonian and celestial mechanics. The gravitational n-body problem remains an active topic for research three centuries after Newton proposed it. Over the years it has served as a stimulus for the development of new mathematics. Three aspects of the problem will be pursued. In brief, they are i) the development of computational algebra techniques for systems of polynomial equations and their application to the problem of relative equilibria of the n-body problem, ii) the study of continuation of families of periodic solutions by a combination of the topological method of isolating blocks and classical variational methods, and iii) the analysis of a simplified model problem which may lead to better understanding of the phenomenon of Arnold diffusion in Hamiltonian systems.The first part of the project concerns the simplest periodic motions of the n-body problem, the so-called relative equilibria. These are special arrangements of the masses such that the gravitational forces can be exactly balanced by centifugal forces when the configuration rotates. For example, three masses arranged in the shape of an equilateral triangle can be rigidly rotated about the center of mass to produce a solution of the three-body problem. For four or more masses, it is a difficult algebraic problem to find or even count these special solutions. New computational algebra methods have met with some success recently the plan is to continue to develop and apply these ideas. Numerical studies show that nearby some of the relative equilibrium solutions there are other interesting families of periodic orbits. The second part of the project is about developing methods to mathematically prove the existence of such solutions. Two different types of proofs will be attempted, one based on qualitative geometrical arguments and another based on a venerable analytical technique. Both of these have been used before in celestial mechanics but the problem to be studied here is essentially different and new ideas are needed. Finally, the third part of the project moves from the study of orderly periodic motions to the realm of chaotic dynamics. It has been known for many years that certain kinds of chaotic motions can lead to instability in mechanical systems. Instability in celestial mechanics can give rise to such phenomena as the slow drifting of the orbital parameters of planets or asteroids. A specific mechanism for such chaotic drifting, known as Arnold diffusion, is currently a very active area of dynamical systems research. However, it is difficult to understand in a system as complex as the gravitational n-body problem. The approach taken in this project will be to study a simplified model problem which is designed to capture the essential features of the real situation but is more amenable to mathematical analysis. Broader impacts of the project include development of scientific software, mentoring of young mathematicians and possible applications to spacecraft mission design.