This is a research project in the general area of dynamical systems theory with applications to problems arising in mechanics, especially celestial mechanics. The gravitational n-body problem remains an active topic for research three centuries after Newton proposed it. In addition to its direct relevance for astronomy and space mission design, it has served as a stimulus for the development of new mathematical techniques. The research is focussed on three main areas: regularization of singularities, topological and variational existence proofs for periodic orbits, and studies of relative equilibrium solutions. One of the characteristic features of celestial mechanics is the presence of collision singularities. It has long been known that simple binary collisions can be ``regularized'' by cleverly chosen changes of coordinates. More complicated singularities like the triple collision in the three-body problem can be ``blown-up'' so as to reveal the quite complicated behavior of near-collision solutions. Part of the proposal is to give a complete regularization and blow-up for the three-body problem which combines the older work on the subject with modern developments in symplectic reduction theory. A similar project for the four-body problem will involve many new ideas and has surprising connections to algebraic surface theory.
Periodic motions have always played an important role in dynamical systems theory. They are often the simplest solutions of the equations of motion and provide a framework for understanding other, more complicated solutions. Recently, variational methods have been used to construct some remarkable symmetric periodic solutions of the n-body problem. This project will develop topological techniques for finding such solutions and will explore the relationship between the variational and topological approaches. The simplest periodic orbits arise from central configurations -- special arrangements of the masses such that the gravitational forces can be exactly balanced by centrifugal forces when the configuration rotates. It is a difficult algebraic problem to find or even count the central configurations. Techniques developed for this problem could be applied to other complicated systems of algebraic equations.
