This project is devoted to research in the general area of dynamical systems theory with emphasis on Hamiltonian and celestial mechanics. Several very different aspects of the subject will be studied. Part of the project deals with the special periodic solutions of the n-body problem arising from central configurations. It is a long-standing open problem to determine how many central configurations are possible, or even if the number is finite. Techniques from algebraic geometry and computational algebra will be used to attack this question. A second part of the project involves the construction of isolating blocks in the phase space of the three-body problem. It is known that it is possible to find simple, explicit isolating blocks at the collinear Lagrange points of the restricted three-body problem. These blocks can be used to study the complicated invariant set nearby. The project is to carry out such a construction near the collinear central configuration of the unrestricted three-body problem. Here the dimension of the phase space is higher and the geometry is much more complicated. A final part of the project is concerned with understanding the mechanism of Arnold diffusion for Hamiltonian systems. The approach taken here is based on the construction of an invariant Cantor set of annuli and subsequent analysis of the resulting dynamics.
The gravitational n-body problem remains an active topic for mathematics research three centuries after Newton proposed it. Over the years it has been a stimulus for the development of new mathematics of wide applicability. It is the classic example of a nonlinear mechanical system and its solutions include orderly cyclical motions, multi-body collisions, and irregular, chaotic behavior. The simplest solutions are the rigidly rotating orbits arising from the central configurations. Central configurations are special arrangements of the masses such that the gravitational forces can be exactly balanced by centrifugal forces when the configuration rotates. Although these solutions are dynamically very simple, the problem of finding or even counting the central configurations turns out to be very difficult when there are four or more masses involved. Part of this project is about how to deal with very complicated algebraic problems such as this, perhaps using the help of computers. The central configurations are important landmarks which provide a starting point for further analysis. It turns out that there are many other interesting solutions near the simple, rigidly rotating ones. One way to trap and study these nearby orbits involves the construction of so-called isolating blocks. The geometry of these blocks provides qualitative information about the solutions inside and can also form the basis of numerical methods for approximating these solutions. Finally, there is the problem of understanding chaotic dynamics in mechanical systems and how such behavior can lead to large-scale instability. Instability in celestial mechanics can give rise to such phenomena as the slow drifting of the orbital parameters of planets or asteroids. It also occurs in a variety of other mechanical systems. This phenomenon of Arnold diffusion is only partially understood at present. The new approach which will be pursued here seems promising but much work remains to be done before it can be applied to complex systems like the n-body problem.
