Title:

Blowing up the N-body Problem

 

 

Abstract:

From the mathematical point of view, one of the special features of the Newtonian N-body problem which makes it so interesting and so difficult is the non-compactness of the phase space. The bodies can fly off to infinity or reach the excluded set of collision singularities. McGehee's blowup method provides a partial compactification which allows one to study solutions near total collision (all bodies collide one point) or parabolic infinity (all bodies tend to infinity with zero limiting velocity). Then, using mostly qualitative geometrical reasoning, one can prove existence of a variety of interesting solutions. My lectures will introduce McGehee's coordinates and discuss several applications with emphasis on low-dimensional cases where one can visualize the results with pictures. Even familiar classical results can be illuminated by blowing things up. Some of the topics to be covered: the blown-up two-body problem -- collisions and parabolic orbits, the shape sphere for the planar three-body problem, spiraling stable and unstable manifolds, solutions bi-asymptotic to triple collision, symbolic dynamics and chaos near triple collision, realizing syzygy sequences and the structure of parabolic solutions.