Compactifying collisions in the N-body problem
Richard Montgomery
2.00 pm-4.00 pm May 13 (MONDAY!)
383N (3rd floor lounge) at STANFORD
Triple collisions act like an essential singularity for the
Newtonian three-body problem: one cannot continue the dynamics through
triple collision in any reasonable way. However, one can prove that triple
collisions are approached asymptotically along one of five possible
limiting shapes -- the central configurations, with analogous results for
collisions in the N-body problem. The basic analytical tool was invented
by R. McGehee in 1974. It is a real blow up of N-tuple collision, combined
with a clever rescaling of velocities and time. The result is the
addition of a manifold with boundary to phase space. This boundary is
called the collision manifold and the flow extends smoothly to it. The
flow on the collision manifold is gradient-like: it admits a Liapanov
function The collision flow helps to organize the dynamics near collision.
The only rest points for the full extended flow lie on the collision
manifold. They are the `central configurations' alluded to above. There
are exactly five central configurations for N = 3, and these were
discovered by Euler and Lagrange. For N = 4 or greater it is an open
problem to show that the central configurations are finite in number. My
goal is to motivate and explain McGehee's coordinates, describe its
underlying geometry, and present some applications of the collision
manifold methods.
Annotated references