Chazy showed that when a solution of the n-body problem tends to total collision then its normalized configuration converges to the set of normalized central configurations. In the planar problem, there are circles of rotationally equivalent central configurations. It's conceivable that by means of an "infinite spin", a total collision solution could converge to such a circle instead of to a particular point on it. Chazy proved that this is not possible if a certain nondegeneracy condition holds. I will discuss joint work with R. Montgomery, where we extended this to the degenerate case, at least if the limiting circle of central configurations is isolated from other circles of central configurations. (It is believed that all central configurations are isolated, even if they are degenerate, but this is not known in general.) Our proof relies on combining the center manifold theorem with the Łojasiewicz gradient inequality. The talk will also describe an explicit example of convergence to a degenerate central configuration of the planar four-body problem discovered by Palmore.
Alain Albouy, Alain Chenciner, Jacques Laskar
