|Location:||MSRI: Baker Board Room|
The planar (n+1)-body problem models the motion of n+1
bodies in the plane under their mutual Newtonian gravitational attraction
forces. When n=3, the question about final motions, that is, what are the
possible limit motions in the planar (n+1)-body problem when time goes to infinity, ceases
to be completely meaningful due to the existence of non-collision singularities.
In this paper we prove the existence of solutions of the planar (n+1)-body
problem which are defined for all forward time and tend to a parabolic motion,
that is, that one of the bodies reaches infinity with zero velocity while the rest
perform a bounded motion.
These solutions are related to whiskered parabolic tori at infinity, that is,
parabolic tori with stable and unstable invariant manifolds which lie at infinity.
These parabolic tori appear in cylinders which can be considered “normally
The existence of these whiskered parabolic tori is a consequence of a general
theorem on parabolic tori developed here. Another application of our theorem
is a conjugation result for a class of skew product maps with a parabolic torus
with its normal form generalizing results of Takens and Voronin.