Realizing All Free Homotopy Classes for the Newtonian Three-Body Problem

Abstract

The configuration space of the planar three-body problem when collisions are excluded has a rich topology which supports a large set of free homotopy classes. Most classes survive modding out by rotations. Those that survive are called the reduced free homotopy classes and have a simple description when projected onto the shape sphere. They are coded by syzygy sequences. We prove that every reduced free homotopy class, and thus every reduced syzygy sequence, is realized by a reduced periodic solution to the Newtonian planar three-body problem. The realizing solutions have nonzero angular momentum, repeatedly come very close to triple collision, and have lots of "stutters"--repeated syzygies of the same type. The heart of the proof is contained in the work by one of us on symbolic dynamics arising out of the central configurations after the triple collision is blown up using McGehee's method.