Realizing all reduced syzygy sequences in the planar three-body problem

Abstract

The configuration space of the planar three-body problem, reduced by rotations and with collisions excluded, has a rich topology which supports a large set of free homotopy classes. These classes have a simple description in terms of syzygy (or eclipse) sequences. Each homotopy class corresponds to a unique ‘reduced’ syzygy sequence. We prove that each reduced syzygy sequence is realized by a periodic solution of the rotation-reduced Newtonian planar three-body problem. The realizing solutions have small, nonzero angular momentum, repeatedly come very close to triple collision, and have lots of ‘stutters’—repeated syzygies of the same type, which cancel out up to homotopy. The heart of the proof stems from 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. We end with a list of open problems.