Shape Space Figure $8$ Solutions of three body problem with two equal masses

Guowei Yu

Published 2015-07-10Version 1

In an unpublished paper by Richard Montgomery, the author claimed to prove the existence of shape space Figure-$8$ solutions of the Newtonian three body problem with two equal masses (they look like a figure $8$'s in the shape space, that is different from the famous Figure-$8$ solution proved by Chenciner and Montgomery). However there is an error in the proof and the problem is still open. In this paper, based on a new local deformation result, we prove the existence of such solutions in weak force potential, $1<\alpha <2$ and in Newtonian potential under an extra condition. This result can be seen as a small step in the program of realizing (reduced) syzygy sequences in the three body problem that were proposed by Montgomery.