Shape Space Figure-$8$ Solution of Three Body Problem with Two Equal Masses

Abstract

In a preprint by Montgomery \cite{Mo99}, the author attempted to prove the existence of a shape space Figure-$8$ solution of the Newtonian three body problem with two equal masses (it looks like a figure $8$ in the shape space, which is different from the famous Figure-$8$ solution with three equal masses \cite{CM00}). Unfortunately there is an error in the proof and the problem is still open. Consider the $\alpha$-homogeneous Newton-type potential, $1/r^{\alpha },$ using action minimization method, we prove the existence of this solution, for $\alpha \in (1,2)$; for $\alpha =1$ (the Newtonian potential), an extra condition is required, which unfortunately seems hard to verify at this moment.

23 pages, 4 figures

References in corpus (3)

• From Brake to Syzygy
• Realizing All Free Homotopy Classes for the Newtonian Three-Body Problem
• Simple choreographies of the planar Newtonian $N$-body Problem

Cited by in corpus (2)

• Simple choreographies of the planar Newtonian $N$-body Problem
• Application of Morse index in weak force $N$-body problem