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

TL;DR

This paper proves the existence of a shape space Figure-8 solution in the three body problem with two equal masses using action minimization, for certain potential exponents, addressing an open problem in celestial mechanics.

Contribution

It establishes the existence of the shape space Figure-8 solution for a range of potentials, extending previous incomplete results and clarifying conditions for the Newtonian case.

Findings

Existence proven for $eta eq 1$ using action minimization.

Additional conditions needed for the Newtonian potential case.

Addresses an open problem in the three body problem literature.

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.