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

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 $α$-homogeneous Newton-type potential, $1/r^α,$ using action minimization method, we prove the existence of this solution, for $α\in (1,2)$; for $α=1$ (the Newtonian potential), an extra condition is required, which unfortunately seems hard to verify at this moment.

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