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arXiv · 1507.02892

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

Guowei Yu
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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.

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Guowei Yu. 2015-07-10. Shape Space Figure-$8$ Solution of Three Body Problem with Two Equal Masses. https://arxiv.org/abs/1507.02892

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