Shooting for the Eight – a Topological Existence Proof for a Figure-eight Orbit of the Three-body Problem

A topological existence proof is given for a figure-eight periodic solution of the equal mass three-body problem. The proof is based on the construction of a Wazewski set W in the phase space. The figure-eight solution is then found by a kind of shooting argument in which symmetrical initial conditions entering W are followed under the flow until they exit W . A linking argument shows that the image of the symmetrical entrance states under this flow map must intersect an appropriate set of symmetrical exit states.