An existence proof of a symmetric periodic orbit in the octahedral six-body problem

TL;DR

This paper proves the existence of a symmetric periodic orbit in the six-body problem with equal masses, characterized by three double collisions per period and no multiple collisions, using variational methods.

Contribution

It provides the first existence proof of such a symmetric periodic orbit in the octahedral six-body problem through action minimization techniques.

Findings

Existence of a symmetric periodic orbit with specific collision properties.

Orbit exhibits three double collisions per period.

No multiple collisions occur in the orbit.

Abstract

We present a proof of the existence of a periodic orbit for the Newtonian six-body problem with equal masses. This orbit has three double collisions each period and no multiple collisions. Our proof is based on the minimization of the Lagrangian action functional on a well chosen class of symmetric loops.