Central configurations of the spatial seven-body problem: two twisted triangles plus one

Abstract

In this work we are interested in the central configurations of the spatial seven-body problem where six of them are at vertices of two congruents equilateral triangles belong to parallel planes and one triangle is a rotation by the angle $\pi/3$ from the other one. We show a existence and uniqueness of a class of central configuration with this shape, namely six bodies with equal masses at the vertice of a regular octahedron with the seventh body in its center. There is not restriction to the mass of the seventh body.