New coordinates for the Four-Body problem

Abstract

A new coordinate system is defined for the Four-Body dynamical problem with general masses, having as its origin of coordinates the center of mass. The transformation from the inertial coordinate system involves a combination of a rotation to the principal axis of inertia, followed by three changes of scale leading the principal moments of inertia to yield a body with three equal moments of inertia, and finally a second rotation that leaves unaltered the equal moments of inertia. These three transformations yield a mass-dependent rigid orthogonal tetrahedron of constant volume in the inertial coordinates. Each of those three linear transformations is a function of three coordinates that produce the nine degrees of freedom of the Four-Body problem, in a coordinate system with the center of mass as origin. The relation between the well known equilateral tetrahedron solution of the gravitational Four-Body problem and the new coordinates is exhibited, and the plane case of central configurations with four different masses is computed numerically in these coordinates.