New coordinates for the four-body problem

Abstract

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