Computing collinear 4-Body Problem central configurations with given masses

TL;DR

This paper presents an algorithm to compute collinear four-body central configurations using an orthocentric tetrahedron model, enabling precise calculations of configurations for given masses and verifying all twelve configurations predicted by Moulton's theorem.

Contribution

The paper introduces a novel geometric algorithm utilizing orthocentric tetrahedra to compute collinear four-body central configurations based on specified masses.

Findings

Algorithm successfully computes all twelve Moulton configurations.

Method accurately determines particle positions in collinear configurations.

Applicable to various mass distributions for four-body problems.

Abstract

An interesting description of a collinear configuration of four particles is found in terms of two spherical coordinates. An algorithm to compute the four coordinates of particles of a collinear Four-Body central configuration is presented by using an orthocentric tetrahedron, which edge lengths are function of given masses. Each mass is placed at the corresponding vertex of the tetrahedron. The center of mass (and orthocenter) of the tetrahedron is at the origin of coordinates. The initial position of the tetrahedron is placed with two pairs of vertices each in a coordinate plan, the lines joining any pair of them parallel to a coordinate axis, the center of masses of each and the center of mass of the four on one coordinate axis. From this original position the tetrahedron is rotated by two angles around the center of mass until the direction of configuration coincides with one axis of coordinates. The four coordinates of the vertices of the tetrahedron along this direction determine the central configuration by finding the two angles corresponding to it. The twelve possible configurations predicted by Moulton's theorem are computed for a particular mass choice.