Pair Space in Classical Mechanics III. Some Four-Body Central Configurations

TL;DR

This paper introduces a pair-space formalism to analyze four-body central configurations in classical mechanics, identifying specific geometric shapes like tetrahedra, kites, and trapeziums based on mass ratios.

Contribution

The paper develops a new vectorial formalism for four-body central configurations and classifies specific non-collinear solutions with equal inter-body distances.

Findings

Identifies tetrahedron, kites, and trapezium as key configurations.

Provides mathematical relations linking shapes to mass ratios.

Formalizes the conditions for these configurations using pair-space equations.

Abstract

We study central configurations in the four body problem, i.e., configurations in which the forces on all the bodies point to a fixed, single point in space. The newly formulated pair-space formalism yields a set of vectorial equations that fully characterize such configurations. We investigate a sub-class of solutions in which at least two pairs of inter-body distances are equal. The only such non-collinear configurations are the tetrahedron (the unique non-planar configuration), kites and the isosceles trapezium. The specific shapes (internal angles) are determined by the ratio of the masses of the bodies. Mathematical expression are given for all these relations.