Central Configurations in Three Dimensions

TL;DR

This paper studies the equilibria of point particles under combined attractive and repulsive central forces, focusing on numerical computation and statistical properties of configurations, revealing regular structures for small numbers and spherical distributions for large systems.

Contribution

It provides numerical analysis and insights into the structure of central configurations under inverse square and linear forces, including new observations on their large-system behavior.

Findings

Small systems form regular convex deltahedra.

Increasing particles leads to multi-shell structures.

Large systems tend to form homogeneous spherical distributions.

Abstract

We consider the equilibria of point particles under the action of two body central forces in which there are both repulsive and attractive interactions, often known as central configurations, with diverse applications in physics, in particular as homothetic time-dependent solutions to Newton's equations of motion and as stationary states in the One Component Plasma model. Concentrating mainly on the case of an inverse square law balanced by a linear force, we compute numerically equilibria and their statistical properties. When all the masses (or charges) of the particles are equal, for small numbers of points they are regular convex deltahedra, which on increasing the number of points give way to a multi-shell structure. In the limit of a large number of points we argue using an analytic model that they form a homogeneous spherical distribution of points, whose spatial distribution appears, from our preliminary investigation, to be similar to that of a Bernal hard-sphere liquid.