Central configurations in three dimensions

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