Minimum Energy Configurations in the $N$-Body Problem and the Celestial Mechanics of Granular Systems

TL;DR

This paper explores minimum energy configurations in celestial mechanics, emphasizing finite density distributions over point masses, and extends the analysis to granular systems, providing specific results for small N and hypotheses for large N.

Contribution

It demonstrates that minimum energy configurations are well-defined for finite density distributions and develops hypotheses for large N granular systems, extending classical celestial mechanics.

Findings

All relative equilibria for N=1,2,3 identified

Minimum energy configurations for N=1,2,3 determined

Hypotheses proposed for large N configurations

Abstract

Minimum energy configurations in celestial mechanics are investigated. It is shown that this is not a well defined problem for point-mass celestial mechanics but well-posed for finite density distributions. This naturally leads to a granular mechanics extension of usual celestial mechanics questions such as relative equilibria and stability. This paper specifically studies and finds all relative equilibria and minimum energy configurations for $N=1,2,3$ and develops hypotheses on the relative equilibria and minimum energy configurations for $N\gg 1$ bodies.