Saari's Conjecture and Variational Minimal Solutions for $N$-Body Problems

TL;DR

This paper proves Saari's conjecture in a specific case and demonstrates that variational minimal solutions in the planar N-body problem are exactly the relative equilibria minimizing a certain function.

Contribution

It introduces a proof of Saari's conjecture for a particular case and characterizes variational minimal solutions as relative equilibria in the planar N-body problem.

Findings

Proof of Saari's conjecture in a specific case

Characterization of variational minimal solutions as relative equilibria

Minimization of the function IU^2 in the configuration space

Abstract

In this paper, we will prove Saari's conjecture in a particular case by using a arithmetic fact, and then, apply it to prove that for any given positive masses, the variational minimal solutions of the N-body problem in ${\mathbb{R}}^2$ are precisely a relative equilibrium solution whose configuration minimizes the function $IU^2$ in ${{\mathbb{R}}}^2$.