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

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 ${{\mathbb{R}}}^2$.