Saari's Conjecture for Elliptical Type $N$-Body Problem and An Application

TL;DR

This paper proves Saari's conjecture for the elliptical type N-body problem using an arithmetic approach and applies it to show that variational minimal solutions are relative equilibria, without assuming finiteness of central configurations.

Contribution

It introduces a novel proof of Saari's conjecture for a specific N-body problem and extends the application to the planetary restricted problem, bypassing previous assumptions.

Findings

Saari's conjecture is proven for the elliptical type N-body problem.

Variational minimal solutions are shown to be relative equilibria.

The proof does not require the finiteness of central configurations.

Abstract

By using an arithmetic fact, we will firstly prove Saari's conjecture in a particular case, which is called the Elliptical Type N-Body Problem, and then we apply it to prove that the variational minimal solution of the planar Newtonian N-body problem is precisely a relative equilibrium solution whose configuration minimizes the function $IU^2$, it's worth noticing that we don't need the hypothesis of Finiteness of Central Configurations. In the Planetary Restricted Problem (which ignore all the mutual gravitational interactions between the planets), the corresponding Saari's conjecture is stated and proved.