Notes on spatial twisted central configurations for $2N$-body problem

TL;DR

This paper investigates spatial central configurations formed by two twisted regular N-polygons in the 2N-body problem, proving size equality under any twist angle and mass ratio.

Contribution

It establishes that the two regular N-polygons must be of equal size regardless of twist angle and mass ratio, advancing understanding of spatial central configurations.

Findings

Sizes of the two N-polygons must be equal for any twist angle.

The result holds for any mass ratio between the polygons.

Provides conditions for spatial twisted central configurations.

Abstract

We study the spatial central configuration formed by two twisted regular $N$-polygons. For any twist angle $\theta$ and any ratio of the masses $b$ in the two regular $N$-polygons, we prove that the sizes of the two regular $N$-polygons must be equal.