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

Abstract

We are concerned with the spatial twisted central configuration formed by two paralleled regular \(N\)-polygons (\(N\geqslant 3\)) with twist angle \(\theta \), and we prove that the values of the masses in each separate regular \(N\)-polygon are equal if and only if \(\theta =0\) or \(\theta =\pi /N\). Moreover, suppose that all of the sides in one regular \(N\)-polygon have length \(a\) and all of the sides in the other regular \(N\)-polygon have length 1, then for the case of the twist angle is \(\theta =\pi /N\), if \(3\leqslant N\leqslant 8\), for any \(a>0\), the spatial twisted central configuration for \(2N\)-body problem exists. Furthermore, whether the twist angle is \(\theta =0\) or \(\theta =\pi /N\): for every \(N\geqslant 3\), if all the masses of the \(2N\) bodies or the lengths of all the sides of the two regular \(N\)-polygons are equal to each other, then there is only one spatial central configuration, and the distance between the two paralleled regular \(N\)-polygons with \(\theta =0\), must be larger than the case of \(\theta =\pi /N\).