Stacked central configurations with a homogeneous potential in $\mathbb{R}^3$

TL;DR

This paper extends the theory of stacked central configurations in three-dimensional space to general homogeneous potentials, analyzing admissible parameters, properties of regular polygons, and the uniqueness of stacked configurations.

Contribution

It generalizes previous results to broader potentials in -dimensional space, identifying admissible parameters and properties of specific configurations.

Findings

Identified admissible  for convex configurations with Newtonian potential

Analyzed properties of regular n-gon co-circular configurations

Showed stacked property is unique to central configurations

Abstract

In this paper we generalize some results in \cite{Yu2021} concerning stacked central configurations. We can deal with the general homogeneous potential $U_{\alpha}$ (containing the vortex case) in $\mathbb{R}^3$. We give the admissible set of $\alpha$ for a convex central configuration (with respect to the Newtonian potential i.e. $\alpha=3$). We discuss some properties of the regular $n$-gon co-circular central configurations. We also find that the stacked property is particular for central configurations by studying the S-balanced configuration case.