Existence of Really Perverse Central Configurations in the Spatial $N$-Body Problem

TL;DR

The paper proves the existence of spatial really perverse central configurations for the Newtonian $N$-body problem with $N$ in the range $27$ to $55$. It constructs a symmetric, polygonal-plus-two-points configuration and reduces the dynamics to a set of algebraic conditions, then shows a root exists for a determinant function $f_n(\alpha)$ when $n\ge 24$, producing positive mass distributions. By eliminating a central mass variable, it opens a path to explicit positive-mass solutions, thereby filling a gap in the literature where only planar examples were known. The results demonstrate that truly spatial really perverse central configurations exist and provide explicit parameter ranges and constructive proof techniques that may inform further bifurcation analyses and energy-manifold studies in celestial mechanics.

Abstract

We construct explicit examples of really perverse central configurations in the spatial Newtonian $N$-body problem. A central configuration is called really perverse if it satisfies the central configuration equations for two distinct mass distributions having the same total mass. While such configurations were previously known only in the planar case for large $N$, we prove the existence of spatial really perverse central configurations for $N=27,\dots,55$.

Figures (2)

• Figure 1: Configuration of \ref{['eqn:symsol']}
• Figure 2: Graphs of $-\tfrac{7}{4}(\tfrac{H_n}{n} - 1)$ and $(\tfrac{2}{(1 + \alpha^2)^{3/2}} - 2)(\tfrac{\alpha^3}{(1 + \alpha^2)^{3/2}} - 1)$.

Theorems & Definitions (2)

• Definition 1
• Theorem 1