Pair Space in Classical Mechanics II. N-Body Central Configurations

TL;DR

This paper explores the properties of N-body central configurations in classical mechanics using the concept of pair space, deriving conditions for both collinear and non-collinear configurations and generalizing known relations.

Contribution

It introduces new geometric and dynamical conditions for N-body central configurations, extending previous results to arbitrary N and linking pair angular momentum conservation to centrality.

Findings

Central configurations conserve all pair angular momenta.

Derived bounds on minimal length for collinear configurations.

Established necessary geometric conditions for non-collinear configurations.

Abstract

A previous work introduced pair space, which is spanned by the center of mass of a system and the relative positions (pair positions) of its constituent bodies. Here, I show that in the $N$-body Newtonian problem, a configuration that does not remain on a fixed line in space is a central configuration if and only if it conserves all pair angular momenta. For collinear systems, I obtain a set of equations for the ratios of the relative distances of the bodies, from which I derive some bounds on the minimal length of the line. For the non-collinear case I derive some geometrical relations, independent of the masses of the bodies. These are necessary conditions for a non-collinear configuration to be central. They generalize, to arbitrary $N$, a consequence of the Dziobek relation, which holds for $N=4$.