Published at: Physica D - Nonlinear Phenomena

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

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