Moments, Equilibrium Equations and Mutual Distances

TL;DR

This paper reviews and extends the classical theory of moments for weighted point configurations, deriving new equilibrium equations based on mutual distances that unify and generalize existing models in arbitrary dimensions.

Contribution

It introduces a unified framework for equilibrium problems using moments, extends classical equations to broader classes, and provides new algebraic equations for mutual distances and configurations.

Findings

Derived homogeneous, isometry-invariant equilibrium equations

Extended $n$-body central configuration equations to new classes

Introduced novel algebraic equations for mutual distances

Abstract

We review and develop the classical theory of moments of configurations of weighted points with a focus on systems with an identically vanishing first moment. The latter condition produces equations for equilibrium configurations of systems of interacting particles under the sole condition that interactions are between pairs of particles and along the lines connecting such pairs. Complying external forces are admitted, so the description of some dynamical equilibrium configurations, such as relative equilibria in Celestial Mechanics, is included in our approach. Moments provide a unified framework for equilibrium problems in arbitrary dimensions. The equilibrium equations are homogeneous and invariant by isometries (for interactions depending only on mutual distances), and are obtained through simple algebraic procedures requiring neither reduction by isometries nor a variational principle for their determination. Our equations include the renowned set of $n$-body central configuration equations by A. Albouy and A. Chenciner. These equations are extended to a rather broad class of equilibrium problems, and new equilibrium equations written in terms of mutual distances are introduced. We also apply moments to the theory of constraints for mutual distances of configurations of fixed dimension, and for co-spherical configurations, thus re-obtaining and adding to classical results by A. Cayley and successors. For the sake of concreteness, novel sets of central configurations equations are provided.