Pair Space in Classical Mechanics I. The Three-Body Problem

TL;DR

This paper introduces a novel pair-space framework for classical mechanics, simplifying the analysis of the three-body problem and deriving new bounds on relative distances.

Contribution

It proposes an extended configuration space called pair-space with auxiliary variables, applied to the three-body problem with generalized potentials, providing new insights and bounds.

Findings

Derived equilateral and collinear solutions easily

Obtained new bounds on relative distances in collinear solutions

Simplified analysis of three-body problem with generalized potential

Abstract

I introduce an extended configuration space for classical mechanical systems, called pair-space, which is spanned by the relative positions of all the pairs of bodies. To overcome the non-independence of this basis, one adds to the Lagrangian a term containing auxiliary variables. As a proof of concept, I apply this representation to the three-body problem with a generalized potential that depends on the distance $r$ between the bodies as $r^{-n}$. I obtain the equilateral and collinear solutions (corresponding to the Lagrange and Euler solutions if $n=1$) in a particularly simple way. In the collinear solution, this representation leads to several new bounds on the relative distances of the bodies.