Published at: Physica D - Nonlinear Phenomena

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

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 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.

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