The planar 3-body problem II:reduction to pure shape and spherical geometry (2nd version)

TL;DR

This paper explores the geometric reduction of the planar three-body problem using equivariant Riemannian geometry, linking shape curves on a sphere to the dynamics of three-body motions.

Contribution

It introduces a geometric framework that reduces three-body trajectories to shape curves on a sphere, enabling reconstruction of motion from shape data.

Findings

Shape curves determine the moduli and time parametrization of three-body motions.

The shape space is a Riemannian cone over a shape sphere, facilitating geometric analysis.

Time parametrization is recoverable from the shape curve and potential function.

Abstract

Geometric reduction of the Newtonian planar three-body problem is investigated in the framework of equivariant Riemannian geometry, which reduces the study of trajectories of three-body motions to the study of their moduli curves, that is, curves which record the change of size and shape, in the moduli space of oriented mass-triangles. The latter space is a Riemannian cone over the shape 2-sphere, and the shape curve is the image curve on this sphere. It is shown that the time parametrized moduli curve is in general determined by the relative geometry of the shape curve and the shape potential function. This also entails the reconstruction of time, namely the geometric shape curve determines the time parametrization of the moduli curve, hence also the three-body motion itself, modulo a fixed rotation of the plane. The first version of this work is an (unpublished) paper from 2012, and the present version is an editorial revision of this.