Symmetric Regularization, Reduction and Blow-Up of the Planar Three-Body Problem

TL;DR

This paper introduces a comprehensive coordinate transformation approach to simplify, regularize, and analyze the planar three-body problem, effectively handling symmetries and singularities to facilitate deeper understanding.

Contribution

It presents a novel sequence of coordinate changes that eliminate symmetries, regularize collisions, and enable a global analysis using modern geometric methods.

Findings

Successfully regularizes all collision singularities.

Provides a global framework using size and shape coordinates.

Derives reduced equations in various local coordinate systems.

Abstract

We carry out a sequence of coordinate changes for the planar three-body problem which successively eliminate the translation and rotation symmetries, regularize all three double collision singularities and blow-up the triple collision. Parametrizing the configurations by the three relative position vectors maintains the symmetry among the masses and simplifies the regularization of binary collisions. Using size and shape coordinates facilitates the reduction by rotations and the blow-up of triple collision while emphasizing the role of the shape sphere. By using homogeneous coordinates to describe Hamiltonian systems whose configurations spaces are spheres or projective spaces, we are able to take a modern, global approach to these familiar problems. We also show how to obtain the reduced and regularized differential equations in several convenient local coordinates systems.