Regularization of the circular restricted three-body problem using 'similar' coordinate systems

TL;DR

This paper introduces a novel regularization method for the three-body problem using 'similar' coordinate systems, involving Hamiltonian transformations and fictitious time, with numerical analysis for the Earth-Moon system.

Contribution

It proposes a new regularization approach based on 'similar' coordinate systems and provides explicit formulas and geometric analysis for the three-body problem.

Findings

Regularized equations derived for both massive and less massive star-centered coordinates

Geometric interpretation of the 'similar' polar angle and Levi-Civita transformation

Numerical comparison of regularized equations for Earth-Moon system

Abstract

The regularization of a new problem, namely the three-body problem, using 'similar' coordinate system is proposed. For this purpose we use the relation of 'similarity', which has been introduced as an equivalence relation in a previous paper (see \cite{rom11}). First we write the Hamiltonian function, the equations of motion in canonical form, and then using a generating function, we obtain the transformed equations of motion. After the coordinates transformations, we introduce the fictitious time, to regularize the equations of motion. Explicit formulas are given for the regularization in the coordinate systems centered in the more massive and the less massive star of the binary system. The 'similar' polar angle's definition is introduced, in order to analyze the regularization's geometrical transformation. The effect of Levi-Civita's transformation is described in a geometrical manner. Using the resulted regularized equations, we analyze and compare these canonical equations numerically, for the Earth-Moon binary system.