On the separation of variables into relative and center of mass motion for two-body system in three-dimensional spaces of constant curvature

TL;DR

This paper introduces biquaternion-based expressions for separating center of mass and relative motions in two-body systems within three-dimensional spaces of constant curvature, highlighting the algebraic challenges due to noncommutativity.

Contribution

It formulates the separation problem using biquaternions and explains the algebraic reasons behind nonseparability in curved spaces.

Findings

Biquaternion expressions for motion variables are derived.

Separation is nontrivial due to noncommutative algebra.

Special cases of separation are analyzed.

Abstract

Expressions for variables of the center of mass and relative motions for two-body system with different and equal masses in three-dimensional spaces of constant curvature are introduced in the terms of biquaternions. The problem of the separation of center mass and relative motion variables for action of two particles into biquaternionic form is formulated. We showed that the algebraic nature of these nonseparable variables follows from the fact that algebra of biquaternions is noncommutative. Some special cases of separation of center mass and relative motion variables are considered.