Some remarks about the centre of mass of two particles in spaces of constant curvature

Luis C García-Naranjo

doi:10.3934/jgm.2020020

Abstract

The concept of centre of mass of two particles in 2D spaces of constant Gaussian curvature is discussed by recalling the notion of "relativistic rule of lever" introduced by Galperin [Comm. Math. Phys. 154 (1993), 63--84] and comparing it with two other definitions of centre of mass that arise naturally on the treatment of the 2-body problem in spaces of constant curvature: firstly as the collision point of particles that are initially at rest, and secondly as the centre of rotation of steady rotation solutions. It is shown that if the particles have distinct masses then these definitions are equivalent only if the curvature vanishes and instead lead to three different notions of centre of mass in the general case.

Highlights

Consider two particles with masses μ1, μ2 > 0 located at q1, q2 ∈ R2
It is well-known that solutions to the 2-body problem satisfying the conditions in C3 do exist for any positive distance between the particles and for any attractive potential
We review the construction of Galperin [6] and extend it to general values of the curvature in Section 3 to establish (1.3)

Summary

Introduction

Consider two particles with masses μ1, μ2 > 0 located at q1, q2 ∈ R2. As is well-known, their centre of mass is the point q ∶=. It is well-known that solutions to the 2-body problem satisfying the conditions in C3 do exist for any positive distance between the particles and for any attractive potential. They are the starting point to consider the planar circular restricted 3-body problem. Galperin’s paper only deals with the values of the curvature κ = ±1, but his construction may be naturally extended to all values of κ ∈ R (see section 3) In this case Eq (1.2) is replaced by the relativistic rule of lever: Generalisation of C1.

Basic working definitions of the spaces of constant curvature

Final remarks