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

Dedicated to James Montaldi

The author acknowledges support for his research from the Program UNAM-DGAPA-PAPIITIN115820 and from the Alexander von Humboldt Foundation for a Georg Forster Experienced Researcher Fellowship that funded a research visit to TU Berlin where part of this work was done

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  • 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 [6] (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.

    Mathematics Subject Classification: Primary: 70F05; Secondary: 70A05.


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  • Figure 1.  Illustration of the centre of mass $ \boldsymbol{\bar {q}} $ according to the characterisations C1, C2 and C3

    Figure 2.  The value of $ r_2 $ as a function of $ \kappa $ according to Eqs. (3), (4) and (5) under the assumption that $ 2\mu_1 = \mu_2 $ and $ r_1 = 1 $. Note that for $ \kappa>0 $ there are two branches for (5) as described in the text. The shaded area corresponds to values of $ (\kappa, r_2) $ that are forbidden since they violate the restriction that $ r = 1+r_2<\pi/ \sqrt{\kappa} $

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