# American Institute of Mathematical Sciences

September  2020, 12(3): 435-446. doi: 10.3934/jgm.2020020

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

 Departamento de Matemáticas y Mecánica, IIMAS, UNAM, Apdo. Postal 20-126, Col. San Angel, Mexico City, 01000, MEXICO

Dedicated to James Montaldi

Received  September 2019 Revised  March 2020 Published  September 2020 Early access  July 2020

Fund Project: 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

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.

Citation: Luis C. García-Naranjo. Some remarks about the centre of mass of two particles in spaces of constant curvature. Journal of Geometric Mechanics, 2020, 12 (3) : 435-446. doi: 10.3934/jgm.2020020
##### References:
 [1] A. V. Borisov, I. S. Mamaev and A. A. Kilin, Two-body problem on a sphere. Reduction, stochasticity, periodic orbits, Regul. Chaotic Dyn., 9 (2004), 265-279.  doi: 10.1070/RD2004v009n03ABEH000280.  Google Scholar [2] A. V. Borisov, L. C. García-Naranjo, I. S. Mamaev and J. Montaldi, Reduction and relative equilibria for the two-body problem on spaces of constant curvature, Celest. Mech. Dyn. Astr., 130 (2018), 36 pp. doi: 10.1007/s10569-018-9835-7.  Google Scholar [3] J. F. Cariñena, M. F. Rañada and M. Santander, Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$, J. Math. Phys., 46 (2005), 052702. doi: 10.1063/1.1893214.  Google Scholar [4] F. Diacu, The non-existence of centre of mass and linear momentum integrals in the curved $N$-body problem, Libertas Math., 32 (2012), 25-37.  doi: 10.14510/lm-ns.v32i1.30.  Google Scholar [5] F. Diacu, E. Pérez-Chavela and J. G. Reyes, An intrinsic approach in the curved $n$-body problem. The negative curvature case, J. Differential Equations, 252 (2012), 4529-4562.  doi: 10.1016/j.jde.2012.01.002.  Google Scholar [6] G. A. Galperin, A concept of the mass center of a system of material points in the constant curvature spaces, Comm. Math. Phys., 154 (1993), 63-84.  doi: 10.1007/BF02096832.  Google Scholar [7] L. C. García-Naranjo, J. C. Marrero, E. Pérez-Chavela and M. Rodríguez-Olmos, Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2, J. Differential Equations, 260 (2016), 6375-6404.  doi: 10.1016/j.jde.2015.12.044.  Google Scholar [8] L. C. García-Naranjo and J. Montaldi, Attracting and repelling 2-body problems on a family of surfaces of constant curvature, J. Dyn. Diff. Equat., (2020). doi: 10.1007/s10884-020-09868-x.  Google Scholar [9] V. V. Kozlov and A. O. Harin, Kepler's problem in constant curvature spaces, Celestial Mech. Dynam. Astronom., 54 (1992), 393-399.  doi: 10.1007/BF00049149.  Google Scholar [10] C. Lim, J. Montaldi and R. M. Roberts, Relative equilibria of point vortices on the sphere, Physica D, 148 (2001), 97-135.  doi: 10.1016/S0167-2789(00)00167-6.  Google Scholar [11] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-0-387-21792-5.  Google Scholar [12] J. Montaldi, R. M. Roberts and I. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Phil. Trans. Roy. Soc. London., 325 (1988), 237-293.  doi: 10.1098/rsta.1988.0053.  Google Scholar [13] J. Montaldi and R. M. Roberts, Relative equilibria of molecules, J. Nonlinear Sci., 9 (1999), 53-88.  doi: 10.1007/s003329900064.  Google Scholar

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##### References:
 [1] A. V. Borisov, I. S. Mamaev and A. A. Kilin, Two-body problem on a sphere. Reduction, stochasticity, periodic orbits, Regul. Chaotic Dyn., 9 (2004), 265-279.  doi: 10.1070/RD2004v009n03ABEH000280.  Google Scholar [2] A. V. Borisov, L. C. García-Naranjo, I. S. Mamaev and J. Montaldi, Reduction and relative equilibria for the two-body problem on spaces of constant curvature, Celest. Mech. Dyn. Astr., 130 (2018), 36 pp. doi: 10.1007/s10569-018-9835-7.  Google Scholar [3] J. F. Cariñena, M. F. Rañada and M. Santander, Central potentials on spaces of constant curvature: The Kepler problem on the two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$, J. Math. Phys., 46 (2005), 052702. doi: 10.1063/1.1893214.  Google Scholar [4] F. Diacu, The non-existence of centre of mass and linear momentum integrals in the curved $N$-body problem, Libertas Math., 32 (2012), 25-37.  doi: 10.14510/lm-ns.v32i1.30.  Google Scholar [5] F. Diacu, E. Pérez-Chavela and J. G. Reyes, An intrinsic approach in the curved $n$-body problem. The negative curvature case, J. Differential Equations, 252 (2012), 4529-4562.  doi: 10.1016/j.jde.2012.01.002.  Google Scholar [6] G. A. Galperin, A concept of the mass center of a system of material points in the constant curvature spaces, Comm. Math. Phys., 154 (1993), 63-84.  doi: 10.1007/BF02096832.  Google Scholar [7] L. C. García-Naranjo, J. C. Marrero, E. Pérez-Chavela and M. Rodríguez-Olmos, Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2, J. Differential Equations, 260 (2016), 6375-6404.  doi: 10.1016/j.jde.2015.12.044.  Google Scholar [8] L. C. García-Naranjo and J. Montaldi, Attracting and repelling 2-body problems on a family of surfaces of constant curvature, J. Dyn. Diff. Equat., (2020). doi: 10.1007/s10884-020-09868-x.  Google Scholar [9] V. V. Kozlov and A. O. Harin, Kepler's problem in constant curvature spaces, Celestial Mech. Dynam. Astronom., 54 (1992), 393-399.  doi: 10.1007/BF00049149.  Google Scholar [10] C. Lim, J. Montaldi and R. M. Roberts, Relative equilibria of point vortices on the sphere, Physica D, 148 (2001), 97-135.  doi: 10.1016/S0167-2789(00)00167-6.  Google Scholar [11] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, 2$^{nd}$ edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-0-387-21792-5.  Google Scholar [12] J. Montaldi, R. M. Roberts and I. Stewart, Periodic solutions near equilibria of symmetric Hamiltonian systems, Phil. Trans. Roy. Soc. London., 325 (1988), 237-293.  doi: 10.1098/rsta.1988.0053.  Google Scholar [13] J. Montaldi and R. M. Roberts, Relative equilibria of molecules, J. Nonlinear Sci., 9 (1999), 53-88.  doi: 10.1007/s003329900064.  Google Scholar
Illustration of the centre of mass $\boldsymbol{\bar {q}}$ according to the characterisations C1, C2 and C3
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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