American Institute of Mathematical Sciences

Advanced
doi: 10.3934/jgm.2020020

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

Luis C. García-Naranjo

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  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.

Keywords: Constant curvature spaces, centre of mass, celestial mechanics, relativistic rule of lever, collision point, centre of steady rotation.
Mathematics Subject Classification: Primary: 70F05; Secondary: 70A05.
Citation: Luis C. García-Naranjo. Some remarks about the centre of mass of two particles in spaces of constant curvature. Journal of Geometric Mechanics, 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

show all references

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

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

Download full-size image

Download as PowerPoint slide

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} $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 533-544. doi: 10.3934/dcdss.2010.3.533
[2]

Nicola Soave, Susanna Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3245-3301. doi: 10.3934/dcds.2012.32.3245
[3]

Ariadna Farrés, Àngel Jorba. On the high order approximation of the centre manifold for ODEs. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 977-1000. doi: 10.3934/dcdsb.2010.14.977
[4]

Nicola Soave, Susanna Terracini. Addendum to: Symbolic dynamics for the $N$-centre problem at negative energies. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3825-3829. doi: 10.3934/dcds.2013.33.3825
[5]

Luca Biasco, Luigi Chierchia. Exponential stability for the resonant D'Alembert model of celestial mechanics. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 569-594. doi: 10.3934/dcds.2005.12.569
[6]

D.J. Georgiev, A. J. Roberts, D. V. Strunin. Nonlinear dynamics on centre manifolds describing turbulent floods: k-$\omega$ model. Conference Publications, 2007, 2007 (Special) : 419-428. doi: 10.3934/proc.2007.2007.419
[7]

Gennadi Sardanashvily. Lagrangian dynamics of submanifolds. Relativistic mechanics. Journal of Geometric Mechanics, 2012, 4 (1) : 99-110. doi: 10.3934/jgm.2012.4.99
[8]

Misha Bialy. On Totally integrable magnetic billiards on constant curvature surface. Electronic Research Announcements, 2012, 19: 112-119. doi: 10.3934/era.2012.19.112
[9]

La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981
[10]

Piotr Gwiazda, Piotr Minakowski, Agnieszka Świerczewska-Gwiazda. On the anisotropic Orlicz spaces applied in the problems of continuum mechanics. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1291-1306. doi: 10.3934/dcdss.2013.6.1291
[11]

Joel Spruck, Ling Xiao. Convex spacelike hypersurfaces of constant curvature in de Sitter space. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2225-2242. doi: 10.3934/dcdsb.2012.17.2225
[12]

Doan The Hieu, Tran Le Nam. The classification of constant weighted curvature curves in the plane with a log-linear density. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1641-1652. doi: 10.3934/cpaa.2014.13.1641
[13]

Yan'e Wang, Jianhua Wu. Stability of positive constant steady states and their bifurcation in a biological depletion model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 849-865. doi: 10.3934/dcdsb.2011.15.849
[14]

Van Duong Dinh. On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1127-1143. doi: 10.3934/dcds.2018047
[15]

Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic & Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015
[16]

Joep H.M. Evers, Sander C. Hille, Adrian Muntean. Modelling with measures: Approximation of a mass-emitting object by a point source. Mathematical Biosciences & Engineering, 2015, 12 (2) : 357-373. doi: 10.3934/mbe.2015.12.357
[17]

Daniele Bartoli, Matteo Bonini, Massimo Giulietti. Constant dimension codes from Riemann-Roch spaces. Advances in Mathematics of Communications, 2017, 11 (4) : 705-713. doi: 10.3934/amc.2017051
[18]

Alfredo Lorenzi, Gianluca Mola. Identification of a real constant in linear evolution equations in Hilbert spaces. Inverse Problems & Imaging, 2011, 5 (3) : 695-714. doi: 10.3934/ipi.2011.5.695
[19]

Rowan Killip, Satoshi Masaki, Jason Murphy, Monica Visan. The radial mass-subcritical NLS in negative order Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 553-583. doi: 10.3934/dcds.2019023
[20]

Ali Hyder, Luca Martinazzi. Conformal metrics on $\mathbb{R}^{2m}$ with constant Q-curvature, prescribed volume and asymptotic behavior. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 283-299. doi: 10.3934/dcds.2015.35.283

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (23)
  • HTML views (24)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences