
-
Previous Article
Characterization of toric systems via transport costs
- JGM Home
- This Issue
-
Next Article
Higher order normal modes
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 |
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 [
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
J. Montaldi and R. M. Roberts,
Relative equilibria of molecules, J. Nonlinear Sci., 9 (1999), 53-88.
doi: 10.1007/s003329900064. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
J. Montaldi and R. M. Roberts,
Relative equilibria of molecules, J. Nonlinear Sci., 9 (1999), 53-88.
doi: 10.1007/s003329900064. |


[1] |
Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete and 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 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] |
Piotr Gwiazda, Piotr Minakowski, Agnieszka Świerczewska-Gwiazda. On the anisotropic Orlicz spaces applied in the problems of continuum mechanics. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1291-1306. doi: 10.3934/dcdss.2013.6.1291 |
[10] |
La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981 |
[11] |
Jörg Weber. Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder. Kinetic and Related Models, 2020, 13 (6) : 1135-1161. doi: 10.3934/krm.2020040 |
[12] |
Joel Spruck, Ling Xiao. Convex spacelike hypersurfaces of constant curvature in de Sitter space. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2225-2242. doi: 10.3934/dcdsb.2012.17.2225 |
[13] |
Doan The Hieu, Tran Le Nam. The classification of constant weighted curvature curves in the plane with a log-linear density. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1641-1652. doi: 10.3934/cpaa.2014.13.1641 |
[14] |
Yan'e Wang, Jianhua Wu. Stability of positive constant steady states and their bifurcation in a biological depletion model. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 849-865. doi: 10.3934/dcdsb.2011.15.849 |
[15] |
Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016 |
[16] |
Gang Li, Xianwen Zhang. A Vlasov-Poisson plasma of infinite mass with a point charge. Kinetic and Related Models, 2018, 11 (2) : 303-336. doi: 10.3934/krm.2018015 |
[17] |
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 |
[18] |
Van Duong Dinh. On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1127-1143. doi: 10.3934/dcds.2018047 |
[19] |
Younghun Hong, Sangdon Jin. Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022010 |
[20] |
Ikuo Arizono, Yasuhiko Takemoto. Statistical mechanics approach for steady-state analysis in M/M/s queueing system with balking. Journal of Industrial and Management Optimization, 2022, 18 (1) : 25-44. doi: 10.3934/jimo.2020141 |
2020 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]