We study the motion of the negative curved symmetric two and three center problem on the Poincaré upper semi plane model for a surface of constant negative curvature $ \kappa $, which without loss of generality we assume $ \kappa = -1 $. Using this model, we first derive the equations of motion for the $ 2 $-and $ 3 $-center problems. We prove that for $ 2 $–center problem, there exists a unique equilibrium point and we study the dynamics around it. For the motion restricted to the invariant $ y $–axis, we prove that it is a center, but for the general two center problem it is unstable. For the $ 3 $–center problem, we show the non-existence of equilibrium points. We study two particular integrable cases, first when the motion of the free particle is restricted to the $ y $–axis, and second when all particles are along the same geodesic. We classify the singularities of the problem and introduce a local and a global regularization of all them. We show some numerical simulations for each situation.
Citation: |
[1] |
S. Alhowaity, F. Diacu and E. Pérez-Chavela, Relative equilibria in curved restricted 4-body problems, Can. Math. Bull., 61 (2018), 673-687.
doi: 10.4153/CMB-2018-019-9.![]() ![]() ![]() |
[2] |
J. Andrade, N. Dávila, E. Pérez-Chavela and C. Vidal, Dynamics and Regularization of the Kepler Problem on Surfaces of Constant Curvature, Can. J. Math., 69 (2017), 961-991.
doi: 10.4153/CJM-2016-014-5.![]() ![]() ![]() |
[3] |
S. V. Bolotin and P. Negrini, Chaotic behavior in the 3-center problem, J. Differ. Equ., 190 (2003), 539-558.
doi: 10.1016/S0022-0396(03)00024-X.![]() ![]() ![]() |
[4] |
W. Bolyai and J. Bolyai, Geometrische Untersuchungen, Teubner, Leipzig-Berlin, 1913.
![]() |
[5] |
F. Diacu, On the singularities of the curved $N$-body problem, Trans. Amer. Math. Soc., 363 (2011), 2249-2264.
doi: 10.1090/S0002-9947-2010-05251-1.![]() ![]() ![]() |
[6] |
F. Diacu, Relative equilibria of the curved $N$-body problem, in ![]() ![]() ![]() |
[7] |
F. Diacu, Relative equilibria of the 3-dimensional curved $n$-body problem, Memoirs Amer. Math. Soc., 228 (2013), 1071.
![]() ![]() |
[8] |
F. Diacu, The curved $N$-body problem: risks and rewards, Math. Intell., 35 (2013), 24-33.
doi: 10.1007/s00283-013-9397-1.![]() ![]() ![]() |
[9] |
F. Diacu and S. Kordlou, Rotopulsators of the curved $N$-body problem, J. Differ. Equ., 255 (2013), 2709-2750.
doi: 10.1016/j.jde.2013.07.009.![]() ![]() ![]() |
[10] |
F. Diacu, R. Martínez, E. Pérez-Chavela and C. Simó, On the stability of tetrahedral relative equilibria in the positively curved 4-body problem, Physica D, 256 (2013), 21-35.
doi: 10.1016/j.physd.2013.04.007.![]() ![]() ![]() |
[11] |
F. Diacu and E. Pérez-Chavela, Homographic solutions of the curved $3$-body problem, J. Differ. Equ., 250 (2011), 340-366.
doi: 10.1016/j.jde.2010.08.011.![]() ![]() ![]() |
[12] |
F. Diacu, E. Pérez-Chavela and M. Santoprete, Saari's conjecture for the collinear $N$-body problem, Trans. Amer. Math. Soc., 357 (2005), 4215-4223.
doi: 10.1090/S0002-9947-04-03606-2.![]() ![]() ![]() |
[13] |
F. Diacu, E. Pérez-Chavela and M. Santoprete, The $N$-body problem in spaces of constant curvature. Part I: Relative equilibria, J. Nonlinear Sci., 22 (2012), 247-266.
doi: 10.1007/s00332-011-9116-z.![]() ![]() ![]() |
[14] |
F. Diacu, E. Pérez-Chavela and M. Santoprete, The $N$-body problem in spaces of constant curvature. Part II: Singularities, J. Nonlinear Sci., 22 (2012), 267-275.
doi: 10.1007/s00332-011-9117-y.![]() ![]() ![]() |
[15] |
F. Diacu, E. Pérez-Chavela and G. Reyes Victoria, An intrinsic approach in the curved $N$-body problem. The negative curvature case, J. Differ. Equ., 252 (2012), 4529-4562.
doi: 10.1016/j.jde.2012.01.002.![]() ![]() ![]() |
[16] |
F. Diacu and §. Popa, All Lagrangian relative equilibria have equal masses, J. Math. Phys., 55 (2014), 112701.
doi: 10.1063/1.4900833.![]() ![]() ![]() |
[17] |
F. Diacu and B. Thorn, Rectangular orbits of the curved 4-body problem, Proc. Amer. Math. Soc., 143 (2015), 1583-1593.
doi: 10.1090/S0002-9939-2014-12326-4.![]() ![]() ![]() |
[18] |
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. Differ. Equ., 260 (2016), 6375-6404.
doi: 10.1016/j.jde.2015.12.044.![]() ![]() ![]() |
[19] |
W. Killing, Die Rechnung in den nichteuklidischen Raumformen, J. Reine Angew. Math., 89 (1880), 265-287.
doi: 10.1515/crll.1880.89.265.![]() ![]() ![]() |
[20] |
W. Killing, Die Mechanik in den nichteuklidischen Raumformen, J. Reine Angew. Math., 98 (1885), 1-48.
doi: 10.1515/crll.1885.98.1.![]() ![]() ![]() |
[21] |
W. Killing, Die Nicht-Euklidischen Raumformen in Analytischer Behandlung, Teubner, Leipzig, 1885.
![]() |
[22] |
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.![]() ![]() ![]() |
[23] |
H. Kragh, Is space Flat? Nineteenth century astronomy and non-Euclidean geometry, J. Astr. Hist. Heritage, 15 (2012), 149-158.
![]() |
[24] |
R. Lipschitz, Extension of the planet-problem to a space of $n$ dimensions and constant integral curvature, Quart. J. Pure Appl. Math., 12 (1873), 349-370.
![]() |
[25] |
N. I. Lobachevsky, The new foundations of geometry with full theory of parallels [in Russian], 1835-1838, in Collected Works, vol. 2, GITTL, Moscow, 1949.
![]() |
[26] |
R. Martínez and C. Simó, On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb S^2$, Discrete Contin. Dyn. Syst. Ser. A, 33 (2013) 1157–1175.
doi: 10.3934/dcds. 2013.33.1157.![]() ![]() ![]() |
[27] |
R. Martínez and C. Simó, Relative equilibria of the restricted 3-body problem in curved spaces, Celes.t Mech. Dyn. Ast.r, 128 (2017), 221-259.
doi: 10.1007/s10569-016-9750-8.![]() ![]() ![]() |
[28] |
E. Pérez-Chavela and J. G. Reyes Victoria, An intrinsic approach in the curved $N$-body problem. The positive curvature case, Trans. Amer. Math. Soc., 364 (2012), 3805-3827.
doi: 10.1090/S0002-9947-2012-05563-2.![]() ![]() ![]() |
[29] |
E. Schering, Die Schwerkraft im Gaussischen Räume, Nachr. Königl. Ges. Wiss. Gött., 15, (1870), 311–321.
![]() |
[30] |
E. Schering, Die Schwerkraft in mehrfach ausgedehnten Gaussischen und Riemmanschen Räumen, Nachr. Königl. Ges. Wiss. Gött., 6 (1873), 149–159.
![]() |
[31] |
A. V. Shchepetilov, Nonintegrability of the two-body problem in constant curvature spaces, J. Phys. A: Math. Gen., 39, (2006), 5787-5806; corrected version at math. DS/0601382.
doi: 10.1088/0305-4470/39/20/011.![]() ![]() ![]() |
[32] |
P. Tibboel, Polygonal homographic orbits in spaces of constant curvature, Proc. Amer. Math. Soc., 141 (2013), 1465-1471.
doi: 10.1090/S0002-9939-2012-11410-8.![]() ![]() ![]() |
[33] |
P. Tibboel, Existence of a class of rotopulsators, J. Math. Anal. Appl., 404 (2013), 185-191.
doi: 10.1016/j.jmaa.2013.02.066.![]() ![]() ![]() |
[34] |
P. Tibboel, Existence of a lower bound for the distance between point masses of relative equilibria in spaces of constant curvature, J. Math. Anal. Appl., 416 (2014), 205-211.
doi: 10.1016/j.jmaa.2014.02.036.![]() ![]() ![]() |
[35] |
T. Vozmischeva, Integrable problems of celestial mechanics in spaces of constant curvature, in Astrophysics and Space Science Library, Volume 295, Springer, 2003.
doi: 10.1007/978-94-017-0303-1.![]() ![]() ![]() |
[36] |
S. Zhu, Eulerian relative equilibria of the curved 3-body problems in $\mathbb S^2$, Proc. Amer. Math. Soc., 142 (2014), 2837-2848.
doi: 10.1090/S0002-9939-2014-11995-2.![]() ![]() ![]() |