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The spatial problem of 2 bodies on a sphere. Reduction and stochasticity. (English) Zbl 1402.70015
Summary: In this paper, we consider in detail the 2-body problem in spaces of constant positive curvature \(S^2\) and \(S^3\). We perform a reduction (analogous to that in rigid body dynamics) after which the problem reduces to analysis of a two-degree-of-freedom system. In the general case, in canonical variables the Hamiltonian does not correspond to any natural mechanical system. In addition, in the general case, the absence of an analytic additional integral follows from the constructed Poincaré section. We also give a review of the historical development of celestial mechanics in spaces of constant curvature and formulate open problems. 121 References.

MSC:
70F15 Celestial mechanics
70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
70-03 History of mechanics of particles and systems
01A55 History of mathematics in the 19th century
01A60 History of mathematics in the 20th century
70F05 Two-body problems
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[1] Albouy, A.; Brezis, H. (ed.); Chang, K.C. (ed.); Li, S.J. (ed.); Rabinowitz, P. (ed.), The underlying geometry of the fixed centers problems, in topological methods, variational methods and their applications (taiyuan, 2002), 11-21, (2003), N.J.
[2] Albouy, A., There is a projective dynamics, Eur. Math. Soc. Newsl., 89, 37-43, (2013) · Zbl 1364.37169
[3] Albouy, A.; Stuchi, T.J., Generalising the classical fixed-centres problem in a non-Hamiltonian way, J. Phys, A, 37, 9109-9123, (2004) · Zbl 1072.70007
[4] Altschuler, E.L.; Williams, T.J.; Ratner, E.R.; Tipton, R.; Stong, R.; Dowla, F.; Wooten, F., Possible global minimum lattice configurations for thomson’s problem of charges on a sphere, Phys. Rev. Lett., 78, 2681-2685, (1997)
[5] Aref, H.; Newton, P.K.; Stremler, M.A.; Tokieda, T.; Vainchtein, D.L., Vortex crystals, Adv. Appl. Mech., 39, 1-79, (2003)
[6] Appell, P., Sur LES lois de forces centrales faisant décrire à leur point d’application une conique quelles que soient LES conditions initiales, Amer. J. Math., 13, 153-158, (1891) · JFM 22.0900.02
[7] Appell, P., De l’homographie en mécanique, Amer. J. Math., 12, 103-114, (1890) · JFM 21.0904.01
[8] Appell, P., Sur une transformation de mouvements, Amer.J. Math., 17, 1-5, (1895) · JFM 26.0824.01
[9] Ball, R.S., Certain problems in the dynamics of a rigid system moving in elliptic space, Trans. Roy. Irish Acad., 28, 159-184, (1881)
[10] Ballesteros, A.; Enciso, A.; Herranz, F.J.; Ragnisco, O., Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of bertrand’s theorem to curved manifolds, Comm. Math. Phys., 290, 1033-1049, (2009) · Zbl 1269.70017
[11] Barut, A.O.; Inomata, A.; Junker, G., Path integral treatment of the hydrogen atom in a curved space of constant curvature, J. Phys. A, 20, 6271-6280, (1987) · Zbl 0628.58005
[12] Bertrand, J., Théorème relatif au mouvement d’un point attiré vers un centre fixe, C.R. Acad. Sci. Paris, 77, 849-853, (1873) · JFM 05.0470.01
[13] Bizyaev, I.A.; Borisov, A.V.; Mamaev, I.S., Figures of equilibrium of an inhomogeneous self-gravitating fluid, Celestial Mech. Dynam. Astronom., 122, 1-26, (2015) · Zbl 1409.76146
[14] Bizyaev, I.A.; Borisov, A.V.; Mamaev, I.S., Superintegrable generalizations of the Kepler and hook problems, Regul. Chaotic Dyn., 19, 415-434, (2014) · Zbl 1309.70020
[15] Boatto, S.; Dritschel, D.G.; Schaefer, R.G., N-body dynamics on closed surfaces: the axioms of mechanics, Proc.R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 472, 20160020, (2016) · Zbl 1371.70035
[16] Bogush, A.A.; Gritsev, V.V.; Kurochkin, Yu.A.; Otchik, V.S., An algebraic treatment of the MIC-Kepler problem on \(S\)3 sphere, Phys. Atomic Nuclei, 65, 1052-1056, (2002) · Zbl 0980.81015
[17] Bogomolov, V.A., Dynamics of the vorticity on a sphere, Fluid Dynam., 12, 863-870, (1977) · Zbl 0442.76020
[18] Borisov, A.V.; Kilin, A.A.; Mamaev, I.S.; Borisov, A.V. (ed.); Kozlov, V.V. (ed.); Mamaev, I.S. (ed.); Sokolovisky, M.A. (ed.), A new integrable problem of motion of point vortices on the sphere, in proc. of the IUTAM symposium on Hamiltonian dynamics, vortex structures, 39-53, (2008), Dordrecht · Zbl 1207.76030
[19] Borisov, A.V.; Kilin, A.A.; Mamaev, I.S., Superintegrable system on a sphere with the integral of higher degree, Regul. Chaotic Dyn., 14, 615-620, (2009) · Zbl 1229.70052
[20] Borisov, A.V.; Kilin, A.A.; Mamaev, I.S., Multiparticle systems: the algebra of integrals and integrable cases, Regul. Chaotic Dyn., 14, 18-41, (2009) · Zbl 1229.37106
[21] Borisov, A.V. and Mamaev, I.S., Poisson Structures and Lie Algebras in Hamiltonian Mechanics, Izhevsk: R&C Dynamics, 1999 (Russian), http://ics.org.ru/upload/iblock/14c/2-1.pdf. · Zbl 1010.70002
[22] Classical Dynamics in Non-Eucledian Spaces, A.V. Borisov, I.S. Mamaev (Eds.), Izhevsk: Institute of Computer Science, 2004 (Russian). · Zbl 1411.70014
[23] Borisov, A.V. and Mamaev, I.S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian). · Zbl 1114.70001
[24] Borisov, A.V.; Mamaev, I.S., Generalized problem of two and four Newtonian centers, Celestial Mech. Dynam. Astronom., 92, 371-380, (2005) · Zbl 1129.70010
[25] Borisov, A.V.; Mamaev, I.S., Superintegrable systems on a sphere, Regul. Chaotic Dyn., 10, 257-266, (2005) · Zbl 1077.37520
[26] Borisov, A.V.; Mamaev, I.S., The restricted two-body problem in constant curvature spaces, Celestial Mech. Dynam. Astronom., 96, 1-17, (2006) · Zbl 1116.70015
[27] Borisov, A.V.; Mamaev, I.S., Relations between integrable systems in plane and curved spaces, Celestial Mech. Dynam. Astronom., 99, 253-260, (2007) · Zbl 1202.70062
[28] Borisov, A.V.; Mamaev, I.S., Symmetries and reduction in nonholonomic mechanics, Regul. Chaotic Dyn., 20, 553-604, (2015) · Zbl 1342.37001
[29] Borisov, A.V.; Mamaev, I.S.; Kilin, A.A., Two-body problem on a sphere: reduction, stochasticity, periodic orbits, Regul. Chaotic Dyn., 9, 265-279, (2004) · Zbl 1065.37058
[30] Borisov, A.V.; Pavlov, A.E., Dynamics and statics of vortices on a plane and a sphere, Regul. Chaotic Dyn., 3, 28-38, (1998) · Zbl 0934.76014
[31] Born, M., Vorlesungen über Atommechanik, Berlin: Springer, 1925. · JFM 51.0727.01
[32] Calogero, F., Solution of a three-body problem in one dimension, J. Math. Phys., 10, 2191-2196, (1969)
[33] Cariñena, J.F.; Rañada, M.F.; Santander, M., Superintegrability on curved spaces, orbits and momentum hodographs: revisiting a classical result by Hamilton, J. Phys. A, 40, 13645-13666, (2007) · Zbl 1124.37036
[34] Cariñena, J.F.; Rañada, M.F.; Santander, M., The Kepler problem and the Laplace-Runge-Lenz vector on spaces of constant curvature and arbitrary signature, Qual. Theory Dyn. Syst., 7, 87-99, (2008) · Zbl 1163.70004
[35] Chernikov, N.A., The relativistic Kepler problem in the Lobachevsky space, Acta Phys. Polon. B, 24, 927-950, (1993)
[36] Clifford, W.K., On the free motion under no forces of a rigid system in an \(n\)-fold homaloid, Proc. London Math. Soc., S1-7, 67-70, (1876) · JFM 08.0597.04
[37] Clifford, W.K.; Tucker, R. (ed.), Motion of a solid in elliptic space, 378-384, (1882), London
[38] Chernikov, N.A., The Kepler problem in the Lobachevsky space and its solution, Acta Phys. Polon. B, 23, 115-122, (1992)
[39] Chernoivan, V.A.; Mamaev, I.S., The restricted two-body problem and the Kepler problem in the constant curvature spaces, Regul. Chaotic Dyn., 4, 112-124, (1999) · Zbl 1137.70339
[40] Darboux, G., Étude d’une question relative au mouvement d’un point sur une surface de révolution, Bull. Soc. Math. France, 5, 100-113, (1877) · JFM 09.0648.02
[41] Darboux, G., Sur une question relative au mouvement d’un point sur une surface de révolution, 467-482, (1886), Paris
[42] Darboux, G., Sur un problème de mécanique, Arch. Néerlandaises Sci., 6, 371-376, (1901) · JFM 32.0725.02
[43] De Francesco, D., Sul moto di un corpo rigido in uno spazio di curvatura costante, Math. Ann., 55, 573-584, (1902) · JFM 33.0766.02
[44] Diacu, F., Relative Equilibria of the Curved N-Body Problem, Paris: Atlantis, 2012. · Zbl 1261.70001
[45] Diacu, F., The curved N-body problem: risks and rewards, Math. Intelligencer, 35, 24-33, (2013) · Zbl 1302.70029
[46] Diacu, F. and Holmes, Ph., Celestial Encounters: The Origins of Chaos and Stability, Princeton: Princeton Univ. Press, 1999. · Zbl 0944.37001
[47] Diacu, F.; Kordlou, Sh., Rotopulsators of the curved N-body problem, J. Differential Equations, 255, 2709-2750, (2013) · Zbl 1323.70070
[48] Diacu, F.; Martínez, R.; Pérez-Chavela, E.; Simó, C., On the stability of tetrahedral relative equilibria in the positively curved 4-body problem, Phys. D, 256/257, 21-35, (2013) · Zbl 1292.70006
[49] Diacu, F. and Senechal, M., Creative Writing in Mathematics and Science, http://www.birs.ca/workshops/2016/16w5093/report16w5093.pdf (BIRS Workshop Reports, 2016, 8 pp.). · Zbl 1072.70007
[50] Dombrowski, P.; Zitterbarth, J., On the planetary motion in the 3-dim. standard spaces M\^{3}_{κ} of constant curvature κ ∈ R, Demonstratio Math., 24, 375-458, (1991) · Zbl 0756.53008
[51] Fehér, L.Gy., Dynamical O(4) symmetry in the asymptotic field of the Prasad-Sommerfield monopole, J. Phys. A, 19, 1259-1270, (1986) · Zbl 0615.58032
[52] Franco-Pérez, L.; Gidea, M.; Levi, M.; Pérez-Chavela, E., Stability interchanges in a curved Sitnikov problem, Nonlinearity, 29, 1056-1079, (2016) · Zbl 1411.70014
[53] Garcia-Gutierrez, L. and Santander, M., Levi-Civita regularization and geodesic flows for the ‘curved’ Kepler problem, Preprint, arXiv:0707.3810v2 (2007), 19 pp. · Zbl 0615.58032
[54] García-Naranjo, L.C.; Marrero, J.C.; Pérez-Chavela, E.; Rodríguez-Olmos, M., Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2, J. Differential Equations, 260, 6375-6404, (2016) · Zbl 1336.37022
[55] Gibbons, G.W.; Warnick, C.M., Hidden symmetry of hyperbolic monopole motion, J. Geom. Phys., 57, 2286-2315, (2007) · Zbl 1138.53059
[56] Grosche, C., The path integral for the Kepler problem on the pseudosphere, Ann. Physics, 204, 208-222, (1990) · Zbl 0722.70017
[57] Grosche, C., On the path integral in imaginary Lobachevsky space, J. Phys. A, 27, 3475-3489, (1994) · Zbl 0832.58009
[58] Heath, R.S., On the dynamics of a rigid body in elliptic space, Phil. Trans.R. Soc. Lond., 175, 281-324, (1884) · JFM 16.0756.01
[59] Higgs, P.W., Dynamical symmetries in a spherical geometry: 1, J. Phys. A, 12, 309-323, (1979) · Zbl 0418.70016
[60] Ikeda, M.; Katayama, N., On generalization of bertrand’s theorem to spaces of constant curvature, Tensor (N.S.), 38, 37-40, (1982) · Zbl 0509.58038
[61] Infeld, L.; Schild, A., A note on the Kepler problem in a space of constant negtive curvature, Phys. Rev., 67, 121-123, (1945) · Zbl 0060.44802
[62] Jovanović, V., A note on the proof of bertrand’s theorem, Theor. Appl. Mech., 42, 27-33, (2015) · Zbl 1397.70003
[63] Kalnins, E.G.; Kress, J.M.; Miller, W., Families of classical subgroup separable superintegrable systems, J. Phys. A, 43, 092001, (2010) · Zbl 1184.37053
[64] Katayama, N., A note on the Kepler problem in a space of constant curvature, Nuovo Cimento B (11), 105, 113-119, (1990)
[65] Katayama, N., On generalized Runge-Lenz vector and conserved symmetric tensor for central-potential systems with a monopole field on spaces of constant curvature, Nuovo Cimento B (11), 108, 657-667, (1993)
[66] Kilin, A.A., Libration points in spaces \(S\)\^{2} and \(L\)\^{2}, Regul. Chaotic Dyn., 4, 91-103, (1999) · Zbl 1008.70007
[67] Killing, W., Die rechnung in den nichteuklidischen raumformen, J. Reine Angew. Math., 1880, 265-287, (1880) · JFM 12.0406.01
[68] Killing, H.W., Die mechanik in den nicht-euklidischen raumformen, J. Reine Angew. Math., 98, 1-48, (1885) · JFM 17.0814.03
[69] Kozlov, V.V.; Kozlov, V.V. (ed.), Problemata nova, ad quorum solutionem mathematici invitantur, in dynamical systems in classical mechanics, 239-254, (1995), R.I. · Zbl 0838.70001
[70] Kozlov, V.V., Dynamics in spaces of constant curvature, Moscow Univ. Math. Bull., 49, 21-28, (1994) · Zbl 1066.83511
[71] Kozlov, V.V., The Newton and ivory theorems of attraction in spaces of constant curvature, Moscow Univ. Math. Bull., 55, 16-20, (2000) · Zbl 1074.37500
[72] Kozlov, V.V.; Fedorov, Yu.N., Integrable systems on the sphere with elastic interaction potentials, Math. Notes, 56, 927-930, (1994) · Zbl 0854.70013
[73] Kozlov, V.V.; Harin, A.O., Kepler’s problem in constant curvature spaces, Celestial Mech. Dynam. Astronom., 54, 393-399, (1992) · Zbl 0765.70007
[74] Kozlov, I.S., Problem of four fixed centers with applications to celestial mechanics, Astron. Zh., 51, 191-198, (1974) · Zbl 0278.70011
[75] Kurakin, L.G., On nonlinear stability of the regular vortex systems on a sphere, Chaos, 14, 1-11, (2004) · Zbl 1080.76031
[76] Kurochkin, Yu.A.; Otchik, V.S., Analogue of the Runge-Lenz vector and energy spectrum in the Kepler problem on a three-dimensional sphere, Dokl. Akad. Nauk BSSR, 23, 987-990, (1979)
[77] Kurochkin, Yu.A., Otchik, V.S., Mardoyan, L.G., Petrosyan, D.R., and Pogosyan, G.S., Kepler Motion on Single-Sheet Hyperboloid, Preprint, arXiv:1603.08139 (2016). · JFM 16.0756.01
[78] Kurochkin, Yu.A.; Otchik, V.S.; Ovsiyuk, E.M.; Shoukavy, D.V., On some integrable systems in the extended Lobachevsky space, Phys. Atomic Nuclei, 74, 944-948, (2011)
[79] Liebmann, H., Die kegelschnitte und die planetenbewegung im nichteuklidischen raum, ber. Königl. Sächs. ges. wiss., Math. Phys. Kl., 54, 393-423, (1902)
[80] Liebmann, H., Über die zentralbewegung in der nichteuklidischen geometrie, Ber. Königl. Sächs. Ges. Wiss., Math. Phys. Kl., 55, 146-153, (1903) · JFM 34.0767.02
[81] Liebmann, H., Nichteuklidische Geometrie, Leipzig: Göschen, 1905. · JFM 36.0520.01
[82] Lindner, J.F.; Roseberry, M.I.; Shai, D.E.; Harmon, N.J.; Olaksen, K.D., Precession and chaos in the classical two-body problem in a spherical universe, Internat.J. Bifur. Chaos Appl. Sci. Engrg., 18, 455-464, (2008) · Zbl 1225.70007
[83] Liouville, J., Sur quelques cas particuliers où LES équations du mouvement d’un point matériel peuvent s’intégrer: 1, J. Math. Pures Appl. (1), 11, 345-378, (1846)
[84] Lipschitz, R., Extension of the planet-problem to a space of n-dimensions and constant integral curvature, Quart. J. Pure Appl. Math., 12, 349-370, (1873) · JFM 05.0442.01
[85] Lobachevsky, N.I., Complete Collected Works: Vol. 2. New Foundations of Geometry with a Complete Theory of Parallels (1835-1838), V.F. Kagan (Ed.), Moscow-Leningrad: GITTL, 1949, pp. 158-159. · Zbl 1302.70029
[86] Maciejewski, A.J.; Przybylska, M., Non-integrability of restricted two body problems in constant curvature spaces, Regul. Chaotic Dyn., 8, 413-430, (2003) · Zbl 1048.37052
[87] Mamaev, I.S., Integrable Problems of Particle Motion in Spaces of Constant Curvature in a Magnetic Monopole Field and in the Field of Two Fixed Newtonian Centers, in The 9th International Workshop on Gravitational Energy and Gravitational Waves (8-12 Dec 1996, Dubna, Russian Federation), Dubna: Joint Inst. Nucl. Res., 1998, pp. 75-78.
[88] Mamaev, I.S., Numerical and Analytical Methods in Dynamical Systems Analysis, PhD Thesis, Moscow: Moscow State Univ., 2000 (Russian).
[89] Miller, W.; Post, S.; Winternitz, P., Classical and quantum superintegrability with applications, J. Phys. A, 46, 423001, (2013) · Zbl 1276.81070
[90] Montanelli, H., Computing hyperbolic choreographies, Regul. Chaotic Dyn., 21, 523-531, (2016) · Zbl 1398.70031
[91] Montanelli, H.; Gushterov, N.I., Computing planar and spherical choreographies, SIAM J. Appl. Dyn. Syst., 15, 235-256, (2016) · Zbl 1398.70031
[92] Morales-Ruiz, J.J.; Ramis, J.-P., Integrability of dynamical systems through differential Galois theory: A practical guide, in differential algebra, complex analysis and orthogonal polynomials, contemp. math., vol. 509, providence, 143-220, (2010) · Zbl 1294.37024
[93] Mordukhai-Boltovskoi, D.D., On some problems of celestial mechanics in non-Euclidean space, Akad. Nauk Ukr. SSR, 1, 47-70, (1932)
[94] Neumann, C., Ausdehnung der kepler’schen gesetze auf den fall, dass die bewegung auf einer kugelfläche stattfindet, Ber. Königl. Sächs. Ges. Wiss., Math. Phys. Kl., 38, 1-2, (1886) · JFM 18.0858.01
[95] Onofri, E.; Pauri, M., Search for periodic Hamiltonian flows: A generalized bertrand’s theorem, J. Math. Phys., 19, 1850-1858, (1978) · Zbl 0422.70024
[96] Otchik, V.S., Symmetry and separation of variables in the two-center Coulomb problem in three dimensional spaces of constant curvature, Dokl. Akad. Nauk BSSR, 35, 420-424, (1991)
[97] Petrosyan, D.; Pogosyan, G., Classical Kepler-Coulomb problem on SO(2, 2) hyperboloid, Phys. Atomic Nuclei, 76, 1273-1283, (2013)
[98] Rañada, M.F., Superintegrable systems with a position dependent mass: Kepler-related and oscillator-related systems, Phys. Lett. A, 380, 2204-2210, (2016) · Zbl 1360.70046
[99] Santoprete, M., Gravitational and harmonic oscillator potentials on surfaces of revolution, J. Math. Phys., 49, 042903, (2008) · Zbl 1152.81602
[100] Schering, E., Die schwerkraft im gaussischen raume, Nachr. Königl. Ges. Wiss. Göttingen, 15, 311-321, (1870) · JFM 02.0688.01
[101] Schering, E., Die schwerkraft in mehrfach ausgedehnten gaussischen und riemannschen Räumen, Nachr. Königl. Ges. Wiss. Göttingen, 1873, 149-159, (1873) · JFM 05.0442.02
[102] Schmidt, D., Central Configurations and Relative Equilibria for the N-Body Problem, in Classical and Celestial Mechanics (Recife, 1993/1999), Princeton, N.J.: Princeton Univ. Press, 2002, pp. 1-33. · Zbl 1181.70016
[103] Schrödinger, E., A method of determining quantum-mechanical eigenvalues and eigenfunctions, Proc. Roy. Irish Acad. Sect. A, 46, 9-16, (1940) · Zbl 0023.08602
[104] Serret, P., Théorie nouvelle géométrique et mécanique des lignes à double courbure, Paris: Mallet-Bachelier, 1860.
[105] Shchepetilov, A.V., Nonintegrability of the two-body problem in constant curvature spaces, J. Phys. A, 39, 5787-5806, (2006) · Zbl 1122.70009
[106] Shchepetilov, A.V., Calculus and mechanics on two-point homogenous Riemannian spaces, (2006), Berlin · Zbl 1115.53017
[107] Shchepetilov, A.V., Reduction of the two-body problem with central interaction on simply connected spaces of constant sectional curvature, J. Phys. A, 31, 6279-6291, (1998) · Zbl 0910.70010
[108] Slawianowski, J.J., Quantized bertrand systems on SO(3, R) and SU(2), Bull. Acad. Pol. Sci. Ser. Sci. Phys. et Astron., 28, 83-94, (1980) · Zbl 0517.70004
[109] Shchepetilov, A.V., Two-body problem on spaces of constant curvature: 1. dependence of the Hamiltonian on the symmetry group and the reduction of the classical system, Theoret. and Math. Phys., 124, 1068-1081, (2000) · Zbl 1112.37312
[110] Szumiński, W.; Maciejewski, A.J.; Przybylska, M., Note on integrability of certain homogeneous Hamiltonian systems, Phys. Lett. A, 379, 2970-2976, (2015) · Zbl 1349.37058
[111] Tremblay, F.; Turbiner, A.V.; Winternitz, P., Periodic orbits for an infinite family of classical superintegrable systems, J. Phys. A, 43, 015202, (2010) · Zbl 1186.37069
[112] Urkunden zur Geschichte der nichteuklidischen Geometrie: Vol. 2. Wolfgang und Johann Bolyai geometrische Untersuchungen: P. 1, 2, P. Stäckel (Ed.), Leipzig: Teubner, 1913. · JFM 21.0904.01
[113] Velpry, C., Kepler’s laws and gravitation in non-Euclidean (classical) mechanics, Acta Phys. Hung. New Ser. Heavy Ion Phys., 11, 131-145, (2000)
[114] Voronec, P.V., Transformation of the equations of motion by means of linear integrals of motion (with an application to the \(n\)-body problem), Kiev. Univ. Izv., 47, 192, (1907)
[115] Vozmischeva, T.G., The Lagrange and two-center problems in the Lobachevsky space, Celestial Mech. Dynam. Astronom., 84, 65-85, (2002) · Zbl 1062.70013
[116] Vozmischeva, T.G., Integrable problems of celestial mechanics in spaces of constant curvature, (2003), Dordrecht · Zbl 1098.70004
[117] Vozmishcheva, T.G.; Oshemkov, A.A., Topological analysis of the two-center problem on a two-dimensional sphere, Sb. Math., 193, 1103-1138, (2002) · Zbl 1036.37021
[118] Weyl, H., Space, Time, Matter, London: Methuen, 1922. · JFM 48.1059.12
[119] Zagryadskii, O.A.; Kudryavtseva, E.A.; Fedoseev, D.A., A generalization of bertrand’s theorem to surfaces of revolution, Sb. Math., 203, 1112-1150, (2012) · Zbl 1408.53115
[120] Zhu, Sh., Eulerian relative equilibria of the curved 3-body problems in \(S\)\^{2}, Proc. Amer. Math. Soc., 142, 2837-2848, (2014) · Zbl 1368.70015
[121] Ziglin, S.L., On non-integrability of the restricted two-body problem on a sphere, Dokl. Ross. Akad. Nauk, 379, 477-478, (2001)
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