Abstract
We consider the problem of two bodies with central interaction that propagate in a simply connected space with a constant curvature and an arbitrary dimension. We obtain the explicit expression for the quantum Hamiltonian via the radial differential operator and generators of the isometry group of a configuration space. We describe the reduced classical mechanical system determined on the homogeneous space of a Lie group in terms of orbits of the coadjoint representation of this group. We describe the reduced classical two-body problem.
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References
J. A. Wolf,Spaces of Constant Curvature, Univ. California Press, Berkeley, CA (1972).
J. Moser,Commun. Pure Appl. Math.,23, 609 (1970).
Yu. S. Osipov,Usp. Mat. Nauk,27, No. 2, 101 (1972).
Y. S. Osipov,Celest. Mech.,16, 191 (1977).
E. A. Belbruno,Celest. Mech.,15, 467 (1977).
A. V. Shchepetilov,J. Phys. A,31, 6279 (1998);32, 1531 (1999).
M. Ikeda and N. Katayama,Tensor,38, 37 (1982).
N. Katayama,Nuovo Cimento B,105, 113 (1990);107, 763 (1992);108, 657 (1993).
Ya. I. Granovskii, A. S. Zhedanov, and I. M. Lutsenko,Theor. Math. Phys.,91, 474, 604 (1992).
V. S. Otchik, “On the two Coulomb centres problem in a spherical geometry,” in:Proc. Intl. Workshop on Symmetry Methods in Physics (A. N. Sissakian, G. S. Pogosyan, and S. I. Vinitsky, eds.), Vol. 2 JINR, Dubna (1994), p. 384.
V. A. Chernoivan and I. S. Mamaev,Regular and Chaotic Dynamics,4, No. 2, 112 (1999).
A. V. Borisov and I. S. Mamaev,Poisson Structures and Lie Algebras in Hamiltonian Mechanics [in Russian], Regular and Chaotic Dynamics (Publ.), Izhevsk (1999).
V. I. Arnold,Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1989); English transl. prev. ed., Springer, Berlin (1978).
A. V. Shchepetilov,Theor. Math. Phys.,118, 197 (1999).
M. Reed and B. Simon,Methods of Modern Mathematical Physics, Vol. 1,Functional Analysis, Acad. Press, New York (1972).
I. M. Oleinik,Mat. Zametki,55, 218 (1994).
A. V. Shchepetilov,Theor. Math. Phys.,109, 1556 (1996).
S. Helgason,Groups and Geometric Analysis, Acad. Press, Orlando, Fla. (1984).
A. A. Kirillov,Elements of the Theory of Representations [in Russian], Nauka, Moscow (1972); English transl., Springer, Berlin (1975).
V. V. Trofimov and A. T. Fomenko,Algebra and Geometry of Integrable Hamiltonian Differential Equations [in Russian], Factorial, Moscow (1995).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 2, pp. 249–264, August, 2000.
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Shchepetilov, A.V. Two-body problem on spaces of constant curvature: I. Dependence of the Hamiltonian on the symmetry group and the reduction of the classical system. Theor Math Phys 124, 1068–1081 (2000). https://doi.org/10.1007/BF02551078
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DOI: https://doi.org/10.1007/BF02551078