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The classical Kepler problem and geodesic motion on spaces of constant curvature. (English) Zbl 0989.70005

Summary: We clarify and generalize previous work by J. Moser [Commun. Pure Appl. Math. 23, 609-636 (1970; Zbl 0193.53803)] and E. A. Belbruno [Celest. Mech. 15, 467-476 (1977; Zbl 0367.70005)] concerning the link between the motions in the classical Kepler problem and geodesic motion on spaces of constant curvature. Both problems can be formulated as Hamiltonian systems, and the phase flow in each system is characterized by the value of the corresponding Hamiltonian and by one other parameter (the mass parameter in Kepler problem and the curvature parameter in geodesic motion problem). Using a canonical transformation, the Hamiltonian vector field for the geodesic motion problem is transformed into one which is proportional to that for the Kepler problem. Within this framework the energy of Kepler problem is equal to (minus) the curvature parameter of the constant curvature space, and the mass parameter is given by the value of Hamiltonian for the geodesic motion problem. We work with the corresponding family of evolution spaces and present a unified treatment which is valid for all values of energy continuously. As a result, there is a correspondence between the constants of motion for both systems, and the Runge-Lenz vector in Kepler problem arises in a natural way from isometries of a space of constant curvature. In addition, the canonical nature of transformation guarantees that the Poisson bracket Lie algebra of constants of motion for classical Kepler problem is identical to that associated with geodesic motion on spaces of constant curvature.

MSC:

70F05 Two-body problems
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
53Z05 Applications of differential geometry to physics
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