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The classical Kepler problem and geodesic motion on spaces of constant curvature

In this paper we clarify and generalise previous work by Moser and Belbruno 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 characterised by the value of the corresponding Hamiltonian and one other parameter (the mass parameter in the Kepler problem and the curvature parameter in the 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 the Kepler problem is equal to (minus) the curvature parameter of the constant curvature space and the mass parameter is given by the value of the 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 the Kepler problem arises in a natural way from the isometries of a space of constant curvature. In addition, the canonical nature of the transformation guarantees that the Poisson bracket Lie algebra of constants of motion for the classical Kepler problem is identical to that associated with geodesic motion on spaces of constant curvature.