Two-body problem on spaces of constant curvature: I. Dependence of the Hamiltonian on the symmetry group and the reduction of the classical system

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.