The classical N-body problem in the context of curved space

We provide the differential equations that generalize the Newtonian N-body problem of celestial mechanics to spaces of constant Gaussian curvature, k, for all k real. In previous studies, the equations of motion made sense only for k nonzero. The system derived here does more than just include the Euclidean case in the limit when k tends to 0: it recovers the classical equations for k=0. This new expression of the laws of motion allows the study of the N-body problem in the context of constant curvature spaces and thus offers a natural generalization of the Newtonian equations that includes the classical case. We end the paper with remarks about the bifurcations of the first integrals.