We provide the differential equations that generalize the Newtonian $N$-body problem of celestial mechanics to spaces of constant Gaussian curvature $\kappa $, for all $\kappa \in \mathbb{R}$. In previous studies, the equations of motion made sense only for $\kappa \ne 0$. The system derived here does more than just include the Euclidean case in the limit $\kappa \to 0;$ it recovers the classical equations for $\kappa =0$. 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 oòers 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.

For the three types of simply connected Riemannian spaces of constant curvature it is shown that the associated spherical functions can be obtained from the corresponding (zonal) spherical functions by application of a differential operator of the form p(i d/dt), where p belongs to a system of orthogonal polynomials: Gegenbauer polynomials, Hahn polynomials or continuous symmetric Hahn polynomials. We give a group theoretic explanation of this phenomenon and relate the properties of the polynomials p to the properties of the corresponding representation. The method is extended to the case of intertwining functions.