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We ask whether Hamiltonian vector fields defined on spaces of constantGaussian curvature $\kappa$ (spheres, for $\kappa>0$, and hyperbolic spheres,for $\kappa<0$), pass continuously through the value $\kappa=0$ if thepotential functions $U_\kappa, \kappa\in\mathbb R$, that define them satisfythe property $\lim_{\kappa\to 0}U_\kappa=U_0$, where $U_0$ corresponds to theEuclidean case. We prove that the answer to this question is positive, both inthe 2- and 3-dimensional cases, which are of physical interest, and then applyour conclusions to the gravitational $N$-body problem.

The continuous transition of Hamiltonian vector fields through manifolds of constant curvature