We ask whether Hamiltonian vector fields defined on spaces of constant Gaussian curvature κ (spheres, for κ > 0, and hyperbolic spheres, for κ < 0) pass continuously through the value κ = 0 if the potential functions Uκ, κ ∈ ℝ, which define them satisfy the property lim κ 0 U κ = U 0 , where U0 corresponds to the Euclidean case. We prove that the answer to this question is positive, both in the 2- and 3-dimensional cases, which are of physical interest, and then apply our conclusions to the gravitational N-body problem.

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