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

Abstract

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 $U_κ, κ\in\mathbb R$, that define them satisfy the property $\lim_{κ\to 0}U_κ=U_0$, where $U_0$ 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.