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 $\kappa$ (spheres, for $\kappa >0$, and hyperbolic spheres, for $\kappa <0$), pass continuously through the value $\kappa =0$ if the potential functions $U_{\kappa },\kappa \in \mathbb{R}$, that define them satisfy the property ${\lim }_{\kappa \to 0}{U}_{\kappa }=U_0$, 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.

12 pages

References in corpus (1)

• The Vlasov-Poisson System for Stellar Dynamics in Spaces of Constant Curvature

Cited by in corpus (2)

• The $N$-Body Problem in Spaces with Uniformly Varying Curvature
• Attracting and repelling 2-body problems on a family of surfaces of constant curvature