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

TL;DR

This paper investigates whether Hamiltonian vector fields on constant curvature spaces transition smoothly through zero curvature, demonstrating positive results in 2D and 3D, with applications to the gravitational N-body problem.

Contribution

It proves the continuous transition of Hamiltonian vector fields across zero curvature for 2D and 3D spaces with applications to gravitational dynamics.

Findings

Hamiltonian vector fields pass continuously through zero curvature.

The results hold for both spheres and hyperbolic spaces.

Applications to gravitational N-body problem are provided.

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.