Extension of a problem of Euler in ${\mathbb{H}}^2$ and in ${\mathbb{S}}^2$

Abstract

In this paper, we extend the notion of stationary curves with respect to the moment of inertia from a point $N$ in the Euclidean plane ${\mathbb{R}}^2$ to the case that the ambient space is either the hyperbolic plane ${\mathbb{H}}^2$ or the sphere ${\mathbb{S}}^2$. We characterize the critical points of this energy in terms of the curvature of the curve and the distance to $N$. In ${\mathbb{H}}^2$, we prove that the only closed stationary curves are circles centered at $N$. In ${\mathbb{S}}^2$, we estimate the value of $\hat{I}\pm$ for closed curves according to the hemisphere of ${\mathbb{S}}^2$ in which the curve lies. In addition, we find the first integrals of the ODEs that describe the parametrizations of stationary curves in both ambient spaces. Finally, we consider the energy minimization problem for curves connecting two points collinear with $N$, in particular solving the case of geodesics.

20 pages, 6 figures