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

TL;DR

The paper extends Euler’s problem of finding stationary curves of the moment-of-inertia-type energy from the Euclidean plane to the hyperbolic plane $\mathbb H^2$ and the sphere $\mathbb S^2$. It derives the Euler–Lagrange equations for $E_\alpha[\gamma]$, establishing the key stationarity condition $\kappa = \alpha\frac{\langle \mathbf n, \xi\rangle}{\mathsf d}$ with $\mathsf d$ the distance to the fixed point $N$ and $\xi$ the unit tangent to the geodesic from $N$ to $\gamma$. The authors classify stationary curves of constant curvature, showing that in $\mathbb H^2$ the only closed $\alpha$-stationary curves are circles centered at $N$, while in $\mathbb S^2$ the sign of $\alpha$ is constrained by hemisphere containment and explicit families such as geodesics through $N$ and circles centered at $N$ arise with $\alpha=-r\cot r$. They also obtain first-integral expressions for non-constant-curvature stationary curves, provide parametrizations, and address energy-minimization problems for curves connecting two points collinear with $N$, including the geodesic as a minimizer in several configurations. These results illuminate how curvature and distance to a reference point govern stationary behavior across constant-curvature geometries.

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 $α$ 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.

Figures (2)

• Figure 1: In the Poincaré model of $\mathbb H^2$, $\alpha$-stationary curves for $\alpha=1$ (left), $\alpha=-1$ (middle) and $\alpha=-3$ (right).
• Figure 2: Examples of $\alpha$-stationary curves in $\mathbb S^2$ for $\alpha=2$ (left), $\alpha=-1$ (middle) and $\alpha=-2$ (rigth).

Theorems & Definitions (28)

• Definition 1.1
• Proposition 2.1
• Proposition 2.2
• Remark 2.3
• Proposition 2.4
• Theorem 2.5
• proof
• Proposition 2.6: maximum principle
• Theorem 2.7
• proof