The hanging chain problem with respect to a circle

TL;DR

This paper investigates the shape of planar curves that extremize potential energy related to their distance from a circle, analyzing cases inside and outside the circle and extending to power-based energies.

Contribution

It characterizes extremal curves for potential energy related to a circle and extends the analysis to power-based energy functions.

Findings

Characterization of extremal curves inside and outside the circle

Extension of the problem to power functions of distance

Descriptions of the shape of energy-minimizing curves

Abstract

Let $\s^1$ be a circle in Euclidean plane. We consider the problem of finding the shape of a planar curve which is an extremal of the potential energy that measures the distance to $\s^1$. We describe the shape of these curves distinguishing if the curves lie in the inside or outside of $\s^1$. We extend the problem for energies that are powers to the distance to $\s^1$.