Thomson problem in one dimension: minimal energy configurations of $N$ charges on a curve

TL;DR

This paper investigates minimal energy configurations of charges on various one-dimensional curves, revealing complex behaviors such as multiple minima and the influence of cusps, with implications for understanding charge distributions on curved geometries.

Contribution

It introduces a study of minimal energy charge configurations on diverse one-dimensional curves, including ellipses, wires, and a cardioid, highlighting the effects of geometry on equilibrium states.

Findings

Multiple minima and unstable equilibria exist for certain geometries.

The cusp of the cardioid significantly affects charge distribution as N becomes large.

Charge configurations are sensitive to the curvature and shape of the underlying curve.

Abstract

We have studied the configurations of minimal energy of $N$ charges on a curve on the plane, interacting with a repulsive potential $V_{ij} = 1/r_{ij}^s$, with $s \geq 1$ and $i,j=1,\dots, N$. Among the examples considered are ellipses of different eccentricity, a straight wire and a cardioid. We have found that, for some geometries, multiple minima are present, as well as points of unstable equilibrium. For the case of the cardioid, we observe that the presence of the cusp has a dramatic effect on the distribution of the charges, in the limit $N \gg 1$.