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

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 $N \gg 1$.