📝 SummarXiv

Abstract

We investigate the low-energy configurations of N mutually repelling charges confined to a spherical cap and interacting via the Coulomb potential. In the continuum limit, this problem was solved by Lord Kelvin, who found a non-uniform charge distribution with an integrable singularity at the boundary. To explore the discrete analogue, we developed an efficient numerical method that enables energy minimization while maintaining the number of charges at the cap's edge fixed. Using this approach we have obtained numerical results for various values of N and cap angular widths. Based on these results, we analyze the emergence and behavior of topological defects as functions of both N and the cap's curvature.