Newton Algorithm on Constraint Manifolds and the 5-Electron Thomson Problem

We give a description of numerical Newton algorithm on a constraint manifold using only the ambient coordinates (usually Euclidean coordinates) and the geometry of the constraint manifold. We apply the numerical Newton algorithm on a sphere in order to find the critical configurations of the 5-electron Thomson problem. As a result, we find a new critical configuration of a regular pentagonal type. We also make an analytical study of the critical configurations found previously and determine their nature using Morse–Bott theory. The last section contains an analytical study of critical configurations for Riesz s -energy of 5-electron on a sphere, and their bifurcation behavior is pointed out.