  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. 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.
