Newton Algorithm on Constraint Manifolds and the 5-electron Thomson problem

TL;DR

This paper develops a Newton algorithm on constraint manifolds using ambient coordinates and applies it to the 5-electron Thomson problem, discovering new critical configurations and analyzing their stability.

Contribution

It introduces a Newton algorithm tailored for constraint manifolds and applies it to find and analyze critical configurations in the 5-electron Thomson problem.

Findings

Discovered a new critical configuration of a regular pentagonal type.

Analyzed the nature and stability of known critical configurations.

Studied bifurcation behavior of Riesz s-energy configurations.

Abstract

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.