Topological Constraints on the Charge Distributions for the Thomson Problem

TL;DR

This paper uses Morse theory to analyze charge distributions on a sphere in the Thomson problem, revealing topological reasons for pentagonal structures that explain previously observed numerical patterns.

Contribution

It introduces a topological approach using Morse theory to explain the formation of pentagonal structures in the Thomson problem, providing a qualitative understanding.

Findings

Pentagonal structures arise due to topological constraints.

Morse theory explains the organization of charges on a sphere.

Accounts for 'pentagonal buttons' observed in numerical studies.

Abstract

The method of Morse theory is used to analyze the distributions of unit charges interacting through a repulsive force and constrained to move on the surface of a sphere -- the Thomson problem. We find that, due to topological reasons, the system may organize itself in the form of pentagonal structures. This gives a qualitative account for the interesting ``pentagonal buttons'' discovered in recent numerical work.