Equilibria of three constrained point charges

TL;DR

This paper analyzes the critical points of Coulomb energy for three point charges under specific geometric constraints, revealing Morse properties, stationary points, and a bifurcation phenomenon similar to buckling.

Contribution

It introduces a detailed study of Coulomb energy critical points under geometric constraints, identifying Morse characteristics and bifurcation behavior for three charges.

Findings

Coulomb energy is a Morse function in the studied settings.

Identification of minima and stationary points of Coulomb energy.

Discovery of a pitchfork bifurcation akin to buckling phenomena.

Abstract

We study the critical points of Coulomb energy considered as a function on configuration spaces associated with certain geometric constraints. Two settings of such kind are discussed in some detail. The first setting arises by considering polygons of fixed perimeter with freely sliding positively charged vertices. The second one is concerned with triples of positive charges constrained to three concentric circles. In each of these cases the Coulomb energy is generically a Morse function. We describe the minima and other stationary points of Coulomb energy and show that, for three charges, a pitchfork bifurcation takes place accompanied by an effect of the Euler's Buckling Beam type.