Point Charges and Polygonal Linkages

TL;DR

This paper studies the Coulomb potential of point charges on polygonal linkages, proving the existence of a unique minimum in convex configurations and demonstrating that two controlling charges suffice to navigate between any two convex states.

Contribution

It establishes the uniqueness of the critical point for positive charges on pentagonal linkages and shows two charges can control configuration transitions.

Findings

Unique critical point in convex configurations for positive charges

Two controlling charges are sufficient for configuration navigation

Proved the absolute minimum of Coulomb potential in convex states

Abstract

We investigate the critical points of Coulomb potential of point charges placed at the vertices of a planar polygonal linkage. It is shown that, for a collection of positive charges on a pentagonal linkage, there is a unique critical point in the set of convex configurations which is the point of absolute minimum. This enables us to prove that two controlling charges are sufficient to navigate between any two convex configurations of a pentagonal linkage.