Point Charges and Polygonal Linkages

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.