Equilibria of Three Point Charges with Quadratic Constraints

Abstract

We discuss equilibrium configurations of the Coulomb potential of positive point charges with positions satisfying certain quadratic constraints in the plane and three-dimensional Euclidean space. The main attention is given to the case of three point charges satisfying a positive definite quadratic constraint in the form of equality or inequality. For a triple of points on the boundary of convex domain, we give a geometric criterion of the existence of positive point charges for which the given triple is an equilibrium configuration. Using this criterion, rather comprehensive results are obtained for three positive charges in the disc, ellipse, and three-dimensional ball. In the case of the circle, we strengthen these results by showing that any configuration consisting of an odd number of points on the circle can be realized as an equilibrium configuration of certain nonzero point charges and give a simple criterion for existence of positive charges with this property. Similar results are obtained for three point charges each of which belongs to one of the three concentric circles. Several related problems and possible generalizations are also discussed.