Equilibria of point charges on convex curves

Abstract

We study the equilibrium positions of three points on a convex curve under influence of the Coulomb potential. We identify these positions as orthotripods, three points on the curve having concurrent normals. This relates the equilibrium positions to the caustic (evolute) of the curve. The concurrent normals can only meet in the core of the caustic, which is contained in the interior of the caustic. Moreover, we give a geometric condition for three points in equilibrium with positive charges only. For the ellipse we show that the space of orthotripods is homeomorphic to a 2-dimensional bounded cylinder.