Equilibria of point charges on convex curves

TL;DR

This paper investigates the equilibrium configurations of three point charges on convex curves, linking these positions to geometric properties like caustics and evolutes, and characterizes the space of such configurations for ellipses.

Contribution

It introduces the concept of orthotripods as equilibrium points, relates them to caustics, and describes the topology of their space for ellipses.

Findings

Orthotripods are points with concurrent normals related to the curve's caustic.

Equilibrium positions with positive charges are constrained by a geometric condition.

The space of orthotripods on an ellipse is a 2D bounded cylinder.

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.