Constrained electrostatic equilibrium

A particle constrained to a surface can have a stable minimum of electrostatic potential restricted to that surface even when the unrestricted equilibrium is unstable. Away from source charges, Laplace's equation makes the full potential Hessian matrix trace zero, which rules out a positive-definite three-dimensional Hessian but does not rule out a positive-definite restriction to two dimensions. Equal positive charges at the corners of a square give a planar minimum at its centre and an unstable perpendicular direction.