To what extent are unstable the maxima of the potential?

Abstract

The classical Lagrange–Dirichlet stability theorem states that, for natural mechanical systems, the strict minima of the potential are dynamically stable. Its converse, i.e., the instability of the maxima of the potential, has been proved by several authors including Liapunov (The general problem of stability of motion, 1892), Hagedorn (Arch Ration Mech Anal 42:281–316, 1971) or Taliaferro (Arch Ration Mech Anal 73(2):183–190, 1980), in various degrees of generality. We complement their theorems by presenting an example of a smooth potential on the plane having an isolated maximum and such that the associated dynamical system has a converging sequence of periodic orbits. This implies that the maximum is not unstable in a stronger sense considered by Siegel and Moser.