Non-uniqueness of solutions of Percival's Euler-Lagrange equation (Q802183)

In [Topology 21, 457-467 (1982; Zbl 0506.58032)] we have studied area preserving twist homeomorphisms of the annulus, using Percival's formalism [\textit{I. C. Percival}, Variational principles for invariant tori and cantori, Symp. on Nonlinear Dynamics and the Beam-Beam Interaction, 302-310 (1979) and J. Phys. A 12, L57-L60 (1979; Zbl 0394.70018)]. We show that there exist \(C^{\infty}\) area preserving twist diffeomorphisms of the annulus, for which there exists at least one solution of Percival's Euler-Lagrange equation where Percival's Lagrangian does not take its maximum. In other words, solutions of Percival's Euler-Lagrange equation need not be unique.