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Non-uniqueness of solutions of Percival's Euler-Lagrange equation

1982; Springer Science+Business Media; Volume: 86; Issue: 4 Linguagem: Inglês

10.1007/bf01214882

1432-0916

John N. Mather,

Mathematics and Applications

Percival [5,6] introduced a Langrangian and an Euler-Lagrange equation for finding quasi-periodic orbits. In [3], we studied area preserving twist homeomorphisms of the annulus, using Percival's formalism. We showed that Percival's Lagrangian has a maximum on a suitable function space, and that a point where it takes its maximum is a solution of Percival's Euler-Lagrange equation. Moreover, in the rigorous interpretation of Percival's formalism which we gave in [3], the solutions of Percival's Euler-Lagrange equation correspond bijectively to a certain class of minimal sets. (We will prove this in Sect. 2.) In [4], we showed that Percival's Lagrangian takes its maximum at only one point. In this paper, we show that there existC ∞ 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.