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Mather sets for plane Hamiltonian systems. (English) Zbl 0641.70014
Recently J. N. Mather [Topology 21, 457-467 (1982; Zbl 0506.58032)] developed a theory for area preserving monotone twist maps of an annulus, the main result of which is a generalization of the notion of an invariant curve of such maps. This theory can be applied to the Poincaré map of plane Hamiltonian system. However, the monotonicity condition is not satisfied in general, for example in the case of a periodic perturbation, the Poincaré map is the period map and the period will not be small enough in general; and thus the theory of monotone twist maps does not apply.
The purpose of this paper is to show the ideas, however, carry over to a more general case leading to a theory for plane periodic Hamiltonian differential equations. This generalization is studied under differentiability assumptions, which are not needed in Mather’s results. Given the necessary differentiability, J. Moser [Ergodic Theory and Dynamical Systems 6, 401-413 (1986; Zbl 0619.49020)] showed that an area preserving monotone twist map can be interpreted by a Hamiltonian system. The methods developed in this paper are similar to those for monotone twist maps. The author provides a reasonably self-contained exposition of Mather’s theory. The theory developed in this paper makes extensive use of the total ordering of the real line, thus it does not generalize to higher order ordinary differential equations.
Reviewer: M.Z.Nashed

##### MSC:
 70H05 Hamilton’s equations 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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##### References:
 [1] N. I. Akhiezer,The Calculus of Variations, Blaisdell Publ. Co., 1962. · Zbl 0119.05604 [2] S. Aubry, P. Y. Le Daeron,The discrete Fraenkel-Kontorova model and its extensions I, Physica 8D, 381-422 (1983). · Zbl 1237.37059 [3] V. Bangert,Mather Sets for Twist Mappings and Geodesics on Tori, in: Dynamics Reported1, Wiley and Teubner, 1987. [4] G. A. Bliss,Calculus of Variations (Carus Monograph1), 6. Aufl. 1971. [5] C. Caratheodory,Variationsrechnung und Partielle Differentialgleichungen erster Ordnung, Teubner 1935. · JFM 61.0547.01 [6] J. Denzler,Studium globaler Minimaler eines Variationsproblems, Diplomarbeit ETH Zürich, February 1987. [7] D. Gilbarg and N. S. Trudinger,Elliptic Partial Differential Equations of Second Order, Springer Grundlehren224 (1977). · Zbl 0361.35003 [8] M. Giaquinta,Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton Univ. Press; Ann. Math. Studies105. · Zbl 0516.49003 [9] O. A. Ladyzhenskaya and N. N. Ural’tseva,Linear and Quasilinear Elliptic Equations, Acad. Press (1968). [10] J. N. Mather,Existence of Quasiperiodic Orbits for Twist Homeomorphisms of the Annulus, Topology21 (1982), 457-467. · Zbl 0506.58032 · doi:10.1016/0040-9383(82)90023-4 [11] J. N. Mather,Destruction of invariant circles, submitted to Ergodic Theory and Dynamical Systems. [12] MacKay, Stark,Lectures on Orbits of Minimal Action for Area Preserving Maps; preprint, University of Warwick, 1985. [13] Ch. B. Morrey,Multiple Integrals in the Calculus of Variations, Springer Grundlehren 130, (1966). · Zbl 0142.38701 [14] J. Moser,Monotone Twist Mappings and the Calculus of Variations, Ergodic Theory and Dynamical Systems6, 401-413 (1968). · Zbl 0619.49020 [15] J. Moser,Recent Developments in the Theory of Hamiltonian Systems, SIAM Reviews28, 459-485 (1986). · Zbl 0606.58022 · doi:10.1137/1028153 [16] J. Moser,Minimal solutions of variational problems on a torus, Ann. Inst. Henri Poincaré,3, 229-272 (1986). · Zbl 0609.49029
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