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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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