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Action minimizing orbits in Hamiltonian systems. (English) Zbl 0822.70011

Graffi, S. (ed.), Transition to chaos in classical and quantum mechanics. Lectures given at the 3rd session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, July 6-13, 1991. Berlin: Springer-Verlag. Lect. Notes Math. 1589, 92-186 (1994).
This is a long and interesting survey on the dynamics of area preserving mappings. The study of this subject dates from the pioneering work of PoincarĂ©. He pointed out the intimate connection between the dynamics of area preserving mappings and the dynamics of two-dimensional Hamiltonian systems. In 1981, the first author discovered a generalization of certain K.A.M. invariant curves [Adv. Math., Suppl. Stud. 7B, 531-562 (1981; Zbl 0505.58027)]. Similar ideas were discovered by S. Aubry and his coworkers, independently, around the same time [NATO ASI Series, Series B: Physics, 118 (1985; Zbl 0661.70002)]. The first author’s original approach was based on a variational principle introduced by I. C. Percival [J. Physics A7, 794-802 (1974; Zbl 0304.70022)]. The purpose of these lectures is to describe this generalization of the K.A.M. invariant curves, the minimisation procedure used to prove their existence, and related constructions. The authors have made no attempt to survey the whole theory of Hamiltonian systems. Instead, their intention is to provide an introduction to the theory developed by the first author and S. Aubry, known as “Aubry-Mather” theory.
For the entire collection see [Zbl 0801.00010].
Reviewer: F.Cardin (Padova)

MSC:

70H05 Hamilton’s equations
70H30 Other variational principles in mechanics
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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