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Recent developments in the theory of Hamiltonian systems. (English) Zbl 0606.58022
Area-preserving mappings of an annulus occur as Poincaré mappings of Hamiltonian systems; they were studied extensively by G. D. Birkhoff. Recently Aubry and Mather investigated the subclass of so-called monotone twist mappings for which they constructed independently closed invariant Cantor sets. Their work led to important new results. Their theory is discussed and related to Hamiltonian systems satisfying a Legendre condition. The connection of this theory with the stability problem, KAM theory, and in particular, the disintegration of invariant tori is discussed. Various constructions of Mather sets are explained, in which minimal solutions of variational problems play a central role. This theory has a close relation to the differential geometric investigations by Morse and Hedlund on geodesics on two-dimensional surfaces.

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37C75 Stability theory for smooth dynamical systems
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