Recent Developments in the Theory of Hamiltonian Systems

Area-preserving mappings of an annulus occur as Poincare 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. T...

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Bibliographic Details
Published inSIAM review Vol. 28; no. 4; pp. 459 - 485
Main Author Moser, Jurgen
Format Journal Article
LanguageEnglish
Published Philadelphia, PA The Society for Industrial and Applied Mathematics 01.12.1986
Society for Industrial and Applied Mathematics
Subjects
Approximation
Cantor set
Curves
Euler equations
Exact sciences and technology
Geometry, differential geometry, and topology
Investigations
Mathematical functions
Mathematical methods in physics
Mathematical monotonicity
Mathematical theorems
Mathematics
Periodic orbits
Physics
Rotation
Systems stability
Vector fields
Hamiltonian system
Iteration
Stability
Dynamic system
Application
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Summary:Area-preserving mappings of an annulus occur as Poincare 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.
ISSN:0036-1445
1095-7200
DOI:10.1137/1028153