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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| Published in | SIAM review Vol. 28; no. 4; pp. 459 - 485 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Philadelphia, PA
The Society for Industrial and Applied Mathematics
01.12.1986
Society for Industrial and Applied Mathematics |
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| Online Access | Get full text |
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| Abstract | 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. |
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| AbstractList | 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. |
| Author | Moser, Jurgen |
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| Cites_doi | 10.1017/S0143385700003588 10.1016/0040-9383(82)90023-4 10.1063/1.524170 10.1007/978-3-322-96656-8_1 10.1007/BF02565865 10.1007/978-1-4899-2689-0 10.24033/bsmf.1182 10.1016/S0294-1449(16)30387-0 10.2307/1968215 10.1063/1.524125 10.1051/jphyslet:01984004509040900 10.1088/0305-4470/12/3/001 10.1007/BF01214882 10.1016/0167-2789(81)90034-8 10.1016/0167-2789(83)90233-6 10.1007/BF02567431 10.1007/BF02401754 10.1016/0040-9383(87)90017-6 10.1007/BF02566283 10.1111/j.1749-6632.1980.tb29678.x 10.1090/S0002-9947-1924-1501263-9 10.1007/BFb0107359 10.1073/pnas.34.2.47 10.1007/978-3-642-87284-6 10.1063/1.1664639 10.1017/S0143385700002455 10.1017/S0143385700002662 |
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| Copyright | Copyright 1986 The Society for Industrial and Applied Mathematics 1987 INIST-CNRS [Copyright] © 1986 Society for Industrial and Applied Mathematics |
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| Keywords | Hamiltonian system Iteration Stability Dynamic system Application |
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| References | Birkhoff G. D. (R4) 1932; 60 R21 R23 R22 Denjoy A. (R8) 1932; 11 R25 R24 R27 R26 R28 Caratheodory C. (R7) 1935 R1 R2 R3 R5 R6 Gelfand I. M. (R9) 1963 Hall Glen Richard (R15) 1984; 4 Herman M. R. (R35) 1983; 103 R30 R10 R31 R12 R34 R11 Moser Jürgen (R29) 1986; 3 R13 R16 R18 R17 R19 Rannou F. (R33) 1974; 31 |
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| Snippet | Area-preserving mappings of an annulus occur as Poincare mappings of Hamiltonian systems; they were studied extensively by G. D. Birkhoff. Recently Aubry and... |
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| SubjectTerms | 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 |
| Title | Recent Developments in the Theory of Hamiltonian Systems |
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