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Monotone twist mappings and the calculus of variations

Published online by Cambridge University Press:  19 September 2008

Jürgen Moser
Jürgen Moser
Affiliation:
Forschungsinstitut für Mathematik, ETH-Zentrum, CH-8092 Zürich, Switzerland
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Abstract

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It is shown that a smooth area-preserving monotone twist mapping ϕ of an annulus A can be interpolated by a flow ϕt which is generated by a t-dependent Hamiltonian in ℝ × A having the period 1 in t and satisfying a Legendre condition. In other words, any such monotone twist mapping can be viewed as a section mapping for the extremals of variational problem on a torus:

where F has period 1 in t and x and satisfies the Legendre condition Fẋẋ>0.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 3 , September 1986 , pp. 401 - 413
Copyright
Copyright © Cambridge University Press 1986

References

REFERENCES

[1]Aubry, S. & Le Daeron, P. Y.. The discrete Frenkel Kontorova model and its applications. Physica 8D, (1983) 381–422.Google Scholar
[2]Katok, A.. Some remarks on Birkhoff and Mather twist theorems. Ergod. Th. & Dynam. Sys. 2, (1982) 185–194.CrossRefGoogle Scholar
[3]Mather, J. N.. Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology 21, (1982) 457–467.CrossRefGoogle Scholar
[4]Moser, J.. Recent developments in the theory of Hamitonian systems. To appear in SIAM Reviews 1986.Google Scholar