2007
DOI: 10.1017/s0143385706000897
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Geometry of KAM tori for nearly integrable Hamiltonian systems

Abstract: We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangean tori by glueing together local KAM conjugacies with help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly-integrable system and an integrable one. This leads to preservation of geometry, which allows us to define all the nontrivial geometric invariants like monodromy or Chern classes of an integrable system also for near integrable systems.

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Cited by 36 publications
(61 citation statements)
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“…proach in the contemporary theory of integrable Hamiltonian dynamical systems described by Cushman and Bates ͑1997͒ and in a wider sense by Bolsinov and Fomenko ͑2004͒ ͓see also Zhilinskií ͑2001a͒, Efstathiou ͑2005͒, and Efstathiou and Sadovskií ͑2005͔͒. The obtained qualitative dynamical characteristics can be then attributed to the original nonintegrable system if the latter satisfies the conditions of the Kolmogorov-Arnol'd-Moser ͑KAM͒ theory and retains sufficiently many invariant tori ͑Rink, 2004; Broer et al, 2007͒. …”
Section: Characteristics Of Individual Perturbationsmentioning
confidence: 98%
“…proach in the contemporary theory of integrable Hamiltonian dynamical systems described by Cushman and Bates ͑1997͒ and in a wider sense by Bolsinov and Fomenko ͑2004͒ ͓see also Zhilinskií ͑2001a͒, Efstathiou ͑2005͒, and Efstathiou and Sadovskií ͑2005͔͒. The obtained qualitative dynamical characteristics can be then attributed to the original nonintegrable system if the latter satisfies the conditions of the Kolmogorov-Arnol'd-Moser ͑KAM͒ theory and retains sufficiently many invariant tori ͑Rink, 2004; Broer et al, 2007͒. …”
Section: Characteristics Of Individual Perturbationsmentioning
confidence: 98%
“…Now the crease (i.e., the singular part of the surface) and the thread correspond to invariant (n + 1)-tori; the smooth part of the surface is associated with the elliptic (n + 2)-tori; the open region above the surface gives rise to invariant Lagrangian (n + 3)-tori. Persistence of these higher-dimensional isotropic tori can be obtained by using 'standard' KAM theory [2,26,55,71,78], for a detailed treatment of this see [18,19].…”
Section: Application: Nearly-integrable Perturbations Of the Lagrangementioning
confidence: 99%
“…Furthermore, it has been conjectured in [7] and then demonstrated rigorously in [32,33] that monodromy 'survives' the breakdown of exact integrability and the Quantum monodromy and its generalizations 2599 onset of 'weak' chaos as long as most of the KAM tori of the perturbed system remain intact. This result is of great consequence to our applications since practically all real atomic and molecular systems are nonintegrable.…”
Section: Hamiltonian Monodromy and Its Generalizationsmentioning
confidence: 99%