Geometry of KAM tori for nearly integrable Hamiltonian systems

Broer, Hendrik; Cushman, Richard; Fassò, Francesco; Takens, Floris

doi:10.1017/s0143385706000897

Cited by 36 publications

(61 citation statements)

References 28 publications

(46 reference 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%

Normalization and global analysis of perturbations of the hydrogen atom

Efstathiou

Sadovskiı́

2010

Rev. Mod. Phys.

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The hydrogen atom perturbed by sufficiently small homogeneous static electric and magnetic fields of arbitrary mutual alignment is a specific perturbation of the Kepler system with three degrees of freedom and three parameters. Normalization of the Keplerian symmetry reveals that the parameter space is stratified into resonant zones of systems, each zone with an internal dynamical stratification of its own ͑Efstathiou, Sadovskií, and Zhilinskií, 2007, Proc. R. Soc. London, Ser. A 463, 1771͒. Based on the fully integrable approximation, the bundle of invariant tori of individual systems within zones is characterized globally and the qualitative dynamical stratification is uncovered. The techniques involved in this analysis are illustrated with the example of the 1:1 resonance zone ͑near orthogonal fields͒ whose structure is known at present. Applications in the corresponding quantum system are also described. 2148 References 2148 2101 K. Efstathiou and D. A. Sadovskií: Normalization and global analysis of …

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Section: Characteristics Of Individual Perturbationsmentioning

confidence: 98%

Normalization and global analysis of perturbations of the hydrogen atom

Efstathiou

Sadovskiı́

2010

Rev. Mod. Phys.

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“…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%

Normal linear stability of quasi-periodic tori

Broer

Hoo

Naudot

2007

Journal of Differential Equations

113

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We consider families of dynamical systems having invariant tori that carry quasi-periodic motions. Our interest is the persistence of such tori under small, nearly-integrable perturbations. This persistence problem is studied in the dissipative, the Hamiltonian and the reversible setting, as part of a more general KAM theory for classes of structure preserving dynamical systems. This concerns the parametrized KAM theory as initiated by Moser [J.K. Moser, On the theory of quasiperiodic motions, SIAM Rev. 8 (2) (1966)145-172; J.K. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann. 169 (1967) 136-176] and further developed in [G.B. Huitema, Unfoldings of quasi-periodic tori, PhD thesis, University of Groningen, 1988; H.W. Broer, G.B. Huitema, F. Takens, Unfoldings of quasi-periodic tori, Mem. Amer. Math. Soc. 83 (421) (1990) 1-82; H.W. Broer, G.B. Huitema, Unfoldings of quasi-periodic tori in reversible systems, J. Dynam. Differential Equations 7 (1) (1995) 191-212]. The corresponding nondegeneracy condition involves certain (trans-)versality conditions on the normal linear, leading, part at the invariant tori. We show that as a consequence, a Cantor family of Diophantine tori with positive Hausdorff measure is persistent under nearly-integrable perturbations. This result extends the above references since presently the case of multiple Floquet exponents is included. Our leading example is the normal 1 : −1 resonance, which occurs a lot in applications, both Hamiltonian and reversible. As an illustration of this we briefly describe the Lagrange top coupled to an oscillator.

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“…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%

Quantum monodromy and its generalizations and molecular manifestations

Sadovskií

Zhilinskii

2006

Molecular Physics

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Quantum monodromy is a non-trivial qualitative characteristic of certain non-regular lattices formed by the joint eigenvalue spectrum of mutually commuting operators. The latter are typically the Hamiltonian (energy) and the momentum operator(s) which label the eigenstates of the system. We give a brief review of known quantum systems with monodromy, which include such fundamental systems as the hydrogen atom in external fields, Fermi resonant vibrations of the CO 2 molecule, and non-rigid triatomic molecules. We emphasize the correspondence between the classical Hamiltonian monodromy and its quantum analogue and discuss possible generalizations of this characteristic in classical integrable Hamiltonian dynamical systems and their quantum counterparts.

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Geometry of KAM tori for nearly integrable Hamiltonian systems

Cited by 36 publications

References 28 publications

Normalization and global analysis of perturbations of the hydrogen atom

Normalization and global analysis of perturbations of the hydrogen atom

Normal linear stability of quasi-periodic tori

Quantum monodromy and its generalizations and molecular manifestations

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