Monodromy in the hydrogen atom in crossed fields

Cushman, Richard; Sadovskií, D. A.

doi:10.1016/s0167-2789(00)00053-1

Cited by 93 publications

(117 citation statements)

References 45 publications

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“…A natural question is whether (nontrivial) monodromy also can be defined for nonintegrable perturbations of the spherical pendulum. Answering this question is of interest in the study of semiclassical versions of such classical systems, see [14,8]. The results of the present paper imply that for an open set of Liouville integrable Hamiltonian systems, under a sufficiently small perturbation, the geometry of the fibration is largely preserved by a Whitney smooth diffeomorphism.…”

Section: Motivationmentioning

confidence: 65%

Geometry of KAM tori for nearly integrable Hamiltonian systems

Broer¹,

Cushman

Fassò³

et al. 2007

Ergod. Th. Dynam. Sys.

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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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Section: Motivationmentioning

confidence: 65%

Geometry of KAM tori for nearly integrable Hamiltonian systems

Broer¹,

Cushman

Fassò³

et al. 2007

Ergod. Th. Dynam. Sys.

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“…We emphasize that the idea of characterizing an aspect of the global geometry of the integrable Hamiltonian fibration by the parallel transport of homology cycles along a closed path is, with appropriate modifications, central to the concept of fractional monodromy and to our approach in the present paper. There are several examples of Hamiltonian systems with non-trivial monodromy and thus no global action coordinates, see [11,[13][14][15]17,23,34,35]. One sufficient condition for a system to have non-trivial monodromy is the existence of focus-focus singularities.…”

Section: The Matrix Of μ([γ ]) Is Called the Monodromy Matrix Along γmentioning

confidence: 99%

Uncovering Fractional Monodromy

Efstathiou

Broer

2013

Commun. Math. Phys.

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Abstract:The uncovering of the role of monodromy in integrable Hamiltonian fibrations has been one of the major advances in the study of integrable Hamiltonian systems in the past few decades: on one hand monodromy turned out to be the most fundamental obstruction to the existence of global action-angle coordinates while, on the other hand, it provided the correct classical analogue for the interpretation of the structure of quantum joint spectra. Fractional monodromy is a generalization of the concept of monodromy: instead of restricting our attention to the toric part of the fibration we extend our scope to also consider singular fibres. In this paper we analyze fractional monodromy for n 1 :(−n 2 ) resonant Hamiltonian systems with n 1 , n 2 coprime natural numbers. We consider, in particular, systems that for n 1 , n 2 > 1 contain one-parameter families of singular fibres which are 'curled tori'. We simplify the geometry of the fibration by passing to an appropriate branched covering. In the branched covering the curled tori and their neighborhood become untwisted thus simplifying the geometry of the fibration: we essentially obtain the same type of generalized monodromy independently of n 1 , n 2 . Fractional monodromy is then recovered by pushing the results obtained in the branched covering back to the original system.

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“…Some perturbed Keplerian systems like the lunar problem [14,40] and the hydrogen (or Rydberg) atom in crossed fields [15,21] display a helpful time scale phenomenon. In both cases the normal form…”

Section: Perturbations That Remove the Degeneracymentioning

confidence: 99%

Perturbations of Superintegrable Systems

Hanβmann

2015

Acta Appl Math

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A superintegrable system has more integrals of motion than degrees d of freedom. The quasi-periodic motions then spin around tori of dimension n < d. Already under integrable perturbations almost all n-tori will break up; in the non-degenerate case the resulting d-tori have n fast and d − n slow frequencies. Such d-parameter families of d-tori do survive Hamiltonian perturbations as Cantor families of d-tori. A perturbation of a superintegrable system that admits a better approximation by a non-degenerate integrable perturbation of the superintegrable system is said to remove the degeneracy. In the minimal case d = n + 1 this can be achieved by means of averaging, but the more integrals of motion the superintegrable system admits the more difficult becomes the perturbation analysis.

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Monodromy in the hydrogen atom in crossed fields

Cited by 93 publications

References 45 publications

Geometry of KAM tori for nearly integrable Hamiltonian systems

Geometry of KAM tori for nearly integrable Hamiltonian systems

Uncovering Fractional Monodromy

Perturbations of Superintegrable Systems

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