Rotation forms and local Hamiltonian monodromy

Abstract : The monodromy of torus bundles associated with completely integrable systems can be computed using geometric techniques (constructing homology cycles) or analytic arguments (computing discontinuities of abelian integrals). In this article, we give a general approach to the computation of monodromy that resembles the analytical one, reducing the problem to the computation of residues of polar 1-forms. We apply our technique to three celebrated examples of systems with monodromy (the champagne bottle, the spherical pendulum, the hydrogen atom) and to the case of non-degenerate focus-focus singularities, re-obtaining the classical results. An advantage of this approach is that the residue-like formula can be shown to be local in a neighborhood of a singularity, hence allowing the definition of monodromy also in the case of non-compact fibers. This idea has been introduced in the literature under the name of scattering monodromy. We prove the coincidence of the two definitions with the monodromy of an appropriately chosen compactification. Published by AIP Publishing.
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Article dans une revue
Journal of Mathematical Physics, American Institute of Physics (AIP), 2017, 58 (2), pp.022902. 〈10.1063/1.4975215〉
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https://hal-univ-bourgogne.archives-ouvertes.fr/hal-01514195
Contributeur : Ub_drive Université de Bourgogne <>
Soumis le : mardi 25 avril 2017 - 18:15:36
Dernière modification le : mardi 23 janvier 2018 - 15:46:28

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K. Efstathiou, A. Giacobbe, Pavao Mardešić, D. Sugny. Rotation forms and local Hamiltonian monodromy. Journal of Mathematical Physics, American Institute of Physics (AIP), 2017, 58 (2), pp.022902. 〈10.1063/1.4975215〉. 〈hal-01514195〉

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