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Abelian Integrals: From the Tangential 16th Hilbert Problem to the Spherical Pendulum

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Mathematical Sciences with Multidisciplinary Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 157))

Abstract

In this chapter we deal with abelian integrals. They play a key role in the infinitesimal version of the 16th Hilbert problem. Recall that 16th Hilbert problem and its ramifications is one of the principal research subject of Christiane Rousseau and of the first author. We recall briefly the definition and explain the role of abelian integrals in 16th Hilbert problem. We also give a simple well-known proof of a property of abelian integrals. The reason for presenting it here is that it serves as a model for more complicated and more original treatment of abelian integrals in the study of Hamiltonian monodromy of fully integrable systems, which is the main subject of this chapter. We treat in particular the simplest example presenting non-trivial Hamiltonian monodromy: the spherical pendulum.

Dedicated to Christiane Rousseau for her mathematical and non-mathematical impact

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Acknowledgements

D. Sugny acknowledges support of the Technische Universität München, Institute for Advanced Study, funded by the German Excellence Initiative and the European Union Seventh Framework Programme under grant agreement number 291763. As a Hans Fisher fellow (partially financed through the Marie Curie COFUND program), D. Sugny also acknowledges the support from the European Union.

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Authors and Affiliations

  1. Institut de Mathématiques de Bourgogne, Université de Bourgogne, UMR CNRS 5584, 9 Avenue Alain Savary, BP 47 870, F-21078, Dijon Cedex, France

    Pavao Mardešić

  2. Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS-Université de Bourgogne, 9 Avenue Alain Savary, BP 47 870, F-21078, Dijon Cedex, France

    Dominique Sugny & Léo Van Damme

  3. Institute for Advanced Study, Technische Universität München, Lichtenbergstrasse 2 a, 85748, Garching, Germany

    Dominique Sugny & Léo Van Damme

Authors
  1. Pavao Mardešić
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  2. Dominique Sugny
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  3. Léo Van Damme
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Correspondence to Pavao Mardešić .

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Editors and Affiliations

  1. Dept of Mathematics and Economics, Virginia State University, Petersburg, Virginia, USA

    Bourama Toni

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Mardešić, P., Sugny, D., Van Damme, L. (2016). Abelian Integrals: From the Tangential 16th Hilbert Problem to the Spherical Pendulum. In: Toni, B. (eds) Mathematical Sciences with Multidisciplinary Applications. Springer Proceedings in Mathematics & Statistics, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-31323-8_15

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