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Geometry of integrable dynamical systems on 2-dimensional surfaces

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Abstract

This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous invariants involved in this classification are the left equivalence classes of period or monodromy functions and the cohomology classes of period cocycles, which can be expressed in terms of Puiseux series. We also study the problem of Hamiltonianization of these integrable vector fields by a compatible symplectic or Poisson structure.

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Acknowledgements

The authors thank Alexey Bolsinov and Andrey Oshemkov for useful discussions about the problem. The work of N.V. Minh is supported by a Ph.D. fellowship from the program “322” of the Ministry of Education and Training of Vietnam.

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  1. Institut de Mathématiques de Toulouse, UMR5219, Université Toulouse 3, Toulouse, France

    Nguyen Tien Zung & Nguyen Van Minh

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  1. Nguyen Tien Zung
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  2. Nguyen Van Minh
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Correspondence to Nguyen Tien Zung.

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Zung, N.T., Minh, N.V. Geometry of integrable dynamical systems on 2-dimensional surfaces. Acta Math Vietnam. 38, 79–106 (2013). https://doi.org/10.1007/s40306-012-0005-9

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