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Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples

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Abstract

This paper deals with non-integrability criteria, based on differential Galois theory and requiring the use of higher order variational equations. A general methodology is presented to deal with these problems. We display a family of Hamiltonian systems which require the use of order k variational equations, for arbitrary values of k, to prove non-integrability. Moreover, using third order variational equations we prove the non-integrability of a non-linear spring-pendulum problem for the values of the parameter that can not be decided using first order variational equations.

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Authors and Affiliations

  1. Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona, Spain

    R. Martínez

  2. Dept. de Matemàtica Aplicada i Anàlisi, Univ. de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    C. Simó

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  1. R. Martínez
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  2. C. Simó
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Correspondence to R. Martínez.

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Martínez, R., Simó, C. Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples. Regul. Chaot. Dyn. 14, 323–348 (2009). https://doi.org/10.1134/S1560354709030010

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