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Noncompactness Property of Fibers and Singularities of Non-Euclidean Kovalevskaya System on Pencil of Lie Algebras

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Abstract

It is shown that Liouville foliations of the family of non-Euclidean analogs of Kovalevskaya integrable system on a pencil of Lie algebras have both compact and noncompact fibers. There exists a bifurcation of their compact common level surface into a noncompact one that has a noncompact singular fiber. In particular, this is true for the non-Euclidean \(e(2,1)\)-analog of the Kovalevskaya case of rigid body dynamics. In the case of nonzero area integral, an effective criterion for existence of a noncompact connected component of the common level surface of first integrals and Casimir functions is proved.

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ACKNOWLEDGMENTS

The author thanks his supervisor A. T. Fomenko for his attention to the work.

Funding

The author receives a scholarship of the Theoretical Physics and Mathematics Advancement Foundation ‘‘BASIS’’ (project no. 18-2-6-51-1).

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Authors and Affiliations

  1. Chair of Differential Geometry and Applications, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    V. A. Kibkalo

  2. Moscow Center for Fundamental and Applied Mathematics, Moscow, Russia

    V. A. Kibkalo

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  1. V. A. Kibkalo
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Correspondence to V. A. Kibkalo.

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Translated by E. Oborin

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Kibkalo, V.A. Noncompactness Property of Fibers and Singularities of Non-Euclidean Kovalevskaya System on Pencil of Lie Algebras. Moscow Univ. Math. Bull. 75, 263–267 (2020). https://doi.org/10.3103/S0027132220060054

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