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A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras

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Abstract

The goal of the paper is to explain why any left-invariant Hamiltonian system on (the cotangent bundle of) a 3-dimensonal Lie group G is Liouville integrable. We derive this property from the fact that the coadjoint orbits of G are two-dimensional so that the integrability of left-invariant systems is a common property of all such groups regardless their dimension.

We also give normal forms for left-invariant Riemannian and sub-Riemannian metrics on 3-dimensional Lie groups focusing on the case of solvable groups, as the cases of SO(3) and SL(2) have been already extensively studied. Our description is explicit and is given in global coordinates on G which allows one to easily obtain parametric equations of geodesics in quadratures.

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Acknowledgments

The authors are very grateful to Alexey Borisov and Ivan Mamaev for valuable comments and stimulating discussions.

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Authors and Affiliations

  1. School of Mathematics, Loughborough University, Loughborough, Leicestershire, LE11 3TU, UK

    Alexey Bolsinov & Jinrong Bao

  2. Faculty of Mechanics and Mathematics, Moscow State University, Moscow, 11992, Russia

    Alexey Bolsinov

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  1. Alexey Bolsinov
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  2. Jinrong Bao
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Correspondence to Alexey Bolsinov or Jinrong Bao.

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Bolsinov, A., Bao, J. A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras. Regul. Chaot. Dyn. 24, 266–280 (2019). https://doi.org/10.1134/S156035471903002X

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  • DOI: https://doi.org/10.1134/S156035471903002X

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