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A note about integrable systems on low-dimensional Lie groups and Lie algebras
journal contributionposted on 2019-07-22, 09:09 authored by Alexey V. Bolsinov, Jinrong Bao
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.
- Mathematical Sciences
Published inRegular and Chaotic Dynamics
Pages266 - 280
CitationBOLSINOV, A.V. and BAO, J., 2019. A note about integrable systems on low-dimensional Lie groups and Lie algebras. Regular and Chaotic Dynamics, 24 (3), pp.266-280.
Publisher© Pleiades Publishing, Ltd.
- AM (Accepted Manuscript)
Publisher statementThis work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/
NotesThis a preprint of the Work accepted for publication in Regular and Chaotic Dynamics, © 2019 Pleiades Publishing, Ltd. The definitive published version is available at https://doi.org/10.1134/S156035471903002X. The publisher's website is at: http://pleiades.online/