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A note about integrable systems on low-dimensional Lie groups and Lie algebras

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journal contribution
posted 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.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Regular and Chaotic Dynamics

Volume

24

Issue

3

Pages

266 - 280

Citation

BOLSINOV, 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.

Version

  • AM (Accepted Manuscript)

Publisher statement

This 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/

Acceptance date

2018-10-20

Publication date

2019-06-03

Notes

This 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/

ISSN

1560-3547

eISSN

1468-4845

Language

  • en