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Regul. Chaotic Dyn., 2019, Volume 24, Issue 3, Pages 266–280 (Mi rcd477)  

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

Alexey Bolsinovab, Jinrong Baob

a Faculty of Mechanics and Mathematics, Moscow State University, 11992 Russia
b School of Mathematics, Loughborough University, Loughborough, Leicestershire, LE11 3TU, United Kingdom

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.

Keywords: Integrable systems, Lie groups, geodesic flow, left-invariant metric, sub-Riemannian structure

Funding Agency Grant Number
Russian Science Foundation 17-11-01303
This work was supported by the Russian Science Foundation (project No. 17-11-01303).


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

References: PDF file   HTML file

Bibliographic databases:

MSC: 37J35, 53B50, 70H06, 70S10
Received: 17.09.2018
Accepted:20.10.2018
Language:

Citation: Alexey Bolsinov, Jinrong Bao, “A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras”, Regul. Chaotic Dyn., 24:3 (2019), 266–280

Citation in format AMSBIB
\Bibitem{BolBao19}
\by Alexey Bolsinov, Jinrong Bao
\paper A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras
\jour Regul. Chaotic Dyn.
\yr 2019
\vol 24
\issue 3
\pages 266--280
\mathnet{http://mi.mathnet.ru/rcd477}
\crossref{https://doi.org/10.1134/S156035471903002X}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85066469876}


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