RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Regul. Chaotic Dyn.: Year: Volume: Issue: Page: Find

 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
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} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000470233800002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85066469876}