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
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.
Integrable systems, Lie groups, geodesic flow, left-invariant metric, sub-Riemannian structure
|Russian Science Foundation
|This work was supported by the Russian Science Foundation (project No. 17-11-01303).
MSC: 37J35, 53B50, 70H06, 70S10
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
\by Alexey Bolsinov, Jinrong Bao
\paper A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras
\jour Regul. Chaotic Dyn.
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