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.
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
\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{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85066469876}
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https://www.mathnet.ru/eng/rcd477
https://www.mathnet.ru/eng/rcd/v24/i3/p266
This publication is cited in the following 5 articles:
Božidar Jovanović, Tijana Šukilović, Srdjan Vukmirović, “Almost multiplicity free subgroups of compact Lie groups and polynomial integrability of sub-Riemannian geodesic flows”, Lett Math Phys, 114:1 (2024)
A.P. Veselov, Y. Ye, “Quantum Bianchi-VII problem, Mathieu functions and arithmetic”, Journal of Geometry and Physics, 189 (2023), 104830
Zlatko Erjavec, Jun-ichi Inoguchi, “J-Trajectories in 4-Dimensional Solvable Lie Group $\mathrm {Sol}_0^4$”, Math Phys Anal Geom, 25:1 (2022)
A. V. Bolsinov, A. P. Veselov, Y. Ye, “Chaos and integrability in $\operatorname{SL}(2,\mathbb R)$-geometry”, Russian Math. Surveys, 76:4 (2021), 557–586
R. El Assoudi-Baikari, E. Zibo, “Elliptic optimal control and symmetric sub-Riemannian spaces”, IFAC PAPERSONLINE, 54:9 (2021), 610–614
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