Аннотация:
The goal of the paper is to explain why any left-invariant Hamiltonian system on (the cotangent bundle of) a 33-dimensonal Lie group GG is Liouville integrable. We derive this property from the fact that the coadjoint orbits of GG 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 33-dimensional Lie groups focusing on the case of solvable groups, as the cases of SO(3)SO(3) and SL(2)SL(2) have been already extensively studied. Our description is explicit and is given in global coordinates on GG which allows one to easily obtain parametric equations of geodesics in quadratures.
Образец цитирования:
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
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\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
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\crossref{https://doi.org/10.1134/S156035471903002X}
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Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/rcd477
https://www.mathnet.ru/rus/rcd/v24/i3/p266
Эта публикация цитируется в следующих 5 статьяx:
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 Sol40Sol40”, Math Phys Anal Geom, 25:1 (2022)
А. В. Болсинов, А. П. Веселов, И. Йе, “Хаос и интегрируемость в SL(2,R)-геометрии”, УМН, 76:4(460) (2021), 3–36; A. V. Bolsinov, A. P. Veselov, Y. Ye, “Chaos and integrability in SL(2,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
Цитирование в формате AMSBIB
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