Loughborough, Leicestershire, LE11 3TU, United Kingdom
School of Mathematics, Loughborough University
Bolsinov A. V., Bao J.
A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras
2019, vol. 24, no. 3, pp. 266-280
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.