A note about integrable systems on low-dimensional Lie groups and Lie algebras
Alexey V. Bolsinov
Jinrong Bao
2134/38371
https://repository.lboro.ac.uk/articles/journal_contribution/A_note_about_integrable_systems_on_low-dimensional_Lie_groups_and_Lie_algebras/9389150
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.
2019-07-22 09:09:30
Integrable systems
Lie groups
Geodesic flow
Left-invariant metric
Sub-Riemannian structure
Mathematical Sciences not elsewhere classified