- Communications in Advanced Mathematical Sciences
- Vol: 3 Issue: 1
- Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Int...
Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable
Authors : Lesfari Ahmed
Pages : 24-52
Doi:10.33434/cams.649612
View : 10 | Download : 5
Publication Date : 2020-03-25
Article Type : Other
Abstract :The aim of this paper is to demonstrate the rich interaction between the Kowalewski-Painlev\'{e} analysis, the properties of algebraic completely integrable (a.c.i.) systems, the geometry of its Laurent series solutions, and the theory of Abelian varieties. We study the classification of metrics for which geodesic flow on the group $SO(n)$ is a.c.i. For $n=3$, the geodesic flow on $SO(3)$ is always a.c.i., and can be regarded as the Euler rigid body motion. For $n=4$, in the Adler-van Moerbeke's classification of metrics for which geodesic flow on $SO(4)$ is a.c.i., three cases come up; two are linearly equivalent to the Clebsch and Lyapunov-Steklov cases of rigid body motion in a perfect fluid, and there is a third new case namely the Kostant-Kirillov Hamiltonian flow on the dual of $so(4)$. Finally, as was shown by Haine, for $n\geq 5$ Manakov's metrics are the only left invariant diagonal metrics on $SO(n)$ for which the geodesic flow is a.c.i.Keywords : Jacobians varieties, Prym varieties, Integrable systems, Topological structure of phase space, Methods of integration