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 : 13 | Download : 11

Publication Date : 2020-03-25

Article Type : Review Paper

Abstract :The aim of this paper is to demonstrate the rich interaction between the Kowalewski-Painlev\`{e} analysis, the properties of algebraic completely integrable insert ignore into journalissuearticles values(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 $SOinsert ignore into journalissuearticles values(n);$ is a.c.i. For $n=3$, the geodesic flow on $SOinsert ignore into journalissuearticles values(3);$ 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 $SOinsert ignore into journalissuearticles values(4);$ 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 $soinsert ignore into journalissuearticles values(4);$. Finally, as was shown by Haine, for $n\geq 5$ Manakov`s metrics are the only left invariant diagonal metrics on $SOinsert ignore into journalissuearticles values(n);$ for which the geodesic flow is a.c.i.
Keywords : Jacobians varieties, Prym varieties, Integrable systems, Topological structure of phase space, Methods of integration