Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable
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 metric...
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| Format: | Article |
| Language: | English |
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Emrah Evren KARA
2020-03-01
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| Series: | Communications in Advanced Mathematical Sciences |
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| Online Access: | https://dergipark.org.tr/tr/download/article-file/1021104 |
| _version_ | 1797294190163394560 |
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| author | Lesfari Ahmed |
| author_facet | Lesfari Ahmed |
| author_sort | Lesfari Ahmed |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-07T21:26:40Z |
| format | Article |
| id | doaj.art-353e68a6daa64e2781a99c6352ad9358 |
| institution | Directory Open Access Journal |
| issn | 2651-4001 |
| language | English |
| last_indexed | 2024-03-07T21:26:40Z |
| publishDate | 2020-03-01 |
| publisher | Emrah Evren KARA |
| record_format | Article |
| series | Communications in Advanced Mathematical Sciences |
| spelling | doaj.art-353e68a6daa64e2781a99c6352ad93582024-02-27T04:36:36ZengEmrah Evren KARACommunications in Advanced Mathematical Sciences2651-40012020-03-0131245210.33434/cams.6496121225Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely IntegrableLesfari Ahmed0Chouaïb Doukkali University, Faculty of Sciences, Dept. of Mathematics, B.P. 20, 24000, El Jadida, Morocco.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.https://dergipark.org.tr/tr/download/article-file/1021104jacobians varietiesprym varietiesintegrable systemstopological structure of phase spacemethods of integration |
| spellingShingle | Lesfari Ahmed Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable Communications in Advanced Mathematical Sciences jacobians varieties prym varieties integrable systems topological structure of phase space methods of integration |
| title | Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable |
| title_full | Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable |
| title_fullStr | Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable |
| title_full_unstemmed | Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable |
| title_short | Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable |
| title_sort | classifying the metrics for which geodesic flow on the group so n is algebraically completely integrable |
| topic | jacobians varieties prym varieties integrable systems topological structure of phase space methods of integration |
| url | https://dergipark.org.tr/tr/download/article-file/1021104 |
| work_keys_str_mv | AT lesfariahmed classifyingthemetricsforwhichgeodesicflowonthegroupsonisalgebraicallycompletelyintegrable |