Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable

Lesfari Ahmed

doi:10.33434/cams.649612

EN

Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable

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

References

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Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Review

Authors

Lesfari Ahmed
0000-0001-6213-4301
Türkiye

Publication Date

March 25, 2020

Submission Date

November 21, 2019

Acceptance Date

February 5, 2020

Published in Issue

Year 2020 Volume: 3 Number: 1

DOI

https://doi.org/10.33434/cams.649612

IZ

https://izlik.org/JA82XH57NR

APA

Ahmed, L. (2020). Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable. Communications in Advanced Mathematical Sciences, 3(1), 24-52. https://doi.org/10.33434/cams.649612

AMA

1.Ahmed L. Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable. Communications in Advanced Mathematical Sciences. 2020;3(1):24-52. doi:10.33434/cams.649612

Chicago

Ahmed, Lesfari. 2020. “Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ Is Algebraically Completely Integrable”. Communications in Advanced Mathematical Sciences 3 (1): 24-52. https://doi.org/10.33434/cams.649612.

EndNote

Ahmed L (March 1, 2020) Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable. Communications in Advanced Mathematical Sciences 3 1 24–52.

IEEE

[1]L. Ahmed, “Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable”, Communications in Advanced Mathematical Sciences, vol. 3, no. 1, pp. 24–52, Mar. 2020, doi: 10.33434/cams.649612.

ISNAD

Ahmed, Lesfari. “Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ Is Algebraically Completely Integrable”. Communications in Advanced Mathematical Sciences 3/1 (March 1, 2020): 24-52. https://doi.org/10.33434/cams.649612.

JAMA

1.Ahmed L. Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable. Communications in Advanced Mathematical Sciences. 2020;3:24–52.

MLA

Ahmed, Lesfari. “Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ Is Algebraically Completely Integrable”. Communications in Advanced Mathematical Sciences, vol. 3, no. 1, Mar. 2020, pp. 24-52, doi:10.33434/cams.649612.

Vancouver

1.Lesfari Ahmed. Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable. Communications in Advanced Mathematical Sciences. 2020 Mar. 1;3(1):24-52. doi:10.33434/cams.649612

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Abstract

Classifying the Metrics for Which Geodesic Flow on the Group $SO(n)$ is Algebraically Completely Integrable

Abstract

Keywords

References

Details

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Subjects

Journal Section

Authors

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

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