Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 11 (2016), 107 -- 134

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This work is licensed under a Creative Commons Attribution 4.0 International License.


Ahmed Lesfari

Abstract. The aim of this survey paper is to investigate the algebraic complete integrability of Euler-Arnold's body description of the four dimensional rigid body, or equivalently of geodesics in SO(4) using left-invariant metrics that arise from inertia tensors, namely non-degenerate maps Λ : so(4)→ so(4)* ≡ so(4) together with the canonical inner product associated to the Killing form. Algebraic complete integrability is motivated by Arnold-Liouville's classical notion of complete integrability : one extends the value of space and time coordinates from ℝ to ℂ, and then the regular invariant manifolds are complex instead of real tori; in addition one demands such complex tori to be projective. Using different methods, as systematized by Adler-Haine-van Moerbeke-Mumford, to study the integrability of the geodesic flow on the rotation group, we will see that the linearization is carried on an abelian surface and each time a Prym variety appears related to this problem.

2010 Mathematics Subject Classification: 37J35, 14H40, 14H70.
Keywords: integrable systems, Jacobians, Prym varieties.

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Ahmed Lesfari
Département de Mathématiques,
Faculté des Sciences, Université Chouaïb Doukkali,
B.P. 20, El-Jadida, Maroc.