On isomorphism of integrable cases of the Euler equations on the bi-hamiltonian manifolds $e(3)$ and $so(4)$
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 18, Tome 317 (2004), pp. 200-212 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The Poisson maps between the Clebsch model and the Schottky system, two Steklov systems, the Kowalevski top and the Neumann system are considered. We prove that these non-canonical transformations of variables are the twisted Poisson maps, which completely define the corresponding pairs of integrable systems.
@article{ZNSL_2004_317_a10,
     author = {A. V. Tsiganov},
     title = {On isomorphism of integrable cases of the {Euler} equations on the bi-hamiltonian manifolds $e(3)$ and $so(4)$},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {200--212},
     year = {2004},
     volume = {317},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_317_a10/}
}
TY  - JOUR
AU  - A. V. Tsiganov
TI  - On isomorphism of integrable cases of the Euler equations on the bi-hamiltonian manifolds $e(3)$ and $so(4)$
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2004
SP  - 200
EP  - 212
VL  - 317
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2004_317_a10/
LA  - ru
ID  - ZNSL_2004_317_a10
ER  - 
%0 Journal Article
%A A. V. Tsiganov
%T On isomorphism of integrable cases of the Euler equations on the bi-hamiltonian manifolds $e(3)$ and $so(4)$
%J Zapiski Nauchnykh Seminarov POMI
%D 2004
%P 200-212
%V 317
%U http://geodesic.mathdoc.fr/item/ZNSL_2004_317_a10/
%G ru
%F ZNSL_2004_317_a10
A. V. Tsiganov. On isomorphism of integrable cases of the Euler equations on the bi-hamiltonian manifolds $e(3)$ and $so(4)$. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 18, Tome 317 (2004), pp. 200-212. http://geodesic.mathdoc.fr/item/ZNSL_2004_317_a10/

[1] A. Clebsch, “Über die Bewegung eines Körpers in einer Flüssigkeit”, Math. Annalen, 3 (1870), 238–262 | DOI | MR

[2] V. A. Steklov, O dvizhenii tverdogo tela v zhidkosti, Kharkov, 1893

[3] F. Schottky, “Über das analytische Problem der Rotation eines starren Körpers in Raume von vier Dimensionen”, Sitzungsberichte drer Königligh preussischen Academie der Wissenschaften zu Berlin, 13 (1981), 227–232

[4] V. A. Steklov, “Sur la mouvement d'un corps solide ayant une cavite de forme ellipsoidale remplie par un liquide incompressible et sur les variations des latitudes”, Annales de la Faculte des Sciences de Toulouse, Ser. 3, 1 (1909), 145–256

[5] V. G. Baryakhtar, E. D. Belokolos, P. I. Golod, Preprint, ITF-84-128R, Kiev, 1984

[6] A. I. Bobenko, “Uravneniya Eilera na algebrakh $e(3)$ i $so(4)$. Izomorfizm integriruemykh sluchaev”, Funkts. analiz i ego pril., 20 (1986), 64–66 | MR | Zbl

[7] S. P. Novikov, “Gamiltonov formalizm i mnogoznachnyi analog teorii Morsa”, UMN, 37 (1982), 3–49 | MR | Zbl

[8] F. Kötter, “Über die Bewegung eines festen Körpers in einer Flussigkeit, I, II”, J. Reine und Angew. Math., 109 (1892), 51–81, 89–111 | DOI

[9] M. Adler, P. van Moerbeke, “The Kowalewski and Hénon–Heiles motion as Manakov geodesic flow on $SO(4)$ and a two-dimensional family of Lax pairs”, Commun. Math. Phys., 113 (1988), 659–700 | DOI | MR

[10] A. V. Bolsinov, “Soglasovannye skobki Puassona na algebrakh Li i polnota semeistv funktsii v involyutsii”, Izv. AN SSSR, ser. mat., 55 (1991), 68–92

[11] A. V. Bolsinov, A. V. Borisov, “Predstavlenie Laksa i soglasovannye skobki Puassona na algebrakh Li”, Mat. zamet., 72 (2002), 11–34 | MR | Zbl

[12] Yu. N. Fedorov, “Integrable systems, Poisson pensils and hyperelliptic Lax pairs”, Reg. Chaotic Dyn., 5 (2000), 171–180 ; A. V. Bolsinov, Yu. N. Fedorov, Steklov–Lyapunov type systems, Preprint, 2003 | DOI | MR | Zbl

[13] L. Haine, E. Horozov, “A Lax pair for the Kowalewski top”, Physica D, 29 (1987), 173–180 | DOI | MR | Zbl

[14] I. V. Komarov, A. V. Tsiganov, “On integration of the Kowalevski gyrostat and the Clebsch problems”, Reg. Chaotic Dyn., 9 (2004), 169–189 | DOI | MR