SIGMA 2 (2006), 012, 9 pages      nlin.SI/0512054      https://doi.org/10.3842/SIGMA.2006.012

On Classical r-Matrix for the Kowalevski Gyrostat on so(4)

Igor V. Komarov and Andrey V. Tsiganov
V.A. Fock Institute of Physics, St. Petersburg State University, St. Petersburg, Russia

Received November 18, 2005, in final form January 19, 2006; Published online January 24, 2006

Abstract
We present the trigonometric Lax matrix and classical r-matrix for the Kowalevski gyrostat on so(4) algebra by using the auxiliary matrix algebras so(3,2) or sp(4).

Key words: Kowalevski top; Lax matrices; classical r-matrix.

pdf (204 kb)   ps (149 kb)   tex (12 kb)

References

1. Babelon O., Viallet C.M., Hamiltonian structures and Lax equations, Phys. Lett. B., 1990, V.237, 411-416.
2. Belavin A.A., Drinfeld V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl., 1982, V.16, 1-29.
   Etingof P., Varchenko A., Geometry and classification of solutions of the classical dynamical Yang-Baxter equation, Comm. Math. Phys, 1998, V.192, 77-120, q-alg/9703040.
3. Bobenko A.I., Kuznetsov V.B., Lax representation for the Goryachev-Chaplygin top and new formulae for its solutions, J. Phys. A: Math. Gen., 1988, V.21, 1999-2006.
4. Eilbeck J.C., Enolskii V.Z., Kuznetsov V.B., Tsiganov A.V., Linear r-matrix algebra for classical separable systems, J. Phys. A: Math. Gen., 1994, V.27, 567-578.
   Sklyanin E.K., Dynamical r-matrices for the elliptic Calogero-Moser model, Algebra i Analiz, 1994, V.6, 227-237 (English transl.: St. Petersburg Math. J., 1995, V.6, 397-406).
5. Kowalevski S., Sur le probléme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 1889, V.12, 177-232.
6. Komarov I.V., Sokolov V.V., Tsiganov A.V., Poisson maps and integrable deformations of Kowalevski top, J. Phys. A: Math. Gen., 2003, V.36, 8035-8048, nlin.SI/0304033.
7. Marshall I.D., The Kowalevski top: its r-matrix interpretation and bi-Hamiltonian formulation, Comm. Math. Phys., 1998, V.191, 723-734.
8. Reyman A.G., Semenov-Tian-Shansky M.A., Group-theoretical methods in the theory of finite-dimensional integrable systems, in Dynamical Systems VII, Editors V.I. Arnold and S.P. Novikov, Encyclopaedia of Mathematical Sciences, V.16, Berlin, Springer, 1994, 116-225.
9. Reyman A.G., Semenov-Tian-Shansky M.A., Lax representation with a spectral parameter for the Kowalewski top and its generalizations, Lett. Math. Phys., 1987, V.14, 55-61.
10. Tsiganov A.V., Integrable deformations of tops related to the algebra so(p,q), Teor. Mat. Fiz., 2004, V.141, 24-37 (English transl.: Theor. Math. Phys., 2004, V.141, 1348-1360).