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ISSN 1088-6826(e) ISSN 0002-9939(p)

The Kowalevski top as a reduction of a Hamiltonian system on $\mathfrak{sp}(4, \mathbb{R} )^*$

Author(s): C. Ivanescu; A. Savu
Journal: Proc. Amer. Math. Soc. 131 (2003), 607-618.
MSC (2000): Primary 70H06; Secondary 37J35, 37J15, 70E17
Posted: May 22, 2002
MathSciNet review: 1933353
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Abstract: We show that the Kowalevski top and Kowalevski gyrostat are obtained as a reduction of a Hamiltonian system on $\mathfrak{sp}(4,\mathbb{R} )^*$ . Therefore the Lax-pair representations for the Kowalevski top and Kowalevski gyrostat are obtained via a direct method by transforming the canonical Lax-pair representation of a system on $\mathfrak{sp}(4, \mathbb{R} )^*$ . Also we show that the nontrivial integral of motion of the Kowalevski top comes from a Casimir function of the Lie-Poisson algebra $\mathfrak{sp}(4, \mathbb{R} )^*$ .

References:

1.: M. Adler, P. van Moerbeke, The Kowalevski and Henon-Heiles motion as Manakov Geodesics flows on - a two-dimensional family of Lax pairs. Commun. Math. Phys. 113 (1988), 659-700. MR 89b:58085
2.: A. Bobenko, A. Reyman, M.A. Semenov-Tian-Shansky, The Kowalevski top 99 years later: a Lax pair, generalizations and explicit solutions. Commun. Math. Phys. 122 (1989), 321-354. MR 91a:58070
3.: H. Freudenthal, H. deVries, Linear Lie groups. Academic Press, New York (1969), 130-131. MR 41:5546
4.: V.V. Golubev, Lectures on integration of the equation of motion of a rigid body about a fixed point. Israel program for scientific translations, Jerusalem, (1960). MR 15e:904
5.: P.A. Griffiths, Linearizing flows and a cohomological interpretation of Lax equations. Amer. J. Math. 107 (1985), 1445-1483. MR 86f:58076
6.: L. Haine, E. Horozov, A Lax-pair for Kowalevski's Top. Physica 29D (1987), 173-180. MR 90a:58068
7.: V. Jurdjevic, Integrable Hamiltonian Systems on Lie Groups: Kowalevski type. Annals of Math. 150 (1999), 605-644. MR 2001i:37092
8.: S. Kowalevski, Sur le probleme de la rotation d'un corps solide autour d'un point fixe. Acta Math. 12 (1889), 177-232.
9.: A. Lesfari, Completely integrable systems: Jacobi's heritage. J. Geom. Phys. 31 no.4, (1999), 265-286. MR 2000f:37077
10.: I.D. Marshall, The Kowalevski Top: its -Matrix Interpretation and Bihamiltonian Formulation. Commun. Math. Phys. 191 (1998), 723-734. MR 99b:58119
11.: A. Reyman, M.A. Semenov-Tian-Shansky, Lax representation with spectral parameter for the Kowalevski top and its generalizations. Lett. Math. Phys. 1 (1987), 55-61. MR 88h:58057

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Additional Information:

C. Ivanescu
Affiliation: Fields Institute, 222 College Street, Toronto, Ontario, Canada M5T 3J1
Email: civanesc@fields.utoronto.ca

A. Savu
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G3
Email: ana@math.toronto.edu

DOI: 10.1090/S0002-9939-02-06541-3
PII: S 0002-9939(02)06541-3
Keywords: Heavy top, Kowalevski top, Lax-pair representation
Received by editor(s): March 1, 2001
Received by editor(s) in revised form: September 6, 2001
Posted: May 22, 2002
Additional Notes: This research was supported by a J.R. Gilkinson Smith and University of Toronto fellowship
Communicated by: Carmen C. Chicone
Copyright of article: Copyright 2002, American Mathematical Society

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