The Kowalevski top as a reduction of a Hamiltonian system on 𝔰𝔭(4,ℝ)*

Ivanescu, C.; Savu, A.

doi:10.1090/S0002-9939-02-06541-3

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

by C. Ivanescu and A. Savu PDF

Proc. Amer. Math. Soc. 131 (2003), 607-618 Request permission

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

Mark Adler and Pierre van Moerbeke, The Kowalewski and Hénon-Heiles motions as Manakov geodesic flows on $\textrm {SO}(4)$—a two-dimensional family of Lax pairs, Comm. Math. Phys. 113 (1988), no. 4, 659–700. MR 923636
A. I. Bobenko, A. G. Reyman, and M. A. Semenov-Tian-Shansky, The Kowalewski top 99 years later: a Lax pair, generalizations and explicit solutions, Comm. Math. Phys. 122 (1989), no. 2, 321–354. MR 994508
Hans Freudenthal and H. de Vries, Linear Lie groups, Pure and Applied Mathematics, Vol. 35, Academic Press, New York-London, 1969. MR 0260926
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Phillip Griffiths, Linearizing flows and a cohomology interpretation of Lax equations, Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983) Math. Sci. Res. Inst. Publ., vol. 2, Springer, New York, 1984, pp. 37–46. MR 765227, DOI 10.1007/978-1-4612-1110-5_{3}
L. Haine and E. Horozov, A Lax pair for Kowalevski’s top, Phys. D 29 (1987), no. 1-2, 173–180. MR 923889, DOI 10.1016/0167-2789(87)90053-4
V. Jurdjevic, Integrable Hamiltonian systems on Lie groups: Kowalewski type, Ann. of Math. (2) 150 (1999), no. 2, 605–644. MR 1726703, DOI 10.2307/121090
S. Kowalevski, Sur le probleme de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12 (1889), 177–232.
P. Erdös, On the distribution of normal point groups, Proc. Nat. Acad. Sci. U.S.A. 26 (1940), 294–297. MR 2000, DOI 10.1073/pnas.26.4.294
I. D. Marshall, The Kowalevski top: its $r$-matrix interpretation and bi-Hamiltonian formulation, Comm. Math. Phys. 191 (1998), no. 3, 723–734. MR 1608519, DOI 10.1007/s002200050285
A. G. Reyman and M. A. Semenov-Tian-Shansky, Lax representation with a spectral parameter for the Kowalewski top and its generalizations, Lett. Math. Phys. 14 (1987), no. 1, 55–61. MR 901700, DOI 10.1007/BF00403470