Lax representation with a spectral parameter for the Kowalewski top and its generalizations.

*(English)*Zbl 0648.70001
Plasma theory and nonlinear and turbulent processes in physics, Proc. Int. Workshop, Kiev/USSR 1987, 135-152 (1988).

[For the entire collection see Zbl 0644.00031.]

The Kowalewski top is the last and the most nontrivial of the three classical integrable cases of the motion of a heavy rigid body with a fixed point. Recently, a new insight into the inegrable problems of analytical mechanics has been provided by the invention of Lax representation technique. Lax representations with spectral parameter have been found for almost all integrable cases of the motion of rigid bodies in arbitrary number of dimensions. However, the Kowalewski top was a striking exception, and its relation to other integrable systems admitting Lax repesentations remained obscure in spite of some interesting observations. In the present note we fill this gap. The Lax pair we propose allows for various generalizations of the Kowalewski top, some of which are new even in the classical setting. In the three dimensional case it leads to a new algebraic curve related to the Kowalewski system which is completely different from the one studied by Kowalewski herself. Eventually this gives new concise formulae for the solutions which are far more explicit than those obtained by S. Kowalewski.

The Kowalewski top is the last and the most nontrivial of the three classical integrable cases of the motion of a heavy rigid body with a fixed point. Recently, a new insight into the inegrable problems of analytical mechanics has been provided by the invention of Lax representation technique. Lax representations with spectral parameter have been found for almost all integrable cases of the motion of rigid bodies in arbitrary number of dimensions. However, the Kowalewski top was a striking exception, and its relation to other integrable systems admitting Lax repesentations remained obscure in spite of some interesting observations. In the present note we fill this gap. The Lax pair we propose allows for various generalizations of the Kowalewski top, some of which are new even in the classical setting. In the three dimensional case it leads to a new algebraic curve related to the Kowalewski system which is completely different from the one studied by Kowalewski herself. Eventually this gives new concise formulae for the solutions which are far more explicit than those obtained by S. Kowalewski.

##### MSC:

70E05 | Motion of the gyroscope |

37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |

37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |

37J99 | Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems |