Topological classification of systems of Kovalevskaya-Yehia type

N. S. Slavina

Sbornik. Mathematics, Tome 205 (2014) no. 1, pp. 101-155

Résumé

All the Fomenko-Zieschang invariants are calculated for the Kovalevskaya-Yehia problem, for all noncritical values of the parameters $g$ and $\lambda$, by constructing admissible systems of coordinates and determining the mutual disposition of the basis cycles. The family of Kovalevskaya-Yehia systems contains 29 pairwise Liouville non-equivalent foliations. These foliations include those that are equivalent to previously known foliations, which arose in the integrable cases of Kovalevskaya and of Kovalevskaya-Yehia for $g=0$, in the Zhukovskiǐ case, and in the Goryachev-Chaplygin-Sretenskiǐ case. Eleven new foliations are included in the 29 foliations, new in the sense that they are not Liouville equivalent to any foliations discovered earlier which arose in the known integrable cases of the rigid body. The topological type of the Liouville foliation for the family of Kovalevskaya-Yehia systems stabilizes at large values of the energy $H$, and this ‘high-energy’ system is roughly Liouville equivalent, at one of the energy levels, to the Goryachev-Chaplygin-Sretenskiǐ integrable case, which is already known. Bibliography: 29 titles.

Détail

Keywords: Liouville equivalent integrable systems.
Mots-clés : Fomenko-Zieschang invariants, Liouville foliation

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