Elliptic Curves and a New Construction of Integrable Systems
arXiv:0901.4743 · doi:10.1134/S1560354709040042
Abstract
A class of elliptic curves with associated Lax matrices is considered. A family of dynamical systems on e(3) parametrized by polynomial a with above Lax matrices are constructed. Five cases from the family are selected by the condition of preserving the standard mea- sure. Three of them are Hamiltonian. It is proved that two other cases are not Hamiltonian in the standard Poisson structure on e(3). Integra- bility of all five cases is proven. Integration procedures are performed in all five cases. Separation of variables in Sklyanin sense is also given. A connection with Hess-Appel'rot system is established. A sort of sep- aration of variables is suggested for the Hess-Appel'rot system.
16 pages
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