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Bi-Hamiltonian representation of Stäckel systems
1Faculty of Physics, A. Mickiewicz University, Umultowska 85, 61-614 Poznań, Poland.
Abstract:
It is shown that linear separation relations are fundamental objects for integration by quadratures of Stäckel-separable Liouville-integrable systems (the so-called Stäckel systems). These relations are further employed for the classification of Stäckel systems. Moreover, we prove that any Stäckel-separable Liouville-integrable system can be lifted to a bi-Hamiltonian system of Gel'fand-Zakharevich type. In conjunction with other known result this implies that the existence of bi-Hamiltonian representation of Liouville-integrable systems is a necessary condition for Stäckel separability.
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