Bi-Hamiltonian representation of Stäckel systems

Details Bi-Hamiltonian representation of Stäckel systems Bi-Hamiltonian representation of Stäckel systems

2009-04-14

arxiv.org/abs/0904.2070v1

Physics

Mathematical Physics

To appear in Physical Review E

Scientific paper

It is shown that a linear separation relations are fundamental objects for integration by quadratures of St\"{a}ckel separable Liouville integrable systems (the so-called St\"{a}ckel systems). These relations are further employed for the classification of St\"{a}ckel systems. Moreover, we prove that {\em any} St\"{a}ckel 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 St\"{a}ckel separability.

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