Bi-Hamiltonian representation of St\"{a}ckel systems

TL;DR

This paper demonstrates that all Stäckel separable Liouville integrable systems can be represented as bi-Hamiltonian systems, highlighting the fundamental role of linear separation relations in their integration and classification.

Contribution

It proves that any Stäckel separable Liouville integrable system admits a bi-Hamiltonian representation of Gel'fand-Zakharevich type, establishing a necessary condition for Stäckel separability.

Findings

Linear separation relations are key for integrating Stäckel systems.

All Stäckel systems can be lifted to bi-Hamiltonian form.

Bi-Hamiltonian representation is necessary for Stäckel separability.

Abstract

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.