Bi-Hamiltonian representation of Stäckel systems

Abstract

It is shown that a 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 {\em 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.

To appear in Physical Review E

References in corpus (5)

• Generalized Stäckel Transform and Reciprocal Transformations for Finite-Dimensional Integrable Systems
• On a Poisson reduction for Gel'fand--Zakharevich manifolds
• Separation of variables in quasi-potential systems of bi-cofactor form
• From bi-Hamiltonian geometry to separation of variables: stationary Harry-Dym and the KdV dressing chain
• Non-Hamiltonian systems separable by Hamilton-Jacobi method

Cited by in corpus (9)

• On bi-integrable natural Hamiltonian systems on the Riemannian manifolds
• The Variety of Integrable Killing Tensors on the 3-Sphere
• Integrable quantum Stäckel systems
• Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem
• Flat minimal quantizations of Stackel systems and quantum separability
• On three-dimensional quasi-Stäckel Hamiltonians
• Bi-presymplectic representation of Liouville integrable systems and related separability theory
• Algebraic curves as a source of separable multi-Hamiltonian systems
• Complex functions and geometric structures associated to the superintegrable Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion