Generalized Lenard Chains, Separation of Variables and Superintegrability

Abstract

We show that the notion of generalized Lenard chains naturally allows formulation of the theory of multi-separable and superintegrable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function defined on a four-dimensional ωN manifold guarantees the separation of variables. As an application, we construct such chains for the Hénon-Heiles systems and for the classical Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler potential are found.

14 pages Revtex

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Cited by in corpus (7)

• Classical and Quantum Superintegrability with Applications
• Simultaneous separation for the Neumann and Chaplygin systems
• Haantjes Structures for the Jacobi-Calogero Model and the Benenti Systems
• Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem
• Partial separability and symplectic-Haantjes manifolds
• On a Trivial Family of Noncommutative Integrable Systems
• A Geometric Study of Superintegrable Systems