Generalized Lenard Chains, Separation of Variables and Superintegrability

TL;DR

This paper demonstrates how generalized Lenard chains facilitate the understanding of superintegrable and multi-separable systems within bi-Hamiltonian geometry, providing new structures and applications to classical systems.

Contribution

It introduces the use of generalized Lenard chains in bi-Hamiltonian geometry to analyze superintegrability and separation of variables, with new structures for classical potentials.

Findings

Generalized Lenard chains guarantee variable separation on -dimensional  N manifolds.

Constructed chains for He9non-Heiles and Smorodinsky-Winternitz systems.

Discovered new bi-Hamiltonian structures for the Kepler potential.

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 \omega N manifold guarantees the separation of variables. As an application, we construct such chains for the H\'enon-Heiles systems and for the classical Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler potential are found.