# Generalized Lenard chains and multi-separability of the Smorodinsky–Winternitz system

@inproceedings{Tondo2014GeneralizedLC, title={Generalized Lenard chains and multi-separability of the Smorodinsky–Winternitz system}, author={Giorgio Tondo}, year={2014} }

We show that the notion of generalized Lenard chains allows to formulate in a natural way the theory of multi-separable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function and by a Nijenhuis tensor defined on a symplectic manifold guarantees the separation of variables. As an application, we construct such a chain for the case I of the classical Smorodinsky–Winternitz model.

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#### References

SHOWING 1-10 OF 37 REFERENCES

Generalized Lenard chains, separation of variables, and superintegrability.

- Mathematics, Physics
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2012

It is proved that the existence of generalized Lenard chains generated by a Hamiltonian function defined on a four-dimensional ωN manifold guarantees the separation of variables. Expand

Generalized Lenard chains and Separation of Variables

- 2005

It is known that integrability properties of soliton equations follow from the existence of Lenard chains of symmetries, constructed by means of a Nijenhuis (i.e. hereditary) operator. In this paper,… Expand

Bihamiltonian structures and Stäckel separability

- Mathematics
- 2000

Abstract It is shown that a class of Stackel separable systems is characterized in terms of a Gel’fand–Zakharevich bihamiltonian structure. This structure arises as an extension of a… Expand

Reduction of bi-Hamiltonian systems and separation of variables: An example from the Boussinesq hierarchy

- Mathematics, Physics
- 2000

We discuss the Boussinesq system with the stationary time t5 within a general framework of stationary flows of n-Gel'fand-Dickey hierarchies. A careful use of the bi-Hamiltonian structure can provide… Expand

Separation of Variables for Bi-Hamiltonian Systems

- Mathematics, Physics
- 2002

We address the problem of the separation of variables for the Hamilton–Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of… Expand

Two degrees of freedom quasi-bi-Hamiltonian systems

- Mathematics
- 1996

Starting from the classical example of the Henon - Heiles integrable Hamiltonian system, we show that it admits a slightly different formulation from the classical bi-Hamiltonian system. We introduce… Expand

On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian

- Physics, Mathematics
- 1998

Abstract It is shown that a class of dynamical systems (encompassing the one recently considered by Calogero [J. Math. Phys. 37 (1996) 1735] is both quasi-bi-Hamiltonian and bi-Hamiltonian. The first… Expand

Bi-Hamiltonian manifolds, quasi-bi-Hamiltonian systems and separation variables

- Mathematics, Physics
- 1999

Abstract We discuss from a bi-Hamiltonian point of view the Hamilton-Jacobi separability of a few dynamical systems. They are shown to admit, in their natural phase space, a quasi-bi-Hamiltonian… Expand

Exact solvability of superintegrable systems

- Physics, Mathematics
- 2001

It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra sl(3). The gauge-rotated Hamiltonians, as well as their integrals of… Expand

Bi-Hamiltonian structure of an integrable Henon-Heiles system

- Physics
- 1991

By making use of the system of coordinates in which the separation of the variables in the Hamilton-Jacobi equation takes place, we find a bi-Hamiltonian stmclure of a two degrees of freedom… Expand