# Separation of Variables for Bi-Hamiltonian Systems

@article{Falqui2002SeparationOV, title={Separation of Variables for Bi-Hamiltonian Systems}, author={Gregorio Falqui and Marco Pedroni}, journal={Mathematical Physics, Analysis and Geometry}, year={2002}, volume={6}, pages={139-179} }

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 bi-Hamiltonian manifolds, called ωN manifolds, to give intrisic tests of separability (and Stäckel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the ωN manifold itself. We apply these results to bi-Hamiltonian systems of the Gel… Expand

#### 106 Citations

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#### References

SHOWING 1-10 OF 52 REFERENCES

A Bi-Hamiltonian Theory for Stationary KdV Flows and their Separability

- Mathematics, Physics
- 2000

We present a fairly new and comprehensive approach to the study of stationary flows of the Korteweg-de Vries hierarchy. They are obtained by means of a double restriction process from a dynamical… Expand

Families of quasi-bi-Hamiltonian systems and separability

- Mathematics, Physics
- 1999

It is shown how to construct an infinite number of families of quasi-bi-Hamiltonian (QBH) systems by means of the constrained flows of soliton equations. Three explicit QBH structures are presented… 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

Bi-Hamiltonian Aspects of the Separability of the Neumann System

- Mathematics, Physics
- 2002

The Neumann system on the two-dimensional sphere is used as a tool to convey some ideas on the bi-Hamiltonian standpoint on separation of variables. We show that from this standpoint, its separation… Expand

Tri-Hamiltonian Vector Fields, Spectral Curves and Separation Coordinates

- Mathematics, Physics
- 2002

We show that for a class of dynamical systems, Hamiltonian with respect to three distinct Poisson brackets (P0,P1,P2), separation coordinates are provided by the common roots of a set of bivariate… Expand

Quasi-bi-Hamiltonian systems and separability

- Mathematics, Physics
- 1997

Two quasi-bi-Hamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover, the most general Pfaffian… Expand

Bihamiltonian geometry and separation of variables for Toda lattices

- Physics
- 2000

We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifolds, we show that these systems can be… Expand

Killing tensors and the separation of the Hamilton-Jacobi equation

- Mathematics
- 1975

This paper investigates the relationship between Killing Tensors and separable systems for the geodesic Hamilton-Jacobi equation in Riemannian and Lorentzian manifolds: locally, a separable system… Expand

A systematic study of the Toda lattice in the context of the Hamilton-Jacobi theory

- Mathematics
- 2001

Abstract. Integrability of the Toda lattice is studied in the framework of the Hamilton-Jacobi theory of separation of variables. It is shown based on the Benenti theory that only the two… Expand

Intrinsic characterization of the variable separation in the Hamilton–Jacobi equation

- Mathematics
- 1997

The nonorthogonal separation of variables in the Hamilton–Jacobi equation corresponding to a natural Hamiltonian H=12gijpipj+V, with a metric tensor of any signature, is intrinsically characterized… Expand