# Separable bi-Hamiltonian systems with quadratic in momenta first integrals

@article{Baszak2003SeparableBS, title={Separable bi-Hamiltonian systems with quadratic in momenta first integrals}, author={Maciej Błaszak}, journal={arXiv: Exactly Solvable and Integrable Systems}, year={2003} }

Geometric separability theory of Gel'fand-Zakharevich bi-Hamiltonian systems on Riemannian manifolds is reviewed and developed. Particular attention is paid to the separability of systems generated by the so-called special conformal Killing tensors, i.e. Benenti systems. Then, infinitely many new classes of separable systems are constructed by appropriate deformations of Benenti class systems. All such systems can be lifted to the Gel'fand-Zakharevich bi-Hamiltonian form.

## 8 Citations

### Systematic Construction of Separable Systems with Quadratic in Momenta First Integrals

- Mathematics
- 2004

Liouville integrable separable systems with quadratic in momenta first integrals are considered. Particular attention is paid to the systems generated by the so-called special conformal Killing…

### Bi-Hamiltonian representation of Stäckel systems.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2009

It is shown that linear separation relations are fundamental objects for integration by quadratures of Stâckel-separable Liouville-integrable systems (the so-called Stäckel systems) and implies that the existence of bi-Hamiltonian representation of LiouVILLE-integRable systems is a necessary condition for StäCkel separability.

### Eigenvalues of Killing Tensors and Separable Webs on Riemannian and Pseudo-Riemannian Manifolds ?

- Mathematics
- 2007

Given a n-dimensional Riemannian manifold of arbitrary signature, we illustrate an algebraic method for constructing the coordinate webs separating the geodesic Hamilton- Jacobi equation by means of…

### A class of Poisson–Nijenhuis structures on a tangent bundle

- Mathematics
- 2004

Equipping the tangent bundle TQ of a manifold with a symplectic form coming from a regular Lagrangian L, we explore how to obtain a Poisson–Nijenhuis structure from a given type (1, 1) tensor field J…

### Structural equations for a special class of conformal killing tensors of arbitrary valence

- Mathematics
- 2008

### Orthogonal Separation of the Hamilton-Jacobi Equation on Spaces of Constant Curvature

- Mathematics
- 2016

We review the theory of orthogonal separation of variables of the Hamilton-Jacobi equation on spaces of constant curvature, highlighting key contributions to the theory by Benenti. This theory…

### Maximal superintegrability of Benenti systems

- Mathematics
- 2005

For a class of Hamiltonian systems, naturally arising in the modern theory of separation of variables, we establish their maximal superintegrability by explicitly constructing the additional…

### Stackel systems: bi-Hamiltonian property and systematic construction

- Mathematics
- 2005

It is shown that separation conditions (separation curves) are fundamental objects of separability theory. They are used for the classification of certain clases of separable systems, for the proof…

## References

SHOWING 1-10 OF 51 REFERENCES

### Separation of Variables for Bi-Hamiltonian Systems

- Mathematics
- 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…

### On the Local Geometry of a Bihamiltonian Structure

- Mathematics
- 1993

We give several examples of bihamiltonian manifolds and show that under very mild assumptions a bihamiltonian structure in “general position” is locally of one of these types. This shows, in…

### Separation of variables in quasi-potential systems of bi-cofactor form

- Physics
- 2002

We perform variable separation in the quasi-potential systems of equations of the form ¨ q =− A −1 ∇k =− ˜ A −1 ∇ ˜ k ,w hereA and ˜ A are Killing tensors, by embedding these systems into a…

### Separable Systems of Stackel

- Mathematics
- 1934

so that the variables are separable, the solution being of the form 2Xi, where Xi is a function of xi alone. In 18932 he showed that when the quadratic differential form 2H 2dxi so determined is…

### 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…

### Linear r-matrix algebra for classical separable systems

- Mathematics
- 1994

We consider a hierarchy of the natural-type Hamiltonian systems of n degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax…

### Quasi-bi-Hamiltonian systems and separability

- 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…