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

@article{Morosi1998OnAC, title={On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian}, author={Carlo Morosi and Giorgio Tondo}, journal={Physics Letters A}, year={1998}, volume={247}, pages={59-64} }

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 formulation entails the separability of these systems; the second one is obtained through a non-canonical map whose form is directly suggested by the associated Nijenhuis tensor.

#### 27 Citations

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

A cohomological obstruction for global quasi-bi-Hamiltonian fields

- Physics
- 2011

Abstract We introduce the notion of integrating factor for a 1-form which is an inner product of a vector fields and a 2-form, and the notion of weakly bi-Hamiltonian field also, which is locally… Expand

Quasi-Bi-Hamiltonian structures of the 2-dimensional Kepler problem

- Mathematics, Physics
- 2016

The existence of quasi-bi-Hamiltonian structures for the Kepler problem is studied. We first relate the superintegrability of the system with the existence of two complex functions endowed with very… Expand

Bi-quasi-Hamiltonian systems

- Mathematics
- 2002

A general notion of bi-quasi-Hamiltonian systems is introduced and is related to previous work on various special cases of such systems.

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

- Mathematics
- 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… Expand

Quasi-bi-Hamiltonian structures and superintegrability: Study of a Kepler-related family of systems endowed with generalized Runge-Lenz integrals of motion

- Physics
- 2021

The existence of quasi-bi-Hamiltonian structures for a two-dimen-sional superintegrable \begin{document}$ (k_1,k_2,k_3) $\end{document} -dependent Kepler-related problem is studied. We make use of an… Expand

An integrable system on the moduli space of rational functions and its variants

- Mathematics, Physics
- 2003

Abstract We study several integrable Hamiltonian systems on the moduli spaces of meromorphic functions on Riemann surfaces (the Riemann sphere, a cylinder and a torus). The action–angle variables and… Expand

Classical Multiseparable Hamiltonian Systems, Superintegrability and Haantjes Geometry

- Physics, Mathematics
- 2020

We show that the theory of classical Hamiltonian systems admitting separation variables can be formulated in the context of (ω,H ) structures. They are essentially symplectic manifolds endowed with a… Expand

Theory of separability of multi-Hamiltonian chains

- Mathematics
- 1999

The theory of separability of one-Casimir bi-Hamiltonian chains is extended onto unsplit multi-Casimir bi-Hamiltonian chains. Multi-Casimir extensions of the known one-Casimir chains are constructed.

Haantjes Structures for the Jacobi-Calogero Model and the Benenti Systems

- Mathematics, Physics
- 2016

In the context of the theory of symplectic-Haantjes manifolds, we construct the Haantjes structures of generalized Stackel systems and, as a particular case, of the quasi-bi-Hamiltonian systems. As… Expand

#### References

SHOWING 1-10 OF 31 REFERENCES

Completely integrable bi-Hamiltonian systems

- Mathematics, Physics
- 1994

We study the geometry of completely integrable bi-Hamiltonian systems and, in particular, the existence of a bi-Hamiltonian structure for a completely integrable Hamiltonian system. We show that… 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

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

Integrable stationary flows: Miura maps and bi-hamiltonian structures

- Physics
- 1987

Abstract We present a Miura map between the finite dimensional phase spaces of stationary flows of integrable nonlinear evolution equations. This is used to construct a finite bi-hamiltonian ladder… Expand

On the euler equation: Bi-Hamiltonian structure and integrals in involution

- Mathematics
- 1996

We propose a bi-Hamiltonian formulation of the Euler equation for the free n-dimensional rigid body moving about a fixed point. This formulation lives on the ‘physical’ phase space so(n), and is… Expand

On the integrability of stationary and restricted flows of the KdV hierarchy

- Mathematics, Physics
- 1995

A bi-Hamiltonian formulation for stationary flows of the KdV hierarchy is derived in an extended phase space. A map between stationary flows and restricted flows is constructed: in one case it… Expand

Separable Hamiltonians and integrable systems of hydrodynamic type

- Mathematics
- 1997

Abstract We exhibit a surprising relationship between separable Hamiltonians and integrable, linearly degenerate systems of hydrodynamic type. This gives a new way of obtaining the general solution… Expand

A new class of integrable systems and its relation to solitons

- Physics
- 1986

We present and study a class of finite-dimensional integrable systems that may be viewed as relativistic generalizations of the Calogero-Moser systems. For special values of the coupling constants we… Expand

Bi-Hamiltonian separable chains on Riemannian manifolds

- Physics
- 1998

Abstract Bi-Hamiltonian separable chains in so-called Nijenhuis coordinates, which are quadratic in the momentum variables, are related with a large classes of Stackel systems.

A Simple model of the integrable Hamiltonian equation

- Mathematics
- 1978

A method of analysis of the infinite‐dimensional Hamiltonian equations which avoids the introduction of the Backlund transformation or the use of the Lax equation is suggested. This analysis is based… Expand