# 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}
}
• Published 1998
• Physics, Mathematics
• Physics Letters A
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
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#### References

SHOWING 1-10 OF 31 REFERENCES
Completely integrable bi-Hamiltonian systems
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 thatExpand
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 introduceExpand
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 PfaffianExpand
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 ladderExpand
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 isExpand
On the integrability of stationary and restricted flows of the KdV hierarchy
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 itExpand
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 solutionExpand
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 weExpand
Bi-Hamiltonian separable chains on Riemannian manifolds
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
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 basedExpand