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

@article{Tondo2016HaantjesSF,
  title={Haantjes Structures for the Jacobi-Calogero Model and the Benenti Systems},
  author={G. Tondo and P. Tempesta},
  journal={Symmetry Integrability and Geometry-methods and Applications},
  year={2016},
  volume={12},
  pages={023}
}
  • G. Tondo, P. Tempesta
  • Published 2016
  • Mathematics, Physics
  • Symmetry Integrability and Geometry-methods and Applications
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 an application, we recover the Haantjes manifolds for the rational Calogero model with three particles and for the Benenti systems. 
Haantjes Algebras of the Lagrange Top
We study a symplectic-Haantjes manifold and a Poisson–Haantjes manifold for the Lagrange top and compute a set of Darboux–Haantjes coordinates. Such coordinates are separation variables for theExpand
Beyond recursion operators
We briefly recall the history of the Nijenhuis torsion of (1, 1)-tensors on manifolds and of the lesser-known Haantjes torsion. We then show how the Haantjes manifolds of Magri and the symplecticExpand
Classical Multiseparable Hamiltonian Systems, Superintegrability and Haantjes Geometry
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 aExpand
Haantjes algebras and diagonalization
Abstract We introduce the notion of Haantjes algebra: It consists of an assignment of a family of operator fields on a differentiable manifold, each of them with vanishing Haantjes torsion. They areExpand
Higher Haantjes Brackets and Integrability
We propose a new, infinite class of brackets generalizing the Frölicher– Nijenhuis bracket. This class can be reduced to a family of generalized Nijenhuis torsions recently introduced. In particular,Expand
A New Class of Generalized Haantjes Tensors and Nilpotency
We propose a new infinite class of generalized binary tensor fields. The first representative of this class is the known Fr\"olicher--Nijenhuis bracket. Also, this new family of tensors reduces toExpand
H O ] 2 4 D ec 2 01 7 Beyond recursion operators
We briefly recall the history of the Nijenhuis torsion of (1, 1)-tensors on manifolds and of the lesser-known Haantjes torsion. We then show how the Haantjes manifolds of Magri and theExpand
A New family of higher-order Generalized Haantjes Tensors, Nilpotency and Integrability
We propose a new infinite class of generalized binary tensor fields, whose first representative of is the known Frolicher--Nijenhuis bracket. This new family of tensors reduces to the generalizedExpand
An approach for obtaining integrable Hamiltonians from Poisson-commuting polynomial families
We discuss a general approach permitting the identification of a broad class of sets of Poisson-commuting Hamiltonians, which are integrable in the sense of Liouville. It is shown that all suchExpand
Constructing a Complete Integral of the Hamilton–Jacobi Equation on Pseudo-Riemannian Spaces with Simply Transitive Groups of Motions
In this work, a method for constructing a complete integral of the geodesic Hamilton-Jacobi equation on pseudo-Riemannian manifolds with simply transitive actions of groups of motions is suggested.Expand
...
1
2
...

References

SHOWING 1-10 OF 55 REFERENCES
Generalized Lenard chains and multi-separability of the Smorodinsky–Winternitz system
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 existenceExpand
On the characterization of integrable systems via the Haantjes geometry
We prove that the existence of a Haantjes structure is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville-Arnold sense. This structure, expressed in termsExpand
Generalized Lenard chains, separation of variables, and superintegrability.
  • P. Tempesta, G. Tondo
  • Mathematics, Physics
  • Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2012
TLDR
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
Separation of Variables for Bi-Hamiltonian Systems
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 ofExpand
Reduction of bi-Hamiltonian systems and separation of variables: An example from the Boussinesq hierarchy
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 provideExpand
Haantjes Manifolds of Classical Integrable Systems
A general theory of classical integrable systems is proposed, based on the geometry of the Haantjes tensor. We introduce the class of symplectic-Haantjes manifolds (or $\omega \mathcal{H}$ manifold),Expand
Bihamiltonian structures and Stäckel separability
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 aExpand
General algebraic identities for the Nijenhuis and Haantjes tensors
We obtain general algebraic identities for the Nijenhuis and Haantjes tensors on an arbitrary manifold?Mn. For n=3 we derive special algebraic identities connected with the Cartan-Killing form?(u,v)H.
Bi-Hamiltonian manifolds, quasi-bi-Hamiltonian systems and separation variables
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-HamiltonianExpand
On a class of dynamical systems both quasi-bi-Hamiltonian and bi-Hamiltonian
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 firstExpand
...
1
2
3
4
5
...