Corpus ID: 119641920

On the characterization of integrable systems via the Haantjes geometry

  title={On the characterization of integrable systems via the Haantjes geometry},
  author={P. Tempesta and G. Tondo},
  journal={arXiv: Mathematical Physics},
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 terms of suitable operators whose Haantjes torsion vanishes, encodes the main features of the notion of integrability, and in particular, under certain hypotheses, allows to solve the problem of determining separation of variables for a given system in an algorithmic way. As an application of the theory… Expand
1 Citations
Haantjes Structures for the Jacobi-Calogero Model and the Benenti Systems
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. AsExpand


Differential-geometric approach to the integrability of hydrodynamic chains: the Haantjes tensor
The integrability of an m-component system of hydrodynamic type, ut = V(u)ux, by the generalized hodograph method requires the diagonalizability of the m ×  m matrix V(u). This condition is known toExpand
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
If a non degenerate completely integrable hamiltonian system in the sense of Arnold-Liouville is bihamiltonian with a splitting condition, then its hamiltonian and its action variables have someExpand
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
A geometrical approach to the nonlinear solvable equations
A geometrical approach to the nonlinear solvable equations, based on the study of the “groups of motion” of special infinite-dimensional manifolds called “symplectic Kahler manifolds”, is suggested.Expand
Integrable systems of classical mechanics and Lie algebras
1. Preliminaries.- 1.1 A Simple Example: Motion in a Potential Field.- 1.2 Poisson Structure and Hamiltonian Systems.- 1.3 Symplectic Manifolds.- 1.4 Homogeneous Symplectic Spaces.- 1.5 The MomentExpand
About the existence of recursion operators for completely integrable Hamiltonian systems near a Liouville torus
It is shown that the existence of a recursion operator near a Liouville torus of a completely integrable Hamiltonian system is not always satisfied and that such an existence implies very strongExpand
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
Necessary conditions for existence of non-degenerate Hamiltonian structures
The necessary criteria are pointed out for the exisence of Hamiltonian and bi-Hamiltonian non-degenerate structures for a nonlinear system of partial differential equations of first order. TheExpand
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