Haantjes Algebras of Classical Integrable Systems

A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based on the geometry of Haantjes tensors. We introduce the class of symplectic-Haantjes manifolds (or $\omega \mathscr{H}$ manifolds), as the natural setting where the notion of integrability can be formulated. We prove that the existence of suitable Haantjes algebras of (1,1) tensor fields with vanishing Haantjes torsion is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville-Arnold sense. We also show that new integrable models arise from the Haantjes geometry. Finally, we present an application of our approach to the study of the Post-Winternitz system and of a stationary flow of the KdV hierarchy.Comment: 31 page

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 Fr\"olicher--Nijenhuis bracket. This new family of tensors reduces to the generalized Nijenhuis torsions of level $m$ recently introduced independently in \cite{KS2017} and \cite{TT2017} and possesses many interesting algebro-geometric properties. We prove that the vanishing of the generalized Nijenhuis torsion of level $(n-1)$ of a nilpotent operator field $A$ over a manifold of dimension $n$ is necessary for the existence of a local chart where the operator field takes a an upper triangular form. Besides, the vanishing of a generalized torsion of level $m$ provides us with a sufficient condition for the integrability of the eigen-distributions of an operator field over an $n$-dimensional manifold. This condition does not require the knowledge of the spectrum and of the eigen-distributions of the operator field. The latter result generalizes the celebrated Haantjes theorem.Comment: 25 page

Generalized Lenard Chains, Separation of Variables and Superintegrability

We show that the notion of generalized Lenard chains naturally allows formulation of the theory of multi-separable and superintegrable systems in the context of bi-Hamiltonian geometry. We prove that the existence of generalized Lenard chains generated by a Hamiltonian function defined on a four-dimensional \omega N manifold guarantees the separation of variables. As an application, we construct such chains for the H\'enon-Heiles systems and for the classical Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler potential are found.Comment: 14 pages Revte

The quasi-bi-Hamiltonian formulation of the Lagrange top

Starting from the tri-Hamiltonian formulation of the Lagrange top in a six-dimensional phase space, we discuss the possible reductions of the Poisson tensors, the vector field and its Hamiltonian functions on a four-dimensional space. We show that the vector field of the Lagrange top possesses, on the reduced phase space, a quasi-bi-Hamiltonian formulation, which provides a set of separation variables for the corresponding Hamilton-Jacobi equation.Comment: 12 pages, no figures, LaTeX, to appear in J. Phys. A: Math. Gen. (March 2002

Classical multiseparable Hamiltonian systems, superintegrability and Haantjes geometry

We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of (omega, H ) structures. They are symplectic manifolds en-dowed with a compatible Haantjes algebra H , namely an algebra of (1,1)tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coor-dinates, will be constructed from the Haantjes algebras associated with a separable sys-tem. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation. We shall prove that a multiseparable system admits as many omega H structures as sepa-ration coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physi-cally relevant systems with three degrees of freedom, possesses multiple Haantjes struc-tures. (C) 2021 Published by Elsevier B.V

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. 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\u2013Haantjes manifold for the Lagrange top and compute a set of Darboux\u2013Haantjes coordinates. Such coordinates are separation variables for the associated Hamilton\u2013Jacobi equation

Bi-Hamiltonian manifolds, quasi-bi-Hamiltonian systems and separation variables

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 formulation of Pfaffian type. This property allows us to straightforwardly recover a set of separation variables for the corresponding Hamilton-Jacobi equation