arXiv subjects

Giorgio Tondo

Publications and source records attributed to Giorgio Tondo.

12 recordsLinked to original sources

arXiv2024

Partial separability and symplectic-Haantjes manifolds

A theory of partial separability for classical Hamiltonian systems is proposed in the context of Haantjes geometry. As a general result, we show that the knowledge of a non-semisimple symplectic-Haantjes manifold for a given Hamiltonian system is sufficient to construct sets of coordinates (called Darboux-Haantjes coordinates) which allow both the partial separability of the associated Hamilton-Jacobi equations and the block-diagonalization of the operators of the corresponding Haantjes algebra. We also introduce a novel class of Hamiltonian systems, characterized by the existence of a generalized Stäckel matrix, which by construction are partially separable. They widely generalize the known families of partially separable Hamiltonian systems. Our systems can be described in terms of semisimple but non-maximal-rank symplectic-Haantjes manifolds.

math-ph↗

arXiv2024

Hamiltonian integrable systems in a magnetic field and Symplectic-Haantjes geometry

We investigate the geometry of classical Hamiltonian systems immersed in a magnetic field in three-dimensional Riemannian configuration spaces. We prove that these systems admit non-trivial symplectic-Haantjes manifolds, which are symplectic manifolds endowed with an algebra of Haantjes (1,1)-tensors. These geometric structures allow us to determine separation variables for known systems algorithmically; besides, the underlying Stäckel geometry is used to construct new families of integrable Hamiltonian models immersed in a magnetic field.

math-ph↗

arXiv2023

Polarization of generalized Nijenhuis torsions

In this work, we introduce the notion of polarization of generalized Nijenhuis torsions and establish several algebraic identities. We prove that these polarizations are relevant in the characterization of Haantjes $C^{\infty}$(M)-modules of operator fields.

math-ph↗

arXiv2022

Generalized Nijenhuis Torsions and block-diagonalization of operator fields

The theory of generalized Nijenhuis torsions, which extends the classical notions due to Nijenhuis and Haantjes, offers new tools for the study of normal forms of operator fields. We propose a general result ensuring that, given a family of commuting operator fields whose generalized Nijenhuis torsion of level $l$ vanishes, there exists a local chart where all operators can be simultaneously block-diagonalized. We also introduce the notion of generalized Haantjes algebra, consisting of operators with a vanishing higher-level torsion, as a new algebraic structure naturally generalizing standard Haantjes algebras.

math-ph↗

arXiv2022

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 ($ω, \mathscr{H}$) structures. They are symplectic manifolds endowed with a compatible Haantjes algebra $\mathscr{H}$, namely an algebra of (1,1)-tensor fields with vanishing Haantjes torsion. A special class of coordinates, called Darboux-Haantjes coordinates, will be constructed from the Haantjes algebras associated with a separable system. These coordinates enable the additive separation of variables of the corresponding Hamilton-Jacobi equation. We shall prove that a multiseparable system admits as many $ω\mathscr{H}$ structures as separation coordinate systems. In particular, we will show that a large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems and some physically relevant systems with three degrees of freedom, possesses multiple Haantjes structures.

math-ph↗

arXiv2021

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, the Haantjes bracket, the first example of our construction, is relevant in the characterization of Haantjes moduli of operators. We shall also prove that the vanishing of a higher-level Nijenhuis torsion of a given operator is a sufficient condition for the integrability of its generalized eigen-distributions. This result (which does not require any knowledge of the spectral properties of the operator) generalizes the celebrated Haantjes theorem. The same vanishing condition also guarantees that the operator can be written, in a local chart, in a block-diagonal form.

math.DG↗

arXiv2021

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 $ω\mathscr{H}$ manifolds), as a 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.

nlin.SI↗

arXiv2020

Haantjes Algebras and Diagonalization

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 are also required to satisfy suitable compatibility conditions. Haantjes algebras naturally generalize several known interesting geometric structures, arising in Riemannian geometry and in the theory of integrable systems. At the same time, as we will show, they play a crucial role in the theory of diagonalization of operators on differentiable manifolds. Assuming that the operators of a Haantjes algebra are semisimple and commute, we shall prove that there exists a set of local coordinates where all operators can be diagonalized simultaneously. Moreover, in the general, non-semisimple case, they acquire simultaneously, in a suitable local chart, a block-diagonal form.

math-ph↗

arXiv2018

Haantjes Algebras of the Lagrange Top

A symplectic-Haantjes manifold and a Poisson-Haantjes manifold for the Lagrange top are studied and a set of Darboux-Haantjes coordinates are computed. Such coordinates are separation variables for the associated Hamilton-Jacobi equation.

math-ph↗

arXiv2016

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 Stäckel 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.

math-ph↗

arXiv2016

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 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, we prove theorems ensuring the existence of a large class of completely integrable systems in the Euclidean plane, constructed starting from a prescribed Haantjes structure. At the same time, we also show that some of the most classical examples of Hamiltonian systems in n dimensions, as for instance the Gantmacher and Stäckel classes, all possess a natural Haantjes structure.

math-ph↗

arXiv2012

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 ωN manifold guarantees the separation of variables. As an application, we construct such chains for the Hénon-Heiles systems and for the classical Smorodinsky-Winternitz systems. New bi-Hamiltonian structures for the Kepler potential are found.

nlin.SI↗