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Giorgio Tondo

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Papers by Giorgio Tondo

Classical Multiseparable Hamiltonian Systems, Superintegrability and Haantjes Geometry

arXiv (Cornell University), Dec 17, 2020

We show that the theory of classical Hamiltonian systems admitting separating variables can be fo... more We show that the theory of classical Hamiltonian systems admitting separating variables can be formulated in the context of (ω, H) structures. They are symplectic manifolds endowed 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 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 ωH 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.

format_quoteHamiltonian with irrational dependence reveals a new perspective on nonseparability, indicating the potential for canonical transformations.format_quote

Generalized Nijenhuis Torsions and block-diagonalization of operator fields

arXiv (Cornell University), May 19, 2022

The theory of generalized Nijenhuis torsions, which extends the classical notions due to Nijenhui... more 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. Contents 1. Introduction 2. Preliminaries on the Nijenhuis and Haantjes geometry 3. The generalized Nijenhuis tensors and block-diagonalization 3.1. Eigen-distributions and spectral properties of generalized Nijenhuis operators 3.2. Block-diagonalization 4. Generalized Haantjes algebras 4.1. Definitions 4.2. Cyclic generalized Haantjes algebras 5. Generalized Nijenhuis torsions and simultaneous block-diagonalization 6. Block-diagonalization of generalized Haantjes algebras 6.1. A level-three generalized Haantjes algebra 6.2. Block-diagonalization 7. A level-four generalized Haantjes algebra 7.1. Spectral analysis 7.2. Block-diagonalization Acknowledgement References

format_quoteVanishing generalized Nijenhuis torsion ensures mutual integrability of eigen-distributions, significant for operator theory on differentiable manifolds.format_quote

Редукция бигамильтоновых систем и разделение переменных: пример из иерархии Буссинеска

Теоретическая и математическая физика, 2000

Higher Haantjes Brackets and Integrability

arXiv (Cornell University), Sep 16, 2018

We propose a new, infinite class of brackets generalizing the Frölicher-Nijenhuis bracket. This c... more 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.

On the relation between the bicomplex and the bilocal formalism for KP systems

From the MR review by B.N.Fil: "The relation between the bilocal (due to Fokas and Santini) ... more From the MR review by B.N.Fil: "The relation between the bilocal (due to Fokas and Santini) and bicomplex approaches for nonlinear integrable equations is discussed. On the basis of the abstract algebraic theory of the Lenard bicomplex, an iterative scheme for the second formulation of the KP chain is explicitly constructed. A "fixed step'' formula, which is the true analogue of the classical recursion formula in (1+1) dimensions, is deduced. From a suitable formulation of the Lenard bicomplex a bilocal formulation of the bicomplex which is closely related to the Fokas Santini scheme is obtained directly.

Partial separability and symplectic-Haantjes manifolds

arXiv (Cornell University), May 11, 2023

A theory of partial separability for classical Hamiltonian systems is proposed in the context of ... more 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.

format_quoteComplete integrals dictate existence of integrable Hamiltonian systems, relating independent functions in involution and solvability of HJ equations.format_quote

Разделение переменных в мультигамильтоновых системах. Применение к лагранжеву волчку

Теоретическая и математическая физика, 2003

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Generalized Nijenhuis Torsions and Block-Diagonalization of Operator Fields

Journal of Nonlinear Science

The theory of generalized Nijenhuis torsions, which extends the classical notions due to Nijenhui... more 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 prove a general result ensuring that, given a family of commuting operator fields whose generalized Nijenhuis torsion of level m 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.

On the theory of polarization of generalized Nijenhuis torsions

Cornell University - arXiv, Sep 26, 2022

The theory of generalized Nijenhuis torsions, recently introduced, offers new powerful tools to d... more The theory of generalized Nijenhuis torsions, recently introduced, offers new powerful tools to detect the Frobenius integrability of operator fields on a differentiable manifold. In this work, we introduce the notion of polarization of generalized Nijenhuis torsion and establish several algebraic identities. We also prove that this new vector-valued 2-form is relevant in the characterization of C ∞ Haantjes moduli of operators.

On the characterization of integrable systems via the Haantjes geometry

arXiv: Mathematical Physics, Aug 19, 2015

We prove that the existence of a Haantjes structure is a necessary and sufficient condition for a... more 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.

Classical Multiseparable Systems, Superintegrability and Haantjes Geometry

arXiv: Mathematical Physics, 2020

We show that the theory of classical Hamiltonian systems admitting separation variables can be fo... more We show that the theory of classical Hamiltonian systems admitting separation variables can be formulated in the context of ($\omega, \mathscr{H}$) structures. They are essentially symplectic manifolds endowed with a 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 large class of multiseparable, superintegrable systems, including the Smorodinsky-Winternitz systems, possesses multiple Haantjes structures.

A New family of higher-order Generalized Haantjes Tensors, Nilpotency and Integrability

arXiv: Differential Geometry, 2018

We propose a new infinite class of generalized binary tensor fields, whose first representative o... more 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 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.

format_quoteThe vanishing of generalized Nijenhuis torsion of a nilpotent operator is necessary for a triangular form in local charts, linking geometry and algebra.format_quote

A New family of higher-order Haantjes Tensors and a Generalized Haantjes Integrability Theorem

We propose a new infinite class of generalized bivariate tensor fields, whose first representativ... more We propose a new infinite class of generalized bivariate tensor fields, whose first representative is the known Frölicher–Nijenhuis bracket. This new family of tensors possesses many interesting algebraic-geometric properties; in particular, it reduces, in the univariate case, to the generalized Nijenhuis torsions of level m∈ℕ\{0} recently introduced independently in [12] and [23]. Among the tensors of the bivariate class, the bivariate Haantjes tensor, a particular instance of our construction, is relevant in the characterization of Haantjes moduli of operator fields. We prove that the vanishing of a generalized torsion of any level m is a sufficient condition for the integrability of the generalized eigen-distributions of an operator field over a differentiable manifold. This new condition, which does not require the knowledge of the spectrum and the eigen-distributions of a given operator field, generalizes the celebrated Haantjes theorem, because it provides us with an integrabi...

format_quoteThe bivariate Haantjes tensor vanishes for commuting semisimple operators, indicating structural geometric properties of operator fields.format_quote

Separation of Variables

It is known that integrability properties of soliton equations follow from the existence of Lenar... more It is known that integrability properties of soliton equations follow from the existence of Lenard chains of symmetries, constructed by means of a Nijenhuis (i.e. hereditary) operator. In this paper, we will discuss the role of such an operator in the framework of separation of variables (SoV) for the classical Hamilton-Jacobi equation. We will show that the existence of generalized Lenard chains, for a given finite-dimensional Hamiltonian system, assures that the corresponding Hamilton-Jacobi equation can be solved by SoV in a set of special coordinates, naturally defined by the Nijenhuis operator. Finally, we will discuss some examples of gener-alized Lenard chains and, in particular, of the so-called quasi-bi-Hamiltonian sys-tems.

format_quoteA new geometric approach to the separation of variables (SoV) offers intrinsic and constructive separability criteria, bridging classical and modern theories.format_quote

4 Haantjes Manifolds and Integrable Systems

A tensorial approach to the theory of classical Hamiltonian integrable systems is proposed, based... more 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 ωH 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.

A New Class of Generalized Haantjes Tensors and Nilpotency

arXiv: Differential Geometry, 2018

We propose a new infinite class of generalized binary tensor fields. The first representative of ... more 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 to the generalized Nijenhuis torsions recently introduced independently in [5] and [12]. We also conjecture that our generalized binary tensors vanish over nilpotent triangular operator fields.

Correction to: Haantjes algebras of classical integrable systems

Annali di Matematica Pura ed Applicata (1923 -), 2021

Higher Haantjes Brackets and Integrability

Communications in Mathematical Physics, 2021

We propose a new, infinite class of brackets generalizing the Frölicher–Nijenhuis bracket. This c... more 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 also prove that the vanishing of a higher-level Nijenhuis torsion of an operator field is a sufficient condition for the integrability of its 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.

Haantjes Manifolds of Classical Integrable Systems

A general theory of classical integrable systems is proposed, based on the geometry of the Haantj... more 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 ωH manifold), as the natural setting where the notion of integrability can be formulated. We prove that the existence of suitable Haantjes structures is a necessary and sufficient condition for a Hamiltonian system to be integrable in the Liouville-Arnold sense. We also prove theorems ensuring the existence of a large family of completely integrable systems, constructed starting from a prescribed Haantjes structure. Furthermore, we propose a novel approach to the theory of separation of variables, intimately related to the geometry of Haantjes manifolds. A special family of coordinates, that we shall call the Darboux-Haantjes coordinates, will be introduced. They are constructed from the Haantjes structure associated with an integrable system, and allow the additive separation of variables of the Hamilton-Jacobi equat...

Haantjes Algebras of the Lagrange Top

Theoretical and Mathematical Physics

A symplectic-Haantjes manifold and a Poisson-Haantjes manifold for the Lagrange top are studied a... more 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. Contents 1. Introduction 1 2. Nijenhuis and Haantjes torsion 2 3. Haantjes algebras 3 4. Poisson-Haantjes manifolds 4 5. The Lagrange top 6 5.1. Euler angles 7 5.2. Euler-Poisson coordinates 8 6. Future Perspectives 13 Acknowledgement 13 References 14