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

Tondo, Giorgio; Tempesta, Piergiulio

doi:10.3842/sigma.2016.023

Cited by 12 publications

(19 citation statements)

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“…While much progress has been made in this direction using Noether symmetries, see for example 6,7 , an explicit approach to quantising (3), allowing for example to display eigenfunctions in configuration space, is still lacking. As to comparison with earlier works, the following remarks are in order: the most general set of Hamiltonians described in this paper, displayed in (20), have been described in various papers within the context of bi-Hamiltonian and quasi-bi-Hamiltonian systems [8][9][10][11] . Results concerning their integrability and their amenability to treatment via separation of variables are obtained there, but the connection to the quite elementary approach presented here is not apparent to the author.…”

Section: Introductionmentioning

confidence: 99%

An approach for obtaining integrable Hamiltonians from Poisson-commuting polynomial families

Leyvraz¹

2017

Journal of Mathematical Physics

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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 such Hamiltonians can be solved explicitly by a separation of variables Ansatz. The method leads in particular to a proof that the so-called "goldfish" Hamiltonian is maximally superintegrable, and leads to an elementary identification of a full set of integrals of motion. The Hamiltonians in involution with the "goldfish" Hamiltonian are also explicitly integrated. New integrable Hamiltonians are identified, among which some have the property of being isochronous, that is, that all their orbits have the same period. Finally, a peculiar structure is identified in the Poisson brackets between the elementary symmetric functions and the set of Hamiltonians commuting with the "goldfish" Hamiltonian: these can be expressed as products between elementary symmetric functions and Hamiltonians. The structure displays an invariance property with respect to one element, and has both a symmetry and a closure property. The meaning of this structure is not altogether clear to the author, but it turns out to be a powerful tool.

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Section: Introductionmentioning

confidence: 99%

An approach for obtaining integrable Hamiltonians from Poisson-commuting polynomial families

Leyvraz¹

2017

Journal of Mathematical Physics

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“…Other relevant examples of Haantjes operators expressed in terms of Killing tensors in a Riemannian manifold can be found in [32].…”

Section: 2mentioning

confidence: 99%

“…We shall present here an example illustrating some interesting aspects of the Haantjes geometry previously discussed. In particular, as an application of the main theorems, we develop the procedure of constructing a Haantjes chart and a cyclic generator for a semisimple Haantjes algebra of rank 2, which has been exploited in the study of the Jacobi-Calogero Hamiltonian system [32].…”

Section: Cyclic Haantjes Operators: Examplementioning

confidence: 99%

“…In other words, although Magri's Haantjes manifolds possess a richer axiomatic structure than our Haantjes algebras, we prefer to articulate our definition on a minimal number of requirements, in order to have a flexible structure, which in a subsequent step can be made suitable for the study of more specific problems, as the separation of variables in the context of integrability or Riemannian geometry. Keeping with this philosophy, we postpone the introduction of additional geometric structures (as Haantjes chains of exact 1-forms, symplectic forms [29], Poisson bivectors [31] or Riemannian metrics [32]) to a further stage of the theory.…”

Section: A Comparison With Other Algebraic Structures: Haantjes Manifolds and Killing-stäckel Algebrasmentioning

confidence: 99%

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Haantjes algebras and diagonalization

Tempesta

Tondo

2021

Journal of Geometry and Physics

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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.

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“…Now, we can verify that it satisfies also Eq. (35), therefore we have got a solution of the problem (27)- (29), which solves also Eqs. (39).…”

Section: Haantjes Algebramentioning

confidence: 99%