Two-Photon Algebra and Integrable Hamiltonian Systems

Ballesteros, Ángel; Herranz, Francisco J.

doi:10.2991/jnmp.2001.8.s.4

Cited by 12 publications

(34 citation statements)

References 8 publications

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“…In this sense the system (3.12) plays a role even more fundamental than the N degrees of freedom Hamiltonian which gives only one additional invariant. This is particularly evident in the case of equation (5.5a) where, due to the presence of the twophoton h 6 coalgebra, one should have constructed the second invariant with other methods, see [7]. In particular, our examples seem to suggest that the integrability properties of an underlying Poisson map are completely governed by those of the associated dynamical system on the generators of the algebra (3.12).…”

Section: Discussionmentioning

confidence: 76%

“…It is well known that out of the 2N − 3 functionally independent invariant it is possible to construct N − 1 functionally independent and commuting with respect to the Poisson bracket (4.4). We show how to construct this set of N −1 commuting invariants with the coalgebra approach, as it was proved in the in the continuum case in [3,7]. This is the content of the following proposition: Proposition 4.2.…”

Section: General Classes Of Additive Differential Systems and Coalgebramentioning

confidence: 90%

“…Furthermore, we give some additional definition to cover the cases when there exist more or less invariants than the number given in Definitions • In the literature on superintegrable systems, see for instance [46], it is often omitted in the definition of superingrability the requirement for the system to be Liouville-Poisson integrable. In fact there are known examples in the literature where it is possible to find a huge number of invariants, but not a full set of commuting ones, see [7]. So, to avoid the situation where a superintegrable system might not be a Liouville-Poisson integrable systems, we prefer to stick to a definition where superintegrability requires Liouville-Poisson integrability.…”

Section: 22mentioning

confidence: 99%

“…So, following the procedure outlined in Section (3), and using the same reasoning in [3,7], from the left and right Casimir functions of such an algebra we derive the following two sets of functionally-independent invariants commuting with respect to the canonical Poisson bracket (4.4):…”

Section: General Classes Of Additive Differential Systems and Coalgebramentioning

confidence: 99%

“…Remark 4.3. In the h 6 Lie-Poisson algebra the element M is central, while the other have the following commutation table [3,7,68]:…”

Section: 2mentioning

confidence: 99%

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Coalgebra symmetry for discrete systems

Gubbiotti¹,

Latini²,

Tapley³

2021

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show abstract

Section: Discussionmentioning

confidence: 76%

Section: General Classes Of Additive Differential Systems and Coalgebramentioning

confidence: 90%

Section: 22mentioning

confidence: 99%

Section: General Classes Of Additive Differential Systems and Coalgebramentioning

confidence: 99%

“…Remark 4.3. In the h 6 Lie-Poisson algebra the element M is central, while the other have the following commutation table [3,7,68]:…”

Section: 2mentioning

confidence: 99%

See 3 more Smart Citations

Coalgebra symmetry for discrete systems

Gubbiotti¹,

Latini²,

Tapley³

2021

Preprint

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Integrable perturbations of the N-dimensional isotropic oscillator

Ballesteros

Blasco

2010

Physics Letters A

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Two new families of completely integrable perturbations of the N -dimensional isotropic harmonic oscillator Hamiltonian are presented. Such perturbations depend on arbitrary functions and N free parameters and their integrals of motion are explicitly constructed by making use of an underlying h 6 -coalgebra symmetry. Several known integrable Hamiltonians in low dimensions are obtained as particular specializations of the general results here presented. An alternative route for the integrability of all these systems is provided by a suitable canonical transformation which, in turn, opens the possibility of adding (N − 1) 'Rosochatius' terms that preserve the complete integrability of all these models.

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Integrable potentials on spaces with curvature from quantum groups

Ballesteros¹,

Herranz²,

Ragnisco³

2005

J. Phys. A: Math. Gen.

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A family of classical integrable systems defined on a deformation of the twodimensional sphere, hyperbolic and (anti-)de Sitter spaces is constructed through Hamiltonians defined on the non-standard quantum deformation of a sl(2) Poisson coalgebra. All these spaces have a non-constant curvature that depends on the deformation parameter z. As particular cases, the analogues of the harmonic oscillator and Kepler-Coulomb potentials on such spaces are proposed. Another deformed Hamiltonian is also shown to provide superintegrable systems on the usual sphere, hyperbolic and (anti-)de Sitter spaces with a constant curvature that exactly coincides with z. According to each specific space, the resulting potential is interpreted as the superposition of a central harmonic oscillator with either two more oscillators or centrifugal barriers. The non-deformed limit z → 0 of all these Hamiltonians can then be regarded as the zero-curvature limit (contraction) which leads to the corresponding (super)integrable systems on the flat Euclidean and Minkowskian spaces. PACS: 02.30.lk 02.20.UwRecently, this integrability-preserving deformation procedure has been used to introduce both superintegrable and integrable free motions on two-dimensional (2D) spaces with curvature, either constant or variable, respectively [12]. Therefore one could expect that potential terms can also be considered, in such a way that the coalgebra approach should provide (super)integrable potentials on curved spaces. The aim of the present paper is to prove this assertion through the construction of some relevant Hamiltonians.In order to make these ideas more explicit, let us consider the non-standard quantum deformation of sl (2) [13] written as a Poisson coalgebra (sl z (2), ∆ z ) with (deformed) Poisson brackets, coproduct and Casimir given by {J 3 , J + } = 2J + cosh zJ − {J 3 , J − } = −2 sinh zJ − z {J − , J + } = 4J 3 (1.1) ∆ z (J − ) = J − ⊗ 1 + 1 ⊗ J − ∆ z (J i ) = J i ⊗ e zJ − + e −zJ − ⊗ J i i = +, 3 (1.2)

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Two-Photon Algebra and Integrable Hamiltonian Systems

Cited by 12 publications

References 8 publications

Coalgebra symmetry for discrete systems

Coalgebra symmetry for discrete systems

Integrable perturbations of the N-dimensional isotropic oscillator

Integrable potentials on spaces with curvature from quantum groups

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