Poisson Algebras and 3D Superintegrable Hamiltonian Systems

Fordy, Allan P.; Huang, Qing

doi:10.3842/sigma.2018.022

Cited by 8 publications

(18 citation statements)

References 13 publications

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“…This paper has continued our work of [2,4,5] on building higher order integrals out of conformal symmetries, and, more generally, building Poisson algebras related to superintegrable systems. We emphasise that whilst all higher order integrals can be built out of Killing vectors in the constant curvature case, there was no such algebraic method available for more general spaces.…”

Section: Reduction To 2 Dimensional Systems: 4d Isometry Algebrasmentioning

confidence: 73%

“…an example of the conformal algebra given in [4] (Table 3), corresponding to the case a 2 = a 3 = 2, a 4 = 0. The subalgebra g 1 , with basis (2a) is just a copy of sl (2).…”

Section: The 3d Euclidean Metric and Its Conformal Algebramentioning

confidence: 99%

“…In this paper we have not considered the addition of potential functions, which can lead to very complicated Poisson algebras in the case of 3 degrees of freedom (see [4]). However, imposing invariance under one or more of the isometries of the kinetic energy can simplify the calculations.…”

Section: Specific Cases Listed In Section 53mentioning

confidence: 99%

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Superintegrable systems on 3 dimensional conformally flat spaces

Fordy

Huang

2020

Journal of Geometry and Physics

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We consider Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. We use the conformal algebra to build additional quadratic first integrals, thus constructing a large class of superintegrable systems and the complete Poisson algebra of first integrals. We then use the isometries to reduce our systems to 2 degrees of freedom. For each isometry algebra we give a universal reduction of the corresponding general Hamiltonian. The superintegrable specialisations reduce, in this way, to systems of Darboux-Koenigs type, whose integrals are reductions of those of the 3 dimensional system. 1 2 n(n + 1) = 6 Killing vectors (when n = 3).The conformal algebra of this 3D metric has dimension 1 2 (n + 1)(n + 2) = 10 (when n = 3). A convenient

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Section: Reduction To 2 Dimensional Systems: 4d Isometry Algebrasmentioning

confidence: 73%

“…an example of the conformal algebra given in [4] (Table 3), corresponding to the case a 2 = a 3 = 2, a 4 = 0. The subalgebra g 1 , with basis (2a) is just a copy of sl (2).…”

Section: The 3d Euclidean Metric and Its Conformal Algebramentioning

confidence: 99%

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Superintegrable systems on 3 dimensional conformally flat spaces

Fordy

Huang

2020

Journal of Geometry and Physics

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“…The remaining blocks are just the corresponding invariant subspaces. In [7] we showed that the 6-dimensional algebra g 1 + g 2 (which here we call g (0) ) has two quadratic Casimirs…”

Section: The 3d Euclidean Metric and Its Conformal Algebramentioning

confidence: 99%

“…In Section 2 we describe the structure of the 10-dimensional conformal algebra in this case, and show that it has a particular decomposition introduced in [7]. We show that this algebra has 4 involutive automorphisms, which play an important role in our calculations.…”

Section: Introductionmentioning

confidence: 99%

Generalised Darboux-Koenigs Metrics and 3-Dimensional Superintegrable Systems

Fordy¹,

Huang²

2019

SIGMA

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The Darboux-Koenigs metrics in 2D are an important class of conformally flat, non-constant curvature metrics with a single Killing vector and a pair of quadratic Killing tensors. In [arXiv:1804.06904] it was shown how to derive these by using the conformal symmetries of the 2D Euclidean metric. In this paper we consider the conformal symmetries of the 3D Euclidean metric and similarly derive a large family of conformally flat metrics possessing between 1 and 3 Killing vectors (and therefore not constant curvature), together with a number of quadratic Killing tensors. We refer to these as generalised Darboux-Koenigs metrics. We thus construct multi-parameter families of super-integrable systems in 3 degrees of freedom. Restricting the parameters increases the isometry algebra, which enables us to fully determine the Poisson algebra of first integrals. This larger algebra of isometries is then used to reduce from 3 to 2 degrees of freedom, obtaining Darboux-Koenigs kinetic energies with potential functions, which are specific cases of the known super-integrable potentials.

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A Kaluza–Klein reduction of super-integrable systems

Fordy¹

2018

Journal of Geometry and Physics

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Given a super-integrable system in n degrees of freedom, possessing an integral which is linear in momenta, we use the "Kaluza-Klein construction" in reverse to reduce to a lower dimensional superintegrable system. We give two examples of a reduction from 3 to 2 dimensions. The constant curvature metric (associated with the kinetic energy) is the same in both cases, but with two different superintegrable extensions. For these, we use different elements of the algebra of isometries of the kinetic energy to reduce to 2−dimensions. Remarkably, the isometries of the reduced space can be derived from those of the 3−dimensional space, even though it requires the use of quadratic expressions in momenta.

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Poisson Algebras and 3D Superintegrable Hamiltonian Systems

Cited by 8 publications

References 13 publications

Superintegrable systems on 3 dimensional conformally flat spaces

Superintegrable systems on 3 dimensional conformally flat spaces

Generalised Darboux-Koenigs Metrics and 3-Dimensional Superintegrable Systems

A Kaluza–Klein reduction of super-integrable systems

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