An Infinite Family of Maximally Superintegrable Systems in a Magnetic Field with Higher Order Integrals

Marchesiello, Antonella; Šnobl, Libor

doi:10.3842/sigma.2018.092

Cited by 18 publications

(37 citation statements)

References 26 publications

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“…Let us also point out that if V 1 (x 1 ) = 0, for constant magnetic field a rotation around x 2 brings the system (7) into (9). Thus, what we will deduce in the following for V 1 = 0 applies also for (9).…”

Section: Constant Magnetic Field and Second Order Polynomial Potentialsmentioning

confidence: 55%

“…Above we have provided a complete answer to the problem of quadratic superintegrability for the considered classes of systems (7) and (9). As we have seen, maximal superintegrability via at most quadratic integrals is very rare in the presence of magnetic field, as opposed to numerous purely scalar maximally superintegrable systems discussed e.g.…”

Section: Superintegrable Systems With Higher Order Integralsmentioning

confidence: 84%

“…A more general 3D infinite family of maximally superintegrable system, including the previous cases (30) and (33) and the one found in [9], corresponds to the caged oscillator…”

Section: Maximal Superintegrable Class Canonically Conjugated To Natumentioning

confidence: 96%

“…The study of superintegrability with magnetic fields was initiated in [1] and subsequently followed in both two spatial dimensions [2,3,4,5] and three spatial dimensions [6,7,8,9,10]. Also a relativistic version of the problem was recently considered, cf.…”

Section: Introductionmentioning

confidence: 99%

“…With these simplifications at hand, the determining equations for the integral can be solved. In section 9.1 we list the superintegrable systems so found; their explicit derivation is rather technical and tedious and we review it in appendices A, B and C. The special case in which the magnetic field is constant and the functions V j in (7) and (9) are at most quadratic polynomials is studied in section 8. Finally, in section 9.2 we discuss the approaches to construction of higher order integrals.…”

Section: Introductionmentioning

confidence: 99%

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Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates

Marchesiello

Šnobl

2020

SIGMA

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We consider superintegrability in classical mechanics in the presence of magnetic fields. We focus on three-dimensional systems which are separable in Cartesian coordinates. We construct all possible minimally and maximally superintegrable systems in this class with additional integrals quadratic in the momenta. Together with the results of our previous paper [J. Phys. A: Math. Theor. 50 (2017) 245202], where one of the additional integrals was by assumption linear, we conclude the classification of three-dimensional quadratically minimally and maximally superintegrable systems separable in Cartesian coordinates. We also describe two particular methods for constructing superintegrable systems with higher order integrals.

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Section: Constant Magnetic Field and Second Order Polynomial Potentialsmentioning

confidence: 55%

Section: Superintegrable Systems With Higher Order Integralsmentioning

confidence: 84%

“…A more general 3D infinite family of maximally superintegrable system, including the previous cases (30) and (33) and the one found in [9], corresponds to the caged oscillator…”

Section: Maximal Superintegrable Class Canonically Conjugated To Natumentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates

Marchesiello

Šnobl

2020

SIGMA

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CN-Smorodinsky–Winternitz system in a constant magnetic field

Shmavonyan

2019

Physics Letters A

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We propose the superintegrable generalization of Smorodinsky-Winternitz system on the Ndimensional complex Euclidian space which is specified by the presence of constant magnetic field. We find out that in addition to 2N Liouville integrals the system has additional functionally independent constants of motion, and compute their symmetry algebra. We perform the Kustaanheimo-Stiefel transformation of C 2 -Smorodinsky-Winternitz system to the (three-dimensional) generalized MICZ-Kepler problem and find the symmetry algebra of the latter one. We observe that constant magnetic field appearing in the initial system has no qualitative impact on the resulting system. * Electronic address: hovhannes.shmavonyan@mail.yerphi.am

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Cylindrical first-order superintegrability with complex magnetic fields

Kubů

Šnobl

2023

Journal of Mathematical Physics

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This article is a contribution to the study of superintegrable Hamiltonian systems with magnetic fields on the three-dimensional Euclidean space E3 in quantum mechanics. In contrast to the growing interest in complex electromagnetic fields in the mathematical community following the experimental confirmation of its physical relevance [Peng et al., Phys. Rev. Lett. 114, 010601 (2015)], they were so far not addressed in the growing literature on superintegrability. Here, we venture into this field by searching for additional first-order integrals of motion to the integrable systems of cylindrical type. We find that already known systems can be extended into this realm by admitting complex coupling constants. In addition to them, we find one new system whose integrals of motion also feature complex constants. All these systems are multiseparable. Rigorous mathematical analysis of these systems is challenging due to the non-Hermitian setting and lost gauge invariance. We proceed formally and pose the resolution of these problems as an open challenge.

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An Infinite Family of Maximally Superintegrable Systems in a Magnetic Field with Higher Order Integrals

Cited by 18 publications

References 26 publications

Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates

Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates

CN-Smorodinsky–Winternitz system in a constant magnetic field

Cylindrical first-order superintegrability with complex magnetic fields

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