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

Marchesiello, Antonella; Šnobl, Libor

doi:10.3842/sigma.2020.015

Cited by 12 publications

(34 citation statements)

References 31 publications

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“…These systems have already been found in e.g. [1,20,23]. For the minimally superintegrable counterparts, the overlap with the spherical case was studied in [1], where we found a new quadratically superintegrable system.…”

Section: Special Quadradic Superintegrability: Circular Parabolicsupporting

confidence: 57%

“…These two systems are already known in the literature, see e.g. [1,20,23]. One should note that the classification of all linearly superintegrable systems for the circular parabolic case was provided in section 6 of the paper [1].…”

Section: Linear Superintegrability: Oblate and Prolate Spheroidalmentioning

confidence: 96%

“…the oblate spheroidal, prolate spheroidal and circular parabolic cases. The first three classes (Cartesian, cylindrical and spherical) belong to the subgroup type and are to be investigated in other papers like [23].…”

Section: Introductionmentioning

confidence: 99%

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On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields

Bertrand,

Kubů,

Šnobl

2020

Preprint

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We extend the investigation of three-dimensional (3D) Hamiltonian systems of non-subgroup type admitting non-zero magnetic fields and an axial symmetry. Three different integrable cases are considered as starting points for superintegrability: the circular parabolic case, the oblate spheroidal case and the prolate spheroidal case. These integrable cases were introduced in [1]. In this paper, we focus on linear and some special cases of quadratic superintegrability. We investigated all possible additional linear integrals of motion for the oblate and the prolate spheroidal cases, separately. For both cases, no previously unknown superintegrable system arises. In addition, we looked for special quadratic integrals of motion for all three cases. We found one new minimally superintegrable system that lies at the intersection of the circular parabolic and cylindrical cases. We also found one new minimally superintegrable system that lies at the intersection of the cylindrical, spherical, oblate spheroidal and prolate spheroidal cases. By imposing additional conditions on these systems, we found for each quadratically minimally superintegrable system a new infinite family of higher-order maximally superintegrable systems linked with the caged and harmonic oscillator without magnetic fields, respectively, through a time-dependent canonical transformation.

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Section: Special Quadradic Superintegrability: Circular Parabolicsupporting

confidence: 57%

Section: Linear Superintegrability: Oblate and Prolate Spheroidalmentioning

confidence: 96%

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On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields

Bertrand,

Kubů,

Šnobl

2020

Preprint

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“…This completes the classification of cylindrical systems with additional quadratic integrals as the systems with first order integrals were classified in O. Kubů's Master thesis [21], where only systems known earlier from [8,12] were found.…”

Section: Introductionmentioning

confidence: 62%

Superintegrability of separable systems with magnetic field: the cylindrical case

Kubů¹,

Marchesiello²,

Šnobl³

2021

J. Phys. A: Math. Theor.

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We present a general method simplifying the search for additional integrals of motion of three dimensional systems with magnetic fields. The method is suitable for systems possessing at least one conserved canonical momentum in a suitable coordinates system. It reduces the problem either to consideration of lower dimensional systems or of particular constrained forms of the hypothetical integral. In particular, it is applicable to all separable systems in the Euclidean space since they are known to possess at least one cyclic coordinates when magnetic field is present.Next, we focus on systems which separate in the cylindrical coordinates. Using our method, we are able to classify all superintegrable systems of this kind under the assumption that all considered integrals are at most second order in the momenta. In addition to already known systems, several new minimally superintegrable systems are found and we show that no quadratically maximally superintegrable ones can exist. We also construct some examples of systems with higher order integrals.

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“…The second (and higher) order symmetries can be related to such nice properties of SE as superintegrability and supersymmetry, see surveys [5] and [6]. We will not discuss them here but mention that searching for such symmetries is still a very popular business, and the modern trends in this field are related to the third order and even arbitrary order integrals of motion [7,8,9], see also paper [10] where the determining equations for such symmetries were presented.…”

Section: Introductionmentioning

confidence: 99%

Symmetries of the Schroedinger-Pauli equations for charged particles and quasirelativistic Schroedinger equations

Nikitin

2021

Preprint

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

Cited by 12 publications

References 31 publications

On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields

On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields

Superintegrability of separable systems with magnetic field: the cylindrical case

Symmetries of the Schroedinger-Pauli equations for charged particles and quasirelativistic Schroedinger equations

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