New classes of quadratically integrable systems in magnetic fields: The generalized cylindrical and spherical cases

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In this paper we present the construction of all nonstandard integrable systems in magnetic ﬁelds whose integrals have leading order structure corresponding to the case (i) of Theorem 1 in [A Marchesiello and L Šnobl 2022 J. Phys. A: Math. Theor. 55 145203]. We ﬁnd that the resulting systems can be written as one family with several parameters. For certain limits of these parameters the system belongs to intersections with already known standard systems separating in Cartesian and / or cylindrical coordinates and the number of independent integrals of motion increases, thus the system becomes minimally superintegrable. These results generalize the particular example presented in section 3 of [A Marchesiello and L Šnobl 2022 J. Phys. A: Math. Theor. 55 145203].

Section: Discussionsupporting

confidence: 71%

Section: Discussionmentioning

confidence: 50%

Section: Introductionmentioning

confidence: 99%

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Family of nonstandard integrable and superintegrable classical Hamiltonian systems in non-vanishing magnetic fields

Hoque

Šnobl

2023

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“…Afterwards, we have searched for new systems in the respective classes. In [27] we have classified systems with integrals generalizing the cylindrical and spherical case, which together with the analysis of the corresponding standard cases in [16,18] completely exhaust these two physically very important cases. In [28] we constructed all integrable systems of one nonstandard case of [26] (case (i) therein), leading to one family of solutions depending on 5 parameters.…”

Section: Introductionmentioning

confidence: 99%

Integrable systems of the ellipsoidal, paraboloidal and conical type with magnetic field

Fazlul Hoque,

Marchesiello,

Šnobl

2024

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We construct integrable Hamiltonian systems with magnetic fields of the ellipsoidal, paraboloidal and conical type, i.e. systems that generalize natural Hamiltonians separating in the respective coordinate systems to include nonvanishing magnetic field. In the ellipsoidal and paraboloidal case each this classification results in three one-parameter families of systems, each involving one arbitrary function of a single variable and a parameter specifying the strength of the magnetic field of the given fully determined form. In the conical case the results are more involved, there are two one-parameter families like in the other cases and one class which is less restrictive and so far resists full classification.

“…We call such integrals nonstandard. To confirm that these classes are not empty, we have to consider the lower order terms in momenta as well in each class; we have done it for some of them [11,12,16]. Quite a few nonstandard integrable systems were found, though a complete study of all classes has not been achieved yet.…”

Section: Introductionmentioning

confidence: 99%

Integrable systems in magnetic fields: the generalized parabolic cylindrical case

Kubů,

Marchesiello,

Šnobl

2024

This article is a contribution to the classification of quadratically integrable systems with vector potentials whose integrals are of the nonstandard, nonseparable type. We focus on generalized parabolic cylindrical case, related to non-subgroup-type coordinates. We find three new systems, two with magnetic fields polynomial in Cartesian coordinates and one with unbounded exponential terms. The limit in the parameters of the integrals yields a new parabolic cylindrical system; the limit of vanishing magnetic fields leads to the free motion. This confirms the conjecture that non-subgroup type integrals can be related to separable systems only in a trivial manner.