${\mathbb{C}}^N$-Smorodinsky-Winternitz system in a constant magnetic field

Abstract

We propose the superintegrable generalization of Smorodinsky-Winternitz system on the $N$-dimensional 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 ${\mathbb{C}}^2$- 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.

arXiv comment9 pages

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Cited by in corpus (9)

• Cylindrical type integrable classical systems in a magnetic field
• New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates
• ${\mathbb{CP}}^N$-Rosochatius system, superintegrability, supersymmetry
• $N$-dimensional Smorodinsky-Winternitz model and related higher rank quadratic algebra $\mathcal{S}\mathcal{W}(N)$
• Symmetries of deformed supersymmetric mechanics on Kähler manifolds
• Quantum SU$(2|1)$ supersymmetric ${\mathbb{C}}^N$ Smorodinsky--Winternitz system
• Hydrogen Atom in Electric and Magnetic Fields: Dynamical Symmetries, Superintegrable and Integrable Systems, Exact Solutions
• SU$(2|1)$ supersymmetric spinning models of chiral superfields
• Superintegrable deformations of oscillator and Coulomb systems