Integrable systems in magnetic fields: the generalized parabolic cylindrical case

Abstract

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 3 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.

20 pages

References in corpus (17)

• Classical and Quantum Superintegrability with Applications
• Integrable and superintegrable quantum systems in a magnetic field
• Integrable Hamiltonian systems with vector potentials
• Superintegrable relativistic systems in spacetime-dependent background fields
• Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials
• General $N^{th}$-order superintegrable systems separating in polar coordinates
• A nonseparable quantum superintegrable system in 2D real Euclidean space
• Higher-order superintegrability of a Holt related potential
• Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates
• Cylindrical type integrable classical systems in a magnetic field
• Haantjes Algebras of Classical Integrable Systems
• Separability and Dynamical Symmetry of Quantum Dots
• New classes of quadratically integrable systems in magnetic fields: the generalized cylindrical and spherical cases
• An Algebraic Geometric Foundation for a Classification of Superintegrable Systems in Arbitrary Dimension
• Family of nonstandard integrable and superintegrable classical Hamiltonian systems in non-vanishing magnetic fields
• New classes of quadratically integrable systems with velocity dependent potentials: non-subgroup type cases
• On superintegrable monopole systems

Cited by in corpus (2)

• Hamiltonian integrable systems in a magnetic field and Symplectic-Haantjes geometry
• Integrability of non-homogeneous Hamiltonian systems with gyroscopic coupling