On rotationally invariant integrable and superintegrable classical systems in magnetic fields with non-subgroup type integrals

Abstract

The aim of the present article is to construct quadratically integrable three dimensional systems in non-vanishing magnetic fields which possess so-called non-subgroup type integrals. The presence of such integrals means that the system possesses a pair of integrals of motion in involution which are (at most) quadratic in momenta and whose leading order terms, that are necessarily elements of the enveloping algebra of the Euclidean algebra, are not quadratic Casimir operators of a chain of its subalgebras. By imposing in addition that one of the integrals has the leading order term $L_z^2$ we can consider three such commuting pairs: circular parabolic, oblate spheroidal and prolate spheroidal. We find all possible integrable systems possessing such structure of commuting integrals and describe their Hamiltonians and their integrals. We show that our assumptions imply the existence of a first order integral $L_z$, i.e. rotational invariance, of all such systems. As a consequence, the Hamilton--Jacobi equation of each of these systems with magnetic field separates in the corresponding coordinate system, as it is known to be the case for all quadratically integrable systems without magnetic field, and in contrast with the subgroup type, i.e. Cartesian, spherical and cylindrical, cases, with magnetic fields. We also look for superintegrable systems within the circular parabolic integrable class. Assuming the additional integral to be first order we demonstrate that only previously known systems exist. However, for a particular second order ansatz for the sought integral ($L^2+\dots$) we find a minimally quadratically superintegrable system. It is not quadratically maximally superintegrable but appears to possess bounded closed trajectories, hinting at hypothetical higher order superintegrability.

References in corpus (17)

• Effects of symmetry breaking in finite quantum systems
• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Screw-symmetric gravitational waves: a double copy of the vortex
• Integrable and superintegrable quantum systems in a magnetic field
• A New Superintegrable Hamiltonian
• Superintegrable Systems with a Third Order Integrals of Motion
• Superintegrability and higher order polynomial algebras I
• Integrable Hamiltonian systems with vector potentials
• Back-reaction on background fields: a coherent state approach
• Fourth order Superintegrable systems separating in Cartesian coordinates I. Exotic quantum potentials
• Algebraic Calculation of the Energy Eigenvalues for the Nondegenerate Three-Dimensional Kepler-Coulomb Potential
• Three-dimensional superintegrable systems in a static electromagnetic field
• Superintegrable relativistic systems in spacetime-dependent background fields
• An Infinite Family of Maximally Superintegrable Systems in a Magnetic Field with Higher Order Integrals
• Quantum integrable systems in three-dimensional magnetic fields: the Cartesian case
• Separability and Dynamical Symmetry of Quantum Dots
• Superintegrable relativistic systems in scalar background fields

Cited by in corpus (6)

• Cylindrical type integrable classical systems in a magnetic field
• On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields
• 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
• Cylindrical first order superintegrability with complex magnetic fields
• Superintegrability in the interaction of two particles with spin