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

TL;DR

This paper investigates superintegrability in 3D axially symmetric Hamiltonian systems with magnetic fields, discovering new minimally superintegrable systems and families of higher-order maximally superintegrable systems through geometric and algebraic analysis.

Contribution

It identifies new minimally superintegrable systems in 3D magnetic fields and constructs infinite families of higher-order superintegrable systems, expanding the classification of such systems.

Findings

No new superintegrable systems in the linear case.

Discovery of a new minimally superintegrable system at the intersection of specific cases.

Construction of infinite families of higher-order maximally superintegrable systems linked to classical oscillators.

Abstract

We extend the investigation of three-dimensional (3D) Hamiltonian systems of non-subgroup type admitting non-zero magnetic fields and an axial symmetry, namely the circular parabolic case, the oblate spheroidal case and the prolate spheroidal case. More precisely, we focus on linear and some special cases of quadratic superintegrability. In the linear case, no new superintegrable system arises. In the quadratic case, we found one new minimally superintegrable system that lies at the intersection of the circular parabolic and cylindrical cases and another one 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. These two systems are linked respectively with the caged and harmonic oscillators without magnetic fields through a time-dependent canonical transformation.