Cylindrical type integrable classical systems in a magnetic field

TL;DR

This paper classifies all second order classical integrable systems of cylindrical type in three-dimensional space with a magnetic field, identifying infinite families and suggesting potential superintegrable cases.

Contribution

It provides a complete classification of second order integrable cylindrical systems with magnetic fields, including explicit forms of Hamiltonians and integrals of motion, and highlights the possibility of superintegrability.

Findings

Infinite families of integrable systems depending on arbitrary functions or parameters

Explicit forms of Hamiltonian and integrals of motion for these systems

Potential existence of superintegrable systems among the classified ones

Abstract

We present all second order classical integrable systems of the cylindrical type in a three dimensional Euclidean space $\mathbb{E}_3$ with a nontrivial magnetic field. The Hamiltonian and integrals of motion have the form $H =\frac{1}{2}\left(\vec{p}+\vec{A}(\vec{x})\right)^2+W(\vec{x})$, $X_1=(p_\phi^A)^2+s_1^r(r, \phi, Z)p_r^A+s_1^\phi(r, \phi, Z)p_\phi^A+s_1^Z(r, \phi, Z)p_Z^A+m_1(r,\phi,Z)$, $X_2=(p_Z^A)^2+s_2^r(r, \phi, Z)p_r^A+s_2^\phi(r, \phi, Z)p_\phi^A+s_2^Z(r, \phi, Z)p_Z^A+m_2(r,\phi,Z)$. Infinite families of such systems are found, in general depending on arbitrary functions or parameters. This leaves open the possibility of finding superintegrable systems among the integrable ones (i.e. systems with 1 or 2 additional independent integrals).