Landau problem on the ellipsoid, hyperboloid and paraboloid of revolution

TL;DR

This paper extends the Landau problem to particles on ellipsoid, hyperboloid, and paraboloid surfaces of revolution, analyzing their Hamiltonians and motion characteristics, including action-angle variables for the ellipsoid case.

Contribution

It introduces a new formulation of the Landau problem on various second-order surfaces of revolution derived from the MICZ-Kepler system.

Findings

Identified parameter conditions for motion similarity to free particles.

Derived Hamiltonians for particles on ellipsoid, hyperboloid, paraboloid.

Constructed action-angle variables for the ellipsoid case.

Abstract

We define the Landau problem on two-dimensional surfaces of revolution of the second order: ellipsoid, hyperboloid and paraboloid. We start form the two-center MICZ-Kepler system Hamiltonian and then making the reduction into the various two-dimensional surfaces listed above we obtain the Hamiltonians of the charged particle moving on the corresponding surface of revolution with the magnetic filed conserving the symmetry of the two-dimensional surface(Landau problem). For each case we figure out at which values of parameters the qualitative character of the moving coincides with that of a free particle moving on the save two-dimensional surface. For the case of finite trajectories (ellipsoid) we construct also the action-angle variables.