Uniform magnetic field with axial harmonic confinement (${\Omega }_c=2e\mu /m,{\Omega }_z=\sqrt{2e\lambda /m}$)

The cylindrical vector potential $A_{\theta }=\mu r$ gives a uniform axial magnetic field $2\mu$, while scalar potential $\lambda z^2$ supplies harmonic axial confinement for positive charge and positive $\lambda$. Transverse motion is cyclotron motion about a fixed guiding centre, and axial motion is an independent harmonic oscillation. A circle centred on the cylindrical origin has $\dot{\theta }=-{\Omega }_c$ for positive charge.

A rational ratio of axial and cyclotron frequencies makes a nondegenerate combined orbit close after an integer number of turns. For ratio $p/q$ in lowest terms it closes after $q$ transverse periods and $p$ axial periods. An integer ratio is a sufficient special case. An irrational ratio instead gives quasiperiodic motion when both component amplitudes are nonzero.