Circular modes of a charged isotropic oscillator (${\omega }_{\pm }=(-{\omega }_c\pm \sqrt{{\omega }_c^2+4{\omega }_0^2})/2$)

An isotropic planar harmonic oscillator of positive charge in a perpendicular magnetic field has two circular normal modes of opposite handedness. With ${\omega }_c=qB/m>0$ and ${\omega }_0^2=k/m$, the signed angular frequencies have the displayed form for trajectories $x+iy\propto e^{i\omega t}$. A static electric field shifts the equilibrium without changing these frequencies. The clockwise mode has the larger frequency magnitude.