Rotating saddle trap ($|\Omega |>\omega$)

A planar quadratic saddle $m{\omega }^2(x^2-y^2)/2$ rotated at angular speed $\Omega$ becomes linearly confining for the displayed strict inequality. Its rotating-frame mode frequencies obey ${ϵ}_{\pm }^2={\Omega }^2\pm {\omega }^2$. Equality has a zero-frequency secular mode and is not robust bounded-motion stability. This is a rotating potential model; a rolling ball on an actual surface has additional constraints.

In rapid rotation, both rotating-frame modes are nearly circular and counter-rotate relative to the axes at frequency near $|\Omega |$. Transforming back to the laboratory leaves a slow nearly harmonic oscillation and small rapid micromotion. For positive rotation the slow ellipse also has higher-order prograde precession at rate ${\omega }^4/(8{\Omega }^3)$. These follow by expanding $\sqrt{{\Omega }^2\pm {\omega }^2}$ and combining the two laboratory circular components.^[ref]