= Solution
A <fictitious force> accounts for acceleration of the chosen <reference frame> rather than an additional interaction. For axes rotating at constant $\boldsymbol\Omega$, differentiating a vector uses $(d/dt)_{\rm lab}=(d/dt)_{\rm rot}+\boldsymbol\Omega\times$. Twice applying this gives
$$
\mathbf a_{\rm lab}=\mathbf a+2\boldsymbol\Omega\times\mathbf v+
\boldsymbol\Omega\times(\boldsymbol\Omega\times\mathbf r),
$$
so moving the last two terms to the <force> side produces the Coriolis and <centrifugal forces>. Here $\mathbf v$ is measured in rotating axes.
The <rotating saddle trap> has potential <force> $(-m\omega^2x,+m\omega^2y)$. Therefore
$$
\ddot x=(\Omega^2-\omega^2)x+2\Omega\dot y,\qquad
\ddot y=(\Omega^2+\omega^2)y-2\Omega\dot x.
$$
The trial solution gives
$$
(\epsilon^2+\Omega^2)^2-\omega^4-4\Omega^2\epsilon^2=0,
\qquad\boxed{\epsilon_\pm^2=\Omega^2\pm\omega^2.}
$$
Both modes are bounded for arbitrary sufficiently small initial displacements and <velocities> precisely when \b[$|\Omega|>\omega$]. At equality there is a zero-frequency secular mode; the exactly stationary solution does not establish stability.
For <rapidly rotating saddle confinement> with $\Omega\gg\omega>0$, real mode ellipses can be written
$$
(x,y)_+=A\left(\cos\epsilon_+t,-\frac\Omega{\epsilon_+}\sin\epsilon_+t\right),\qquad
(x,y)_-=B\left(\cos\epsilon_-t,-\frac{\epsilon_-}\Omega\sin\epsilon_-t\right).
$$
They are nearly circular clockwise motions at rates $\epsilon_\pm\simeq\Omega\pm\omega^2/(2\Omega)$. Rotating these axes back to the laboratory cancels the large angular rate: \b[the leading laboratory motion is a slow approximately harmonic ellipse with <frequency> $\omega^2/(2\Omega)$], plus small rapid micromotion. It results from the changing saddle <force>, not a static potential minimum. At next order the ellipse precesses prograde at $\omega^4/(8\Omega^3)$. The rotating-potential model and this guiding-centre distinction are analysed in the primary work cited here.<Rotating saddle trap as Foucault's pendulum by Oleg N. Kirillov and Mark Levi>[preprint pp. 2 and 6–7, equations (12)–(18)]{ref}
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2014/paper-1-rotating-saddle.png]
{title=Original exact trajectories in rotating and laboratory coordinates for a chosen rapid saddle rotation}
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