= Gravity-restored centrifugal governor bifurcation
{title2=$\cos\theta_{\rm eq}=\omega_0^2/\Omega^2,\quad\omega_0^2=g/l$}
For total ball mass $m$, arm length $l$ and imposed constant shaft speed $\Omega$, the reduced effective potential is $V_{\rm eff}/(ml^2)=-\Omega^2\sin^2\theta/2-\omega_0^2\cos\theta$. Its equilibrium equation is $\sin\theta(\omega_0^2-\Omega^2\cos\theta)=0$. The downward state is stable for $|\Omega|<\omega_0$ and loses linear stability above it; nonzero stable openings then satisfy the displayed relation. Extending the angle to signed values gives a <supercritical pitchfork bifurcation>. Squared <angular frequencies> of <small oscillations> are $\omega_0^2-\Omega^2$ at the downward state and $\Omega^2-\omega_0^4/\Omega^2$ at the open states. At threshold the quadratic term vanishes but the quartic potential is positive, so zero linear frequency does not imply finite-amplitude free motion.
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