= Small nutation about a conical pendulum
{title2=$\omega_\theta^2=\frac gl\frac{1+3\cos^2\theta_0}{\cos\theta_0}$}
For a <spherical pendulum> with nonzero conserved vertical <angular momentum> $h$, eliminate the azimuthal <angular velocity> using $h=ml^2\sin^2\theta\dot\phi$. The polar <effective potential> is $V_{\mathrm{eff}}=h^2/(2ml^2\sin^2\theta)-mgl\cos\theta$. A <conical pendulum> at $0<\theta_0<\pi/2$ is a minimum satisfying $h^2=m^2gl^3\sin^4\theta_0/\cos\theta_0$. <Linearization> at fixed $h$ yields polar <angular frequency> $\omega_\theta^2=V_{\mathrm{eff}}''(\theta_0)/(ml^2)=\Omega^2(1+3\cos^2\theta_0)$, where $\Omega^2=g/(l\cos\theta_0)$ is the conical revolution frequency squared. Holding $\dot\phi$ fixed during the perturbation would describe a different, externally driven constraint. The polar perturbation is a <nutation>; the exactly planar $h=0$ limit requires separate treatment.
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