= Axial angular velocity of a symmetric top
{title2=$n=\boldsymbol\omega\cdot\mathbf e$}
For a <symmetric top> with axial <unit vector> $\mathbf e$ and <angular velocity> $\boldsymbol\omega$, the classical axial spin is the scalar $n=\boldsymbol\omega\cdot\mathbf e$. The identity $\dot{\mathbf e}=\boldsymbol\omega\times\mathbf e$ implies $\boldsymbol\omega=\mathbf e\times\dot{\mathbf e}+n\mathbf e$. If the transverse and axial <principal moments of inertia> are $A$ and $C$, then the <angular momentum> about the fixed vertex is $\mathbf L=A\mathbf e\times\dot{\mathbf e}+Cn\mathbf e$. A <torque> perpendicular to $\mathbf e$ conserves $n$. During steady <precession> at rate $\Omega$ and inclination $\alpha$, the general <angular velocity> is $\Omega\mathbf k+(n-\Omega\cos\alpha)\mathbf e$; the <precession> rate by itself does not fix $n$.
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