= Torque-free precession of a symmetric top
{title2=$\dot\phi=L/I_\perp$}
For an axially symmetric <rigid body> with $I_1=I_2=I_\perp$ and $I_3=I_\parallel$, no external <torque> means that <angular momentum> $\mathbf L$ is fixed in an <inertial frame>. The axial <angular velocity> $\omega_3$ is constant by <Euler equations for a torque-free rigid body>, and the symmetry axis obeys
$$
\dot{\mathbf e}_3=\frac{\mathbf L}{I_\perp}\times\mathbf e_3.
$$
Its angle to $\mathbf L$ is constant and it precesses in space at $L/I_\perp$. The transverse angular-velocity components instead rotate in body coordinates at signed rate $(I_\parallel-I_\perp)\omega_3/I_\perp$. These are different frame descriptions and must not be substituted for one another.
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