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Put $L_z=L_3(1-{ϵ}^2/2)$ and seek $\theta =O(ϵ)$. Using $\cos \theta =1-{\theta }^2/2+O({\theta }^4)$ and ${\sin }^2\theta ={\theta }^2+O({\theta }^4)$, the $\theta$-dependent part of the effective potential isIt has a local minimum precisely whenwhich is the gyroscopic stabilization of an inverted symmetric top condition. Differentiating the displayed approximation givesand henceTaking $L_3>0$ without loss of orientation, the steady precession rate isso

The generalized momenta conjugate to the two ignorable coordinates areandSince the Lagrangian contains neither $\varphi$ nor $\psi$, both momenta are constant in time.

Eliminating the two angular velocities givesConservation of energy then yields the heavy symmetric top reductionwhere

As $\theta \to 0$, the first term diverges unlessWriting their common value as $p$, the small-angle expansion isBy the effective potential stability criterion, the vertical state is stable exactly whenAt the vertical state $p=I_3(\dot{\psi }+\dot{\varphi })$, so this is the gyroscopic stabilization of an inverted symmetric top by sufficiently rapid spin.