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Solving the two momentum equations givesSubstitution into the conserved energy reduces the heavy symmetric top to one-dimensional effective potential motion:whereThis is the heavy symmetric top reduction.

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.