Uniform precession of a heavy symmetric top ($I_1\cos \theta {\Omega }^2-I_3{\omega }_3\Omega +Mgl=0$)

A heavy symmetric top precesses uniformly when its inclination is constant and its azimuthal speed is constant. For a nonvertical inclination, the condition is $I_1\cos \theta {\Omega }^2-I_3{\omega }_3\Omega +Mgl=0$. The real-root condition depends on whether the body axis points above or below the horizontal.

When $s^2\gg 4mg\ell I_1\cos \theta$, the two nonvertical steady-precession roots are approximately $mg\ell /s$ and $s/(I_1\cos \theta )-mg\ell /s$. These asymptotic descriptions do not apply at a singular denominator without taking the original quadratic limit.