Heavy symmetric top

A heavy symmetric top is a symmetric top with one point fixed in a uniform gravitational field. Its vertical and body-axis rotational symmetries give two cyclic Euler angles.

Writing the unit symmetry axis as $n$ and axial spin momentum as $s=I_3{\omega }_3$, angular momentum is $L=I_{1}n\times \dot{n}+sn$. Taking its derivative and applying gravitational torque gives $I_{1}n\times \ddot{n}+s\dot{n}=-mg\ell n\times \hat{z}$.

Variation in the inclination of a rotating body's axis, distinguished from its azimuthal precession. For a heavy symmetric top, the cyclic angular momenta reduce the motion to ${\dot{\mu }}^2=F(\mu )$ with $\mu =\cos \theta$ and $F$ cubic. Its admissible interval and turning points describe the nutation.

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.