At the cost of some complication a model without the fast top assumption may be constructed.
Joos [12] considers a symmetrical top that has been set
into rotation about its figure axis, which is designated as the
axis. After the initial spinup no moments are applied about
the figure axis insuring that is constant. He then
writes the angular velocity vector as the sum of a component
along the figure axis of the top , and a component
normal to the figure axis.
The angular momentum vector
is the moment of inertia about an axis perpendicular to the
figure axis.
Differentiating
Noting that
we also have
.
Solving for the derivative of ,
With the derivatives of and in hand (equations 19 and 20) one may now integrate forward the motion of the top by standard differential equation solvers. This has been done with the torque , in the direction, and the motion of the center of mass determined by equation (2), and the model successfully predicts the observed high and the low frequency spin speed stability limits to within 20%. It also illustrates the mechanism of loss. At the low frequency limit the top tips over enough so the magnetic field gradient no longer supports it. At the high frequency limit the trapping in the horizontal plane gets softer and softer, and the top eventually wanders away.