A spinning top has angular momentum L = I3 w3 along its axis. Gravity acts at the centre of mass a distance r from the pivot and gives a torque tau = r x Mg, horizontal and perpendicular to L. Since dL/dt = tau, L does not fall - its tip moves sideways, which is precession. For fast spin the rate is Omega = M g r / (I3 w3), independent of the tilt angle.

The page integrates the full heavy symmetric top (Lagrange equations) with RK4, using the conserved momenta p_phi and w3: I1 theta'' = I1 phi'^2 sin(theta)cos(theta) - I3 w3 phi' sin(theta) + M g r sin(theta), phi' = (p_phi - I3 w3 cos(theta)) / (I1 sin^2(theta)). Released from rest, the axis first dips (nutation) then precesses; a steady start removes the nutation. Moments of inertia use a uniform disc: I3 = MR^2/2, I1 = I3/2 + M r^2 (parallel axis).

Simplifications: no friction or air drag (so the spin never decays), rod mass neglected, the table is a hard stop at the horizontal. Numeric checks are shown live below the readouts.