Euler's equations of rigid-body motion
/ OY-ler /
Euler's equations are Newton's second law for rotation, written in the one frame where it stays simple: the body's own rotating frame. The problem they solve is that in the fixed lab frame the inertia tensor of a tumbling body keeps changing as the body reorients, which is awful to work with. Ride along with the body instead and the inertia tensor is frozen -- at the price that the frame is now rotating, which the equations must account for.
The starting point is that torque equals the rate of change of angular momentum in an inertial frame, N = (dL/dt)_inertial. Converting the time derivative into the rotating body frame gives (dL/dt)_body + omega x L = N. Choosing the principal axes as the body frame, so that L_i = I_i omega_i, yields the three Euler equations: I_1 omega_1_dot - (I_2 - I_3) omega_2 omega_3 = N_1, I_2 omega_2_dot - (I_3 - I_1) omega_3 omega_1 = N_2, I_3 omega_3_dot - (I_1 - I_2) omega_1 omega_2 = N_3. They are coupled and nonlinear because of the omega x L term.
These are the workhorse equations for every tumbling rigid body: spacecraft attitude control, the free rotation of asteroids, the wobble of a thrown club. Set the torques to zero and they describe torque-free motion; linearizing about steady spin about each principal axis reveals that rotation about the largest and smallest moment is stable while the intermediate axis is unstable -- the tennis-racket theorem falls straight out of them.
For a torque-free symmetric top (I_1 = I_2, N = 0), Euler's third equation gives omega_3_dot = 0, so the spin about the symmetry axis is constant; the first two then become omega_1_dot = -Omega omega_2 and omega_2_dot = +Omega omega_1 with Omega = (I_3 - I_1) omega_3 / I_1, describing a steady circular precession of omega in the body frame.
Euler's equations reduce a torque-free symmetric top to constant spin plus steady body-frame precession.
The omega components in Euler's equations are measured in the rotating body frame, not the lab frame. Solving them gives omega in the body frame; you still need the Euler angles to reconstruct the actual orientation in space.