Rigid-body rotation is simplest in a coordinate frame attached to the body's principal axes. In that rotating frame, angular momentum has components $$L_i=I_i\omega_i.$$ The relation between a vector's derivative in an inertial frame and in the body frame is $$\left(\frac{d\mathbf L}{dt}\right){!\text{inertial}} =\left(\frac{d\mathbf L}{dt}\right){!\text{body}}+\boldsymbol\omega\times\mathbf L.$$ Using $d\mathbf L/dt=\boldsymbol\tau$ gives Euler's equations: $$I_1\dot\omega_1+(I_3-I_2)\omega_2\omega_3=\tau_1,$$ $$I_2\dot\omega_2+(I_1-I_3)\omega_3\omega_1=\tau_2,$$ $$I_3\dot\omega_3+(I_2-I_1)\omega_1\omega_2=\tau_3.$$

Even with zero external torque, the body-frame components of $\boldsymbol\omega$ need not be constant. The angular-momentum vector is fixed in inertial space while the body rotates beneath it.

For torque-free rotation exactly about a principal axis, two components vanish and the remaining $\omega_i$ is constant. Small disturbances reveal an important stability result: rotation about the axes with the smallest or largest principal moment is stable, while rotation about the intermediate principal axis is unstable. This is the tennis-racket theorem.

Euler's equations explain why three-dimensional rigid-body motion is richer than $\tau=I\alpha$. They connect the inertia tensor, rotating coordinates and conservation of angular momentum, forming the basis for gyroscopes, spacecraft attitude dynamics and molecular rotation.