 Euler's equations (rigid body dynamics)

In principal body coordinates, Euler's equations express angular momentum balance as $I\dot{\omega }+\omega \times (I\omega )=\tau$, with applied torque $\tau$. Setting the torque to zero gives the following torque-free specialization.

Euler's equations in the context of rigid body dynamics describe the rotation of a rigid body about a fixed point. When dealing with the motion of a rigid body, it's often useful to consider it as a system of particles and apply Newtonian mechanics. However, for rotating bodies, Euler's equations provide a more efficient approach.