An alternative to the Euler equation of rigid body rotational dynamics

The Euler equation provides a convenient framework for studying the rotational dynamics of rigid bodies in solid mechanics. While this equation is written from the point of view of an inertial observer, it is implemented in a non-inertial ancillary coordinate system attached to the rigid body and the equations of the rotation are consequently expressed in this ancillary system. We examine how the rotational dynamics of rigid bodies can be described by the inertial observer directly in the inertial coordinate system (instead of employing an ancillary non-inertial frame), and derive the differential equations of the rotation in this inertial system. This approach can have advantages in situations where the rigid body has both translational motion in addition to rotational motion.