Euler's moment equations

Euler’s Moment Equations in terms of the principle axes is given by

In order to derive these equations, we start with the angular momentum of a rigid body

Since the vector is in the body frame and we want the Moment in an inertial frame we need to use the transport theorem since our body is in a non-inertial reference frame to express the derivative of the angular momentum vector in this frame. So the Moment is given by

Since we are assuming the inertia tensor is expressed using the principal axes of the body the products of inertia are zero

and using the shorter notation

Also since the moments of inertia are constant, when we take the derivative of the Inertia Tenser it is zero, so

Carrying out the matrix multiplication

after evaluating the cross product, we are left with adding the vectors

Once we add these vectors we are left with Euler’s Moment Equations