The rotational part

The six equations derived so far are valid even if ${\cal A}$ rotates with respect to the inertial frame. They are just the translational part of the motion. The rotational part can be decoupled from the translational part by using the translating frame. All translational aspects of the motion have already been considered. Imagine that ${\cal A}$ is only rotating while its center of mass remains fixed. Once the rotational part of the motion has been determined, it can be combined with the translational part by simply viewing things from the inertial frame. Therefore, the motion of ${\cal A}$ is now considered with respect to the translating frame, which makes it appear to be pure rotation.

Unfortunately, characterizing the rotational part of the motion is substantially more complicated than the translation case and the 2D rotation case. This should not be surprising in light of the difficulties associated with 3D rotations in Chapters 3 and 4.

Following from Newton's second law, the change in the moment of momentum is

$${N}(u) = {d{E}\over dt} .$$ (13.82)

The remaining challenge is to express the right-hand side of (13.82) in a form that can be inserted into the state transition equation.

Figure 13.11: The angular velocity is defined as a rotation rate of the coordinate frame about an axis.