Suppose now that the body-fixed frame is rotating in the space-fixed
frame with angular velocity
. Then the total torque on the rigid
body becomes [272]
If the body-fixed frame is aligned with the principal axes of rotation (§B.4.16), then the mass moment of inertia tensor is diagonal, say
so that the term
Substituting this result into Eq.(B.30), we obtain the following equations of angular motion for an object rotating in the body-fixed frame defined by its three principal axes of rotation:
These are call Euler's
equations:B.29Since these equations are in the body-fixed frame,
is the mass
moment of inertia about principal axis
, and
is the
angular velocity about principal axis
.