#### Euler's Equations for Rotations in the Body-Fixed Frame

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 [270]

Similarly, the total external forces on the center of mass become

*cf.*Eq.(B.15))

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.29}Since 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 .

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Examples

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Angular Motion in the Space-Fixed Frame