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#### 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 (B.30)

Similarly, the total external forces on the center of mass become 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 diag . In this frame, the angular momentum is simply so that the term becomes (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.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 .

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