#### Body-Fixed and Space-Fixed Frames of Reference

Rotation is always about some (instantaneous) axis of rotation that is free to change over time. It is convenient to express rotations in a coordinate system having its origin ( ) located at the center-of-mass of the rigid body (§B.4.1), and its coordinate axes aligned along the principal directions for the body (§B.4.16). This body-fixed frame then moves within a stationary space-fixed frame (or star frame'').

In Eq.(B.29) above, we wrote down Newton's second law for angular motion in the body-fixed frame, i.e., the coordinate system having its origin at the center of mass. Furthermore, it is simplest ( is diagonal) when its axes lie along principal directions (§B.4.16).

As an example of a local body-fixed coordinate system, consider a spinning top. In the body-fixed frame, the vertical'' axis coincides with the top's axis of rotation (spin). As the top loses rotational kinetic energy due to friction, the top's rotation-axis precesses around a circle, as observed in the space-fixed frame. The other two body-fixed axes can be chosen as any two mutually orthogonal axes intersecting each other (and the spin axis) at the center of mass, and lying in the plane orthogonal to the spin axis. The space-fixed frame is of course that of the outside observer's inertial frameB.28in which the top is spinning.

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Angular Motion in the Space-Fixed Frame
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Positive Definiteness of the Moment of Inertia Tensor