#### 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 frame^{B.28}in which the top is spinning.

**Next Section:**

Angular Motion in the Space-Fixed Frame

**Previous Section:**

Positive Definiteness of the Moment of Inertia Tensor