Euler's equations

is only valid in an inertial frame. However, we have seen that is most simply expressed in a frame of reference whose axes are aligned along the principal axes of rotation of the body. Such a frame of reference rotates with the body, and is, therefore, non-inertial. Thus, it is helpful to define two Cartesian coordinate systems with the same origins. The first, with coordinates , , , is a fixed inertial frame; let us denote this the

Here, is the time derivative in the fixed frame, and the time derivative in the body frame. Combining Equations (8.23) and (8.24), we obtain

In the body frame, let

(8.26) |

where , and are the principal moments of inertia. Hence, in the body frame, the components of Equation (8.25) yield

where . Here, we have made use of the fact that the moments of inertia of a rigid body are constant in time in the co-rotating body frame. The preceding three equations are known as

Consider a body that is freely rotating; that is, in the absence of external torques. Furthermore, let the body be rotationally symmetric about the -axis. It follows that . Likewise, we can write . In general, however, . Thus, Euler's equations yield

Clearly, is a constant of the motion. Equation (8.30) and (8.31) can be written

(8.33) | ||||||

and | (8.34) |

where . As is easily demonstrated, the solution to these equations is

(8.35) | ||||||

and | (8.36) |

where is a constant. Thus, the projection of the angular velocity vector onto the - plane has the fixed length , and rotates steadily about the -axis with angular velocity . It follows that the length of the angular velocity vector, , is a constant of the motion. Clearly, the angular velocity vector subtends some constant angle, , with the -axis, which implies that and . Hence, the components of the angular velocity vector are

(8.37) | ||||||

(8.38) | ||||||

and | (8.39) |

where

We conclude that, in the body frame, the angular velocity vector precesses about the symmetry axis (i.e., the -axis) with the angular frequency . Now, the components of the angular momentum vector are

(8.41) | ||||||

(8.42) | ||||||

and | (8.43) |

Thus, in the body frame, the angular momentum vector is also of constant length, and precesses about the symmetry axis with the angular frequency . Furthermore, the angular momentum vector subtends a constant angle with the symmetry axis, where

The angular momentum vector, the angular velocity vector, and the symmetry axis all lie in the same plane; that is,