is only valid in an

Here, is the time derivative in the fixed frame, and the time derivative in the body frame. Combining Equations (501) and (502), we obtain

Now, in the body frame let and . It follows that , where , and are the principal moments of inertia. Hence, in the body frame, the components of Equation (503) yield

where . Here, we have made use of the fact that the moments of inertia of a rigid body are

Consider a rigid body which is constrained to rotate about a fixed
axis with *constant* angular velocity. It follows that
.
Hence, Euler's equations, (504)-(506), reduce to

These equations specify the components of the steady (in the body frame) torque exerted on the body by the constraining supports. The steady (in the body frame) angular momentum is written

It is easily demonstrated that . Hence, the torque is perpendicular to both the angular velocity and the angular momentum vectors. Note that if the axis of rotation is a principal axis then two of the three components of are zero (in the body frame). It follows from Equations (507)-(509) that all three components of the torque are zero. In other words,

Suppose that the body is *freely rotating*: *i.e.*, there are no 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 (511) and (512) can be written

(514) | |||

(515) |

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

(516) | |||

(517) |

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 makes some constant angle, , with the -axis, which implies that and . Hence, the components of the angular velocity vector are

(518) | |||

(519) | |||

(520) |

where

We conclude that, in the body frame, the angular velocity vector

(522) | |||

(523) | |||

(524) |

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 makes a constant angle with the symmetry axis, where

Note that the angular momentum vector, the angular velocity vector, and the symmetry axis all lie in the

Let us now consider the most general motion of a freely rotating
*asymmetric* rigid body, as seen in the body frame. Since a freely rotating
body experiences no external torques, its angular momentum vector
is a constant of the motion in the inertial fixed frame. In general, the direction of this vector varies with time in the non-inertial body frame, but its
length remains fixed. This can be seen from Equation (502):
if then the scalar product of this equation
with implies that . It follows from Equation (510) that

This constraint can also be derived directly from Euler's equations. We conclude that, in the body frame, the components of must simultaneously satisfy the two constraints (526) and (527). These constraints are the equations of two ellipsoids whose principal axes coincide with the principal axes of the body, and whose principal radii are in the ratio and , respectively. In general, the intersection of these two ellipsoids is a