Principal axes of rotation

for .

The directions of the three mutually orthogonal unit vectors
define the three so-called *principal axes
of rotation* of the rigid body under investigation. These axes are special because when the body rotates about
one of them (i.e., when
**
**
is parallel to one of them) the angular momentum vector
becomes parallel to the angular velocity vector
**
**
.
This can be seen from a comparison of Equation (8.13) and Equation (8.19).

Suppose that we reorient our Cartesian coordinate
axes so they coincide with the mutually orthogonal principal axes of rotation. In this new reference frame, the eigenvectors of
are the unit vectors,
,
, and
, and the eigenvalues
are the moments of inertia about these axes,
,
, and
, respectively. These latter quantities are referred to as the
*principal moments of inertia*.
The products of inertia are all zero in the new
reference frame. Hence, in this frame, the moment
of inertia tensor takes the form of a diagonal matrix:

(8.20) |

Incidentally, it is easy to verify that , , and are indeed the eigenvectors of this matrix, with the eigenvalues , , and , respectively, and that

When expressed in our new coordinate system, Equation (8.13) yields

(8.21) |

whereas Equation (8.18) reduces to

(8.22) |

In conclusion, there are many great simplifications to be had by choosing a coordinate system whose axes coincide with the principal axes of rotation of the rigid body under investigation.