Principal Axes of Rotation

A principal axis of rotation (or principal direction) is an eigenvector of the mass moment of inertia tensor (introduced in the previous section) defined relative to some point (typically the center of mass). The corresponding eigenvalues are called the principal moments of inertia. Because the moment of inertia tensor is defined relative to the point $\underline{x}=\underline{0}$ in the space, the principal axes all pass through that point (usually the center of mass).

As derived above (§B.4.14), the angular momentum vector is given by the moment of inertia tensor times the angular-velocity vector:

$$\underline{L}\eqsp \mathbf{I}\,\underline{\omega}$$

If $\underline {\omega }$ is an eigenvector of $\mathbf{I}$ , then we have

$$\underline{L}\eqsp \mathbf{I}\,\underline{\omega}\eqsp \lambda\underline{\omega}$$

where the (scalar) eigenvalue $\lambda$ is called a principal moment of inertia. If we set the rigid body assocated with $\mathbf{I}$ rotating about the axis $\underline {\omega }$ , then $\lambda$ is the mass moment of inertia of the body for that rotation. As will become clear below, there are always three mutually orthogonal principal axes of rotation, and three corresponding principal moments of inertia (in 3D space, of course).