# What is the physical meaning of the principal axes of inertia?

What is the physical meaning of the principal axes of inertia? I used to think that the axes of inertia are, in some sense, the only axes about which the body can rotate without the angular momentum "slipping" to other axes. In other words, I thought they are the only axes around which the body can have a motion of simple rotation about an axis, and any attempt to rotate the body around a non-principal axis wil result in a complex motion, consisting of a superposition of rotations around more than one principal axes (to put it differently, I thought principal axes are analogous to normal modes in vibrating systems, where the system can vibrate in a single frequency only if it's a a normal mode).

However, torque-free precession - or the general motion of a symmetric, non-spherical top - shows this is not the case. A symmetric, non-spherical top in general has a spin around its top axis, plus an additional spin around an axis that can make an angle with the top axis - and is, in general, non-principal. So what do the principal axes of inertia mean? What is their physical interpretation (since every book I read just says they are the eigenspaces of the inertia tensor, which is a statement that lacks any physical meaning)?

One way to say it: you do not need to apply any external torque to keep an object rotating about a principal axis. To maintain constant angular velocity around any axis through the center of mass which cannot be defined as a principle axis, torque is required.

Consider an ideal barbell, with equal point masses separated by a massless rod. You can make it rotate with constant angular velocity about any axis you like; for example, it could be spun about an axis through the midpoint of the rod which makes an angle of $45^\circ$ with the rod. Each of the masses would require a centripetal force to keep it moving in its circle. Since the circles are not coplanar, this pair of forces constitute a torque. As soon as you stop supplying this torque, the barbell will switch to rotating around an axis perpendicular to the bar (which is a principle axis).

If an object is rotating about one of its principal axes, then the direction of its angular velocity coincides with the direction of its angular momentum.

Because $\vec L = I \vec\omega$, $\vec \omega$ being an eigenvector of $I$ is equivalent to $\vec L$ being parallel to $\vec\omega$.

• Yeah, I know that... This still doesn't have a clear physical interpretation, since angular momentum itself is a rather abstract concept without a clear physical picture. Feb 28, 2018 at 17:16
• Remember that angular momentum is conserved, not angular velocity. If the two do not coincide, then the angular velocity will generically change with time (since the inertia tensor of a rotating object is typically time-dependent for an external observer). This is the torque-free precession you mentioned. Feb 28, 2018 at 17:27
• In other words, rotation about a principal axis implies that the angular velocity is constant. Feb 28, 2018 at 17:28

Consider the rotation of a rigid body in the absence of any external forces or torques. If the angular momentum satisfies $$\vec{L} \propto I \vec{L},$$ then the angular momentum $\vec{L}$ will remain constant in time (in the frame of the body). If $\vec{L}$ is the eigenvector with the largest or smallest eigenvalue, this is a stable equilibrium. For the intermediate eigenvalue, it's unstable.

The simplest demonstration of this is to spin a book (or a cell phone if you're careful) about each of its primary axes, which are exactly what you would guess they are. The book stably rotates about two of the axes (the one pointing out of the pages of the book and the one pointing up along the page), but not the third (the one pointing across the page).

The non-principal direction of rotation of a symmetric non-spherical top comes from its symmetry. Because two of the eigenvalues are the same, there are no longer six fixed points (two for each axis) for the evolution of $\vec{L}$. Instead, there are two, corresponding to rotation about the primary axis. If $\vec{L}$ is not at one of these fixed points, it traces a circle. Another way of saying this is that the magnitude of the projection of $\vec{L}$ onto the two-dimensional eigenspace of $I$ is a constant.