# Does an irregular rigid body can only rotate in three directions?

Suppose that at a certain instant the angular momentum with respect to the center of mass is not parallel to the angular velocity. Does this necessarily imply that the angular momentum is rotating around the axis of rotation? If so, then an isolated body must necessarily have angular momentum with respect to the center of mass parallel angular velocity (otherwise the angular momentum varies and the system is not isolated).  However, this gives me a perplexity: an irregular rigid body has only 3 axes such that $$\vec{L}_{cm}$$ and $$\vec{\omega}$$ are parallel. Does this mean that the irregular rigid body can only rotate in three directions? It seems completely absurd to me, but where is the mistake?

A general rigid body has 3 principal axis. Suppose $$I_1. If it rotates around $$I_1$$ or $$I_3$$ without external torques, the angular velocity doesn't change and is parallel to the angular momentum. Theoretically the same happens with rotation around $$I_2$$, but in a unstable mode. Any deviation and it starts to wobble (tennis racket effect).

The angular momentum is conserved if there is no applied torques, but the angular velocity can varies chaotically. https://www.youtube.com/watch?v=WvrbejgDL3A

• So the answer to my question is that the non-parallelism does not create problems because, if the system is not free, the two vectors can rotate in many ways, if the system is "free" from forces (like "free faling" asteroids) it can only rotate the angular velocity. However a free body can have a simple rotation (I mean in turn to a specific axis without the rotation axis changing) only if rotating around a principal axis of inertia. Right?
– user291161
May 4, 2022 at 8:53
• Yes, I think you are correct for a free body (no torques) with $\vec \omega$ along a principal axis. Note: A rigid body can be forced (by applying torque) to rotate about a fixed axis that is not a principal axis. In this case $\vec \omega$ is fixed in direction, but $\vec L$ is not fixed in direction. The body is not dynamically balanced. May 4, 2022 at 11:52

You state "an irregular rigid body has only 3 axes such that $$\vec{L}_{cm}$$ and $$\vec{\omega}$$ are parallel." I presume you mean the principal axes?

In terms of the Cartesian principal axes, the angular momentum is $$\vec L = I_1\omega_1\hat i + I_2\omega_2\hat j + I_3\omega_3\hat k$$ where the I's are the moments of inertia and $$\hat i$$, etc. are axis unit vectors. The angular velocity is $$\vec \omega = \omega_1\hat i + \omega_2\hat j + \omega_3\hat k$$. $$\vec L$$ and $$\vec \omega$$ are not necessarily parallel using the principal axes.

If the axis of rotation is any principal axis, say axes 3, then $$\vec L = I_3\omega \hat k$$ where $$\vec \omega = \omega_3\hat k$$, then $$\vec L$$ and $$\vec \omega$$ are parallel, but this is a special case. It is not required that $$\vec \omega$$ lie along a principal axis.

• Yes I mean principal axes
– user291161
May 4, 2022 at 8:54