Inertia Tensor as a Transformation

[uliuli]

Hi everyone,

I was thinking about the relationship between angular velocity and angular momentum for a rigid body: $I \omega = L$. In particular, I'm trying to gain a little bit of intuition as to what transformations $I$ can perform on $\omega$.

Let's use a reference frame at the center of mass of the body. We can rewrite the inertia tensor as: $I = R I_0 R^T$, where $I_0$ is the diagonalized inertia tensor and $R$ rotates the coordinate system from a principle-axes-aligned frame to the current frame. When I now apply $I$ to $\omega$, I am equivalently rotating $\omega$ by some arbitrary rotation, scaling each component by a positive (and in general, different for each component) number, and rotating by the inverse of the first rotation. In general, what does this concatenation of transformations give me?

Thanks!

 

[uliuli]

To reply to my own thread :):

The more I think about this, the less I think one can say about the family of transforms that $I$ can perform. I can 'design' $I_0$ to be arbitrarily close to a projection onto a cartesian axis, I can make $R$ any rotation I want... there is a lot of freedom.

About the only concrete restriction on the transform I've come up with is that $\omega$ and $L$ have a positive inner product: $\omega^T L = \omega^T R I_0 R^T \omega = (R^T \omega)^T I_0 (R^T \omega) = \tilde{\omega}^T I_0 \tilde{\omega} > 0$ due to the positive definiteness of $I_0$.

Ohh well!