Angular Momentum and Principal Axes of Inertia

[jpas]

Hi

I´m self-studying Alonso and Finn´s Mechanics and I have a question about this subject.

Let a body rotate about an arbitrary axis P having angular momentum $$\vec L$$.
Consider a referential with three perpendicular axes, $$X_{0} , Y_{0} , Z_{0}$$ , which are also principal axes of inertia.
The book says we can write $$\vec L$$ as

$$\vec L = \vec u_{x} I_1 \omega_{x0} + \vec u_{y} I_2 \omega_{y0} + \vec u_{z} I_3 \omega_{z0}$$

Does anybody how to derive this formula? The book usually explains things, but perhaps this is supost to be obvious.

By the way, I already know how to derive $$\vec L = I \vec \omega$$ for a body rotating about a principal axis of inertia but I don´t know how to derive this one.

Thank you​

 

[Galileo]

Generally, if a rigid body is rotating about an arbitrary axis, the angular momentum need not point in the same direction as the rotation axis, as it does when $\vec L = I \vec \omega$ (for rotation about a principal axis).

The relation between $\vec L$ and $\omega$ is still linear, and I is generally a tensor quantity (the inertia tensor).
An object always has three principal axes and in that coordinate system the inertia tensor is diagonal. This leads directly to:
$$\vec L = \vec u_{x} I_1 \omega_{x0} + \vec u_{y} I_2 \omega_{y0} + \vec u_{z} I_3 \omega_{z0}$$
It's really the only thing it can be if you know $\vec L = I \vec \omega$ holds for principal axes, there are three principal axes and the correspondence between $\vec w$ and $\L$ is linear.

 

[jpas]

Hello Galileo,

Thanks for the answer. Unfortunately, I couldn´t follow it because I don´t know what a tensor is. I´m still a high school student. I guess I´ll just have to use it without knowing how to derive it. which is something I really hate.