# Angular Momentum and Principal Axes of Inertia

 P: 45 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
 Sci Advisor HW Helper P: 2,002 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.