Tensor of Inertia: Investigating a Puzzling Physics Problem

[ralqs]

This is a page from my textbook: http://tinypic.com/r/wbq4k9/7

It says that the products of inertia are unchanged to first order...but they clearly aren't! This is important, because the authors return to this idea repeatedly: that the tensor of inertia remains unchanged to first order for small angular displacements about the principal axes.

Consider: http://tinypic.com/r/b4g3s9/7

This is the author's derivation of Euler's equations. I would think that there should be a contribution from the product of inertia, but there apparently isn't.

So who's wrong: me, or the authors (along with, apparently, Euler and most of the physics community)? I suspect I know, but I can't find any fault with my logic.

 

[LightHero]

The answer here is that the authors and Euler are not wrong. The products of inertia do remain unchanged to first order for small angular displacements about the principal axes, but the derivation of Euler's equations does not account for those changes. This is because Euler's equations are derived using the principle of conservation of angular momentum, which does not depend on the products of inertia.