If $z_G$ is a principal axis that goes through the center of mass, are every other axes $z$ parralel to $z_G$ also principal axes?

Question

I know that : If $y$ is a principal axis $\iff$ you can express the angular momentum with respect to a point $Y$ on that axis (if the solid is rotating around that axis of course) with the formula : $\vec{L_Y} = I_y\vec{\omega}$, where $I_y$ is the moment of inertia with respect to the axis $y$ (else, you would have to use a tensor of inertia)(at least, that's what I learnt at school).

Now consider that $z_G$ is a principal axis that goes through the center of mass $G$ of a solid (of mass $m$) and let $I_{z_G}$ be the moment of inertia with respect to that axis. The parallel axis theorem gives us the moment of inertia with respect to any axes $z$ parallel to $z_G$ : $I_z = I_{z_G} + md^2$, where $d$ is the perpendicular distance between the two axes. So if the solid is rotating around one of those axes $z$ with an angular velocity $\omega$, the angular momentum with respect to a point $Z$ on $z$ would be $\vec{L_Z} = I_z\vec{\omega}$, which would mean $z$ is a principal axis of inertia. Am I right ?

Thanks.

Answer 1

Consider the tensor form of the Huygens-Steiner theorem, i.e. $$I_\Omega = I_{\text{CM}} + M\Vert\mathbf a\Vert^2(I-\hat{\mathbf a}\otimes\hat{\mathbf a}),$$ where $\mathbf a$ is the displacement vector between the centre of mass and the new pole $\Omega$. Suppose that $(\mathbf e_1,\mathbf e_2,\mathbf e_3)$ is an orthonormal system of principal axes with respect to the centre of mass. In order for $\mathbf e_i$, for a fixed $i$, to remain a principal axis, we have to ensure that $$(\mathbf e_i,I\mathbf e_j) = (\mathbf e_j,I\mathbf e_i) = 0$$ for any $j\neq i$ (the equality in the middle follows from the symmetry of the tensor $I$). Hence you have the conditions $$(\mathbf e_i\cdot\hat{\mathbf a})(\mathbf e_j\cdot\hat{\mathbf a}) = 0,\qquad\forall j\neq i$$ which implies either $$\mathbf e_i\cdot\hat{\mathbf a} = 0$$ or $$\mathbf e_j\cdot\hat{\mathbf a} = 0$$ for any $j\neq i$. Therefore the new pole must be a translation of the centre of mass either on the plane perpendicular to $e_i$ or along the direction $\mathbf e_i$ in order for $\mathbf e_i$ to remain a principal axis of inertia.