Intuition Behind Intermediate Axis Theorem in an Ideal Setting

[deuteron]

TL;DR Summary

What is the physical intuition behind the intermediate axis theorem? Is the rotation about the intermediate axis unstable even under an ideal condition, if yes, physically why?

For a rigid body with three principal axis with distinct moments of inertia, would the principal axis with the intermediate moment of inertia still be unstable in ideal conditions, e.g. no gravity, no friction etc.? From the mathematical derivation I deduce that it should be unstable, since we make no assumptions about the external conditions to derive the intermediate axis theorem, but physically it makes no sense why the intermediate axis is unstable.

With mathematical derivation, I mean the following:
For ##I_1<I_2<I_3##, consider Euler's equations of rotation:
$$\begin{align}
I_1\dot\omega_1=(I_2-I_3)\omega_2\omega_3\\
I_2\dot\omega_2=(I_3-I_1)\omega_3\omega_1\\
I_3\dot\omega_3=(I_1-I_2)\omega_1\omega_2
\end{align} $$
Assuming an initial rotation along the axis with ##I_2## and therefore assuming ##\omega_1=\omega_3=0##, we get:

$$\begin{align}
\dot\omega_2=0\quad\Rightarrow\omega_2=\text{const.}\\
\Rightarrow\begin{matrix} \dot\omega_1=\frac{I_2-I_3}{I_1}\omega_2\omega_3=K_1\omega_3\\ \dot\omega_3=\frac {I_1-I_2}{I_3}\omega_1\omega_2=K_3\omega_1\end{matrix}\\ \Rightarrow\begin{matrix}\ddot\omega_1=K_1\dot\omega_3=K_1K_2\omega_1=\lambda\omega_1\\ \ddot\omega_3=K_3\dot\omega_1=K_3K_1\omega_3=\lambda\omega_3\end{matrix}\quad\text{with}\quad K_1K_3>0\\ \Rightarrow\begin{matrix} \omega_1=c_1e^{\sqrt{\lambda}t}+c_2e^{\sqrt{\lambda}t}\\ \omega_3=c_1e^{\sqrt{\lambda}t}+c_2e^{\sqrt{\lambda}t}\end{matrix}
\end{align}$$

which means that the angular velocities on both the first and the third axes tend to exponentially grow with time, until they are large enough to cause rotations, which causes unstability of the intermediate axis

 

[Juanda]

TL;DR Summary: What is the physical intuition behind the intermediate axis theorem? Is the rotation about the intermediate axis unstable even under an ideal condition, if yes, physically why?

This video is trying to provide exactly that. To provide an intuitive explanation of it. Or that's the goal as the creator says. I hope it helps.

 

[Swamp Thing]

A nice demo of the phenomenon:

https://www.physicsforums.com/threads/a-flip-of-the-earth-on-its-axis.994061/post-6398725

(Though it doesn't address the OP's request for an explanation, it's fun to watch, so...)

 

[deuteron]

This video is trying to provide exactly that. To provide an intuitive explanation of it. Or that's the goal as the creator says. I hope it helps.

Thank you! I just watched it, however, here the intermediate axis theorem is derived intuitionally after assuming a small deviation from the axis with the intermediate moment of inertia. What confuses me is whether the rotation would still be unstable if there were no initial deviations from the ##2##nd axis to begin with. Mathematically, I think, yes; but I haven't seen an explanation of the theorem that doesn't assume the initial deviation.

 

[Swamp Thing]

whether the rotation would still be unstable if there were no initial deviations from the 2nd axis to begin with.

It would be metastable, like an idealized pencil standing on its sharpened point.

 

[A.T.]

Thank you! I just watched it,

Here is a previous thread on that Veritasium video, where some of its shortcomings are discussed
https://www.physicsforums.com/threads/intermediate-axis-theorem-intuitive-explanation.977692/
Here a continuation with more videos:
https://www.physicsforums.com/threads/intermediate-axis-theorem-intuitive-explanation.1006800/

I liked this one in particular:

however, here the intermediate axis theorem is derived intuitionally after assuming a small deviation from the axis with the intermediate moment of inertia.

In this context, "instability" means that small deviations get larger.

What confuses me is whether the rotation would still be unstable if there were no initial deviations from the ##2##nd axis to begin with.

In an idealized case, where it rotates exactly around the 2nd axis, it would not deviate due to symmetry (how would it know which way to deviate).

 

[Juanda]

I liked this one in particular:

That was a nice watch. Thanks for sharing.
Often looking at the same thing from a different perspective helps in understanding it.
Adding just another perspective that could be interesting, here is a video that, instead of using rigid body equations, builds a "rigid" body using point masses and very stiff springs.


Rigid body equations can be really hard (at least for me they are). I think it's easier to grasp spring forces that make the shape of the body almost constant if the springs are stiff enough. Trying to solve such a system would be awful in the past but it's easily achievable for computers and, in my opinion, even simpler to program it into code.