# Intuition Behind Intermediate Axis Theorem in an Ideal Setting

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• deuteron
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

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deuteron said:
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.

Filip Larsen and berkeman
Juanda said:
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.

deuteron said:
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.

vanhees71 and Filip Larsen
deuteron said:
Thank you! I just watched it,
Here is a previous thread on that Veritasium video, where some of its shortcomings are discussed
Here a continuation with more videos:

I liked this one in particular:

deuteron said:
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.

deuteron said:
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).

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Swamp Thing, Juanda and vanhees71
A.T. said:
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.

## What is the Intermediate Axis Theorem?

The Intermediate Axis Theorem, also known as the Tennis Racket Theorem, describes the rotational behavior of a rigid body with three distinct principal moments of inertia. It states that rotation around the axis with the intermediate moment of inertia is unstable, while rotation around the axes with the largest and smallest moments of inertia is stable.

## Why is the rotation around the intermediate axis unstable?

Rotation around the intermediate axis is unstable due to the nature of the distribution of mass and the resulting moments of inertia. Small perturbations in the rotational motion around this axis can grow over time, leading to a tumbling motion. This instability arises because the intermediate axis does not provide the same level of resistance to changes in rotational motion as the axes with the largest and smallest moments of inertia.

## Can you provide an intuitive example of the Intermediate Axis Theorem?

An intuitive example is the rotation of a tennis racket. When you toss a tennis racket into the air with a spin, it tends to flip unpredictably if spun around its intermediate axis (the handle). However, if you spin it around the axis parallel to the handle or the axis perpendicular to the face of the racket, it rotates smoothly and predictably.

## How does the Intermediate Axis Theorem apply to real-world objects?

The Intermediate Axis Theorem applies to any rigid body with three distinct principal moments of inertia, including satellites, space stations, and tools in microgravity environments. Understanding this theorem helps in designing and controlling the orientation and stability of such objects during rotation.

## What is the mathematical basis for the Intermediate Axis Theorem?

The mathematical basis for the Intermediate Axis Theorem lies in the equations of rotational motion, particularly Euler's equations for a rigid body. These equations describe how the angular velocity components evolve over time. For the intermediate axis, small deviations in angular velocity components can lead to exponential growth in the deviation, resulting in instability, as shown through linear stability analysis.

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