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deuteron

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- 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

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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