= Normalized intermediate-axis model
{title2=$(I_1,I_2,I_3)=(1,2,3)$}
Choose an inertia unit $I_*$ and an <angular speed> unit $\Omega_*>0$, so physical <principal moments of inertia> are $I_*(1,2,3)$, physical <angular velocity> is $\Omega_*\boldsymbol\omega$, and normalized <time> is $t=\Omega_*t_{\mathrm{physical}}$. The <Euler equations for a torque-free rigid body> become
$$
\dot\omega_1=-\omega_2\omega_3,\qquad
\dot\omega_2=\omega_3\omega_1,\qquad
\dot\omega_3=-\frac13\omega_1\omega_2.
$$
This is a physically admissible ideal planar <mass> distribution: $I_3=I_1+I_2$ is the <perpendicular axis theorem>. A book of finite thickness need not have these exact ratios.
For an original comparison, start with <angular velocity> $(1,\varepsilon,0)$, $(\varepsilon,1,0)$ or $(\varepsilon,0,1)$, respectively. With $\varepsilon=0.01$, the transverse <Euclidean norm> stays small about axes one and three, but becomes order one about axis two. For the intermediate-axis start, its <linearization> gives
$$
\omega_1\simeq\varepsilon\cosh(t/\sqrt3),\qquad
\omega_3\simeq-\frac{\varepsilon}{\sqrt3}\sinh(t/\sqrt3),\qquad
\omega_2\simeq1.
$$
The <hyperbolic functions> contain a growing <exponential function> with rate $\lambda_2=1/\sqrt3$. Reducing $\varepsilon$ tenfold delays reaching the same small transverse amplitude by approximately $\sqrt3\log 10\simeq3.99$ normalized <time> units, once the growing term dominates. That delays the instability. Once the transverse <angular velocity> is comparable to the main component, the <linearization> ceases to describe the motion.
\Image[/intermediate-axis-perturbation-growth.png]
{title=Small <perturbations> about the three <principal axes>}
{description=Original <Runge-Kutta method> calculations for the <normalized intermediate-axis model>, with initial <angular velocity> $(1,0.01,0)$, $(0.01,1,0)$ and $(0.01,0,1)$. Left: transverse <Euclidean norm> relative to each initial <principal axis>. Right: full <Euler equations for a torque-free rigid body> compared with the displayed <linearization>, also using $(0.001,1,0)$. Dashed curves are linear predictions. These are model curves, not measured throws.}
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