= Torque-free tumbling
{title2=$c_2=\hat e_2\cdot\hat L$}
A useful readout of a tumble is the alignment of the intermediate <principal axis> $\hat e_2$ with the conserved <angular momentum> direction $\hat L$. In the <principal body frame>, write $L_i=I_i\omega_i$ and $L=|\mathbf L|$. Then
$$
c_2(t)=\hat e_2(t)\cdot\hat L=\frac{L_2(t)}L.
$$
A change from $c_2\simeq+1$ to $c_2\simeq-1$ means that the directed body axis has turned nearly $180^\circ$ relative to $\mathbf L$. It does not mean that $\mathbf L$ reversed in the <fixed space frame>. The unsigned line through that <principal axis> hides this distinction, so a marked end matters in an actual demonstration.
For the <normalized intermediate-axis model> with $\boldsymbol\omega(0)=(0.01,1,0)$, the <rotational kinetic energy> and squared <angular momentum> are $E=1.00005$ and $L^2=4.0001$. Hence $L^2-2EI_2=-0.0001$: this initial condition lies close to, but off, the <intermediate-axis separatrix>. An independent calculation evolves $\dot{\mathbf L}_{\mathrm{body}}=\mathbf L_{\mathrm{body}}\times I^{-1}\mathbf L_{\mathrm{body}}$ and a <rotation matrix> taking body components to space components. The intermediate <principal axis> crosses $c_2=0$ at normalized <times> approximately $10.3773$ and $31.1319$, alternating long nearly aligned intervals with quicker reversals.
\Image[/intermediate-axis-body-alignment-and-phase.png]
{title=Torque-free reversals of a nearly aligned body axis}
{description=Original independent <Runge-Kutta method> calculation in <angular momentum> and <rotation matrix> variables, starting with $(L_1,L_2,L_3)=(0.01,2,0)$ in the <normalized intermediate-axis model>. Left: body-frame orbit projected onto $(L_1/L,L_3/L)$; dashed lines are the <intermediate-axis separatrix> projections $L_3=\pm\sqrt3L_1$. The red point is the initial condition; the inset resolves its separation from those lines. Right: $c_2=L_2/L$, with dotted lines at the first two zero crossings. The full <angular momentum> vector remains fixed in space.}
Both calculations use step size $0.005$ without rescaling either invariant. Over $0\le t\le50$, the tumbling calculation changes <rotational kinetic energy> and squared <angular momentum> by less than $4\times10^{-13}$ relative to their initial values. Reconstructing $\mathbf L$ in the <fixed space frame> gives relative error below $3\times10^{-12}$. Halving the step changes the body components and <rotation matrix> entries by less than $3\times10^{-10}$. These checks distinguish physical tumbling from numerical drift. On the exact <intermediate-axis separatrix>, approach to an intermediate-axis rotation takes infinite <time>; the recurrent off-separatrix example here is different.
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