A torque-free motion of a symmetric rigid-material model with constant angular momentum and energy whose body angular velocity is not constant: the invariants do not force a uniform rotation

Use the numerical SI convention of Two explicit free-rotation models checked: principal-axis spins, a non-uniform symmetric-top motion with its invariants, and the linearized frequencies of an asymmetric top: $t=T\mathrm{/}{T}_{\ast }$ with $T_{\ast }=1\mathrm{s}$, inertia entries in $M_{\ast }{D}_{\ast }^2$ with $M_{\ast }=1\mathrm{k}\mathrm{g}$ and $D_{\ast }=1\mathrm{m}$, ${\omega }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(T)=\omega (T\mathrm{/}{T}_{\ast })\mathrm{/}{T}_{\ast }$ and energy in $M_{\ast }{D}_{\ast }^2\mathrm{/}{T}_{\ast }^2$. Thus $t=\pi \mathrm{/}2$ means $T=\pi T_{\ast }\mathrm{/}2$; constancy or nonconstancy is preserved by these fixed nonzero scales. Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), carried from the supplier for its integral and quaternion constructions. No measured data enter this counterexample (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

Given: The model S of Two explicit free-rotation models checked: principal-axis spins, a non-uniform symmetric-top motion with its invariants, and the linearized frequencies of an asymmetric top, namely the rigid-material model with body inertia matrix $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(1,1,2)$ in the declared basis, used torque-free, and the path $\omega (t)=(\frac{1}{2}\cos t,\frac{1}{2}\sin t,1)$.

Model data: $\nu$ is the centred reference mass distribution of model S with body inertia matrix $I=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(1,1,2)$ in the declared orthonormal basis, and the model is nondegenerate (Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope clauses (i)-(ii), Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model clause (a), Two explicit free-rotation models checked: principal-axis spins, a non-uniform symmetric-top motion with its invariants, and the linearized frequencies of an asymmetric top).

The path $\omega$ is the body angular velocity of a torque-free motion of model S; along that motion $L_c$ is constant with $C=\parallel L_c{\parallel }_2^2=\frac{17}{4}$ by Torque-free motion of the rigid-material model: conservation of the angular momentum about the centre of mass, the energy-momentum invariants, the principal-axis equations and global existence of the spin dynamics clause 1, and the rotational kinetic energy is the constant $E=\frac{9}{8}$; and $\omega$ satisfies the torque-free Euler equations $l^{\mathrm{\prime }}=l\times (I^{-1}l)$ with $l=I\omega =(\frac{1}{2}\cos t,\frac{1}{2}\sin t,2)$ (Two explicit free-rotation models checked: principal-axis spins, a non-uniform symmetric-top motion with its invariants, and the linearized frequencies of an asymmetric top claim 2, Torque-free motion of the rigid-material model: conservation of the angular momentum about the centre of mass, the energy-momentum invariants, the principal-axis equations and global existence of the spin dynamics clauses 2-3).

Values of the path: $\omega (0)=(\frac{1}{2},0,1)$ and $\omega (\frac{\pi }{2})=(0,\frac{1}{2},1)$, using $\cos 0=\sin \frac{\pi }{2}=1$ and $\sin 0=\cos \frac{\pi }{2}=0$ with the sine and cosine of Sine and cosine defined by their real power series, the definition of $\pi$ in Pi as twice the smallest positive zero of cosine, and the exact quarter-turn values of Quarter-turn values and shifts by pi/2 and pi together with the zero values of The derivatives of sine and cosine are cosine and minus sine; the Euclidean norm of the difference is computed with The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ and Square roots exist: a unique $\sqrt{a}\ge 0$ with $(\sqrt{a}{)}^2=a$; the positives are $\{x^2:x\mathrm{\ne }0\}$.