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

The hypotheses of [L2] are met: $\nu$ is a centred mass distribution with finite second moment by [L1] and is not supported on any line through the origin by the nondegeneracy hypothesis; $I_0$ has at least two points; $c$ is differentiable and $R$ twice differentiable because the configuration path is of class $C^2$; and the balance hypothesis $(Rl{)}^{\mathrm{\prime }}=\tau$ holds with $\tau \equiv 0$ because $Rl=L_c$ by [L1] and $L_c^{\mathrm{\prime }}=0$ by [L3]. Hence clause 2 of [L2] gives $l^{\mathrm{\prime }}=l\times (I^{-1}l)$ and the constancy of $C$ and $E$, and clause 3 of [L2] gives their principal-basis expressions and the component equations.

By [L3] the continuous function $L_c$ has zero derivative on $I_0$, so it is constant on $I_0$; hence $\parallel L_c{\parallel }_2$ is constant, and $\parallel L_c(t){\parallel }_2^2=⟨l(t),l(t)⟩=C$ by [L1]. This is clause 1.

The field $F(l)=l\times (I^{-1}l)$ is locally Lipschitz in the state variable: by bilinearity and additivity of the cross product [L6], for all $x,y\in {\mathbb{R}}^3$ one has $F(x)-F(y)=(x-y)\times (I^{-1}x)+y\times (I^{-1}(x-y))$, so with the bound $\parallel I^{-1}z{\parallel }_2\le \parallel z{\parallel }_2\mathrm{/}{I}_{\min }$ valid in the principal basis [L4] and the norm bound of [L6], $\parallel F(x)-F(y){\parallel }_2\le \parallel x-y{\parallel }_2\parallel I^{-1}x{\parallel }_2+\parallel y{\parallel }_2\parallel I^{-1}(x-y){\parallel }_2\le (\parallel x{\parallel }_2+\parallel y{\parallel }_2)\parallel x-y{\parallel }_2\mathrm{/}{I}_{\min }$; on the ball $\parallel z{\parallel }_2\le r$ this is at most $2r\parallel x-y{\parallel }_2\mathrm{/}{I}_{\min }$, a finite state-Lipschitz constant on every compact cylinder.

Writing $\omega ={\sum }_j{\omega }_j{e}_j$ in the principal basis of [L4], one has ${\omega }^{\mathsf{T}}I\omega ={\sum }_j{I}_j{\omega }_j^2\ge I_{\min }{\sum }_j{\omega }_j^2=I_{\min }\parallel \omega {\parallel }_2^2$, so $\parallel \omega (t){\parallel }_2^2\le 2E\mathrm{/}{I}_{\min }$ with $E=\frac{1}{2}{\omega }^{\mathsf{T}}I\omega$ constant by step 1.1 and $I_{\min }>0$ by [L4]; together with $l=I\omega$ this bounds $\parallel l{\parallel }_2^2=C$ on $I_0$. The identification of $E$ with the rotational term of the kinetic energy is clause 3 of [L5]. This is clause 3.

Every solution of $l^{\mathrm{\prime }}=F(l)$ has a unique maximal extension defined on all of $\mathbb{R}$: by [L7] and step 1.3 a unique maximal solution exists through every datum, with open domain $(a,b)$; along it $\frac{d}{dt}\parallel l(t){\parallel }_2^2=2⟨l^{\mathrm{\prime }}(t),l(t)⟩=2⟨l(t)\times (I^{-1}l(t)),l(t)⟩=0$ because the cross product is orthogonal to each factor [L6], so by [L3], applied to this continuous differentiable scalar on $(a,b)$, $\parallel l(t){\parallel }_2^2\equiv \parallel l(t_0){\parallel }_2^2$ for any datum time $t_0\in (a,b)$ and its values over $(a,b)$ lie in the sphere $\{z:\parallel z{\parallel }_2=\parallel l(t_0){\parallel }_2\}\subseteq {\mathbb{R}}^3$, which is closed and bounded and therefore compact by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line. If $b<+\mathrm{\infty }$, then near $b$ the solution stays in that compact subset of the state domain, and clause 2 of [L7] extends it past $b$, contradicting maximality; hence $b=+\mathrm{\infty }$. If $a>-\mathrm{\infty }$, put $z(s):=l(-s)$ on $(-b,-a)$. By the chain rule [L8], $z\mathrm{\prime }(s)=-F(z(s))$. The field $-F$ is continuous and locally Lipschitz with the same constants as $F$, and $z$ stays in the same compact sphere near its finite right endpoint $-a$. Applying the autonomous compact continuation of [L7] to $-F$ extends $z$ past $-a$; reflecting this extension back extends $l$ past $a$, contradicting maximality. Hence $a=-\mathrm{\infty }$. Every original solution is a restriction of this unique global maximal solution by the uniqueness assertion of [L7].

The $l$-path of a torque-free motion is a solution of $l^{\mathrm{\prime }}=F(l)$ on $I_0$ by step 1.1, hence by step 2.2 it is the restriction of a solution defined on all of $\mathbb{R}$ (uniqueness), its maximal extension satisfies $\parallel l(t){\parallel }_2=\parallel l(t_0){\parallel }_2$ for any datum time $t_0$ and every $t$, and it takes values in $\mathrm{\Lambda }(C,E)$, which is compact as the intersection of the closed sphere and the closed level set $\{z:⟨z,I^{-1}z⟩=2E\}$ and nonempty because it contains $l(t_0)$. This is clause 4.