Euler's equations with torque for the rigid-material model, the rotational power identity, the torque-free reduction and frame covariance

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), used only through the integrals of Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion; the theorems below neither use nor imply any other choice principle. Adopt a model of the rigid-material branch Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope with the external-torque branch Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass: a reference mass distribution $\nu$ with total mass $m$ and centred reference coordinates, a time window $I_0$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), an inertial frame, a motion $x(t,\rho )=c(t)+R(t)\rho$ with configuration path $(c,R)$ of class $C^2$ (Configuration, configuration space and orientation of a rigid-material model in an inertial frame), body angular velocity $\omega$ (Body and spatial angular velocity of a rigid-material motion, and the material velocity field) and body inertia matrix $I$ (Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model), and a continuous external torque datum $\tau :I_0\to {\mathbb{R}}^3$ with respect to which the adopted balance of Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass clause (iii) holds along the motion. Then:

1. Euler's equations with torque. Writing $l:=I\omega$, one has $l^{\mathrm{\prime }}=R^{\mathsf{T}}\tau -\omega \times I\omega$ and $I{\omega }^{\mathrm{\prime }}(t)+\omega (t)\times I\omega (t)=R(t{)}^{\mathsf{T}}\tau (t)(t\in I_0)\mathrm{.}$
2. Principal-axis components. If $(e_1,e_2,e_3)$ is a positively oriented orthonormal eigenbasis of $I$ with $I{e}_j=I_j{e}_j$ (Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values clause (a)) and $\omega ={\sum }_j{\omega }_j{e}_j$, $R^{\mathsf{T}}\tau ={\sum }_j{\tau }_j{e}_j$, then the equations are componentwise $I_1{\omega }_1^{\mathrm{\prime }}=(I_2-I_3){\omega }_2{\omega }_3+{\tau }_1,I_2{\omega }_2^{\mathrm{\prime }}=(I_3-I_1){\omega }_3{\omega }_1+{\tau }_2,I_3{\omega }_3^{\mathrm{\prime }}=(I_1-I_2){\omega }_1{\omega }_2+{\tau }_3\mathrm{.}$
3. Rotational power identity. The rotational kinetic energy $K_{\mathrm{r}\mathrm{o}\mathrm{t}}(t):=\frac{1}{2}\omega (t{)}^{\mathsf{T}}I\omega (t)$ is differentiable on $I_0$ with $\frac{d}{dt}{K}_{\mathrm{r}\mathrm{o}\mathrm{t}}(t)=\omega (t)\cdot R(t{)}^{\mathsf{T}}\tau (t)(t\in I_0);$ in particular, if $\tau \equiv 0$, then $K_{\mathrm{r}\mathrm{o}\mathrm{t}}$ is constant on $I_0$, and $K_{\mathrm{r}\mathrm{o}\mathrm{t}}$ coincides with the rotational term $\frac{1}{2}{\omega }^{\mathsf{T}}I\omega$ of the model's kinetic energy in clause 3 of Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion.
4. Torque-free reduction and nondegeneracy. Suppose $\tau \equiv 0$ and the model is nondegenerate in the sense of Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model clause (a), so that $I$ is invertible with $I^{-1}l=\omega$. Then, writing $\omega$ as a function of $l$, $l^{\mathrm{\prime }}=l\times (I^{-1}l),{\omega }^{\mathrm{\prime }}=F(\omega ):=-I^{-1}(\omega \times I\omega );$ the second form exhibits the torque-free body angular velocity as a solution of the autonomous field $F$ on ${\mathbb{R}}^3$.
5. Frame covariance. Let a second inertial frame be related to the first by $t^{\mathrm{\prime }}=t+b$, $x^{\mathrm{\prime }}=Qx+ut+a$ with $b\in \mathbb{R}$, $a,u\in {\mathbb{R}}^3$, $Q\in SO(3)$ (Inertial frames, stationary observers and the adopted Galilean transformations between frames clause (iv)), and let $\tilde{x}(t^{\mathrm{\prime }},\rho ):=Qx(t^{\mathrm{\prime }}-b,\rho )+u(t^{\mathrm{\prime }}-b)+a$, $\tilde{\tau }(t^{\mathrm{\prime }}):=Q\tau (t^{\mathrm{\prime }}-b)$ be the transformed motion and datum of Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass clause (vii). Then the balance holds in the second frame, and $I{\tilde{\omega }}^{\mathrm{\prime }}(t^{\mathrm{\prime }})+\tilde{\omega }(t^{\mathrm{\prime }})\times I\tilde{\omega }(t^{\mathrm{\prime }})=\tilde{R}(t^{\mathrm{\prime }}{)}^{\mathsf{T}}\tilde{\tau }(t^{\mathrm{\prime }})(t^{\mathrm{\prime }}\in I_0+b),$ where $\tilde{\omega }(t^{\mathrm{\prime }})=\omega (t^{\mathrm{\prime }}-b)$ and $\tilde{R}(t^{\mathrm{\prime }})=QR(t^{\mathrm{\prime }}-b)$; the body-frame torque is unchanged: $\tilde{R}(t^{\mathrm{\prime }}{)}^{\mathsf{T}}\tilde{\tau }(t^{\mathrm{\prime }})=R(t^{\mathrm{\prime }}-b{)}^{\mathsf{T}}\tau (t^{\mathrm{\prime }}-b)$.

The statement is conditional on the adopted model and torque balance; it asserts no force law for $\tau$, no conservation of $L_c$ and no measured value.

• What is a theorem and what is an assumption. The balance $L_c^{\mathrm{\prime }}=\tau$ is adopted in Adoption of the external-torque branch of the rigid-material model: declared torque data and the balance of angular momentum about the centre of mass; clauses 1-4 here are its consequences in the body frame and in the torque-free case, and clause 5 records that the formulation is consistent with the framework's frame changes. No clause produces $\tau$ from a force system.

• Where the nondegeneracy is used. Clauses 1-3 and 5 hold for every model of the branch, including degenerate ones; only clause 4 uses the invertibility of $I$, and the items that invert $I$ on this page state that hypothesis explicitly.

• The rotational power identity. Clause 3 says that the rate of change of the rotational kinetic energy equals the power $\omega \cdot R^{\mathsf{T}}\tau$ developed by the external torque; in the torque-free branch the rotational kinetic energy is a constant of the motion. This is a consequence of the adopted balance, not a separate assumption, and it does not by itself determine the motion.