Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), used only to make the integrals below finite. Adopt the model and notation of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope: a reference mass distribution $\nu$ with total mass $m$ and centred reference coordinates, a fixed inertial frame, and a motion $x(t,\rho )=c(t)+R(t)\rho$ with configuration path $(c,R)$ of class $C^2$ on the time window $I_0$, 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). Define the total momentum, kinetic energy and angular momentum about the centre of mass of the model along the motion by the componentwise integrals $P(t):=\int {\mathrm{\partial }}_{t}x(t,\rho )d\nu (\rho ),K(t):=\frac{1}{2}\int \parallel {\mathrm{\partial }}_{t}x(t,\rho ){\parallel }_2^{2}d\nu (\rho ),$ $L_c(t):=\int (x(t,\rho )-c(t))\times ({\mathrm{\partial }}_{t}x(t,\rho )-c^{\mathrm{\prime }}(t))d\nu (\rho ),$ with dimensions $\mu \mathrm{\ell }{\tau }^{-1}$, $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ and $\mu {\mathrm{\ell }}^2{\tau }^{-1}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Then for every $t\in I_0$:

1. the three integrands are $\nu$-integrable, so $P(t)$, $K(t)$ and $L_c(t)$ are a well-defined finite vector and finite real numbers;
2. $P(t)=m{c}^{\mathrm{\prime }}(t)$;
3. $K(t)={\displaystyle \frac{1}{2}}m\parallel c^{\mathrm{\prime }}(t){\parallel }_2^2+{\displaystyle \frac{1}{2}}\omega (t{)}^{\mathsf{T}}I\omega (t)$;
4. $L_c(t)=R(t)I\omega (t)$.

The identities hold for every $C^2$ configuration path; no equation of motion, force law or conservation statement is used or asserted.

• Instantaneous identities. The three displayed identities are identities at each time for the stated motion; they do not say that either term of the kinetic splitting is conserved, and no equation of motion for $(c,R)$ enters. The terms are the model's translational kinetic energy $\frac{1}{2}m\parallel c^{\mathrm{\prime }}{\parallel }_2^2$ and rotational kinetic energy $\frac{1}{2}{\omega }^{\mathsf{T}}I\omega$.

• Frame dependence of the two terms. The centre-of-mass velocity $c^{\mathrm{\prime }}$ and hence the translational term depend on the inertial frame; the rotational term $\frac{1}{2}{\omega }^{\mathsf{T}}I\omega$ is built from the body angular velocity and the body inertia matrix and is therefore unchanged by a change of inertial frame. The precise transformation laws are proved in Behaviour of the rigid-material configuration, angular velocity, inertia data and inertial quantities under a change of inertial frame on this page.

• The countable-choice hypothesis. ${\mathrm{A}\mathrm{C}}_{\omega }$ is used only to make the integrals finite through [L1] and [L3]; the algebraic identities themselves are choice-free once the integrals exist. A consumer that needs the identities under weak hypotheses must carry this assumption.

• Degenerate shapes remain allowed. The identities hold for every mass distribution of the branch, including those supported on a line; in that case $I\omega$ is not injective in $\omega$, so the angular momentum does not determine the body angular velocity, as the companion page's counterexample exhibits.