Behaviour of the rigid-material configuration, angular velocity, inertia data and inertial quantities under a change of inertial frame

Adopt the model of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope and notation of Configuration, configuration space and orientation of a rigid-material model in an inertial frame, Body and spatial angular velocity of a rigid-material motion, and the material velocity field and Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model: a reference mass distribution $\nu$ with total mass $m$ and centred reference coordinates, a body inertia matrix $I$, a motion $x(t,\rho )=c(t)+R(t)\rho$ in one inertial frame with configuration path $(c,R)$ of class $C^2$ and body angular velocity $\omega$, and the quantities of Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion, $P(t)=\int {\mathrm{\partial }}_{t}x(t,\rho )d\nu (\rho ),K(t)=\frac{1}{2}\int \parallel {\mathrm{\partial }}_{t}x(t,\rho ){\parallel }_2^{2}d\nu (\rho ),L_c(t)=\int (x(t,\rho )-c(t))\times ({\mathrm{\partial }}_{t}x(t,\rho )-c^{\mathrm{\prime }}(t))d\nu (\rho )\mathrm{.}$ Let a second inertial frame be related to the first by the transformation $t^{\mathrm{\prime }}=t+b$, $x^{\mathrm{\prime }}=Qx+ut+a$ of Inertial frames, stationary observers and the adopted Galilean transformations between frames clause (iv), with $b\in \mathbb{R}$, $a,u\in {\mathbb{R}}^3$ and $Q\in SO(3)$; the rotation $Q$ and the boost velocity $u$ are the parameters written $R$ and $v$ in Inertial frames, stationary observers and the adopted Galilean transformations between frames, renamed here to avoid clashes with the orientation path $R(t)$ and the material velocity field $v(t,p)$ of Configuration, configuration space and orientation of a rigid-material model in an inertial frame. Let $\tilde{x}(t^{\mathrm{\prime }},\rho ):=Qx(t^{\mathrm{\prime }}-b,\rho )+u(t^{\mathrm{\prime }}-b)+a$ be the description of the same motion in the second frame. Then:

1. $\tilde{x}(t^{\mathrm{\prime }},\rho )=\tilde{c}(t^{\mathrm{\prime }})+\tilde{R}(t^{\mathrm{\prime }})\rho$ with $\tilde{c}(t^{\mathrm{\prime }})=Qc(t^{\mathrm{\prime }}-b)+u(t^{\mathrm{\prime }}-b)+a$ and $\tilde{R}(t^{\mathrm{\prime }})=QR(t^{\mathrm{\prime }}-b)$; the maps $\tilde{c}$ and $\tilde{R}$ are of class $C^2$, $\tilde{R}$ takes values in $SO(3)$, and $\tilde{x}$ is a motion of the same model with the same reference mass distribution;
2. the body angular velocities satisfy $\tilde{\omega }(t^{\mathrm{\prime }})=\omega (t^{\mathrm{\prime }}-b)$, and the spatial angular velocities satisfy ${\tilde{\omega }}_{\mathrm{s}}(t^{\mathrm{\prime }})=Q{\omega }_{\mathrm{s}}(t^{\mathrm{\prime }}-b)$;
3. the material velocity fields satisfy ${\mathrm{\partial }}_{t^{\mathrm{\prime }}}\tilde{x}(t^{\mathrm{\prime }},\rho )=Q{\mathrm{\partial }}_{t}x(t^{\mathrm{\prime }}-b,\rho )+u$; equivalently, for a spatial point $p$ of the first frame with image $p^{\mathrm{\prime }}=Qp+ut+a$, $\tilde{v}(t^{\mathrm{\prime }},p^{\mathrm{\prime }})=Qv(t,p)+u$;
4. the body inertia matrix computed from $\nu$ is the same matrix $I$ in either frame, and the spatial inertia tensors satisfy $\tilde{I}(t^{\mathrm{\prime }})=QI(t^{\mathrm{\prime }}-b)Q^{\mathsf{T}}$; consequently $\tilde{\omega }(t^{\mathrm{\prime }}{)}^{\mathsf{T}}I\tilde{\omega }(t^{\mathrm{\prime }})=\omega (t^{\mathrm{\prime }}-b{)}^{\mathsf{T}}I\omega (t^{\mathrm{\prime }}-b)$ and ${\tilde{\omega }}_{\mathrm{s}}(t^{\mathrm{\prime }}{)}^{\mathsf{T}}\tilde{I}(t^{\mathrm{\prime }}){\tilde{\omega }}_{\mathrm{s}}(t^{\mathrm{\prime }})={\omega }_{\mathrm{s}}(t^{\mathrm{\prime }}-b{)}^{\mathsf{T}}I(t^{\mathrm{\prime }}-b){\omega }_{\mathrm{s}}(t^{\mathrm{\prime }}-b)$;
5. the inertial quantities satisfy $\tilde{P}(t^{\mathrm{\prime }})=QP(t^{\mathrm{\prime }}-b)+mu,{\tilde{L}}_c(t^{\mathrm{\prime }})=Q{L}_c(t^{\mathrm{\prime }}-b),$ $\tilde{K}(t^{\mathrm{\prime }})=\frac{1}{2}m\parallel Q{c}^{\mathrm{\prime }}(t^{\mathrm{\prime }}-b)+u{\parallel }_2^2+\frac{1}{2}\omega (t^{\mathrm{\prime }}-b{)}^{\mathsf{T}}I\omega (t^{\mathrm{\prime }}-b)\mathrm{.}$

The statements are kinematic and inertial identities for the model; no equation of motion, no conservation law and no measured value is asserted.

• Body quantities are invariant, spatial quantities rotate. The body angular velocity and the body inertia matrix are unchanged by the change of inertial frame; the spatial angular velocity and the spatial inertia tensor transform by $Q$ and conjugation. This is the precise sense in which the rotational term $\frac{1}{2}{\omega }^{\mathsf{T}}I\omega$ of the kinetic splitting is frame-independent while the translational term $\frac{1}{2}m\parallel c^{\mathrm{\prime }}{\parallel }_2^2$ is not.

• Consistency with the particle branch. For a body whose mass is concentrated at its centre of mass the rotational term vanishes and the laws of clause 5 reduce to the transformation of the kinetic quantities of a particle of mass $m$; the transformation law for particle systems is the subject of Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy, and the present item is its rigid-material analogue. The two are consistent where both apply, and neither is used to prove the other.

• What is not asserted. No transformation law for forces, torques or energies beyond the displayed inertial quantities is claimed; no equation of motion is transformed, because the branch has none at this stage.