Rigid placements, body and spatial angular velocity, and material velocities

Let $I\subseteq \mathbb{R}$ be an interval with at least two points (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), let $\nu$ be a mass distribution on ${\mathbb{R}}^3$ with total mass $m$ (Mass distributions on ${\mathbb{R}}^3$: total mass, first moment and inertia matrix), let $c:I\to {\mathbb{R}}^3$ be differentiable and let $R:I\to SO(3)$ be differentiable (The orthogonal group $O(3)$ and the rotation group $SO(3)$), differentiability meaning entrywise differentiability of $R$ in the sense of The derivative $f^{\mathrm{\prime }}(c)={\lim }_{x\to c}\frac{f(x)-f(c)}{x-c}$ of $f:A\to \mathbb{R}$ at a point $c\in A$ that is a limit point of $A$, and differentiability on a set with derivative $R^{\mathrm{\prime }}$; write $c^{\mathrm{\prime }}$ for the derivative of $c$.

The rigid placement with reference shape $\nu$, translation $c$ and rotation $R$ is the map $x:I\times {\mathbb{R}}^3⟶{\mathbb{R}}^3,x(t,\rho ):=c(t)+R(t)\rho ,$ so that the material point labelled by $\rho$ (a point of the reference shape) is carried to $x(t,\rho )$ at time $t$; the map $x(t,\cdot )$ is the placement at time $t$, and the pair $(c(t),R(t))$ is its configuration. The material velocity of the label $\rho$ at time $t$ is the derivative ${\mathrm{\partial }}_{t}x(t,\rho ):=c^{\mathrm{\prime }}(t)+R^{\mathrm{\prime }}(t)\rho ,$ which exists by differentiability of $c$ and of the entries of $R$.

Body and spatial angular velocity. By Derivative rules for differentiable matrix-valued curves, and skewness of $R^{\mathsf{T}}{R}^{\mathrm{\prime }}$ for a curve in $SO(3)$ the matrix $R(t{)}^{\mathsf{T}}{R}^{\mathrm{\prime }}(t)$ is skew-symmetric at every $t\in I$. By The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$ there is therefore a unique vector $\omega (t)\in {\mathbb{R}}^3$ with $R(t{)}^{\mathsf{T}}{R}^{\mathrm{\prime }}(t)=[\omega (t){]}_{\times },$ and the map $\omega :I\to {\mathbb{R}}^3$ is the body angular velocity of the placement; the spatial angular velocity is ${\omega }_{\mathrm{s}}(t):=R(t)\omega (t)$. Explicitly $\omega (t)=((R^{\mathsf{T}}{R}^{\mathrm{\prime }}{)}_{32},(R^{\mathsf{T}}{R}^{\mathrm{\prime }}{)}_{13},(R^{\mathsf{T}}{R}^{\mathrm{\prime }}{)}_{21})(t)$, a finite sum of products of entries of $R$ and $R^{\mathrm{\prime }}$; consequently, if $R$ is twice differentiable, then $\omega$ is differentiable and its derivative is the body angular acceleration.

Equivalently, by the conjugation clause of The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$, $R^{\mathrm{\prime }}(t)R(t{)}^{\mathsf{T}}=[{\omega }_{\mathrm{s}}(t){]}_{\times }$.

• The reference shape is a label space. $\rho$ is not the position of the material point at time $t$; the position is $x(t,\rho )$. The mass distribution $\nu$ is a measure on the label space, and no invariance of $\nu$ under translations or rotations is assumed.

• Existence and uniqueness of $\omega$ are discharged, not assumed. Skew-symmetry of $R^{\mathsf{T}}{R}^{\mathrm{\prime }}$ is the differentiated orthogonality relation of the curve $R$; the vector whose cross-product matrix it is exists and is unique by the explicit bijection of the two cited lemmas. No smoothness beyond differentiability of the entries of $R$ is used to define $\omega$.

• This is mathematics. No physical law is asserted: not that a physical body is rigid, not that its configuration is described by a curve in $SO(3)$, and not that $\omega$ is measured by any apparatus. The physical reading is supplied by the consuming classical-mechanics items.