1.4.1. Rigid motion
Rigid motion preserves distance between any pair of points, and thus angles. The motion of two material points \(P\), \(Q\) performing a rigid motion obeys
(1.1)\[\vec{v}_P - \vec{v}_Q = \vec{\omega} \times (P - Q) \ ,\]
being \(\vec{v}_P\), \(\vec{v}_Q\) the velocity of the points and \(\vec{\omega}\) the angular velocity of the rigid motion. Taking a point \(Q\) as the reference point of the motion, the velocity of all other points can be found
\[\vec{v}_P = \vec{v}_Q + \vec{\omega} \times (P-Q) \ ,\]
as a function of the velocity of \(Q\), the angular velocity of the rigid motion, and the relative position \(P-Q\).
Given 3 points \(P(t)\), \(Q(t)\), \(R(t)\), the distance bewteen each pair of points is constant and thus its time derivative zero,
\[0 = \dfrac{d}{dt} |P(t)-Q(t)|^2 = 2 \left(P - Q\right) \cdot \left( \vec{v}_P - \vec{v}_Q \right) \quad \rightarrow \quad \Delta \vec{v}_{QP} = \vec{\omega}_{QP} \times \Delta \vec{r}_{QP}\]
\[0 = \dfrac{d}{dt} |P(t)-R(t)|^2 = 2 \left(P - R\right) \cdot \left( \vec{v}_P - \vec{v}_R \right) \quad \rightarrow \quad \Delta \vec{v}_{RP} = \vec{\omega}_{RP} \times \Delta \vec{r}_{RP}\]
\[\begin{split}\begin{aligned}
0 = \dfrac{d}{dt} \left[ (P-Q) \cdot (P-R) \right]
& = \Delta \vec{v}_{QP} \cdot \Delta \vec{r}_{RP} + \Delta \vec{r}_{QP} \cdot \Delta \vec{v}_{RP} = \\
& = \vec{\omega}_{QP} \times \Delta \vec{r}_{QP} \cdot \Delta \vec{r}_{RP} + \Delta \vec{r}_{QP} \cdot \Delta \vec{\omega}_{RP} \times \Delta \vec{r}_{RP} = \\
& = \Delta \vec{r}_{QP} \times \Delta \vec{r}_{RP} \cdot ( \vec{\omega}_{QP} - \vec{\omega}_{RP} ) \ ,
\end{aligned}\end{split}\]
and since \(\Delta \vec{r}_{QP}\), \(\Delta \vec{r}_{RP}\) are arbitrary it follows that the vector \(\vec{\omega} = \vec{\omega}_{QP} = \vec{\omega}_{RP}\) is unique for all the points performing a rigid motion.
The configuration of a material vector \(\vec{a}\) undergoing a rotation is described by the product of the rotation tensor \(\mathbb{R}\) by the reference configuration \(\vec{a}^0\),
\[
\vec{a} = \mathbb{R} \cdot \vec{a}^0
\qquad , \qquad
\vec{b} = \mathbb{R} \cdot \vec{b}^0 \ .
\]
In order to preserve distance, and angles
\[\begin{split}\begin{cases}
|\vec{a}|^2 = & \vec{a} \cdot \vec{a} = \vec{a}^0 \cdot \mathbb{R}^T \cdot \mathbb{R} \cdot \vec{a}^0 = \vec{a}^0 \cdot \vec{a}^0 = |\vec{a}|^2 \\
& \vec{a} \cdot \vec{b} = \vec{a}^0 \cdot \mathbb{R}^T \cdot \mathbb{R} \cdot \vec{b}^0 = \vec{a}^0 \cdot \vec{a}^0
\end{cases}
\qquad \rightarrow \qquad \mathbb{R}^T \cdot \mathbb{R} = \mathbb{I}
\end{split}\]
the rotation tensor is unitary
(1.2)\[\mathbb{I} = \mathbb{R}^T \cdot \mathbb{R} = \mathbb{R} \cdot \mathbb{R}^T\]
Note
From relation (1.2), it follows that
\[1 = |\mathbb{I}| = |\mathbb{R}^T| |\mathbb{R}| = |\mathbb{R}|^2 \ ,\]
and thus \(|\mathbb{R}| = \mp 1\). If \(|\mathbb{R}| = 1\), \(\mathbb{R}\) represents a rotation, and implies conservation of orientation of space; if \(|\mathbb{R}| = -1\) represents a reflection w.r.t. a plane, and implies inversion of orientation of space.
Orientation of space is determined by the transformation of a RHS triad of vectors: if the transform triad is RHS, then orientation of space is preserved; if it becomes LHS, then orientation of space is inverted.
Note
Rotation tensor \(\mathbb{R}\) is not singular and its determinant equals \(|\mathbb{R}| = 1\). Thus, \(\mathbb{R}^T \cdot \mathbb{R} = \mathbb{I}\) implies \(\mathbb{R} \cdot \mathbb{R}^T = \mathbb{I}\). Multiplying (1.2) by \(\mathbb{R}\) on the left
\[\mathbb{0} = \mathbb{R} \cdot \mathbb{R}^T \cdot \mathbb{R} - \mathbb{R} \cdot \mathbb{I} = (\mathbb{R} \cdot \mathbb{R}^T - \mathbb{I}) \cdot \mathbb{R} \ ,\]
and since \(\mathbb{R}\) is non-singular, it follows that \(\mathbb{R} \cdot \mathbb{R}^T = \mathbb{I}\).
Time derivative of the relation (1.2) reads
\[\mathbb{0} = \dfrac{d}{dt} \left( \mathbb{R} \cdot \mathbb{R}^T \right) = \dot{\mathbb{R}} \cdot \mathbb{R}^T + \mathbb{R} \cdot \dot{\mathbb{R}}^T\]
It follows that the 2-nd order tensor \(\dot{\mathbb{R}} \cdot \mathbb{R}^T = - \mathbb{R} \cdot \dot{\mathbb{R}}^T\) is anti-symmetric, and thus it can be written as
(1.3)\[\dot{\mathbb{R}} \cdot \mathbb{R}^T =: \vec{\omega}_{\times} \ ,\]
being the vector \(\vec{\omega}\) the angular velocity. Since \(\mathbb{R}\) is unitary by (1.2), multiplying (1.3) with the dot-product on the right by \(\mathbb{R}\), it follows
\[\dot{\mathbb{R}} = \vec{\omega}_{\times} \cdot \mathbb{R} \ ,\]
and the expression of the time derivative of a material vector \(\vec{a}\),
(1.4)\[\dfrac{d \vec{a}}{d t} = \dot{\mathbb{R}} \cdot \vec{a}^0 = \vec{\omega}_{\times} \cdot \mathbb{R} \cdot \vec{a}^0 = \vec{\omega}_{\times} \cdot \vec{a} = \vec{\omega} \times \vec{a} \ .\]
Position and Orientation.
The most general rigid motion is the combination of the translation of a reference point \(Q\) and the rotation w.r.t. this point of other material points,
(1.5)\[\begin{split}\begin{aligned}
\vec{r}_P
& = \vec{r}_Q + (P - Q) = \\
& = \vec{r}_Q + \mathbb{R} \cdot (P-Q)^0
\end{aligned}\end{split}\]
Velocity and Angular velocity.
Time derivative of the relation (1.5) between positions of material points gives again (1.1)
(1.6)\[\begin{split}\begin{aligned}
\vec{v}_P
& = \vec{v}_Q + \dot{\mathbb{R}} \cdot (P-Q)^0 = \\
& = \vec{v}_Q + \vec{\omega}_{\times} \mathbb{R} \cdot (P-Q)^0 = \\
& = \vec{v}_Q + \vec{\omega} \times (P-Q)
\end{aligned}\end{split}\]
Acceleration and Angular acceleration.
Time derivatives of the relation {eq}`eq:rigid:vel} gives
\[\begin{split}\begin{aligned}
\vec{a}_P
& = \vec{a}_Q + \vec{\alpha} \times (P-Q) + \vec{\omega} \times \left( \vec{v}_P - \vec{Q} \right) = \\
& = \vec{a}_Q + \vec{\alpha} \times (P-Q) + \vec{\omega} \times \left[ \, \vec{\omega} \times (P - Q) \, \right] \ .
\end{aligned}\end{split}\]
