Rigid body

Remarks

Suppose the motion is continuous in $t$ and the positions at a time $t_0 \in I$ do not all lie in one plane. Then for each $t$ there are a unique $R(t)$ in the special orthogonal group $SO(3)$ and a unique $b(t) \in \R^3$ with $x_X(t) = R(t)\, x_X(t_0) + b(t)$ for all $X \in B$. So the configuration of a rigid body is a point of $SE(3)$, a space of dimension $6$.

Example: let $X$ range over the unit cube and let $R_z(t)$ be the rotation by the angle $t$ about the $x_3$-axis. Then $x_X(t) = R_z(t)\, X + (0, 0, t)$ is a rigid body moving by a screw motion; the distances are kept because $R_z(t)$ is orthogonal.

Non-example: $x_X(t) = (1 + t) X$ for $t \ge 0$ is a uniform expansion; two points at distance $d$ at time $0$ are at distance $(1 + t) d$ at time $t$, so this is not a rigid body. In continuum mechanics, a body is rigid when its deformation gradient is a rotation at every point and time.