The transformation of the affine velocity and its application to a rotating disc

The aim of the article is to find a transformation that links the local affine velocity of a non-rigid body in the laboratory inertial reference frame $S$ with the centroaffine velocity of this body in the accompanying nonrotating reference frame $k$. This paper is based on the kinematics of a continuous medium and the generalized Lorentz transformation. In the paper we show the 3D transformation of the velocity linking the reference system $S$ and the reference system $k$, which moves without the rotation. Wherein the motion of various points of the rigid system $k$ is inhomogeneous. Using these formulas, we obtain the desired direct and inverse transformations of the local affine velocity. Important special cases of this transformation are considered. They are the motion of particles in a uniform force field and the precession of Thomas. As an example of using the transformation of the affine velocity in $S$, accelerated rotation of a disk is considered and the local angular velocity and the magnitude of the deformation of its points are calculated. The calculated stretching coefficient is consistent with the known one, and the formula found for the angular velocity is more general than the earlier result obtained for uniform rotation of a disk.