# The transformation of affine velocity and its application to a rotating disk

## TL;DR

This paper derives a velocity transformation linking local affine velocity in an inertial frame to centro-affine velocity in an accelerated frame, with applications to rotating disks and special cases like Thomas precession.

## Contribution

It introduces a new 3D velocity transformation formula connecting inertial and accelerated frames, extending previous results to non-uniform motions and deformations.

## Key findings

- Derived a general velocity transformation formula.
- Applied the transformation to a rotating disk example.
- Confirmed the stretching coefficient matches known results.

## Abstract

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 centro-affine velocity of motion of this body in the accompanying accelerated frame $ k $. This paper is based on the kinematics of a continuous medium and the generalized Lorentz transformation. In this paper we show the 3D transformation of velocity linking the reference system $ S $ and the reference system $ k $, which moves without rotation. Wherein the motion of various points of the rigid system $ k $ is inhomogeneous. Using these formulas, we obtain the desired direct and inverse transformation 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 affine velocity in $ S $, accelerated rotation of the disk was considered and the local angular velocity and the magnitude of the deformation of its points were calculated. Wherein, 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 the disk.

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Source: https://tomesphere.com/paper/1905.13029