The theory of screws derived from a module over the dual numbers

TL;DR

This paper establishes a new mathematical framework linking screw theory in mechanics to modules over dual numbers, providing a canonical way to represent screw vectors and deepening the understanding of affine space geometry.

Contribution

It proves the converse that any rank 3 free module over dual numbers with certain properties corresponds to screw vectors in Euclidean space, unifying screw theory with dual number modules.

Findings

Provides a canonical construction of Euclidean space from dual number modules.

Shows that screw theory results can be derived from D-module geometry.

Offers a new perspective on affine space as vector space over dual numbers.

Abstract

The theory of screws clarifies many analogies between apparently unrelated notions in mechanics, including the duality between forces and angular velocities. It is known that the real 6-dimensional space of screws can be endowed with an operator E, E^2 = 0, that converts it into a rank 3 free module over the dual numbers. In this paper we prove the converse, namely, given a rank 3 free module over the dual numbers, endowed with orientation and a suitable scalar product (D-module geometry), we show that it is possible to define, in a canonical way, a Euclidean space so that each element of the module is represented by a screw vector field over it. The new approach has the effectiveness of motor calculus while being independent of any reduction point. It gives insights into the transference principle by showing that affine space geometry is basically vector space geometry over the dual numbers. The main results of screw theory are then recovered by using this point of view.