Affine Tensors in Mechanics of Freely Falling Particles and Rigid Bodies

TL;DR

This paper extends an affine tensor framework to describe the mechanics of freely falling particles and rigid bodies, emphasizing an intrinsic affine covariant derivative and torsors that unify momentum representations.

Contribution

It introduces a novel application of affine tensors and the affine covariant derivative to model the mechanics of particles and rigid bodies without metric notions.

Findings

Torsors unify mass, linear, and angular momentum.

Affine covariant derivative characterizes free-fall motion.

Rigid body dynamics are described within this affine tensor framework.

Abstract

In a previous paper, we proposed an approach for the dynamics of 3D bodies and shells based on the use of affine tensors. This new theoretical frame is very large and the applications are not limited to the mechanics of continua. In the present paper, we show how it can be also applied to the description of the mechanics of freely falling particles and rigid bodies. The mass, the linear and angular momenta are structured as a single object called torsor. Excluding all metric notions, we define the torsors as skew-symmetric bilinear mappings operating on the linear space of the affine functions. Torsors are a particular family of affine tensors. On this ground, we define an intrinsic differential operator called the affine covariant derivative. Next, we claim that the torsor characterizing the behavior of a freely falling particle is affine covariant derivative free, that allows recovering both laws of linear and angular momentum. Finally, it is shown how the motion of rigid bodies can be describe within this frame.