Affine tensors in mechanics of freely falling particles and rigid bodies

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 a 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 behaviour 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 described within this frame.