Cosserat media in dynamics

TL;DR

This paper develops a unified geometric framework for analyzing the dynamics of various material bodies using affine connections and torsors, extending classical mechanics to higher-dimensional and relativistic contexts.

Contribution

It introduces a general covariant approach for the dynamics of d-dimensional bodies using affine Cartan connections and torsors, applicable across different dimensions and relativity frameworks.

Findings

Formulation of 10 balance equations from covariant divergence principles

Application of the framework to bodies of dimensions 1 to 4 in Galilean relativity

Establishment of a unified geometric approach for diverse material bodies

Abstract

Our aim is to develop a general approach for the dynamics of material bodies of dimension d represented by a mater manifold of dimension (d + 1) embedded into the space-time. It can be declined for d = 0 (pointwise object), d = 1 (arch if it is a solid, flow in a pipe or jet if it is a fluid), d = 2 (plate or shell if it is a solid, sheet of fluid), d = 3 (bulky bodies). We call torsor a skew-symmetric bilinear map on the vector space of affine real functions on the affine tangent space to the space-time. We use the affine connections as originally developed by \'Elie Cartan, that is the connections associated to the affine group. We introduce a general principle of covariant divergence free torsor from which we deduce 10 balance equations. We show the relevance of this general principle by applying it for d from 1 to 4 in the context of the Galilean relativity.