Parametrization of Cosserat Equations

TL;DR

This paper demonstrates that the stress and couple-stress Cosserat equations can be parametrized by a specific number of arbitrary functions using advanced algebraic analysis techniques.

Contribution

It introduces a novel application of homological algebra to establish a first order parametrization of Cosserat equations, a previously unresolved problem.

Findings

Cosserat equations admit a first order parametrization.

The parametrization involves n^2(n^2-1)/4 arbitrary functions.

The approach uses deep results from algebraic analysis and homological algebra.

Abstract

As a matter of fact, the solution space of many systems of ordinary or partial differential equations in engineering or mathematical physics "can/cannot" be parametrized by a certain number of arbitrary functions behaving like potentials. For example, such a parametrization exists for a control system if and only if it is controllable and may be of high order. The first set of Maxwell equations admits a first order parametrization by the 4-potential. However, Einstein equations in vacuum do not admit any parametrization. Finally, the stress equations in continuum mechanics admit a second order parametrization by means of n^2(n^2-1)/12 arbitrary functions, the case n=2 being the Airy function. The purpose of this paper is to use unexpected deep results of homological algebra and algebraic analysis in order to prove for the first time that the stress/couple-stress Cosserat equations admit a first order parametrization by mens of n^2(n^2-1)/4 arbitrary functions