Parametrization of Cosserat equations.

The solution space of many systems of ordinary differential (OD) or partial differential (PD) equations in engineering or mathematical physics 'can/cannot' be parameterized by a certain number of arbitrary functions behaving like potentials. Among the well-known OD examples, we recall that a classical control system is parametrizable if and only if it is controllable. One interesting PD case is concerned with continuum mechanics where the first-order stress equations admit a rather simple second-order parametrization by means of the single Airy function in dimension 2. The purpose of this paper is to apply the new techniques of 'Algebraic Analysis', a mixture of differential geometry and homological algebra, in order to give a positive and explicit answer concerning the possibility to obtain a first-order parametrization of the stress/couple-stress equations met in the study of Cosserat media. At the same time, as a corollary of the homological test involved, we shall give for the first time the reason for which the compatibility conditions (CC) for the deformation tensor in classical elasticity theory are 'second order' while the corresponding CC for the Cosserat fields are only 'first order' and explain why this order is equal to the order of the corresponding parametrization. Finally, a close link existing between Cosserat elasticity and Weyl electromagnetism is also provided within a conformal framework. [ABSTRACT FROM AUTHOR]