Parametrization of Cosserat Equations

As a matter of fact, the solution space of many systems of ordinary differential (OD) or partial differential (PD) equations in engineering or mathematical physics ”can/cannot” be parametrized by a certain number of arbitrary functions behaving like ”potentials”. In view of the explicit examples to be met later on, it must be noticed that the parametrizing operator, though often of the first order, may be, on the contrary, of arbitrarily large order. Among the well known examples, we recall that a classical OD control system is parametrizable if and only if it is controllable (Kalman test of 1969 in [9]). Among PD systems, the electromagnetic (EM) field, solution of the first set of 4 Maxwell equations, admits a well known first order parametrization by means of the 4-potential while the EM induction, solution of the second set of 4 Maxwell equations (in vacuum), also admits a first order parametrization by means of the so-called 4-pseudopotential. On the contrary, it is now known that, contrary to the EM situation, the set of 10 second order linearized Einstein equations (in vacuum) cannot be parametrized and cannot therefore be considered as field equations (see [18,24] for more details; see also [30] and http://wwwb.math.rwth-aachen.de/OreModules for a computer algebra solution). One among the best interesting and useful cases is concerned with continuum mechanics where the first order stress equations (in vacuum) admits a rather simple second order parametrization by means of the single Airy function in dimension 2 and it is not so well known (!) that a much more complicated second order parametrization can be achieved in dimension n ≥ 2 by means of n(n − 1)/12 arbitrary functions.