A PEDESTRIAN APPROACH TO COSSERAT / MAXWELL / WEYL THEORY

Jean-François Pommaret (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

The purpose of this short notice is to present an elementary summary of a few recent results obtained through the application of the formal theory of systems of partial differential equations and Lie pseudo groups to engineering (elasticity theory, electromagnetism, coupling phenomena) and mathematical (gauge theory, general relativity) physics along the following scheme: 1) Lie groups of transformations may be considered as Lie pseudo groups of transformations but no action type method can be used as parameters never appear any longer. 2) The work of Cartan is superseded by the use of the canonical Spencer sequence while the work of Vessiot is superseded by the use of the canonical Janet sequence but the link between these two sequences and thus these two works is not known today. 3)Using duality theory, the formal adjoint of the Spencer operator for the conformal group of transformations of space-time provides the Cosserat equations, the Maxwell equations and the Weyl equations on equal footing but such a result leads to deep contradictions. Accordingly, the results thus obtained prove that the foundations of engineering and mathematical physics must be revisited within the framework of jet theory though striking it may look like for established theories.

Mots clés: Lie groups; Lie pseudo groups; gauge theory; general relativity; Spencer operator; formal adjoint; Cosserat equations; Maxwell equations; conformal group

Domaines: Equations aux dérivées partielles [math.AP]; Géométrie différentielle [math.DG]; Physique mathématique [math-ph]; Théorie des groupes [math.GR]

Jean-François Pommaret. A PEDESTRIAN APPROACH TO COSSERAT / MAXWELL / WEYL THEORY. 2012. ⟨hal-00740314⟩