Cauchy, Cosserat, Clausius, Maxwell, Weyl Equations Revisited

Abstract

The Cauchy stress equations (1823), the Cosserat couple-stress equations (1909), the Clausius virial equation (1870), the Maxwell/Weyl equations (1873,1918) are among the most famous partial differential equations that can be found today in any textbook dealing {\it separately and/or successively} with elasticity theory, continuum mechanics, thermodynamics, electromagnetism and electrodynamics. Over a manifold of dimension $n$, their respective numbers are $n, n(n-1)/2, 1, n$ with a total of $(n+1)(n+2)/2$, that is $15$ when $n= 4$ for space-time. As a matter of fact, this is just the number of parameters of the Lie group of conformal transformations with $n$ translations, $n(n-1)/2$ rotations, $1$ dilatation and $n$ highly non-linear elations introduced by E. Cartan in $1922$. The purpose of this short but difficult paper is to prove that the form of these equations only depends on the structure of the conformal group for $n\geq 1$ arbitrary because they are described {\it as a whole} by the (formal) adjoint of the first Spencer operator existing in the Spencer differential sequence. Such a group theoretical implication is obtained for the first time by totally new differential geometric methods. Meanwhile, these equations can be all parametrized by the adjoint of the second Spencer operator through $ n(n^2 - 1)(n+2)/4$ potentials.This result brings the need to revisit the mathematical foundations of Electromagnetism and Gauge Theory according to a clever but rarely quoted paper of H. Poincar\'{e} (1901).