Minimum Parametrization of the Cauchy Stress Operator

Abstract

When ${\cal{D}}:\xi \rightarrow \eta$ is a linear differential operator, a "direct problem " is to find the generating compatibility conditions (CC) in the form of an operator ${\cal{D}}_1:\eta \rightarrow \zeta$ such that ${\cal{D}}\xi=\eta$ implies ${\cal{D}}_1\eta=0$. When ${\cal{D}}$ is involutive, the procedure provides successive first order involutive operators ${\cal{D}}_1, ... , {\cal{D}}_n$ when the ground manifold has dimension $n$. Conversely, when ${\cal{D}}_1$ is given, a more difficult " inverse problem " is to look for an operator ${\cal{D}}: \xi \rightarrow \eta$ having the generating CC ${\cal{D}}_1\eta=0$. If this is possible, that is when the differential module defined by ${\cal{D}}_1$ is torsion-free, one shall say that the operator ${\cal{D}}_1$ is parametrized by ${\cal{D}}$ and there is no relation in general between ${\cal{D}}$ and ${\cal{D}}_2$. The parametrization is said to be " minimum " if the differential module defined by ${\cal{D}}$ has a vanishing differential rank and is thus a torsion module. The parametrization of the Cauchy stress operator in arbitrary dimension $n$ has attracted many famous scientists (G.B. Airy in 1863 for $n=2$, J.C. Maxwell in 1863, G. Morera and E. Beltrami in 1892 for $n=3$, A. Einstein in 1915 for $n=4$) . This paper proves that all these works are using the Einstein operator and not the Ricci operator. As a byproduct, they are all based on a confusion between the so-called $div$ operator induced from the Bianchi operator ${\cal{D}}_2$ and the Cauchy operator which is the formal adjoint of the Killing operator ${\cal{D}}$ parametrizing the Riemann operator ${\cal{D}}_1$ for an arbitrary $n$. Like the Michelson and Morley experiment, it is an open historical problem to know whether Einstein was aware of these previous works or not, as the comparison needs no comment.