The purpose of this paper is to revisit the well known potentials, also called
stress functions, needed in order to study the parametrizations of the stress equations, respectively provided by G.B. Airy (1863) for 2-dimensional elasticity, then by E. Beltrami (1892), J.C. Maxwell (1870) for 3-dimensional elasticity, finally by A. Einstein (1915) for 4-dimensional elasticity, both with a variational procedure introduced by C. Lanczos (1949, 1962) in order to relate potentials to Lagrange multipliers. Using the methods of Algebraic Analysis, namely mixing differential geometry with homological algebra and combining the
double duality test involved with the
Spencer cohomology, we shall be able to extend these results to an arbitrary situation with an arbitrary dimension n. We shall also explain why double duality is perfectly adapted to variational calculus with differential constraints as a way to eliminate the corresponding Lagrange multipliers. For example, the
canonical parametrization of the stress equations is just described by the formal adjoint of the
components of the linearized Riemann tensor considered as a linear second order differential operator but the minimum number of potentials needed is equal to
for any
minimal parametrization, the Einstein parametrization being “
in between” with
potentials. We provide all the above results without even using indices for writing down explicit formulas in the way it is done in any textbook today, but it could be
strictly impossible to obtain them without using the above methods. We also revisit the
possibility (Maxwell equations of electromagnetism) or the
impossibility (Einstein equations of gravitation) to obtain canonical or minimal parametrizations for various equations of physics. It is nevertheless important to notice that, when n and the algorithms presented are known, most of the calculations can be achieved by using computers for the corresponding symbolic computations. Finally, though the paper is mathematically oriented as it aims providing new insights towards the mathematical foundations of general relativity, it is written in a rather self-contained way.