  The main purpose of this paper is to revisit the well known potentials,
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) and G. Morera
(1892) 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 n2(n2 -- 1)/12 components of the
linearized Riemann tensor considered as a linear second order differential
operator but the minimum number of potentials needed in elasticity theory is
equal to n(n -- 1)/2 for any minimal parametrization. Meanwhile, we can provide
all the above results without even using indices for writing down explicit
formulas in the way it is done in any textbook today. The example of
relativistic continuum mechanics with n = 4 is provided in order to prove that
it could be strictly impossible to obtain such results without using the above
methods. We also revisit the possibility (Maxwell equations of electromag-
netism) or the impossibility (Einstein equations of gravitation) to obtain
canonical or minimal parametrizations for various other 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 elasticity theory and mathematical physics, it is
written in a rather self-contained way.
