Mathematics – Analysis of PDEs
Scientific paper
2011-02-24
Mathematics
Analysis of PDEs
Though a few of the results presented are proved in the recent references provided, the way they are combined with others and
Scientific paper
The Spencer operator, introduced by D.C. Spencer fifty years ago, is rarely used in mathematics today and, up to our knowledge, has never been used in engineering applications or mathematical physics. The main purpose of this paper, an extended version of a lecture at the second workshop on Differential Equations by Algebraic Methods (DEAM2, february 9-11, 2011, Linz, Austria) is to prove that the use of the Spencer operator constitutes the common secret of the three following famous books published about at the same time in the beginning of the last century, though they do not seem to have anything in common at first sight as they are successively dealing with elasticity theory, commutative algebra, electromagnetism and general relativity: (C) E. and F. COSSERAT: "Th\'eorie des Corps D\'eformables", Hermann, Paris, 1909. (M) F.S. MACAULAY: "The Algebraic Theory of Modular Systems", Cambridge University Press, 1916. (W) H. WEYL: "Space, Time, Matter", Springer, Berlin, 1918 (1922, 1958; Dover, 1952). Meanwhile, we shall point out the importance of (M) for studying control identifiability and of (C)+(W) for the group theoretical unification of finite elements in engineering sciences, recovering in a purely mathematical way well known field-matter coupling phenomena (piezzoelectricity, photoelasticity, streaming birefringence, viscosity, ...). As a byproduct and though disturbing it could be, we shall prove that these unavoidable new diferential and homological methods contradict the mathematical foundations of both engineering (continuum mechanics,electromagnetism) and mathematical (gauge theory, general relativity) physics.
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