Abstract:
In 1870, R. Clausius found the virial theorem which amounts to introduce the trace of the stress tensor when studying the foundations of thermodynamics, as a way to relate the absolute temperature of an ideal gas to the mean kinetic energy of its molecules. In 1901, H. Poincar{\'e} introduced a duality principle in analytical mechanics in order to study lagrangians invariant under the action of a Lie group of transformations. In 1909, the brothers E. and F. Cosserat discovered another approach for studying the same problem though using quite different equations. In 1916, H. Weyl considered again the same problem for the conformal group of transformations, obtaining at the same time the Maxwell equations and an additional specific equation also involving the trace of the impulsion-energy tensor. Finally, having in mind the space-time formulation of electromagnetism and the Maurer-Cartan equations for Lie groups, gauge theory has been created by C.N. Yang and R.L. Mills in 1954 as a way to introduce in physics the differential geometric methods available at that time, independently of any group action, contrary to all the previous approaches. The main purpose of this paper is to revisit the mathematical foundations of thermodynamics and gauge theory by using new differential geometric methods coming from the formal theory of systems of partial differential equations and Lie pseudogroups, mostly developped by D.C Spencer and coworkers around 1970. In particular, we justify and extend the virial theorem, showing that the Clausius/Cosserat/Maxwell/Weyl equations are nothing else but the formal adjoint of the Spencer operator appearing in the canonical Spencer sequence for the conformal group of space-time and are thus totally dependent on the group action. The duality principle also appeals to the formal adjoint of a linear differential operator used in differential geometry and to the extension modules used in homological algebra.

Abstract:
In 1953, the physicists E. Inon\"u and E.P. Wigner introduced the concept of deformation of a Lie algebra by claiming that the limit $1/c \rightarrow 0$, when c is the speed of light, of the composition law $(u,v) \rightarrow (u+v)/(1+(uv/c^2))$ of speeds in special relativity (Poincar\'e group) should produce the composition law $(u,v) \rightarrow u + v $ used in classical mechanics (Galil\'ee group). However, the dimensionless composition law $(u'=u/c,v'=v/c) \rightarrow (u'+v')/(1+u'v')$ does not contain any longer a perturbation parameter. Nevertheless, this idea brought the birth of the " deformation theory of algebraic structures", culminating in the use of the Chevalley-Eilenberg cohomology of Lie algebras and one of the first applications of computer algebra in the seventies. One may also notice that the main idea of general relativity is to deform the Minkowski metric of space-time by means of the small dimensionless parameter $\phi/c^2$ where $\phi=GM/r$ is the gravitational potential at a distance r of a central attractive mass M with gravitational constant G. A few years later, a " deformation theory of geometric structures " on manifolds of dimension n was introduced and one may quote riemannian, symplectic or complex analytic structures. Though often conjectured, the link between the two approaches has never been exhibited and the aim of this paper is to provide the solution of this problem by new methods. The key tool is made by the " Vessiot structure equations " (1903) for Lie groups or Lie pseudogroups of transformations, which, contrary to the " Cartan structure equations ", are still unknown today and contain " structure constants " which, like in the case of constant riemannian curvature, have in general nothing to do with any Lie algebra. The main idea is then to introduce the purely differential Janet sequence $0 \rightarrow \Theta \rightarrow T \rightarrow F_0 \rightarrow F_1 \rightarrow ... \rightarrow F_n \rightarrow 0$ as a resolution of the sheaf $\Theta \subset T$ of infinitesimal transformations and to induce a purely algebraic " deformation sequence " with finite dimensional vector spaces and linear maps, even if $\Theta$ is infinite dimensional. The infinitesimal equivalence problem for geometric structures has to do with the local exactness at $ F_0 $ of the Janet sequence while the deformation problem for algebraic structures has to do with the exactness of the deformation sequence at the invariant sections of $F_1 $, that is ONE STEP FURTHER ON in the sequence and this unexpected result explains why the many tentatives

Abstract:
In the last chapter of his book "The Algebraic Theory of Modular Systems " published in 1916, F. S. Macaulay developped specific techniques for dealing with " unmixed polynomial ideals " by introducing what he called " inverse systems ". The purpose of this paper is to extend such a point of view to differential modules defined by linear multidimensional systems, that is by linear systems of ordinary differential (OD) or partial differential (PD) equations of any order, with any number of independent variables, any number of unknowns and even with variable coefficients in a differential field. The first and main idea is to replace unmixed polynomial ideals by " pure differential modules ". The second idea is to notice that a module is 0-pure if and only if it is torsion-free and thus if and only if it admits an " absolute parametrization " by means of arbitrary potential like functions, or, equivalently, if it can be embedded into a free module by means of an " absolute localization ". The third idea is to refer to a difficult theorem of algebraic analysis saying that an r-pure module can be embedded into a module of projective dimension equal to r, that is a module admitting a projective resolution with exactly r operators. The fourth and final idea is to establish a link between the use of extension modules for such a purpose and specific formal properties of the underlying multidimensional system through the use of involution and a "relative localization " leading to a "relative parametrization ", that is to the use of potential-like functions satisfying a kind of "minimum differential constraint " limiting, in some sense, the number of independent variables appearing in these functions, in a way similar to the situation met in the Cartan-K\"ahler theorem of analysis. The paper is written in a rather effective self-contained way and we provide many explicit examples that should become test examples for a future use of computer algebra.

Abstract:
The purpose of this short notice is to present an elementary summary of a few recent results obtained through the application of the formal theory of systems of partial differential equations and Lie pseudo groups to engineering (elasticity theory, electromagnetism, coupling phenomena) and mathematical (gauge theory, general relativity) along the following scheme: 1) Lie groups of transformations may be considered as Lie pseudo groups of transformations but no action type method can be used as parameters never appear any longer. 2) The work of Cartan is superseded by the use of the canonical Spencer sequence while the work of Vessiot is superseded by the use of the canonical Janet sequence but the link between these two sequences and thus these two works is not known today. 3)Using duality theory, the formal adjoint of the Spencer operator for the conformal group of transformations of space-time provides the Cosserat equations, the Maxwell equations and the Weyl equations on equal footing but such a result leads to deep contradictions. Accordingly, the results thus obtained prove that the foundations of engineering and mathematical physics must be revisited within the framework of jet theory though striking it may look like for established theories.

Abstract:
The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide. The paper is therefore divided into three parts corresponding to the different formal methods used. 1) CARTAN VERSUS VESSIOT: The quadratic terms appearing in the " Riemann tensor " according to the " Vessiot structure equations " must not be identified with the quadratic terms appearing in the well known " Cartan structure equations " for Lie groups and a similar comment can be done for the " Weyl tensor ". In particular, " curvature+torsion" (Cartan) must not be considered as a generalization of "curvature alone" (Vessiot). Roughly, Cartan and followers have not been able to " quotient down to the base manifold ", a result only obtained by Spencer in 1970 through the "nonlinear Spencer sequence" but in a way quite different from the one followed by Vessiot in 1903 for the same purpose and still ignored. 2) JANET VERSUS SPENCER: The " Ricci tensor " only depends on the nonlinear transformations (called " elations " by Cartan in 1922) that describe the "difference " existing between the Weyl group (10 parameters of the Poincar\'e subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined by a canonical splitting, that is to say without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly, the Spencer sequence for the conformal Killing system and its formal adjoint fully describe the Cosserat/Maxwell/Weyl theory but General Relativity is not coherent at all with this result. 3) ALGEBRAIC ANALYSIS: Contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be " parametrized ", that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical foundations of mathematical physics must be revisited within this formal framework,

Abstract:
The purpose of this paper is to revisit the infinite Lie group theoretical framework of hydrodynamics developped by V. Arnold in 1966. First of all, we extend this approach from the Lie pseudogroup of volume preserving transformations to an arbitrary Lie pseudogroup. Then we prove that, contrary to what could be believed from the work of Arnold which is of a purely analytical nature, the same results can be obtained from a purely formal point of view. Finally, we provide the analogue for both the so-called "body" and "space" dynamical equations. We conclude by showing that even this new approach can be superseded by dynamics on Lie groupoids, along ideas pioneered by the brothers E. and F. Cosserat or H. Weyl at the beginning of the last century, on the condition to change the underlying philosophy.

Abstract:
As a matter of fact, the solution space of many systems of ordinary or partial differential equations in engineering or mathematical physics "can/cannot" be parametrized by a certain number of arbitrary functions behaving like potentials. For example, such a parametrization exists for a control system if and only if it is controllable and may be of high order. The first set of Maxwell equations admits a first order parametrization by the 4-potential. However, Einstein equations in vacuum do not admit any parametrization. Finally, the stress equations in continuum mechanics admit a second order parametrization by means of n^2(n^2-1)/12 arbitrary functions, the case n=2 being the Airy function. The purpose of this paper is to use unexpected deep results of homological algebra and algebraic analysis in order to prove for the first time that the stress/couple-stress Cosserat equations admit a first order parametrization by mens of n^2(n^2-1)/4 arbitrary functions

Abstract:
Since its original publication in 1916 under the title "The Algebraic Theory of Modular Systems", the book by F. S. Macaulay has attracted a lot of scientists with a view towards pure mathematics (D. Eisenbud,...) or applications to control theory (U. Oberst,...).However, a carefull examination of the quotations clearly shows that people have hardly been looking at the last chapter dealing with the so-called "inverse systems", unless in very particular situations. The purpose of this paper is to provide for the first time the full explanation of this chapter within the framework of the formal theory of systems of partial differential equations (Spencer operator on sections, involution,...) and its algebraic counterpart now called "algebraic analysis" (commutative and homological algebra, differential modules,...). Many explicit examples are fully treated and hints are given towards the way to work out computer algebra packages.

Abstract:
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.

Abstract:
The first purpose of this paper is to point out a curious result announced by Macaulay on the Hilbert function of a differential module in his famous book The Algebraic Theory of Modular Systems published in 1916. Indeed, on page 78/79 of this book, Macaulay is saying the following: " A polynomial ideal $\mathfrak{a} \subset k[{\chi}\_1$,..., ${\chi}\_n]=k[\chi]$ is of the {\it principal class} and thus {\it unmixed} if it has rank $r$ and is generated by $r$ polynomials. Having in mind this definition, a primary ideal $\mathfrak{q}$ with associated prime ideal $\mathfrak{p} = rad(\mathfrak{q})$ is such that any ideal $\mathfrak{a}$ of the principal class with $\mathfrak{a} \subset \mathfrak{q}$ determines a primary ideal of greater {\it multiplicity} over $k$. In particular, we have $dim\_k(k[\chi]/({\chi}\_1$,...,${\chi}\_n)^2)=n+1$ because, passing to a system of PD equations for one unknown $y$, the parametric jets are \{${y,y\_1, ...,y\_n}$\} but any ideal $\mathfrak{a}$ of the principal class with $\mathfrak{a}\subset ({\chi}\_1,{…},{\chi}\_n)^2$ is contained into a {\it simple} ideal, that is a primary ideal $\mathfrak{q}$ such that $rad(\mathfrak{q})=\mathfrak{m}\in max(k[\chi])$ is a maximal and thus prime ideal with $dim\_k(M)=dim\_k(k[\chi]/\mathfrak{q})=2^n$ at least. Accordingly, any primary ideal $\mathfrak{q}$ may not be a member of the primary decomposition of an unmixed ideal $\mathfrak{a} \subseteq \mathfrak{q}$ of the principal class. Otherwise, $\mathfrak{q}$ is said to be of the {\it principal noetherian class} ". Our aim is to explain this result in a modern language and to illustrate it by providing a similar example for $n=4$. The importance of such an example is that it allows for the first time to exhibit symbols which are $2,3,4$-acyclic without being involutive. Another interest of this example is that it has properties quite similar to the ones held by the system of conformal Killing equations which are still not known. For this reason, we have put all the examples at the end of the paper and each one is presented in a rather independent way though a few among them are quite tricky. Meanwhile, the second purpose is to prove that the methods developped by Macaulay in order to study {\it unmixed polynomial ideals} are only particular examples of new formal differential geometric techniques that have been introduced recently in order to study {\it pure differential modules}. However these procedures are based on the formal theory of systems of ordinary differential (OD) or partial differential (PD) equations, in particular on a