**Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics**

J.F. Pommaret *CERMICS, Ecole Nationale des Ponts et Chaussées, France* 

#### **1. Introduction**

Let us revisit briefly the foundation of n-dimensional elasticity theory as it can be found today in any textbook, restricting our study to *n* = 2 for simplicity. If *x* = (*x*1, *x*2) is a point in the plane and *ξ* = (*ξ*1(*x*), *ξ*2(*x*)) is the displacement vector, lowering the indices by means of the Euclidean metric, we may introduce the "small" deformation tensor *�* = (*�ij* = *�ji* = (1/2)(*∂iξ<sup>j</sup>* + *∂jξi*)) with *n*(*n* + 1)/2 = 3 (independent) *components* (*�*11, *�*<sup>12</sup> = *�*21, *�*22). If we study a part of a deformed body, for example a thin elastic plane sheet, by means of a variational principle, we may introduce the local density of free energy *ϕ*(*�*) = *ϕ*(*�ij*|*i* ≤ *<sup>j</sup>*) = *<sup>ϕ</sup>*(*�*11, *�*12, *�*22) and vary the total free energy *<sup>F</sup>* = *<sup>ϕ</sup>*(*�*)*dx* with *dx* <sup>=</sup> *dx*<sup>1</sup> <sup>∧</sup> *dx*<sup>2</sup> by introducing *<sup>σ</sup>ij* <sup>=</sup> *∂ϕ*/*∂�ij* for *<sup>i</sup>* <sup>≤</sup> *<sup>j</sup>* in order to obtain *<sup>δ</sup><sup>F</sup>* <sup>=</sup> (*σ*11*δ�*<sup>11</sup> + *σ*12*δ�*<sup>12</sup> + *σ*22*δ�*22)*dx*. Accordingly, the "decision" to define the stress tensor *σ* by a symmetric matrix with *σ*<sup>12</sup> = *σ*<sup>21</sup> is purely artificial within such a variational principle. Indeed, the usual Cauchy device (1828) assumes that each element of a boundary surface is acted on by a surface density of force *σ* with a linear dependence *σ* = (*σir*(*x*)*nr*) on the outward normal unit vector *n* = (*nr*) and does not make any assumption on the stress tensor. It is only by an equilibrium of forces and couples, namely the well known *phenomenological static torsor equilibrium*, that one can "prove" the symmetry of *σ*. However, even if we assume this symmetry, *we now need the different summation σijδ�ij* = *σ*11*δ�*<sup>11</sup> + 2*σ*12*δ�*<sup>12</sup> + *σ*22*δ�*<sup>22</sup> = *σir∂rδξi*. An integration by parts and a change of sign produce the volume integral (*∂rσir*)*δξidx* leading to the stress equations *∂rσir* = 0. *The classical approach to elasticity theory, based on invariant theory with respect to the group of rigid motions, cannot therefore describe equilibrium of torsors by means of a variational principle where the proper torsor concept is totally lacking*.

There is another equivalent procedure dealing with a *variational calculus with constraint*. Indeed, as we shall see in Section 7, the deformation tensor is not any symmetric tensor as it must satisfy *<sup>n</sup>*2(*n*<sup>2</sup> <sup>−</sup> <sup>1</sup>)/12 compatibility conditions (CC), that is only *<sup>∂</sup>*22*�*<sup>11</sup> <sup>+</sup> *<sup>∂</sup>*11*�*<sup>22</sup> <sup>−</sup> 2*∂*12*�*<sup>12</sup> = 0 when *n* = 2. In this case, introducing the *Lagrange multiplier* −*φ* for convenience, *we have to vary* (*ϕ*(*�*) − *φ*(*∂*22*�*<sup>11</sup> + *∂*11*�*<sup>22</sup> − 2*∂*12*�*12))*dx for an arbitrary �*. A double integration by parts now provides the parametrization *<sup>σ</sup>*<sup>11</sup> <sup>=</sup> *<sup>∂</sup>*22*φ*, *<sup>σ</sup>*<sup>12</sup> <sup>=</sup> *<sup>σ</sup>*<sup>21</sup> <sup>=</sup> <sup>−</sup>*∂*12*φ*, *<sup>σ</sup>*<sup>22</sup> <sup>=</sup> *<sup>∂</sup>*11*<sup>φ</sup>* of the stress equations by means of the Airy function *φ* and the *formal adjoint* of the CC, *on the condition to observe that we have in fact* <sup>2</sup>*σ*<sup>12</sup> <sup>=</sup> <sup>−</sup>2*∂*12*<sup>φ</sup>* as another way to understand the deep meaning of the factor "2" in the summation. In arbitrary dimension, it just remains to notice

**2. From Lie groups to Lie pseudogroups**

*is said to be equivariant if f*(*ax*) = *a f*(*x*), ∀*x* ∈ *X*, ∀*a* ∈ *G.*

the *source* while the second factor is called the *target*.

*Lie*" that will be of constant use in the sequel ([26]):

*a principal homogeneous space (PHS) for G.*

of coordinates, a section transforms like ¯

*<sup>∂</sup>x*¯*<sup>r</sup>* (*ϕ*(*x*))*∂iϕ<sup>r</sup>*

*∂* ¯ *f l*

*ρ*(*x*)*ω<sup>ρ</sup>*

satisfying:

*∂x<sup>i</sup>*

like:

effective.

/*∂a<sup>σ</sup>* = *θ<sup>i</sup>*

Evariste Galois (1811-1832) introduced the word "*group*" for the first time in 1830. Then the group concept slowly passed from algebra (groups of permutations) to geometry (groups of transformations). It is only in 1880 that Sophus Lie (1842-1899) studied the groups of transformations depending on a finite number of parameters and now called *Lie groups of transformations*. Let *X* be a manifold with local coordinates *x* = (*x*1, ..., *xn*) and *G* be a *Lie group*, that is another manifold with local coordinates *a* = (*a*1, ..., *ap*) called *parameters* with a *composition G* <sup>×</sup> *<sup>G</sup>* <sup>→</sup> *<sup>G</sup>* : (*a*, *<sup>b</sup>*) <sup>→</sup> *ab*, an *inverse G* <sup>→</sup> *<sup>G</sup>* : *<sup>a</sup>* <sup>→</sup> *<sup>a</sup>*−<sup>1</sup> and an *identity e* <sup>∈</sup> *<sup>G</sup>*

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 3

(*ab*)*<sup>c</sup>* <sup>=</sup> *<sup>a</sup>*(*bc*) = *abc*, *aa*−<sup>1</sup> <sup>=</sup> *<sup>a</sup>*−1*<sup>a</sup>* <sup>=</sup> *<sup>e</sup>*, *ae* <sup>=</sup> *ea* <sup>=</sup> *<sup>a</sup>*, <sup>∀</sup>*a*, *<sup>b</sup>*, *<sup>c</sup>* <sup>∈</sup> *<sup>G</sup>*

**Definition 2.1.** *G is said to act on X if there is a map X* × *G* → *X* : (*x*, *a*) → *y* = *ax* = *f*(*x*, *a*) *such that* (*ab*)*x* = *a*(*bx*) = *abx*, ∀*a*, *b* ∈ *G*, ∀*x* ∈ *X and, for simplifying the notations, we shall use global notations even if only local actions are existing. The set Gx* = {*a* ∈ *G* | *ax* = *x*} *is called the isotropy subgroup of G at x* ∈ *X. The action is said to be effective if ax* = *x*, ∀*x* ∈ *X* ⇒ *a* = *e. A subset S* ⊂ *X is said to be invariant under the action of G if aS* ⊂ *S*, ∀*a* ∈ *G and the orbit of x* ∈ *X is the invariant subset Gx* = {*ax* | *a* ∈ *G*} ⊂ *X. If G acts on two manifolds X and Y, a map f* : *X* → *Y*

For reasons that will become clear later on, it is often convenient to introduce the *graph X* × *G* → *X* × *X* : (*x*, *a*) → (*x*, *y* = *ax*) of the action. In the product *X* × *X*, the first factor is called

**Definition 2.2.** *The action is said to be free if the graph is injective and transitive if the graph is surjective. The action is said to be simply transitive if the graph is an isomorphism and X is said to be*

In order to fix the notations, we quote without any proof the "*Three Fundamental Theorems of*

**First fundamental theorem**: The orbits *x* = *f*(*x*0, *a*) satisfy the system of PD equations

*generators* of the action and are linearly independent over the constants when the action is

If *X* is a manifold, we denote as usual by *T* = *T*(*X*) the *tangent bundle* of *X*, by *T*∗ = *T*∗(*X*) the *cotangent bundle*, by <sup>∧</sup>*rT*<sup>∗</sup> the *bundle of r-forms* and by *SqT*<sup>∗</sup> the *bundle of q-symmetric tensors*.

*i* = 1, ..., *n* and *k* = 1, ..., *m* simply denoted by (*x*, *y*), *projection π* : E → *X* : (*x*, *y*) → (*x*) and changes of local coordinates *x*¯ = *ϕ*(*x*), *y*¯ = *ψ*(*x*, *y*). If E and F are two fibered manifolds over *X* with respective local coordinates (*x*, *y*) and (*x*, *z*), we denote by E×*X*F the *fibered product* of E and F over *X* as the new fibered manifold over *X* with local coordinates (*x*, *y*, *z*). We denote by *f* : *X* → E : (*x*) → (*x*, *y* = *f*(*x*)) a global *section* of E, that is a map such that *π* ◦ *f* = *idX* but local sections over an open set *U* ⊂ *X* may also be considered when needed. Under a change

*<sup>∂</sup>x<sup>i</sup>* (*x*, *<sup>f</sup>*(*x*)) + *∂ψ<sup>l</sup>*

*<sup>ρ</sup>*(*x*)*∂<sup>i</sup>* are called *infinitesimal*

*f*(*ϕ*(*x*)) = *ψ*(*x*, *f*(*x*)) and the derivatives transform

*<sup>∂</sup>y<sup>k</sup>* (*x*, *<sup>f</sup>*(*x*))*∂<sup>i</sup> <sup>f</sup> <sup>k</sup>*(*x*)

, *yk*) for

*<sup>σ</sup>*(*a*) with *det*(*ω*) �<sup>=</sup> 0. The vector fields *θρ* <sup>=</sup> *<sup>θ</sup><sup>i</sup>*

More generally, let <sup>E</sup> be a *fibered manifold*, that is a manifold with local coordinates (*x<sup>i</sup>*

(*x*) = *∂ψ<sup>l</sup>*

that the above compatibility conditions are nothing else but the linearized Riemann tensor in Riemannan geometry, a crucial mathematical tool in the theory of general relativity.

It follows that the only possibility to revisit the foundations of engineering and mathematical physics is to use new mathematical methods, namely the theory of systems of partial differential equations and Lie pseudogroups developped by D.C. Spencer and coworkers during the period 1960-1975. In particular, Spencer invented the first order operator now wearing his name in order to bring in a canonical way the formal study of systems of ordinary differential (OD) or partial differential (PD) equations to that of equivalent first order systems. However, despite its importance, the *Spencer operator* is rarely used in mathematics today and, up to our knowledge, has never been used in engineering or mathematical physics. The main reason for such a situation is that the existing papers, largely based on hand-written lecture notes given by Spencer to his students (the author was among them in 1969) are quite technical and the problem also lies in the only "accessible" book "Lie equations" he published in 1972 with A. Kumpera. Indeed, the reader can easily check by himself that *the core of this book has nothing to do with its introduction* recalling known differential geometric concepts on which most of physics is based today.

The first and technical purpose of this chapter, 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 recall briefly its definition, both in the framework of systems of linear ordinary or partial differential equations and in the framework of differential modules. The local theory of Lie pseudogroups and the corresponding non-linear framework are also presented for the first time in a rather elementary manner though it is a difficult task.

The second and central purpose 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 the foundations of elasticity theory, commutative algebra, electromagnetism (EM) and general relativity (GR):

[C] E. and F. COSSERAT: "Théorie des Corps Déformables", Hermann, Paris, 1909. [M] F.S. MACAULAY: "The Algebraic Theory of Modular Systems", Cambridge, 1916. [W] H. WEYL: "Space, Time, Matter", Springer, Berlin, 1918 (1922, 1958; Dover, 1952).

Meanwhile we shall point out the striking importance of the second book for studying *identifiability* in control theory. We shall also obtain from the previous results the group theoretical unification of finite elements in engineering sciences (elasticity, heat, electromagnetism), solving the *torsor problem* and recovering in a purely mathematical way known *field-matter coupling phenomena* (piezzoelectricity, photoelasticity, streaming birefringence, viscosity, ...).

As a byproduct and though disturbing it may be, the third and perhaps essential purpose is to prove that *these unavoidable new differential and homological methods contradict the existing mathematical foundations of both engineering (continuum mechanics, electromagnetism) and mathematical (gauge theory, general relativity) physics*.

Many explicit examples will illustate this chapter which is deliberately written in a rather self-contained way to be accessible to a large audience, which does not mean that it is elementary in view of the number of new concepts that must be patched together. However, the reader must never forget that *each formula* appearing in this new general framework has been used explicitly or implicitly in [C], [M] and [W] for a mechanical, mathematical or physical purpose.

#### **2. From Lie groups to Lie pseudogroups**

2 Will-be-set-by-IN-TECH

that the above compatibility conditions are nothing else but the linearized Riemann tensor in

It follows that the only possibility to revisit the foundations of engineering and mathematical physics is to use new mathematical methods, namely the theory of systems of partial differential equations and Lie pseudogroups developped by D.C. Spencer and coworkers during the period 1960-1975. In particular, Spencer invented the first order operator now wearing his name in order to bring in a canonical way the formal study of systems of ordinary differential (OD) or partial differential (PD) equations to that of equivalent first order systems. However, despite its importance, the *Spencer operator* is rarely used in mathematics today and, up to our knowledge, has never been used in engineering or mathematical physics. The main reason for such a situation is that the existing papers, largely based on hand-written lecture notes given by Spencer to his students (the author was among them in 1969) are quite technical and the problem also lies in the only "accessible" book "Lie equations" he published in 1972 with A. Kumpera. Indeed, the reader can easily check by himself that *the core of this book has nothing to do with its introduction* recalling known differential geometric concepts on which

The first and technical purpose of this chapter, 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 recall briefly its definition, both in the framework of systems of linear ordinary or partial differential equations and in the framework of differential modules. The local theory of Lie pseudogroups and the corresponding non-linear framework are also presented for the

The second and central purpose 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 the foundations of elasticity theory, commutative

Meanwhile we shall point out the striking importance of the second book for studying *identifiability* in control theory. We shall also obtain from the previous results the group theoretical unification of finite elements in engineering sciences (elasticity, heat, electromagnetism), solving the *torsor problem* and recovering in a purely mathematical way known *field-matter coupling phenomena* (piezzoelectricity, photoelasticity, streaming

As a byproduct and though disturbing it may be, the third and perhaps essential purpose is to prove that *these unavoidable new differential and homological methods contradict the existing mathematical foundations of both engineering (continuum mechanics, electromagnetism) and*

Many explicit examples will illustate this chapter which is deliberately written in a rather self-contained way to be accessible to a large audience, which does not mean that it is elementary in view of the number of new concepts that must be patched together. However, the reader must never forget that *each formula* appearing in this new general framework has been used explicitly or implicitly in [C], [M] and [W] for a mechanical, mathematical or

[C] E. and F. COSSERAT: "Théorie des Corps Déformables", Hermann, Paris, 1909. [M] F.S. MACAULAY: "The Algebraic Theory of Modular Systems", Cambridge, 1916. [W] H. WEYL: "Space, Time, Matter", Springer, Berlin, 1918 (1922, 1958; Dover, 1952).

first time in a rather elementary manner though it is a difficult task.

algebra, electromagnetism (EM) and general relativity (GR):

*mathematical (gauge theory, general relativity) physics*.

Riemannan geometry, a crucial mathematical tool in the theory of general relativity.

most of physics is based today.

birefringence, viscosity, ...).

physical purpose.

Evariste Galois (1811-1832) introduced the word "*group*" for the first time in 1830. Then the group concept slowly passed from algebra (groups of permutations) to geometry (groups of transformations). It is only in 1880 that Sophus Lie (1842-1899) studied the groups of transformations depending on a finite number of parameters and now called *Lie groups of transformations*. Let *X* be a manifold with local coordinates *x* = (*x*1, ..., *xn*) and *G* be a *Lie group*, that is another manifold with local coordinates *a* = (*a*1, ..., *ap*) called *parameters* with a *composition G* <sup>×</sup> *<sup>G</sup>* <sup>→</sup> *<sup>G</sup>* : (*a*, *<sup>b</sup>*) <sup>→</sup> *ab*, an *inverse G* <sup>→</sup> *<sup>G</sup>* : *<sup>a</sup>* <sup>→</sup> *<sup>a</sup>*−<sup>1</sup> and an *identity e* <sup>∈</sup> *<sup>G</sup>* satisfying:

$$(ab)c = a(bc) = abc, \qquad \qquad aa^{-1} = a^{-1}a = e, \qquad \qquad ae = ea = a, \qquad \qquad \forall a, b, c \in G$$

**Definition 2.1.** *G is said to act on X if there is a map X* × *G* → *X* : (*x*, *a*) → *y* = *ax* = *f*(*x*, *a*) *such that* (*ab*)*x* = *a*(*bx*) = *abx*, ∀*a*, *b* ∈ *G*, ∀*x* ∈ *X and, for simplifying the notations, we shall use global notations even if only local actions are existing. The set Gx* = {*a* ∈ *G* | *ax* = *x*} *is called the isotropy subgroup of G at x* ∈ *X. The action is said to be effective if ax* = *x*, ∀*x* ∈ *X* ⇒ *a* = *e. A subset S* ⊂ *X is said to be invariant under the action of G if aS* ⊂ *S*, ∀*a* ∈ *G and the orbit of x* ∈ *X is the invariant subset Gx* = {*ax* | *a* ∈ *G*} ⊂ *X. If G acts on two manifolds X and Y, a map f* : *X* → *Y is said to be equivariant if f*(*ax*) = *a f*(*x*), ∀*x* ∈ *X*, ∀*a* ∈ *G.*

For reasons that will become clear later on, it is often convenient to introduce the *graph X* × *G* → *X* × *X* : (*x*, *a*) → (*x*, *y* = *ax*) of the action. In the product *X* × *X*, the first factor is called the *source* while the second factor is called the *target*.

**Definition 2.2.** *The action is said to be free if the graph is injective and transitive if the graph is surjective. The action is said to be simply transitive if the graph is an isomorphism and X is said to be a principal homogeneous space (PHS) for G.*

In order to fix the notations, we quote without any proof the "*Three Fundamental Theorems of Lie*" that will be of constant use in the sequel ([26]):

**First fundamental theorem**: The orbits *x* = *f*(*x*0, *a*) satisfy the system of PD equations *∂x<sup>i</sup>* /*∂a<sup>σ</sup>* = *θ<sup>i</sup> ρ*(*x*)*ω<sup>ρ</sup> <sup>σ</sup>*(*a*) with *det*(*ω*) �<sup>=</sup> 0. The vector fields *θρ* <sup>=</sup> *<sup>θ</sup><sup>i</sup> <sup>ρ</sup>*(*x*)*∂<sup>i</sup>* are called *infinitesimal generators* of the action and are linearly independent over the constants when the action is effective.

If *X* is a manifold, we denote as usual by *T* = *T*(*X*) the *tangent bundle* of *X*, by *T*∗ = *T*∗(*X*) the *cotangent bundle*, by <sup>∧</sup>*rT*<sup>∗</sup> the *bundle of r-forms* and by *SqT*<sup>∗</sup> the *bundle of q-symmetric tensors*. More generally, let <sup>E</sup> be a *fibered manifold*, that is a manifold with local coordinates (*x<sup>i</sup>* , *yk*) for *i* = 1, ..., *n* and *k* = 1, ..., *m* simply denoted by (*x*, *y*), *projection π* : E → *X* : (*x*, *y*) → (*x*) and changes of local coordinates *x*¯ = *ϕ*(*x*), *y*¯ = *ψ*(*x*, *y*). If E and F are two fibered manifolds over *X* with respective local coordinates (*x*, *y*) and (*x*, *z*), we denote by E×*X*F the *fibered product* of E and F over *X* as the new fibered manifold over *X* with local coordinates (*x*, *y*, *z*). We denote by *f* : *X* → E : (*x*) → (*x*, *y* = *f*(*x*)) a global *section* of E, that is a map such that *π* ◦ *f* = *idX* but local sections over an open set *U* ⊂ *X* may also be considered when needed. Under a change of coordinates, a section transforms like ¯ *f*(*ϕ*(*x*)) = *ψ*(*x*, *f*(*x*)) and the derivatives transform like:

$$\frac{\partial f^l}{\partial \mathfrak{x}^r}(\varphi(\mathfrak{x})) \partial\_i \varphi^r(\mathfrak{x}) = \frac{\partial \psi^l}{\partial \mathfrak{x}^i}(\mathfrak{x}, f(\mathfrak{x})) + \frac{\partial \psi^l}{\partial y^k}(\mathfrak{x}, f(\mathfrak{x})) \partial\_i f^k(\mathfrak{x})$$

The *Lie derivative* of an *r*-form with respect to a vector field *ξ* ∈ *T* is the linear first order operator L(*ξ*) linearly depending on *j*1(*ξ*) and uniquely defined by the following three

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 5

It can be proved that L(*ξ*) = *i*(*ξ*)*d* + *di*(*ξ*) where *i*(*ξ*) is the *interior multiplication* (*i*(*ξ*)*ω*)*i*1...*ir* =

Using crossed-derivatives in the PD equations of the First Fundamental Theorem and

invariant vector fields, we obtain the *compatibility conditions* (CC) expressed by the following

Applying *d* to the MC equations and substituting, we obtain the *integrability conditions* (IC): **Third fundamental theorem** For any Lie algebra G defined by structure constants satisfying :

> *μρc μ στ* + *c<sup>λ</sup> μσc μ τρ* + *c<sup>λ</sup> μτc μ ρσ* = 0

**Example 2.1.** *Considering the affine group of transformations of the real line y* = *a*1*x* + *a*2*, we obtain <sup>θ</sup>*<sup>1</sup> <sup>=</sup> *<sup>x</sup>∂x*, *<sup>θ</sup>*<sup>2</sup> <sup>=</sup> *<sup>∂</sup><sup>x</sup>* <sup>⇒</sup> [*θ*1, *<sup>θ</sup>*2] = <sup>−</sup>*θ*<sup>2</sup> *and thus <sup>ω</sup>*<sup>1</sup> = (1/*a*1)*da*1, *<sup>ω</sup>*<sup>2</sup> <sup>=</sup> *da*<sup>2</sup> <sup>−</sup> (*a*2/*a*1)*da*<sup>1</sup> <sup>⇒</sup> *<sup>d</sup>ω*<sup>1</sup> <sup>=</sup>

Only ten years later Lie discovered that the Lie groups of transformations are only particular examples of a wider class of groups of transformations along the following definition where

**Definition 2.6.** *A Lie pseudogroup of transformations* Γ ⊂ *aut*(*X*) *is a group of transformations solutions of a system of OD or PD equations such that, if y* = *f*(*x*) *and z* = *g*(*y*) *are two solutions, called finite transformations, that can be composed, then z* <sup>=</sup> *<sup>g</sup>* ◦ *<sup>f</sup>*(*x*) = *<sup>h</sup>*(*x*) *and x* <sup>=</sup> *<sup>f</sup>* <sup>−</sup>1(*y*) = *<sup>g</sup>*(*y*)

The underlying system may be nonlinear and of high order as we shall see later on. We shall speak of an *algebraic pseudogroup* when the system is defined by *differential polynomials* that is polynomials in the derivatives. In the case of Lie groups of transformations the system is obtained by differentiating the action law *y* = *f*(*x*, *a*) with respect to *x* as many times as necessary in order to eliminate the parameters. Looking for transformations "close" to the identity, that is setting *y* = *x* + *tξ*(*x*) + ... when *t* � 1 is a small constant parameter and passing to the limit *t* → 0, we may linearize the above nonlinear *system of finite Lie equations* in order to obtain a linear *system of infinitesimal Lie equations* of the same order for vector fields. Such a system has the property that, if *ξ*, *η* are two solutions, then [*ξ*, *η*] is also a solution. Accordingly, the set Θ ⊂ *T* of solutions of this new system satifies [Θ, Θ] ⊂ Θ and can

*<sup>σ</sup>*(*a*)*da<sup>σ</sup>* both with the matrix *<sup>α</sup>* = *<sup>ω</sup>*−<sup>1</sup> of right

*ρσω<sup>ρ</sup>* <sup>∧</sup> *<sup>ω</sup><sup>σ</sup>* <sup>=</sup> <sup>0</sup> *or the equivalent*

*ωii*1...*ir* and that [L(*ξ*),L(*η*)] = L(*ξ*) ◦ L(*η*) − L(*η*) ◦ L(*ξ*) = L([*ξ*, *η*]), ∀*ξ*, *η* ∈ *T*.

*<sup>∂</sup><sup>i</sup> <sup>f</sup>* , <sup>∀</sup> *<sup>f</sup>* ∈ ∧0*T*<sup>∗</sup> <sup>=</sup> *<sup>C</sup>*∞(*X*).

3. L(*ξ*)(*α* ∧ *β*)=(L(*ξ*)*α*) ∧ *β* + *α* ∧ (L(*ξ*)*β*), ∀*α*, *β* ∈ ∧*T*∗.

introducing the family of 1-forms *ω<sup>τ</sup>* = *ω<sup>τ</sup>*

*ρσατ.*

*cτ ρσ* + *<sup>c</sup><sup>τ</sup>*

*are also solutions while y* = *x is a solution.*

therefore be considered as the Lie algebra of Γ.

corollary where one must care about the sign used:

**Corollary 2.1.** *One has the Maurer-Cartan (MC) equations dω<sup>τ</sup>* + *c<sup>τ</sup>*

one can construct an analytic group *G* such that G = *Te*(*G*).

*aut*(*X*) denotes the group of all local diffeomorphisms of *X*:

*σρ* = 0, *<sup>c</sup><sup>λ</sup>*

0, *<sup>d</sup>ω*<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*<sup>1</sup> <sup>∧</sup> *<sup>ω</sup>*<sup>2</sup> <sup>=</sup> <sup>0</sup> <sup>⇔</sup> [*α*1, *<sup>α</sup>*2] = <sup>−</sup>*α*<sup>2</sup> *with <sup>α</sup>*<sup>1</sup> <sup>=</sup> *<sup>a</sup>*1*∂*<sup>1</sup> <sup>+</sup> *<sup>a</sup>*2*∂*2, *<sup>α</sup>*<sup>2</sup> <sup>=</sup> *<sup>∂</sup>*2*.*

properties:

*ξi*

1. <sup>L</sup>(*ξ*)*<sup>f</sup>* <sup>=</sup> *<sup>ξ</sup>*. *<sup>f</sup>* <sup>=</sup> *<sup>ξ</sup><sup>i</sup>*

2. L(*ξ*)*d* = *d*L(*ξ*).

*relations* [*αρ*, *ασ*] = *c<sup>τ</sup>*

We may introduce new coordinates (*x<sup>i</sup>* , *yk*, *y<sup>k</sup> <sup>i</sup>* ) transforming like:

$$
\overline{y}\_r^l \partial\_i \varphi^r(\mathbf{x}) = \frac{\partial \psi^l}{\partial \mathbf{x}^i}(\mathbf{x}\_\prime y) + \frac{\partial \psi^l}{\partial y^k}(\mathbf{x}\_\prime y) y\_i^k
$$

We shall denote by *Jq*(E) the *q-jet bundle* of <sup>E</sup> with local coordinates (*x<sup>i</sup>* , *yk*, *y<sup>k</sup> <sup>i</sup>* , *<sup>y</sup><sup>k</sup> ij*, ...) = (*x*, *yq*) called *jet coordinates* and sections *fq* : (*x*) <sup>→</sup> (*x*, *<sup>f</sup> <sup>k</sup>*(*x*), *<sup>f</sup> <sup>k</sup> <sup>i</sup>* (*x*), *<sup>f</sup> <sup>k</sup> ij*(*x*), ...)=(*x*, *fq*(*x*)) transforming like the sections *jq*(*f*) : (*x*) <sup>→</sup> (*x*, *<sup>f</sup> <sup>k</sup>*(*x*), *<sup>∂</sup><sup>i</sup> <sup>f</sup> <sup>k</sup>*(*x*), *<sup>∂</sup>ij <sup>f</sup> <sup>k</sup>*(*x*), ...)=(*x*, *jq*(*f*)(*x*)) where both *fq* and *jq*(*f*) are over the section *f* of E. Of course *Jq*(E) is a fibered manifold over *X* with projection *<sup>π</sup><sup>q</sup>* while *Jq*+*r*(E) is a fibered manifold over *Jq*(E) with projection *<sup>π</sup>q*+*<sup>r</sup> <sup>q</sup>* , <sup>∀</sup>*<sup>r</sup>* <sup>≥</sup> 0.

**Definition 2.3.** *A system of order q on* E *is a fibered submanifold* R*<sup>q</sup>* ⊂ *Jq*(E) *and a solution of* R*<sup>q</sup> is a section f of* E *such that jq*(*f*) *is a section of* R*q.*

**Definition 2.4.** *When the changes of coordinates have the linear form x*¯ = *ϕ*(*x*), *y*¯ = *A*(*x*)*y, we say that* E *is a vector bundle over X and denote for simplicity a vector bundle and its set of sections by the same capital letter E. When the changes of coordinates have the form x*¯ = *ϕ*(*x*), *y*¯ = *A*(*x*)*y* + *B*(*x*) *we say that* E *is an affine bundle over X and we define the associated vector bundle E over X by the local coordinates* (*x*, *v*) *changing like x*¯ = *ϕ*(*x*), *v*¯ = *A*(*x*)*v.*

**Definition 2.5.** *If the tangent bundle T*(E) *has local coordinates* (*x*, *<sup>y</sup>*, *<sup>u</sup>*, *<sup>v</sup>*) *changing like <sup>u</sup>*¯*<sup>j</sup>* <sup>=</sup> *∂iϕ<sup>j</sup>* (*x*)*u<sup>i</sup>* , *<sup>v</sup>*¯*<sup>l</sup>* <sup>=</sup> *∂ψ<sup>l</sup> <sup>∂</sup>xi* (*x*, *<sup>y</sup>*)*u<sup>i</sup>* <sup>+</sup> *∂ψ<sup>l</sup> <sup>∂</sup>yk* (*x*, *<sup>y</sup>*)*vk, we may introduce the vertical bundle V*(E) <sup>⊂</sup> *<sup>T</sup>*(E) *as a vector bundle over* E *with local coordinates* (*x*, *y*, *v*) *obtained by setting u* = 0 *and changes <sup>v</sup>*¯*<sup>l</sup>* <sup>=</sup> *∂ψ<sup>l</sup> <sup>∂</sup>yk* (*x*, *<sup>y</sup>*)*vk. Of course, when* <sup>E</sup> *is an affine bundle with associated vector bundle E over X, we have V*(E) = E ×*<sup>X</sup> E.*

For a later use, if <sup>E</sup> is a fibered manifold over *<sup>X</sup>* and *<sup>f</sup>* is a section of <sup>E</sup>, we denote by *<sup>f</sup>* <sup>−</sup>1(*V*(E)) the *reciprocal image* of *V*(E) by *f* as the vector bundle over *X* obtained when replacing (*x*, *y*, *v*) by (*x*, *f*(*x*), *v*) in each chart. It is important to notice in variational calculus that a *variation δ f* of *f* is such that *δ f*(*x*), as a vertical vector field not necessary "small", is a section of this vector bundle and that (*f* , *δ f*) is nothing else than a section of *V*(E) over *X*.

We now recall a few basic geometric concepts that will be constantly used. First of all, if *<sup>ξ</sup>*, *<sup>η</sup>* <sup>∈</sup> *<sup>T</sup>*, we define their *bracket* [*ξ*, *<sup>η</sup>*] <sup>∈</sup> *<sup>T</sup>* by the local formula ([*ξ*, *<sup>η</sup>*])*<sup>i</sup>* (*x*) = *ξr*(*x*)*∂rη<sup>i</sup>* (*x*) − *ηs*(*x*)*∂sξ<sup>i</sup>* (*x*) leading to the *Jacobi identity* [*ξ*, [*η*, *ζ*]] + [*η*, [*ζ*, *ξ*]] + [*ζ*, [*ξ*, *η*]] = 0, ∀*ξ*, *η*, *ζ* ∈ *T* allowing to define a *Lie algebra* and to the useful formula [*T*(*f*)(*ξ*), *T*(*f*)(*η*)] = *T*(*f*)([*ξ*, *η*]) where *T*(*f*) : *T*(*X*) → *T*(*Y*) is the tangent mapping of a map *f* : *X* → *Y*.

**Second fundamental theorem**: If *θ*1, ..., *θ<sup>p</sup>* are the infinitesimal generators of the effective action of a lie group *G* on *X*, then [*θρ*, *θσ*] = *c<sup>τ</sup> ρσθτ* where the *c<sup>τ</sup> ρσ* are the *structure constants* of a Lie algebra of vector fields which can be identified with G = *Te*(*G*).

When *<sup>I</sup>* <sup>=</sup> {*i*<sup>1</sup> <sup>&</sup>lt; ... <sup>&</sup>lt; *ir*} is a multi-index, we may set *dx<sup>I</sup>* <sup>=</sup> *dxi*<sup>1</sup> <sup>∧</sup>...<sup>∧</sup> *dxir* for describing <sup>∧</sup>*rT*<sup>∗</sup> and introduce the *exterior derivative d* : <sup>∧</sup>*rT*<sup>∗</sup> → ∧*r*+1*T*<sup>∗</sup> : *<sup>ω</sup>* <sup>=</sup> *<sup>ω</sup>Idx<sup>I</sup>* <sup>→</sup> *<sup>d</sup><sup>ω</sup>* <sup>=</sup> *<sup>∂</sup>iωIdx<sup>i</sup>* <sup>∧</sup> *dx<sup>I</sup>* with *<sup>d</sup>*<sup>2</sup> <sup>=</sup> *<sup>d</sup>* ◦ *<sup>d</sup>* <sup>≡</sup> 0 in the *Poincaré sequence*:

$$\wedge^0 T^\* \stackrel{d}{\longrightarrow} \wedge^1 T^\* \stackrel{d}{\longrightarrow} \wedge^2 T^\* \stackrel{d}{\longrightarrow} \dots \stackrel{d}{\longrightarrow} \wedge^n T^\* \longrightarrow 0$$

The *Lie derivative* of an *r*-form with respect to a vector field *ξ* ∈ *T* is the linear first order operator L(*ξ*) linearly depending on *j*1(*ξ*) and uniquely defined by the following three properties:

1. <sup>L</sup>(*ξ*)*<sup>f</sup>* <sup>=</sup> *<sup>ξ</sup>*. *<sup>f</sup>* <sup>=</sup> *<sup>ξ</sup><sup>i</sup> <sup>∂</sup><sup>i</sup> <sup>f</sup>* , <sup>∀</sup> *<sup>f</sup>* ∈ ∧0*T*<sup>∗</sup> <sup>=</sup> *<sup>C</sup>*∞(*X*). 2. L(*ξ*)*d* = *d*L(*ξ*). 3. L(*ξ*)(*α* ∧ *β*)=(L(*ξ*)*α*) ∧ *β* + *α* ∧ (L(*ξ*)*β*), ∀*α*, *β* ∈ ∧*T*∗.

4 Will-be-set-by-IN-TECH

transforming like the sections *jq*(*f*) : (*x*) <sup>→</sup> (*x*, *<sup>f</sup> <sup>k</sup>*(*x*), *<sup>∂</sup><sup>i</sup> <sup>f</sup> <sup>k</sup>*(*x*), *<sup>∂</sup>ij <sup>f</sup> <sup>k</sup>*(*x*), ...)=(*x*, *jq*(*f*)(*x*)) where both *fq* and *jq*(*f*) are over the section *f* of E. Of course *Jq*(E) is a fibered manifold over *X* with projection *<sup>π</sup><sup>q</sup>* while *Jq*+*r*(E) is a fibered manifold over *Jq*(E) with projection *<sup>π</sup>q*+*<sup>r</sup> <sup>q</sup>* , <sup>∀</sup>*<sup>r</sup>* <sup>≥</sup> 0.

**Definition 2.3.** *A system of order q on* E *is a fibered submanifold* R*<sup>q</sup>* ⊂ *Jq*(E) *and a solution of* R*<sup>q</sup>*

**Definition 2.4.** *When the changes of coordinates have the linear form x*¯ = *ϕ*(*x*), *y*¯ = *A*(*x*)*y, we say that* E *is a vector bundle over X and denote for simplicity a vector bundle and its set of sections by the same capital letter E. When the changes of coordinates have the form x*¯ = *ϕ*(*x*), *y*¯ = *A*(*x*)*y* + *B*(*x*) *we say that* E *is an affine bundle over X and we define the associated vector bundle E over X by the*

**Definition 2.5.** *If the tangent bundle T*(E) *has local coordinates* (*x*, *<sup>y</sup>*, *<sup>u</sup>*, *<sup>v</sup>*) *changing like <sup>u</sup>*¯*<sup>j</sup>* <sup>=</sup>

*as a vector bundle over* E *with local coordinates* (*x*, *y*, *v*) *obtained by setting u* = 0 *and changes*

For a later use, if <sup>E</sup> is a fibered manifold over *<sup>X</sup>* and *<sup>f</sup>* is a section of <sup>E</sup>, we denote by *<sup>f</sup>* <sup>−</sup>1(*V*(E)) the *reciprocal image* of *V*(E) by *f* as the vector bundle over *X* obtained when replacing (*x*, *y*, *v*) by (*x*, *f*(*x*), *v*) in each chart. It is important to notice in variational calculus that a *variation δ f* of *f* is such that *δ f*(*x*), as a vertical vector field not necessary "small", is a section of this vector

We now recall a few basic geometric concepts that will be constantly used. First of all, if

allowing to define a *Lie algebra* and to the useful formula [*T*(*f*)(*ξ*), *T*(*f*)(*η*)] = *T*(*f*)([*ξ*, *η*])

**Second fundamental theorem**: If *θ*1, ..., *θ<sup>p</sup>* are the infinitesimal generators of the effective

When *<sup>I</sup>* <sup>=</sup> {*i*<sup>1</sup> <sup>&</sup>lt; ... <sup>&</sup>lt; *ir*} is a multi-index, we may set *dx<sup>I</sup>* <sup>=</sup> *dxi*<sup>1</sup> <sup>∧</sup>...<sup>∧</sup> *dxir* for describing <sup>∧</sup>*rT*<sup>∗</sup> and introduce the *exterior derivative d* : <sup>∧</sup>*rT*<sup>∗</sup> → ∧*r*+1*T*<sup>∗</sup> : *<sup>ω</sup>* <sup>=</sup> *<sup>ω</sup>Idx<sup>I</sup>* <sup>→</sup> *<sup>d</sup><sup>ω</sup>* <sup>=</sup> *<sup>∂</sup>iωIdx<sup>i</sup>* <sup>∧</sup> *dx<sup>I</sup>*

(*x*) leading to the *Jacobi identity* [*ξ*, [*η*, *ζ*]] + [*η*, [*ζ*, *ξ*]] + [*ζ*, [*ξ*, *η*]] = 0, ∀*ξ*, *η*, *ζ* ∈ *T*

*ρσθτ* where the *c<sup>τ</sup>*

−→ ... *<sup>d</sup>* −→ ∧*nT*<sup>∗</sup> −→ <sup>0</sup>

*<sup>∂</sup>yk* (*x*, *<sup>y</sup>*)*vk. Of course, when* <sup>E</sup> *is an affine bundle with associated vector bundle E over X, we*

*<sup>∂</sup>x<sup>i</sup>* (*x*, *<sup>y</sup>*) + *∂ψ<sup>l</sup>*

*<sup>i</sup>* ) transforming like:

*<sup>∂</sup>y<sup>k</sup>* (*x*, *<sup>y</sup>*)*y<sup>k</sup>*

*i*

*<sup>∂</sup>yk* (*x*, *<sup>y</sup>*)*vk, we may introduce the vertical bundle V*(E) <sup>⊂</sup> *<sup>T</sup>*(E)

*<sup>i</sup>* (*x*), *<sup>f</sup> <sup>k</sup>*

, *yk*, *y<sup>k</sup> <sup>i</sup>* , *<sup>y</sup><sup>k</sup>*

(*x*) = *ξr*(*x*)*∂rη<sup>i</sup>*

*ρσ* are the *structure constants*

(*x*) −

*ij*(*x*), ...)=(*x*, *fq*(*x*))

*ij*, ...) =

, *yk*, *y<sup>k</sup>*

(*x*) = *∂ψ<sup>l</sup>*

We shall denote by *Jq*(E) the *q-jet bundle* of <sup>E</sup> with local coordinates (*x<sup>i</sup>*

(*x*, *yq*) called *jet coordinates* and sections *fq* : (*x*) <sup>→</sup> (*x*, *<sup>f</sup> <sup>k</sup>*(*x*), *<sup>f</sup> <sup>k</sup>*

We may introduce new coordinates (*x<sup>i</sup>*

*y*¯ *l r∂iϕ<sup>r</sup>*

*is a section f of* E *such that jq*(*f*) *is a section of* R*q.*

*∂iϕ<sup>j</sup>* (*x*)*u<sup>i</sup>*

*<sup>v</sup>*¯*<sup>l</sup>* <sup>=</sup> *∂ψ<sup>l</sup>*

*ηs*(*x*)*∂sξ<sup>i</sup>*

, *<sup>v</sup>*¯*<sup>l</sup>* <sup>=</sup> *∂ψ<sup>l</sup>*

*have V*(E) = E ×*<sup>X</sup> E.*

*local coordinates* (*x*, *v*) *changing like x*¯ = *ϕ*(*x*), *v*¯ = *A*(*x*)*v.*

bundle and that (*f* , *δ f*) is nothing else than a section of *V*(E) over *X*.

*<sup>ξ</sup>*, *<sup>η</sup>* <sup>∈</sup> *<sup>T</sup>*, we define their *bracket* [*ξ*, *<sup>η</sup>*] <sup>∈</sup> *<sup>T</sup>* by the local formula ([*ξ*, *<sup>η</sup>*])*<sup>i</sup>*

where *T*(*f*) : *T*(*X*) → *T*(*Y*) is the tangent mapping of a map *f* : *X* → *Y*.

of a Lie algebra of vector fields which can be identified with G = *Te*(*G*).

<sup>∧</sup>0*T*<sup>∗</sup> *<sup>d</sup>* −→ ∧1*T*<sup>∗</sup> *<sup>d</sup>* −→ ∧2*T*<sup>∗</sup> *<sup>d</sup>*

*<sup>∂</sup>xi* (*x*, *<sup>y</sup>*)*u<sup>i</sup>* <sup>+</sup> *∂ψ<sup>l</sup>*

action of a lie group *G* on *X*, then [*θρ*, *θσ*] = *c<sup>τ</sup>*

with *<sup>d</sup>*<sup>2</sup> <sup>=</sup> *<sup>d</sup>* ◦ *<sup>d</sup>* <sup>≡</sup> 0 in the *Poincaré sequence*:

It can be proved that L(*ξ*) = *i*(*ξ*)*d* + *di*(*ξ*) where *i*(*ξ*) is the *interior multiplication* (*i*(*ξ*)*ω*)*i*1...*ir* = *ξi ωii*1...*ir* and that [L(*ξ*),L(*η*)] = L(*ξ*) ◦ L(*η*) − L(*η*) ◦ L(*ξ*) = L([*ξ*, *η*]), ∀*ξ*, *η* ∈ *T*.

Using crossed-derivatives in the PD equations of the First Fundamental Theorem and introducing the family of 1-forms *ω<sup>τ</sup>* = *ω<sup>τ</sup> <sup>σ</sup>*(*a*)*da<sup>σ</sup>* both with the matrix *<sup>α</sup>* = *<sup>ω</sup>*−<sup>1</sup> of right invariant vector fields, we obtain the *compatibility conditions* (CC) expressed by the following corollary where one must care about the sign used:

**Corollary 2.1.** *One has the Maurer-Cartan (MC) equations dω<sup>τ</sup>* + *c<sup>τ</sup> ρσω<sup>ρ</sup>* <sup>∧</sup> *<sup>ω</sup><sup>σ</sup>* <sup>=</sup> <sup>0</sup> *or the equivalent relations* [*αρ*, *ασ*] = *c<sup>τ</sup> ρσατ.*

Applying *d* to the MC equations and substituting, we obtain the *integrability conditions* (IC):

**Third fundamental theorem** For any Lie algebra G defined by structure constants satisfying :

$$c^{\tau}\_{\rho\sigma} + c^{\tau}\_{\sigma\rho} = 0,\qquad c^{\lambda}\_{\mu\rho}c^{\mu}\_{\sigma\tau} + c^{\lambda}\_{\mu\sigma}c^{\mu}\_{\tau\rho} + c^{\lambda}\_{\mu\tau}c^{\mu}\_{\rho\sigma} = 0$$

one can construct an analytic group *G* such that G = *Te*(*G*).

**Example 2.1.** *Considering the affine group of transformations of the real line y* = *a*1*x* + *a*2*, we obtain <sup>θ</sup>*<sup>1</sup> <sup>=</sup> *<sup>x</sup>∂x*, *<sup>θ</sup>*<sup>2</sup> <sup>=</sup> *<sup>∂</sup><sup>x</sup>* <sup>⇒</sup> [*θ*1, *<sup>θ</sup>*2] = <sup>−</sup>*θ*<sup>2</sup> *and thus <sup>ω</sup>*<sup>1</sup> = (1/*a*1)*da*1, *<sup>ω</sup>*<sup>2</sup> <sup>=</sup> *da*<sup>2</sup> <sup>−</sup> (*a*2/*a*1)*da*<sup>1</sup> <sup>⇒</sup> *<sup>d</sup>ω*<sup>1</sup> <sup>=</sup> 0, *<sup>d</sup>ω*<sup>2</sup> <sup>−</sup> *<sup>ω</sup>*<sup>1</sup> <sup>∧</sup> *<sup>ω</sup>*<sup>2</sup> <sup>=</sup> <sup>0</sup> <sup>⇔</sup> [*α*1, *<sup>α</sup>*2] = <sup>−</sup>*α*<sup>2</sup> *with <sup>α</sup>*<sup>1</sup> <sup>=</sup> *<sup>a</sup>*1*∂*<sup>1</sup> <sup>+</sup> *<sup>a</sup>*2*∂*2, *<sup>α</sup>*<sup>2</sup> <sup>=</sup> *<sup>∂</sup>*2*.*

Only ten years later Lie discovered that the Lie groups of transformations are only particular examples of a wider class of groups of transformations along the following definition where *aut*(*X*) denotes the group of all local diffeomorphisms of *X*:

**Definition 2.6.** *A Lie pseudogroup of transformations* Γ ⊂ *aut*(*X*) *is a group of transformations solutions of a system of OD or PD equations such that, if y* = *f*(*x*) *and z* = *g*(*y*) *are two solutions, called finite transformations, that can be composed, then z* <sup>=</sup> *<sup>g</sup>* ◦ *<sup>f</sup>*(*x*) = *<sup>h</sup>*(*x*) *and x* <sup>=</sup> *<sup>f</sup>* <sup>−</sup>1(*y*) = *<sup>g</sup>*(*y*) *are also solutions while y* = *x is a solution.*

The underlying system may be nonlinear and of high order as we shall see later on. We shall speak of an *algebraic pseudogroup* when the system is defined by *differential polynomials* that is polynomials in the derivatives. In the case of Lie groups of transformations the system is obtained by differentiating the action law *y* = *f*(*x*, *a*) with respect to *x* as many times as necessary in order to eliminate the parameters. Looking for transformations "close" to the identity, that is setting *y* = *x* + *tξ*(*x*) + ... when *t* � 1 is a small constant parameter and passing to the limit *t* → 0, we may linearize the above nonlinear *system of finite Lie equations* in order to obtain a linear *system of infinitesimal Lie equations* of the same order for vector fields. Such a system has the property that, if *ξ*, *η* are two solutions, then [*ξ*, *η*] is also a solution. Accordingly, the set Θ ⊂ *T* of solutions of this new system satifies [Θ, Θ] ⊂ Θ and can therefore be considered as the Lie algebra of Γ.

We shall now describe the second part with more details as it has been (and still is !) the crucial tool used in both engineering (analytical and continuum mechanics) and mathematical (gauge theory and general relativity) physics in an absolutely contradictory manner. We shall try to use the least amount of mathematics in order to prepare the reader for the results presented in the next sections. For this let us start with an elementary experiment that will link at once continuum mechanics and gauge theory in an unusual way. Let us put a thin elastic rectilinear rubber band along the *x* axis on a flat table and translate it along itself. The band will remain identical as no deformation can be produced by this constant translation. However, if we move each point continuously along the same direction but in a point depending way, for example fixing one end and pulling on the other end, there will be of course a deformation of the elastic band according to the Hooke law. It remains to notice that a constant translation can be written in the form *y* = *x* + *a*<sup>2</sup> as in Example 2.1 while a point depending translation can be written in the form *y* = *x* + *a*2(*x*) which is written in any textbook of continuum mechanics in the form *y* = *x* + *ξ*(*x*) by introducing the *displacement vector ξ*. However nobody could even imagine to make *a*<sup>1</sup> also point depending and to consider *y* = *a*1(*x*)*x* + *a*2(*x*) as we shall do in Example 7.2.We also notice that the movement of a rigid body in space may be written in the form *y* = *a*(*t*)*x* + *b*(*t*) where now *a*(*t*) is a time depending orthogonal matrix and *b*(*t*) is a time depending vector. What makes all the difference between the two examples is that the group is *acting* on *x* in the first but *not acting* on *t* in the second. Finally, a point depending rotation or dilatation is not accessible to intuition and the general theory must be done in the

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 7

If *X* is a manifold and *G* is a lie group *not acting necessarily* on *X*, let us consider maps *a* : *X* → *G* : (*x*) → (*a*(*x*)) or equivalently sections of the trivial (principal) bundle *X* × *G* over *X*. If *x* + *dx* is a point of *X* close to *x*, then *T*(*a*) will provide a point *a* + *da* = *a* + *<sup>∂</sup><sup>a</sup>*

close to *a* on *G*. We may bring *a* back to *e* on *G* by acting on *a* with *a*−1, *either on the left or on the right*, getting therefore a 1-form *a*−1*da* = *A* or *daa*−<sup>1</sup> = *B*. As *aa*−<sup>1</sup> = *e* we also get *daa*−<sup>1</sup> <sup>=</sup> <sup>−</sup>*ada*−<sup>1</sup> <sup>=</sup> <sup>−</sup>*b*−1*db* if we set *<sup>b</sup>* <sup>=</sup> *<sup>a</sup>*−<sup>1</sup> as a way to link *<sup>A</sup>* with *<sup>B</sup>*. When there is an action *y* = *ax*, we have *x* = *a*−1*y* = *by* and thus *dy* = *dax* = *daa*−1*y*, a result leading through

*<sup>i</sup>* (*x*)*dx<sup>i</sup>* <sup>=</sup> <sup>−</sup>*ω<sup>τ</sup>*

*<sup>i</sup>* (*x*)*dx<sup>i</sup>* <sup>=</sup> *<sup>ω</sup><sup>τ</sup>*

the 2-form *dA* <sup>−</sup> [*A*, *<sup>A</sup>*] = *<sup>F</sup>* ∈ ∧2*T*<sup>∗</sup> ⊗ G by the local formula (care to the sign):

linear operator *<sup>d</sup>* : <sup>∧</sup>0*T*<sup>∗</sup> ⊗G→∧1*T*<sup>∗</sup> ⊗ G : (*λτ*(*x*)) <sup>→</sup> (*∂iλτ*(*x*)) leading to:

*<sup>i</sup>* (*x*) <sup>−</sup> *<sup>c</sup><sup>τ</sup>*

*<sup>X</sup>* <sup>×</sup> *<sup>G</sup>* −→ *<sup>T</sup>*<sup>∗</sup> ⊗ G *MC* −→ ∧2*T*<sup>∗</sup> ⊗ G *<sup>a</sup>* −→ *<sup>a</sup>*−1*da* <sup>=</sup> *<sup>A</sup>* −→ *dA* <sup>−</sup> [*A*, *<sup>A</sup>*] = *<sup>F</sup>*

Choosing *a* "close" to *e*, that is *a*(*x*) = *e* + *tλ*(*x*) + ... and linearizing as usual, we obtain the

Introducing the induced bracket [*A*, *A*](*ξ*, *η*)=[*A*(*ξ*), *A*(*η*)] ∈ G, ∀*ξ*, *η* ∈ *T*, we may define

*ρσA<sup>ρ</sup>*

*<sup>i</sup>* (*x*)*A<sup>σ</sup>*

*σ*(*b*(*x*))*∂ib<sup>σ</sup>*(*x*)*dx<sup>i</sup>*

*σ*(*a*(*x*))*∂ia<sup>σ</sup>*(*x*)*dx<sup>i</sup>*

*<sup>j</sup>* (*x*) = *<sup>F</sup><sup>τ</sup>*

)

)

*ij*(*x*)

the First Fundamental Theorem of Lie to the equivalent formulas:

*a*−1*da* = *A* = (*A<sup>τ</sup>*

*daa*−<sup>1</sup> = *B* = (*B<sup>τ</sup>*

*<sup>j</sup>* (*x*) <sup>−</sup> *<sup>∂</sup>jA<sup>τ</sup>*

*∂iA<sup>τ</sup>*

and obtain from the second fundamental theorem: **Theorem 3.1.** *There is a nonlinear gauge sequence:*

*<sup>∂</sup><sup>x</sup> dx*

following manner.

Though the collected works of Lie have been published by his student F. Engel at the end of the 19*th* century, these ideas did not attract a large audience because the fashion in Europe was analysis. Accordingly, at the beginning of the 20*th* century and for more than fifty years, only two frenchmen tried to go further in the direction pioneered by Lie, namely Elie Cartan (1869-1951) who is quite famous today and Ernest Vessiot (1865-1952) who is almost ignored today, each one deliberately ignoring the other during his life for a precise reason that we now explain with details as it will surprisingly constitute the heart of this chapter. (The author is indebted to Prof. Maurice Janet (1888-1983), who was a personal friend of Vessiot, for the many documents he gave him, partly published in [25]). Roughly, the idea of many people at that time was to extend the work of Galois along the following scheme of increasing difficulty:

1) *Galois theory* ([34]): Algebraic equations and permutation groups.

2) *Picard-Vessiot theory* ([17]): OD equations and Lie groups.

3) *Differential Galois theory* ([24],[37]): PD equations and Lie pseudogroups.

In 1898 Jules Drach (1871-1941) got and published a thesis ([9]) with a jury made by Gaston Darboux, Emile Picard and Henri Poincaré, the best leading mathematicians of that time. However, despite the fact that it contains ideas quite in advance on his time such as the concept of a "differential field" only introduced by J.-F. Ritt in 1930 ([31]), the jury did not notice that the main central result was wrong: Cartan found the counterexamples, Vessiot recognized the mistake, Paul Painlevé told it to Picard who agreed but Drach never wanted to acknowledge this fact and was supported by the influent Emile Borel. As a byproduct, everybody flew out of this "affair", never touching again the Galois theory. After publishing a prize-winning paper in 1904 where he discovered for the first time that the differential Galois theory must be a theory of (irreducible) PHS for algebraic pseudogroups, Vessiot remained alone, trying during thirty years to correct the mistake of Drach that we finally corrected in 1983 ([24]).

#### **3. Cartan versus Vessiot : The structure equations**

We study first the work of Cartan which can be divided into two parts. The first part, for which he invented exterior calculus, may be considered as a tentative to extend the MC equations from Lie groups to Lie pseudogroups. The idea for that is to consider the system of order *q* and its *prolongations* obtained by differentiating the equations as a way to know certain derivatives called *principal* from all the other arbitrary ones called *parametric* in the sense of Janet ([13]). Replacing the derivatives by jet coordinates, we may try to copy the procedure leading to the MC equations by using a kind of "composition" and "inverse" on the jet coordinates. For example, coming back to the last definition, we get successively:

$$
\frac{\partial h}{\partial \mathbf{x}} = \frac{\partial \mathbf{g}}{\partial y} \frac{\partial f}{\partial \mathbf{x'}} \qquad \frac{\partial^2 h}{\partial \mathbf{x'}^2} = \frac{\partial^2 g}{\partial y^2} \frac{\partial f}{\partial \mathbf{x}} \frac{\partial f}{\partial \mathbf{x}} + \frac{\partial g}{\partial y} \frac{\partial^2 f}{\partial \mathbf{x}^2} \dots
$$

Now if *<sup>g</sup>* <sup>=</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup> then *<sup>g</sup>* ◦ *<sup>f</sup>* <sup>=</sup> *id* and thus *<sup>∂</sup><sup>g</sup> ∂y ∂ f <sup>∂</sup><sup>x</sup>* = 1, ... while the new identity *idq* = *jq*(*id*) is made by the successive derivatives of *y* = *x*, namely (1, 0, 0, ...). These *awfully complicated computations* bring a lot of structure constants and have been so much superseded by the work of Donald C. Spencer (1912-2001) ([11],[12],[18],[33]) that, in our opinion based on thirty years of explicit computations, this tentative has only been used for classification problems and is not useful for applications compared to the results of the next sections. In a single concluding sentence, *Cartan has not been able to "go down" to the base manifold X while Spencer did succeed fifty years later*.

6 Will-be-set-by-IN-TECH

Though the collected works of Lie have been published by his student F. Engel at the end of the 19*th* century, these ideas did not attract a large audience because the fashion in Europe was analysis. Accordingly, at the beginning of the 20*th* century and for more than fifty years, only two frenchmen tried to go further in the direction pioneered by Lie, namely Elie Cartan (1869-1951) who is quite famous today and Ernest Vessiot (1865-1952) who is almost ignored today, each one deliberately ignoring the other during his life for a precise reason that we now explain with details as it will surprisingly constitute the heart of this chapter. (The author is indebted to Prof. Maurice Janet (1888-1983), who was a personal friend of Vessiot, for the many documents he gave him, partly published in [25]). Roughly, the idea of many people at that time was to extend the work of Galois along the following scheme of increasing difficulty:

In 1898 Jules Drach (1871-1941) got and published a thesis ([9]) with a jury made by Gaston Darboux, Emile Picard and Henri Poincaré, the best leading mathematicians of that time. However, despite the fact that it contains ideas quite in advance on his time such as the concept of a "differential field" only introduced by J.-F. Ritt in 1930 ([31]), the jury did not notice that the main central result was wrong: Cartan found the counterexamples, Vessiot recognized the mistake, Paul Painlevé told it to Picard who agreed but Drach never wanted to acknowledge this fact and was supported by the influent Emile Borel. As a byproduct, everybody flew out of this "affair", never touching again the Galois theory. After publishing a prize-winning paper in 1904 where he discovered for the first time that the differential Galois theory must be a theory of (irreducible) PHS for algebraic pseudogroups, Vessiot remained alone, trying during thirty years to correct the mistake of Drach that we finally corrected in

We study first the work of Cartan which can be divided into two parts. The first part, for which he invented exterior calculus, may be considered as a tentative to extend the MC equations from Lie groups to Lie pseudogroups. The idea for that is to consider the system of order *q* and its *prolongations* obtained by differentiating the equations as a way to know certain derivatives called *principal* from all the other arbitrary ones called *parametric* in the sense of Janet ([13]). Replacing the derivatives by jet coordinates, we may try to copy the procedure leading to the MC equations by using a kind of "composition" and "inverse" on the jet coordinates. For

> *<sup>∂</sup>x*<sup>2</sup> <sup>=</sup> *<sup>∂</sup>*2*<sup>g</sup> ∂y*<sup>2</sup> *∂ f ∂x ∂ f ∂x* + *∂g ∂y*

> > *∂y ∂ f*

is made by the successive derivatives of *y* = *x*, namely (1, 0, 0, ...). These *awfully complicated computations* bring a lot of structure constants and have been so much superseded by the work of Donald C. Spencer (1912-2001) ([11],[12],[18],[33]) that, in our opinion based on thirty years of explicit computations, this tentative has only been used for classification problems and is not useful for applications compared to the results of the next sections. In a single concluding sentence, *Cartan has not been able to "go down" to the base manifold X while Spencer did succeed*

*∂*<sup>2</sup> *f <sup>∂</sup>x*<sup>2</sup> , ...

*<sup>∂</sup><sup>x</sup>* = 1, ... while the new identity *idq* = *jq*(*id*)

1) *Galois theory* ([34]): Algebraic equations and permutation groups.

3) *Differential Galois theory* ([24],[37]): PD equations and Lie pseudogroups.

2) *Picard-Vessiot theory* ([17]): OD equations and Lie groups.

**3. Cartan versus Vessiot : The structure equations**

example, coming back to the last definition, we get successively:

, *<sup>∂</sup>*2*<sup>h</sup>*

*∂h <sup>∂</sup><sup>x</sup>* <sup>=</sup> *<sup>∂</sup><sup>g</sup> ∂y ∂ f ∂x*

Now if *<sup>g</sup>* <sup>=</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup> then *<sup>g</sup>* ◦ *<sup>f</sup>* <sup>=</sup> *id* and thus *<sup>∂</sup><sup>g</sup>*

1983 ([24]).

*fifty years later*.

We shall now describe the second part with more details as it has been (and still is !) the crucial tool used in both engineering (analytical and continuum mechanics) and mathematical (gauge theory and general relativity) physics in an absolutely contradictory manner. We shall try to use the least amount of mathematics in order to prepare the reader for the results presented in the next sections. For this let us start with an elementary experiment that will link at once continuum mechanics and gauge theory in an unusual way. Let us put a thin elastic rectilinear rubber band along the *x* axis on a flat table and translate it along itself. The band will remain identical as no deformation can be produced by this constant translation. However, if we move each point continuously along the same direction but in a point depending way, for example fixing one end and pulling on the other end, there will be of course a deformation of the elastic band according to the Hooke law. It remains to notice that a constant translation can be written in the form *y* = *x* + *a*<sup>2</sup> as in Example 2.1 while a point depending translation can be written in the form *y* = *x* + *a*2(*x*) which is written in any textbook of continuum mechanics in the form *y* = *x* + *ξ*(*x*) by introducing the *displacement vector ξ*. However nobody could even imagine to make *a*<sup>1</sup> also point depending and to consider *y* = *a*1(*x*)*x* + *a*2(*x*) as we shall do in Example 7.2.We also notice that the movement of a rigid body in space may be written in the form *y* = *a*(*t*)*x* + *b*(*t*) where now *a*(*t*) is a time depending orthogonal matrix and *b*(*t*) is a time depending vector. What makes all the difference between the two examples is that the group is *acting* on *x* in the first but *not acting* on *t* in the second. Finally, a point depending rotation or dilatation is not accessible to intuition and the general theory must be done in the following manner.

If *X* is a manifold and *G* is a lie group *not acting necessarily* on *X*, let us consider maps *a* : *X* → *G* : (*x*) → (*a*(*x*)) or equivalently sections of the trivial (principal) bundle *X* × *G* over *X*. If *x* + *dx* is a point of *X* close to *x*, then *T*(*a*) will provide a point *a* + *da* = *a* + *<sup>∂</sup><sup>a</sup> <sup>∂</sup><sup>x</sup> dx* close to *a* on *G*. We may bring *a* back to *e* on *G* by acting on *a* with *a*−1, *either on the left or on the right*, getting therefore a 1-form *a*−1*da* = *A* or *daa*−<sup>1</sup> = *B*. As *aa*−<sup>1</sup> = *e* we also get *daa*−<sup>1</sup> <sup>=</sup> <sup>−</sup>*ada*−<sup>1</sup> <sup>=</sup> <sup>−</sup>*b*−1*db* if we set *<sup>b</sup>* <sup>=</sup> *<sup>a</sup>*−<sup>1</sup> as a way to link *<sup>A</sup>* with *<sup>B</sup>*. When there is an action *y* = *ax*, we have *x* = *a*−1*y* = *by* and thus *dy* = *dax* = *daa*−1*y*, a result leading through the First Fundamental Theorem of Lie to the equivalent formulas:

$$a^{-1}da = A = (A\_i^{\tau}(\mathbf{x})d\mathbf{x}^i = -\omega\_\sigma^\tau(b(\mathbf{x}))\partial\_i b^\sigma(\mathbf{x})d\mathbf{x}^i),$$

$$dda^{-1} = B = (B\_i^{\mathsf{T}}(\mathfrak{x})d\mathfrak{x}^i = \omega\_\sigma^{\mathsf{T}}(a(\mathfrak{x}))\partial\_i a^\sigma(\mathfrak{x})d\mathfrak{x}^i)^\*$$

Introducing the induced bracket [*A*, *A*](*ξ*, *η*)=[*A*(*ξ*), *A*(*η*)] ∈ G, ∀*ξ*, *η* ∈ *T*, we may define the 2-form *dA* <sup>−</sup> [*A*, *<sup>A</sup>*] = *<sup>F</sup>* ∈ ∧2*T*<sup>∗</sup> ⊗ G by the local formula (care to the sign):

$$
\partial\_i A\_j^\tau(\mathbf{x}) - \partial\_j A\_i^\tau(\mathbf{x}) - c\_{\rho\sigma}^\tau A\_i^\rho(\mathbf{x}) A\_j^\sigma(\mathbf{x}) = F\_{ij}^\tau(\mathbf{x})
$$

and obtain from the second fundamental theorem:

**Theorem 3.1.** *There is a nonlinear gauge sequence:*

$$\begin{array}{ccccc} X \times G \longrightarrow & T^\* \otimes \mathcal{G} & \stackrel{M\mathbb{C}}{\longrightarrow} & \wedge^2 T^\* \otimes \mathcal{G} \\ a & \longrightarrow a^{-1} da = A \longrightarrow dA - [A, A] = F \end{array}$$

Choosing *a* "close" to *e*, that is *a*(*x*) = *e* + *tλ*(*x*) + ... and linearizing as usual, we obtain the linear operator *<sup>d</sup>* : <sup>∧</sup>0*T*<sup>∗</sup> ⊗G→∧1*T*<sup>∗</sup> ⊗ G : (*λτ*(*x*)) <sup>→</sup> (*∂iλτ*(*x*)) leading to:

Let us now turn to the other way proposed by Vessiot in 1903 ([36]) and 1904 ([37]). Our purpose is only to sketch the main results that we have obtained in many books ([23-26], we do not know other references) and to illustrate them by a series of specific examples, asking

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 9

1. If E = *X* × *X*, we shall denote by Π*<sup>q</sup>* = Π*q*(*X*, *X*) the open subfibered manifold of

2. The Lie algebra Θ ⊂ *T* of infinitesimal transformations is then obtained by linearization, setting *y* = *x* + *tξ*(*x*) + ... and passing to the limit *t* → 0 in order to obtain the linear

3. Passing from source to target, we may *prolong* the vertical infinitesimal transformations

+ ( *<sup>∂</sup>*2*η<sup>k</sup> <sup>∂</sup>yr∂y<sup>s</sup> <sup>y</sup><sup>r</sup>*

where we have replaced *jq*(*f*)(*x*) by *yq*, each component beeing the "formal" derivative of

the differential invariants. Keeping in mind the well known property of the Jacobian determinant while passing to the finite point of view, any (local) transformation *y* = *f*(*x*) can be lifted to a (local) transformation of the differential invariants between themselves of the form *u* → *λ*(*u*, *jq*(*f*)(*x*)) allowing to introduce a *natural bundle* F over *X* by patching changes of coordinates *x*¯ = *ϕ*(*x*), *u*¯ = *λ*(*u*, *jq*(*ϕ*)(*x*)). A section *ω* of F is called a *geometric object* or *structure* on *X* and transforms like *ω*¯(*f*(*x*)) = *λ*(*ω*(*x*), *jq*(*f*)(*x*)) or simply *ω*¯ = *jq*(*f*)(*ω*). This is a way to generalize vectors and tensors (*q* = 1) or even connections (*<sup>q</sup>* <sup>=</sup> 2). As a byproduct we have <sup>Γ</sup> <sup>=</sup> { *<sup>f</sup>* <sup>∈</sup> *aut*(*X*)|Φ*ω*(*jq*(*f*)) = *jq*(*f*)−1(*ω*) = *<sup>ω</sup>*} as a new way to write out the Lie form and we may say that Γ *preserves ω*. We also obtain

*ft* = *exp*(*tξ*) ∈ *aut*(*X*), ∀*ξ* ∈ *T*, we may define the *ordinary Lie derivative* with value in

*<sup>q</sup>* (*ω*) = *ω*}. Coming back to the infinitesimal point of view and setting

4. As [Θ, Θ] ⊂ Θ, we may use the Frobenius theorem in order to find a generating fundamental set of *differential invariants* {Φ*τ*(*yq*)} up to order *<sup>q</sup>* which are such that <sup>Φ</sup>*τ*(*y*¯*q*) = <sup>Φ</sup>*τ*(*yq*) by using the chain rule for derivatives whenever *<sup>y</sup>*¯ <sup>=</sup> *<sup>g</sup>*(*y*) <sup>∈</sup> <sup>Γ</sup> acting now on *Y*. Of course, in actual practice *one must use sections of Rq instead of solutions* but it is only in section 6 that we shall see why the use of the Spencer operator will be crucial for this purpose. Specializing the <sup>Φ</sup>*<sup>τ</sup>* at *idq*(*x*) we obtain the *Lie form* <sup>Φ</sup>*τ*(*yq*) = *<sup>ω</sup>τ*(*x*) of <sup>R</sup>*q*. 5. The main discovery of Vessiot, fifty years in advance, has been to notice that the

*i ys <sup>j</sup>* <sup>+</sup> *∂η<sup>k</sup> <sup>∂</sup>y<sup>r</sup> <sup>y</sup><sup>r</sup> ij*) *<sup>∂</sup> ∂y<sup>k</sup> ij* + ...

*<sup>∂</sup>yk* to the jet coordinates up to order *<sup>q</sup>* in order to obtain:

*<sup>q</sup>* (*V*(R*q*)) ⊂ *Jq*(*T*) by reciprocal image with Θ = {*ξ* ∈

(*x*) *<sup>∂</sup>*

*<sup>∂</sup>xi* commutes with the

*<sup>∂</sup>yk* , exchanging therefore

*projection α<sup>q</sup>* : Π*<sup>q</sup>* → *X* : (*x*, *yq*) → (*x*) and *target projection β<sup>q</sup>* : Π*<sup>q</sup>* → *X* : (*x*, *yq*) → (*y*). We shall sometimes introduce a copy *Y* of *X* with local coordinates (*y*) in order to avoid any confusion between the source and the target manifolds. Let us start with a Lie pseudogroup Γ ⊂ *aut*(*X*) defined by a system R*<sup>q</sup>* ⊂ Π*<sup>q</sup>* of order *q*. In all the sequel we shall suppose that the system is involutive (see next section) and that Γ is *transitive* that is ∀*x*, *y* ∈ *X*, ∃ *f* ∈ Γ, *y* = *f*(*x*) or, equivalently, the map (*αq*, *βq*) : R*<sup>q</sup>* → *X* × *X* : (*x*, *yq*) →

*<sup>i</sup>* ) �= 0 with *source*

*Jq*(*<sup>X</sup>* <sup>×</sup> *<sup>X</sup>*) defined independently of the coordinate system by *det*(*y<sup>k</sup>*

the reader to imagine any link with what has been said.

(*x*, *y*) is surjective.

*T*|*jq*(*ξ*) ∈ *Rq*}.

the previous one .

<sup>R</sup>*<sup>q</sup>* <sup>=</sup> { *fq* <sup>∈</sup> <sup>Π</sup>*q*<sup>|</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup>

*η* = *ηk*(*y*) *<sup>∂</sup>*

involutive system *Rq* = *id*−<sup>1</sup>

*<sup>η</sup>k*(*y*) *<sup>∂</sup>*

*<sup>∂</sup>y<sup>k</sup>* <sup>+</sup> *∂η<sup>k</sup> <sup>∂</sup>y<sup>r</sup> <sup>y</sup><sup>r</sup> i ∂ ∂y<sup>k</sup> i*

prolongation at order *q* of any horizontal vector field *ξ* = *ξ<sup>i</sup>*

prolongation at order *q* of any vertical vector field *η* = *ηk*(*y*) *<sup>∂</sup>*

**Corollary 3.1.** *There is a linear gauge sequence:*

$$\wedge^0 T^\* \otimes \mathcal{G} \stackrel{d}{\longrightarrow} \wedge^1 T^\* \otimes \mathcal{G} \stackrel{d}{\longrightarrow} \wedge^2 T^\* \otimes \mathcal{G} \stackrel{d}{\longrightarrow} \dots \stackrel{d}{\longrightarrow} \wedge^n T^\* \otimes \mathcal{G} \longrightarrow 0$$

*which is the tensor product by* G *of the Poincaré sequence:*

**Remark 3.1.** *When the physicists C.N. Yang and R.L. Mills created (non-abelian) gauge theory in 1954 ([38],[39]), their work was based on these results which were the only ones known at that time, the best mathematical reference being the well known book by S. Kobayashi and K. Nomizu ([15]). It follows that the only possibility to describe elecromagnetism (EM) within this framework was to call A the Yang-Mills potential and F the Yang-Mills field (a reason for choosing such notations) on the condition to have dim*(*G*) = 1 *in the abelian situation c* = 0 *and to use a Lagrangian depending on F* = *dA* − [*A*, *A*] *in the general case. Accordingly the idea was to select the unitary group U*(1)*, namely the unit circle in the complex plane with Lie algebra the tangent line to this circle at the unity* (1, 0)*. It is however important to notice that the resulting Maxwell equations dF* = 0 *have no equivalent in the non-abelian case c* �= 0*.*

Just before Albert Einstein visited Paris in 1922, Cartan published many short Notes ([5]) announcing long papers ([6]) where he selected *G* to be the Lie group involved in the Poincaré (conformal) group of space-time preserving (up to a function factor) the Minkowski metric *<sup>ω</sup>* = (*dx*1)<sup>2</sup> + (*dx*2)<sup>2</sup> + (*dx*3)<sup>2</sup> <sup>−</sup> (*dx*4)<sup>2</sup> with *<sup>x</sup>*<sup>4</sup> <sup>=</sup> *ct* where *<sup>c</sup>* is the speed of light. In the first case *F* is decomposed into two parts, the *torsion* as a 2-form with value in translations on one side and the *curvature* as a 2-form with value in rotations on the other side. This result was looking coherent *at first sight* with the Hilbert variational scheme of general relativity (GR) introduced by Einstein in 1915 ([21],[38]) and leading to a Lagrangian depending on *F* = *dA* − [*A*, *A*] as in the last remark.

In the meantime, Poincaré developped an invariant variational calculus ([22]) which has been used again without any quotation, successively by G. Birkhoff and V. Arnold (compare [4], 205-216 with [2], 326, Th 2.1). A particular case is well known by any student in the analytical mechanics of rigid bodies. Indeed, using standard notations, the movement of a rigid body is described in a fixed Cartesian frame by the formula *x*(*t*) = *a*(*t*)*x*<sup>0</sup> + *b*(*t*) where *a*(*t*) is a 3 × 3 time dependent orthogonal matrix (rotation) and *b*(*t*) a time depending vector (translation) as we already said. Differentiating with respect to time by using a dot, the *absolute speed* is *v* = *x*˙(*t*) = *a*˙(*t*)*x*<sup>0</sup> + ˙ *b*(*t*) and we obtain the *relative speed a*−1(*t*)*v* = *a*−1(*t*)*a*˙(*t*)*x*<sup>0</sup> + *a*−1(*t*)˙ *b*(*t*) by projection in a frame fixed in the body. Having in mind Example 2.1, it must be noticed that the so-called *Eulerian speed v* = *v*(*x*, *t*) = *aa*˙ <sup>−</sup>1*x* + ˙ *<sup>b</sup>* <sup>−</sup> *aa*˙ <sup>−</sup>1*<sup>b</sup>* only depends on the 1-form *B* = (*aa*˙ <sup>−</sup>1, ˙ *<sup>b</sup>* <sup>−</sup> *aa*˙ <sup>−</sup>1*b*). The Lagrangian (kinetic energy in this case) is thus a quadratic function of the 1-form *A* = (*a*−1*a*˙, *a*−<sup>1</sup> ˙ *<sup>b</sup>*) where *<sup>a</sup>*−1*a*˙ is a 3 <sup>×</sup> 3 skew symmetric time depending matrix. Hence, "*surprisingly*", this result is not coherent at all with EM where the Lagrangian is the quadratic expression (*�*/2)*E*<sup>2</sup> <sup>−</sup> (1/2*μ*)*B*<sup>2</sup> because the electric field *E* and the magnetic field *B* are combined in the EM field *F* as a 2-form satisfying the first set of Maxwell equations *dF* = 0. The dielectric constant *�* and the magnetic constant *μ* are leading to the electric induction *D* = *� E* and the magnetic induction *H* = (1/*μ*) *B* in the second set of Maxwell equations. In view of the existence of well known field-matter couplings such as piezoelectricity and photoelasticity that will be described later on, such a situation is contradictory as it should lead to put on equal footing 1-forms and 2-forms contrary to any unifying mathematical scheme but no other substitute could have been provided at that time.

8 Will-be-set-by-IN-TECH

**Remark 3.1.** *When the physicists C.N. Yang and R.L. Mills created (non-abelian) gauge theory in 1954 ([38],[39]), their work was based on these results which were the only ones known at that time, the best mathematical reference being the well known book by S. Kobayashi and K. Nomizu ([15]). It follows that the only possibility to describe elecromagnetism (EM) within this framework was to call A the Yang-Mills potential and F the Yang-Mills field (a reason for choosing such notations) on the condition to have dim*(*G*) = 1 *in the abelian situation c* = 0 *and to use a Lagrangian depending on F* = *dA* − [*A*, *A*] *in the general case. Accordingly the idea was to select the unitary group U*(1)*, namely the unit circle in the complex plane with Lie algebra the tangent line to this circle at the unity* (1, 0)*. It is however important to notice that the resulting Maxwell equations dF* = 0 *have no equivalent in the*

Just before Albert Einstein visited Paris in 1922, Cartan published many short Notes ([5]) announcing long papers ([6]) where he selected *G* to be the Lie group involved in the Poincaré (conformal) group of space-time preserving (up to a function factor) the Minkowski metric *<sup>ω</sup>* = (*dx*1)<sup>2</sup> + (*dx*2)<sup>2</sup> + (*dx*3)<sup>2</sup> <sup>−</sup> (*dx*4)<sup>2</sup> with *<sup>x</sup>*<sup>4</sup> <sup>=</sup> *ct* where *<sup>c</sup>* is the speed of light. In the first case *F* is decomposed into two parts, the *torsion* as a 2-form with value in translations on one side and the *curvature* as a 2-form with value in rotations on the other side. This result was looking coherent *at first sight* with the Hilbert variational scheme of general relativity (GR) introduced by Einstein in 1915 ([21],[38]) and leading to a Lagrangian depending on

In the meantime, Poincaré developped an invariant variational calculus ([22]) which has been used again without any quotation, successively by G. Birkhoff and V. Arnold (compare [4], 205-216 with [2], 326, Th 2.1). A particular case is well known by any student in the analytical mechanics of rigid bodies. Indeed, using standard notations, the movement of a rigid body is described in a fixed Cartesian frame by the formula *x*(*t*) = *a*(*t*)*x*<sup>0</sup> + *b*(*t*) where *a*(*t*) is a 3 × 3 time dependent orthogonal matrix (rotation) and *b*(*t*) a time depending vector (translation) as we already said. Differentiating with respect to time by using a dot, the *absolute speed* is

by projection in a frame fixed in the body. Having in mind Example 2.1, it must be noticed

Hence, "*surprisingly*", this result is not coherent at all with EM where the Lagrangian is the

are combined in the EM field *F* as a 2-form satisfying the first set of Maxwell equations *dF* = 0. The dielectric constant *�* and the magnetic constant *μ* are leading to the electric induction *D* =

the existence of well known field-matter couplings such as piezoelectricity and photoelasticity that will be described later on, such a situation is contradictory as it should lead to put on equal footing 1-forms and 2-forms contrary to any unifying mathematical scheme but no other

*b*(*t*) and we obtain the *relative speed a*−1(*t*)*v* = *a*−1(*t*)*a*˙(*t*)*x*<sup>0</sup> + *a*−1(*t*)˙

*<sup>b</sup>*) where *<sup>a</sup>*−1*a*˙ is a 3 <sup>×</sup> 3 skew symmetric time depending matrix.

*B* in the second set of Maxwell equations. In view of

*<sup>b</sup>* <sup>−</sup> *aa*˙ <sup>−</sup>1*b*). The Lagrangian (kinetic energy in this case) is thus a quadratic function

*b*(*t*)

*B*

*<sup>b</sup>* <sup>−</sup> *aa*˙ <sup>−</sup>1*<sup>b</sup>* only depends on the 1-form

*E* and the magnetic field

−→ ... *<sup>d</sup>* −→ ∧*nT*<sup>∗</sup> ⊗ G −→ <sup>0</sup>

**Corollary 3.1.** *There is a linear gauge sequence:*

*non-abelian case c* �= 0*.*

*v* = *x*˙(*t*) = *a*˙(*t*)*x*<sup>0</sup> + ˙

of the 1-form *A* = (*a*−1*a*˙, *a*−<sup>1</sup> ˙

*B* = (*aa*˙ <sup>−</sup>1, ˙

*�*

*F* = *dA* − [*A*, *A*] as in the last remark.

that the so-called *Eulerian speed v* = *v*(*x*, *t*) = *aa*˙ <sup>−</sup>1*x* + ˙

*E* and the magnetic induction *H* = (1/*μ*)

substitute could have been provided at that time.

quadratic expression (*�*/2)*E*<sup>2</sup> <sup>−</sup> (1/2*μ*)*B*<sup>2</sup> because the electric field

*which is the tensor product by* G *of the Poincaré sequence:*

<sup>∧</sup>0*T*<sup>∗</sup> ⊗ G *<sup>d</sup>* −→ ∧1*T*<sup>∗</sup> ⊗ G *<sup>d</sup>* −→ ∧2*T*<sup>∗</sup> ⊗ G *<sup>d</sup>*

Let us now turn to the other way proposed by Vessiot in 1903 ([36]) and 1904 ([37]). Our purpose is only to sketch the main results that we have obtained in many books ([23-26], we do not know other references) and to illustrate them by a series of specific examples, asking the reader to imagine any link with what has been said.


$$\eta^k(y)\frac{\partial}{\partial y^k} + \frac{\partial \eta^k}{\partial y^r} y^r\_i \frac{\partial}{\partial y^k\_i} + (\frac{\partial^2 \eta^k}{\partial y^r \partial y^s} y^r\_i y^s\_j + \frac{\partial \eta^k}{\partial y^r} y^r\_{ij})\frac{\partial}{\partial y^k\_{ij}} + \dots$$

where we have replaced *jq*(*f*)(*x*) by *yq*, each component beeing the "formal" derivative of the previous one .


This result proves that the MC equations are only examples of the Vessiot structure equations. We finally explain the name given to this structure ([26], p 296). Indeed, when *X* is a PHS for a lie group *G*, the graph of the action is an isomorphism and we obtain a map *X* × *X* → *G* :

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 11

In order to produce a Lie form, let us first notice that the general solution of the system of infinitesimal equations is *ξ* = *λτθτ* with *λ* = *cst*. Introducing the inverse matrix (*ω*)=(*ω<sup>τ</sup>*

of the *reciprocal distribution α* = {*ατ*} made by n vectors commuting with {*θτ*}, we obtain

**Example 3.2.** *(Affine and projective structures of the real line) In Example 2.1 with n* = 1*, the special Lie equations are* Φ(*y*2) ≡ *yxx*/*yx* = 0 ⇒ *∂xxξ* = 0 *with q* = 2 *and we let the reader check as an*

+ *ω*(*y*)*yx* = *ω*(*x*) ⇒ *∂xxξ* + *ω*(*x*)*∂xξ* + *ξ∂xω*(*x*) = 0

*<sup>x</sup>* = *ω*(*x*) ⇒ *∂xxxξ* + 2*ω*(*x*)*∂xξ* + *ξ∂xω*(*x*) = 0

*We could study in the same way the group of projective transformations of the real line*

**Example 3.3.** *<sup>n</sup>* <sup>=</sup> 2, *<sup>q</sup>* <sup>=</sup> 1, <sup>Γ</sup> <sup>=</sup> {*y*<sup>1</sup> <sup>=</sup> *<sup>f</sup>*(*x*1), *<sup>y</sup>*<sup>2</sup> <sup>=</sup> *<sup>x</sup>*2/(*<sup>∂</sup> <sup>f</sup>*(*x*1)/*∂x*1)} *where f is an arbitrary*

<sup>1</sup> <sup>=</sup> *<sup>x</sup>*2,

<sup>2</sup> = 0,

*<sup>∂</sup>*(*x*1, *<sup>x</sup>*2) <sup>≡</sup> *<sup>y</sup>*<sup>1</sup>

*We obtain* <sup>F</sup> <sup>=</sup> *<sup>T</sup>*∗×*X*∧2*T*<sup>∗</sup> *and <sup>ω</sup>* = (*α*, *<sup>β</sup>*) *where <sup>α</sup> is a* <sup>1</sup>*-form and <sup>β</sup> is a* <sup>2</sup>*-form with special section <sup>ω</sup>* = (*x*2*dx*1, *dx*<sup>1</sup> <sup>∧</sup> *dx*2)*. It follows that dα*/*<sup>β</sup> is a well defined scalar because <sup>β</sup>* �<sup>=</sup> <sup>0</sup>*. The Vessiot structure equation is dα* = *cβ with a single structure constant c which cannot have anything to do with a Lie algebra. Considering the other section <sup>ω</sup>*¯ = (*dx*1, *dx*<sup>1</sup> <sup>∧</sup> *dx*2)*, we get <sup>c</sup>*¯ <sup>=</sup> <sup>0</sup>*. As c* <sup>=</sup> <sup>−</sup><sup>1</sup> *and*

**Example 3.4.** *(Symplectic structure) With n* <sup>=</sup> <sup>2</sup>*p*, *<sup>q</sup>* <sup>=</sup> <sup>1</sup> *and* <sup>F</sup> <sup>=</sup> <sup>∧</sup>2*T*∗*, let <sup>ω</sup> be a closed* <sup>2</sup>*-form of maximum rank, that is dω* = 0, *det*(*ω*) �= 0*. The equivalence problem is nothing else than the Darboux problem in analytical mechanics giving the possibility to write locally ω* = ∑ *dp* ∧ *dq by*

**Example 3.5.** *(Contact structure) With n* <sup>=</sup> 3, *<sup>q</sup>* <sup>=</sup> 1, *<sup>w</sup>* <sup>=</sup> *dx*<sup>1</sup> <sup>−</sup> *<sup>x</sup>*3*dx*<sup>2</sup> <sup>⇒</sup> *<sup>w</sup>* <sup>∧</sup> *dw* <sup>=</sup> *dx*<sup>1</sup> <sup>∧</sup> *dx*<sup>2</sup> <sup>∧</sup>

*<sup>j</sup>*1(*f*)−1(*dw*) = *dj*1(*f*)−1(*w*) = *<sup>ρ</sup>dw* <sup>+</sup> *<sup>d</sup><sup>ρ</sup>* <sup>∧</sup> *<sup>w</sup>* <sup>⇒</sup> *<sup>j</sup>*1(*f*)−1(*<sup>w</sup>* <sup>∧</sup> *dw*) = *<sup>ρ</sup>*2(*<sup>w</sup>* <sup>∧</sup> *dw*)

*dx*3*, let us consider* <sup>Γ</sup> <sup>=</sup> { *<sup>f</sup>* <sup>∈</sup> *aut*(*X*)|*j*1(*f*)−1(*w*) = *<sup>ρ</sup>w*}*. This is not a Lie form but we get:*

1*y*2 <sup>2</sup> <sup>−</sup> *<sup>y</sup>*<sup>1</sup> 2*y*2 <sup>1</sup> = 1

*There is an isomorphism J*1(F*aff*) � F*aff* <sup>×</sup>*X*F*proj* : *<sup>j</sup>*1(*ω*) <sup>→</sup> (*ω*, *<sup>γ</sup>* <sup>=</sup> *<sup>∂</sup>x<sup>ω</sup>* <sup>−</sup> (1/2)*ω*2)*.*

*y* = (*ax* + *b*)/(*cx* + *d*) *and get with more work the general lie equations:*

<sup>Φ</sup>1(*y*1) <sup>≡</sup> *<sup>y</sup>*2*y*<sup>1</sup>

<sup>Φ</sup>2(*y*1) <sup>≡</sup> *<sup>y</sup>*2*y*<sup>1</sup>

<sup>Φ</sup>3(*y*1) <sup>≡</sup> *<sup>∂</sup>*(*y*1, *<sup>y</sup>*2)

*thus <sup>c</sup>*¯ �<sup>=</sup> *c, the equivalence problem j*1(*f*)−1(*ω*) = *<sup>ω</sup>*¯ *cannot even be solved formally.*

*using canonical conjugate coordinates* (*q*, *p*)=(*position*, *momentum*)*.*

)<sup>2</sup> + *ω*(*y*)*y*<sup>2</sup>

*<sup>∂</sup><sup>x</sup>* (*x*, *a*(*x*, *y*)).

*i* )

(*x*, *<sup>y</sup>*) <sup>→</sup> (*a*(*x*, *<sup>y</sup>*)) leading to a first order system of finite Lie equations *yx* <sup>=</sup> *<sup>∂</sup> <sup>f</sup>*

*λ* = *cst* ⇔ [*ξ*, *α*] = 0 ⇔ L(*ξ*)*ω* = 0.

*exercise that the general Lie equations are: yxx yx*

*with no IC. The special section is ω*(*x*) = 0*.*

− 3 2 ( *yxx yx*

*invertible map. The involutive Lie form is:*

*yxxx yx*

*<sup>ω</sup>*−1(*V*(F)) by the formula :

$$\mathcal{D}\mathfrak{F} = \mathcal{D}\_{\omega}\mathfrak{F} = \mathcal{L}(\mathfrak{F})\omega = \frac{d}{dt}j\_{\mathfrak{f}}(f\_t)^{-1}(\omega)|\_{t=0} \Rightarrow \Theta = \{\mathfrak{F} \in T|\mathcal{L}(\mathfrak{f})\omega = 0\}$$

while we have *<sup>x</sup>* <sup>→</sup> *<sup>x</sup>* <sup>+</sup> *<sup>t</sup>ξ*(*x*) + ... <sup>⇒</sup> *<sup>u</sup><sup>τ</sup>* <sup>→</sup> *<sup>u</sup><sup>τ</sup>* <sup>+</sup> *<sup>t</sup>∂μξkLτμ <sup>k</sup>* (*u*) + ... where *μ* = (*μ*1, ..., *μn*) is a multi-index as a way to write down the system of infinitesimal Lie equations in the *Medolaghi form*:

$$\Omega^{\mathsf{T}} \equiv (\mathcal{L}(\xi)\omega)^{\mathsf{T}} \equiv -L\_k^{\mathsf{T}\mu}(\omega(\mathsf{x})) \partial\_{\mu}\xi^k + \xi^r \partial\_r \omega^\mathsf{T}(\mathsf{x}) = 0$$


**Example 3.1.** *(Principal homogeneous structure) When* Γ *is made by the translations y<sup>i</sup>* = *x<sup>i</sup>* + *a<sup>i</sup> , the Lie form is* Φ*<sup>k</sup> <sup>i</sup>* (*y*1) <sup>≡</sup> *<sup>y</sup><sup>k</sup> <sup>i</sup>* <sup>=</sup> *<sup>δ</sup><sup>k</sup> <sup>i</sup> (Kronecker symbol) and the linearization is <sup>∂</sup>iξ<sup>k</sup>* <sup>=</sup> <sup>0</sup>*. The natural bundle is* <sup>F</sup> <sup>=</sup> *<sup>T</sup>*∗×*X*...×*XT*<sup>∗</sup> *(n times) with det*(*ω*) �<sup>=</sup> <sup>0</sup> *and the general Medolaghi form is <sup>ω</sup><sup>τ</sup> <sup>r</sup> ∂iξ<sup>r</sup>* + *ξr∂rω<sup>τ</sup> <sup>i</sup>* <sup>=</sup> <sup>0</sup> <sup>⇔</sup> [*ξ*, *ατ*] = <sup>0</sup> *with <sup>τ</sup>* <sup>=</sup> 1, ..., *n if <sup>α</sup>* = (*α<sup>i</sup> <sup>τ</sup>*) = *<sup>ω</sup>*−1*. Using crossed derivatives, one finally gets the zero order equations:*

$$(\xi^r \partial\_r (\alpha^i\_\rho(\mathbf{x}) \alpha^j\_\sigma(\mathbf{x}) (\partial\_i \omega^\tau\_j(\mathbf{x}) - \partial\_j \omega^\tau\_i(\mathbf{x}))) = 0$$

*leading therefore (up to sign) to the n*2(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)/2 *Vessiot structure equations:*

$$
\partial\_i \omega\_j^\tau(\mathfrak{x}) - \partial\_j \omega\_i^\tau(\mathfrak{x}) = c\_{\rho \sigma}^\tau \omega\_i^\rho(\mathfrak{x}) \omega\_j^\sigma(\mathfrak{x}),
$$

This result proves that the MC equations are only examples of the Vessiot structure equations. We finally explain the name given to this structure ([26], p 296). Indeed, when *X* is a PHS for a lie group *G*, the graph of the action is an isomorphism and we obtain a map *X* × *X* → *G* : (*x*, *<sup>y</sup>*) <sup>→</sup> (*a*(*x*, *<sup>y</sup>*)) leading to a first order system of finite Lie equations *yx* <sup>=</sup> *<sup>∂</sup> <sup>f</sup> <sup>∂</sup><sup>x</sup>* (*x*, *a*(*x*, *y*)). In order to produce a Lie form, let us first notice that the general solution of the system of infinitesimal equations is *ξ* = *λτθτ* with *λ* = *cst*. Introducing the inverse matrix (*ω*)=(*ω<sup>τ</sup> i* ) of the *reciprocal distribution α* = {*ατ*} made by n vectors commuting with {*θτ*}, we obtain *λ* = *cst* ⇔ [*ξ*, *α*] = 0 ⇔ L(*ξ*)*ω* = 0.

**Example 3.2.** *(Affine and projective structures of the real line) In Example 2.1 with n* = 1*, the special Lie equations are* Φ(*y*2) ≡ *yxx*/*yx* = 0 ⇒ *∂xxξ* = 0 *with q* = 2 *and we let the reader check as an exercise that the general Lie equations are:*

$$\frac{\partial \mathbf{x}}{\partial \mathbf{x}} + \omega(\mathbf{y}) y\_{\mathbf{x}} = \omega(\mathbf{x}) \Rightarrow \partial\_{\mathbf{x}\mathbf{x}} \mathfrak{F} + \omega(\mathbf{x}) \partial\_{\mathbf{x}} \mathfrak{F} + \mathfrak{F} \partial\_{\mathbf{x}} \omega(\mathbf{x}) = 0$$

*with no IC. The special section is ω*(*x*) = 0*.*

10 Will-be-set-by-IN-TECH

is a multi-index as a way to write down the system of infinitesimal Lie equations in the

6. By analogy with "special" and "general" relativity, we shall call the given section *special* and any other arbitrary section *general*. The problem is now to study the formal properties of the linear system just obtained with coefficients only depending on *j*1(*ω*), exactly like L.P. Eisenhart did for F = *S*2*T*<sup>∗</sup> when finding the constant Riemann curvature condition for a metric *ω* with *det*(*ω*) �= 0 ([26], Example 10, p 249). Indeed, if any expression involving *ω* and its derivatives is a scalar object, it must reduce to a constant because Γ is assumed to be transitive and thus cannot be defined by any zero order equation. Now one can prove that the CC for *ω*¯ , thus for *ω* too, only depend on the Φ and take the quasi-linear symbolic form *v* ≡ *I*(*u*1) ≡ *A*(*u*)*ux* + *B*(*u*) = 0, allowing to define an affine subfibered manifold B<sup>1</sup> ⊂ *J*1(F) over F. Now, if one has two sections *ω* and *ω*¯ of F, the *equivalence problem* is to look for *<sup>f</sup>* <sup>∈</sup> *aut*(*X*) such that *jq*(*f*)−1(*ω*) = *<sup>ω</sup>*¯ . When the two sections satisfy the same CC, the problem is sometimes locally possible (Lie groups of transformations, Darboux

7. Instead of the CC for the equivalence problem, let us look for the *integrability conditions* (IC) for the system of infinitesimal Lie equations and suppose that, for the given section, all the equations of order *q* + *r* are obtained by differentiating *r* times only the equations of order *q*, then it was claimed by Vessiot ([36] with no proof, see [26], p 209) that such a property is held if and only if there is an equivariant section *c* : F→F<sup>1</sup> : (*x*, *u*) → (*x*, *u*, *v* = *c*(*u*)) where F<sup>1</sup> = *J*1(F)/B<sup>1</sup> is a natural vector bundle over F with local coordinates (*x*, *u*, *v*). Moreover, any such equivariant section depends on a finite number of constants *c* called *structure constants* and the IC for the *Vessiot structure equations I*(*u*1) = *c*(*u*) are of

8. Finally, when *Y* is no longer a copy of *X*, a system A*<sup>q</sup>* ⊂ *Jq*(*X* × *Y*) is said to be an *automorphic system* for a Lie pseudogroup <sup>Γ</sup> <sup>⊂</sup> *aut*(*Y*) if, whenever *<sup>y</sup>* <sup>=</sup> *<sup>f</sup>*(*x*) and *<sup>y</sup>*¯ <sup>=</sup> ¯

**Example 3.1.** *(Principal homogeneous structure) When* Γ *is made by the translations y<sup>i</sup>* = *x<sup>i</sup>* + *a<sup>i</sup>*

*<sup>i</sup>* (*x*) = *<sup>c</sup><sup>τ</sup>*

and can be implemented on computer in the differential algebraic framework.

*bundle is* <sup>F</sup> <sup>=</sup> *<sup>T</sup>*∗×*X*...×*XT*<sup>∗</sup> *(n times) with det*(*ω*) �<sup>=</sup> <sup>0</sup> *and the general Medolaghi form is <sup>ω</sup><sup>τ</sup>*

*σ*(*x*)(*∂iω<sup>τ</sup>*

are two solutions, then there exists one and only one transformation *y*¯ = *g*(*y*) ∈ Γ such

*f* = *g* ◦ *f* . Explicit tests for checking such a property formally have been given in [24]

*<sup>j</sup>* (*x*) <sup>−</sup> *<sup>∂</sup>jω<sup>τ</sup>*

*ρσω<sup>ρ</sup>*

*<sup>i</sup>* (*x*)*ω<sup>σ</sup> <sup>j</sup>* (*x*)

*<sup>i</sup> (Kronecker symbol) and the linearization is <sup>∂</sup>iξ<sup>k</sup>* <sup>=</sup> <sup>0</sup>*. The natural*

*<sup>i</sup>* (*x*))) = 0

*<sup>τ</sup>*) = *<sup>ω</sup>*−1*. Using crossed derivatives, one*

*<sup>k</sup>* (*ω*(*x*))*∂μξ<sup>k</sup>* <sup>+</sup> *<sup>ξ</sup><sup>r</sup>*

*dt jq*(*ft*)−1(*ω*)|*t*=<sup>0</sup> <sup>⇒</sup> <sup>Θ</sup> <sup>=</sup> {*<sup>ξ</sup>* <sup>∈</sup> *<sup>T</sup>*|L(*ξ*)*<sup>ω</sup>* <sup>=</sup> <sup>0</sup>}

*<sup>k</sup>* (*u*) + ... where *μ* = (*μ*1, ..., *μn*)

*f*(*x*)

*,*

*<sup>r</sup> ∂iξ<sup>r</sup>* +

*∂rωτ*(*x*) = 0

*<sup>ω</sup>*−1(*V*(F)) by the formula :

a polynomial form *J*(*c*) = 0.

*<sup>i</sup>* (*y*1) <sup>≡</sup> *<sup>y</sup><sup>k</sup>*

*finally gets the zero order equations:*

*<sup>i</sup>* <sup>=</sup> *<sup>δ</sup><sup>k</sup>*

*ξr ∂r*(*α<sup>i</sup>*

*<sup>i</sup>* <sup>=</sup> <sup>0</sup> <sup>⇔</sup> [*ξ*, *ατ*] = <sup>0</sup> *with <sup>τ</sup>* <sup>=</sup> 1, ..., *n if <sup>α</sup>* = (*α<sup>i</sup>*

*∂iω<sup>τ</sup>*

*ρ*(*x*)*α<sup>j</sup>*

*leading therefore (up to sign) to the n*2(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)/2 *Vessiot structure equations:*

*<sup>j</sup>* (*x*) <sup>−</sup> *<sup>∂</sup>jω<sup>τ</sup>*

that ¯

*the Lie form is* Φ*<sup>k</sup>*

*ξr∂rω<sup>τ</sup>*

*Medolaghi form*:

<sup>D</sup>*<sup>ξ</sup>* <sup>=</sup> <sup>D</sup>*ωξ* <sup>=</sup> <sup>L</sup>(*ξ*)*<sup>ω</sup>* <sup>=</sup> *<sup>d</sup>*

while we have *<sup>x</sup>* <sup>→</sup> *<sup>x</sup>* <sup>+</sup> *<sup>t</sup>ξ*(*x*) + ... <sup>⇒</sup> *<sup>u</sup><sup>τ</sup>* <sup>→</sup> *<sup>u</sup><sup>τ</sup>* <sup>+</sup> *<sup>t</sup>∂μξkLτμ*

<sup>Ω</sup>*<sup>τ</sup>* <sup>≡</sup> (L(*ξ*)*ω*)*<sup>τ</sup>* ≡ −*Lτμ*

problem in analytical mechanics,...) but sometimes not ([23], p 333).

*We could study in the same way the group of projective transformations of the real line y* = (*ax* + *b*)/(*cx* + *d*) *and get with more work the general lie equations:*

$$\frac{y\_{\text{xxx}}}{y\_{\text{x}}} - \frac{3}{2} (\frac{y\_{\text{xxx}}}{y\_{\text{x}}})^2 + \omega(y)y\_{\text{x}}^2 = \omega(\mathbf{x}) \Rightarrow \partial\_{\text{xxx}}\xi + 2\omega(\mathbf{x})\partial\_{\text{x}}\xi + \xi\partial\_{\text{x}}\omega(\mathbf{x}) = 0$$

*There is an isomorphism J*1(F*aff*) � F*aff* <sup>×</sup>*X*F*proj* : *<sup>j</sup>*1(*ω*) <sup>→</sup> (*ω*, *<sup>γ</sup>* <sup>=</sup> *<sup>∂</sup>x<sup>ω</sup>* <sup>−</sup> (1/2)*ω*2)*.*

**Example 3.3.** *<sup>n</sup>* <sup>=</sup> 2, *<sup>q</sup>* <sup>=</sup> 1, <sup>Γ</sup> <sup>=</sup> {*y*<sup>1</sup> <sup>=</sup> *<sup>f</sup>*(*x*1), *<sup>y</sup>*<sup>2</sup> <sup>=</sup> *<sup>x</sup>*2/(*<sup>∂</sup> <sup>f</sup>*(*x*1)/*∂x*1)} *where f is an arbitrary invertible map. The involutive Lie form is:*

$$\begin{aligned} \Phi^1(y\_1) &\equiv y^2 y\_1^1 = x^2, \\ \Phi^2(y\_1) &\equiv y^2 y\_2^1 = 0, \\ \Phi^3(y\_1) &\equiv \frac{\partial(y^1, y^2)}{\partial(x^1, x^2)} \equiv y\_1^1 y\_2^2 - y\_2^1 y\_1^2 = 1. \end{aligned}$$

*We obtain* <sup>F</sup> <sup>=</sup> *<sup>T</sup>*∗×*X*∧2*T*<sup>∗</sup> *and <sup>ω</sup>* = (*α*, *<sup>β</sup>*) *where <sup>α</sup> is a* <sup>1</sup>*-form and <sup>β</sup> is a* <sup>2</sup>*-form with special section <sup>ω</sup>* = (*x*2*dx*1, *dx*<sup>1</sup> <sup>∧</sup> *dx*2)*. It follows that dα*/*<sup>β</sup> is a well defined scalar because <sup>β</sup>* �<sup>=</sup> <sup>0</sup>*. The Vessiot structure equation is dα* = *cβ with a single structure constant c which cannot have anything to do with a Lie algebra. Considering the other section <sup>ω</sup>*¯ = (*dx*1, *dx*<sup>1</sup> <sup>∧</sup> *dx*2)*, we get <sup>c</sup>*¯ <sup>=</sup> <sup>0</sup>*. As c* <sup>=</sup> <sup>−</sup><sup>1</sup> *and thus <sup>c</sup>*¯ �<sup>=</sup> *c, the equivalence problem j*1(*f*)−1(*ω*) = *<sup>ω</sup>*¯ *cannot even be solved formally.*

**Example 3.4.** *(Symplectic structure) With n* <sup>=</sup> <sup>2</sup>*p*, *<sup>q</sup>* <sup>=</sup> <sup>1</sup> *and* <sup>F</sup> <sup>=</sup> <sup>∧</sup>2*T*∗*, let <sup>ω</sup> be a closed* <sup>2</sup>*-form of maximum rank, that is dω* = 0, *det*(*ω*) �= 0*. The equivalence problem is nothing else than the Darboux problem in analytical mechanics giving the possibility to write locally ω* = ∑ *dp* ∧ *dq by using canonical conjugate coordinates* (*q*, *p*)=(*position*, *momentum*)*.*

**Example 3.5.** *(Contact structure) With n* <sup>=</sup> 3, *<sup>q</sup>* <sup>=</sup> 1, *<sup>w</sup>* <sup>=</sup> *dx*<sup>1</sup> <sup>−</sup> *<sup>x</sup>*3*dx*<sup>2</sup> <sup>⇒</sup> *<sup>w</sup>* <sup>∧</sup> *dw* <sup>=</sup> *dx*<sup>1</sup> <sup>∧</sup> *dx*<sup>2</sup> <sup>∧</sup> *dx*3*, let us consider* <sup>Γ</sup> <sup>=</sup> { *<sup>f</sup>* <sup>∈</sup> *aut*(*X*)|*j*1(*f*)−1(*w*) = *<sup>ρ</sup>w*}*. This is not a Lie form but we get:*

$$j\_1(f)^{-1}(dw) = dj\_1(f)^{-1}(w) = \rho dw + d\rho \wedge w \Rightarrow j\_1(f)^{-1}(w \wedge dw) = \rho^2 (w \wedge dw)$$

*This is a new geometric object of order* 2 *allowing to obtain, as in Example 3.2, an isomorphism*

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 13

<sup>1</sup> = *g*1*:*

*li*(*x*) + *<sup>γ</sup><sup>r</sup>*

*<sup>j</sup> ωli*(*x*))

*ir*(*x*)*∂jξ<sup>r</sup>* <sup>−</sup> *<sup>γ</sup><sup>r</sup>*

*lj*(*x*)*γ<sup>k</sup>*

*lij* <sup>=</sup> <sup>0</sup> <sup>⇒</sup> *<sup>ρ</sup>klij* <sup>=</sup> *<sup>ω</sup>krρ<sup>r</sup>*

*ri*(*x*) <sup>−</sup> *<sup>γ</sup><sup>r</sup>*

*ij*(*x*)*∂rξ<sup>k</sup>* <sup>+</sup> *<sup>ξ</sup>r∂rγ<sup>k</sup>*

*li*(*x*)*γ<sup>k</sup>*

*lij* = *ρijkl.*

*rj*(*x*)

*lij* = 0 *are only linear*

*irj*(*x*) = *ρji*(*x*)

*ij*(*x*) =

*rj*(*x*)*∂iξ<sup>r</sup>* <sup>+</sup> *<sup>γ</sup><sup>k</sup>*

*lj*(*x*) <sup>−</sup> *<sup>∂</sup>jγ<sup>k</sup>*

*is still a first order geometric object and even a tensor as a section of* <sup>∧</sup>2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *T satisfying the*

*kij* <sup>+</sup> *<sup>ω</sup>krρ<sup>r</sup>*

*<sup>i</sup> <sup>ω</sup>lj*(*x*) <sup>−</sup> *<sup>δ</sup><sup>k</sup>*

*describing the constant Riemannian curvature condition of Eisenhart [10]. Finally, as we have*

*by contracting indices and the scalar curvature ρ*(*x*) = *ωij*(*x*)*ρij*(*x*) *in order to obtain ρ*(*x*) = *n*(*n* − 1)*c. It remains to obtain all these results in a purely formal way, for example to prove that the number of components of the Riemann tensor is equal to n*2(*n*<sup>2</sup> <sup>−</sup> <sup>1</sup>)/12 *without dealing with*

**Remark 3.2.** *Comparing the various Vessiot structure equations containing structure constants, we discover at once that the many c appearing in the MC equations are absolutely on equal footing with the only c appearing in the other examples. As their factors are either constant, linear or quadratic, any identification of the quadratic terms appearing in the Riemann tensor with the quadratic terms appearing in the MC equations is definitively not correct or, in an equivalent but more abrupt way, the Cartan structure equations have nothing to do with the Vessiot structure equations. As we shall see,*

**Remark 3.3.** *Let us consider again Example 3.2 with ∂xx f*(*x*)/*∂<sup>x</sup> f*(*x*) = *ω*¯(*x*) *and introduce a variation η*(*f*(*x*)) = *δ f*(*x*) *as in analytical or continuum mechanics. We get similarly δ∂<sup>x</sup> f* =

*involved is computed over the target. Let us now pass from the target to the source by introducing*

*simple variation δω*¯ = L(*ξ*)*ω*¯ *over the soure. As another example of this general variational procedure, let us compare with the similar variations on which classical finite elasticity theory is based.*

*target. Passing now from the target to the source as before, we find the particularly simple variation δω*¯ = L(*ξ*)*ω*¯ *over the source. For "small" deformations, source and target are of course identified but it is not true that the infinitesimal deformation tensor is in general the limit of the finite deformation*

*<sup>∂</sup><sup>y</sup> ∂<sup>x</sup> f and so on, a result leading to δω*¯(*x*) = *∂<sup>x</sup> f*L(*η*)*ω*(*f*(*x*)) *where the Lie derivative*

*<sup>∂</sup><sup>y</sup> ∂<sup>x</sup> f* = *∂xξ∂<sup>x</sup> f* + *ξ∂xx f and so on, a result leading to the particularly*

(*x*)(L(*η*)*ω*)*kl*(*f*(*x*)) *where the Lie derivative involved is computed over the*

(*x*) = *ω*¯*ij*(*x*)*, where ω is the Euclidean metric, we obtain*

*ri*(*x*) = <sup>0</sup>*, we can only introduce the Ricci tensor <sup>ρ</sup>ij*(*x*) = *<sup>ρ</sup><sup>r</sup>*

*<sup>j</sup>*1(*ω*) � (*ω*, *<sup>γ</sup>*) *and the second order equations with f* <sup>−</sup><sup>1</sup>

*<sup>l</sup>* (*y<sup>l</sup> ij* <sup>+</sup> *<sup>γ</sup><sup>l</sup>*

*ij* <sup>≡</sup> (L(*ξ*)*γ*)*<sup>k</sup>*

*ijl* <sup>+</sup> *<sup>ρ</sup><sup>k</sup>*

*rs*(*y*)*y<sup>r</sup> i ys j* ) = *γ<sup>k</sup> ij*(*x*)

*ij* <sup>≡</sup> *<sup>∂</sup>ijξ<sup>k</sup>* <sup>+</sup> *<sup>γ</sup><sup>k</sup>*

*lij*(*x*) <sup>≡</sup> *<sup>∂</sup>iγ<sup>k</sup>*

*jli* <sup>=</sup> 0, *<sup>ω</sup>rlρ<sup>l</sup>*

*combinations of the previous ones and we get the Vessiot structure equations:*

*lij*(*x*) = *<sup>c</sup>*(*δ<sup>k</sup>*

*ρk*

*most of mathematical physics today is based on such a confusion.*

*Accordingly, the IC must express that the new first order equations* (L(*ξ*)*ρ*)*<sup>k</sup>*

*ij*) *is a section of S*2*T*<sup>∗</sup> ⊗ *T. Surprisingly, the following expression:*

*Lie form g<sup>k</sup>*

*purely algebraic relations : ρk lij* <sup>+</sup> *<sup>ρ</sup><sup>k</sup>*

*Riemann tensor ρ<sup>k</sup>*

*rj*(*x*) <sup>−</sup> *<sup>∂</sup>jγ<sup>r</sup>*

*Medolaghi form* Γ*<sup>k</sup>*

0

*ρr*

*indices.*

*<sup>∂</sup>x<sup>δ</sup> <sup>f</sup>* = *∂η*

*<sup>η</sup>* <sup>=</sup> *ξ∂<sup>x</sup> <sup>f</sup>* <sup>⇒</sup> *∂η*

(*δω*¯)*ij*(*x*) = *∂<sup>i</sup> f <sup>k</sup>*(*x*)*∂<sup>j</sup> f <sup>l</sup>*

*Starting now with ωkl*(*f*(*x*))*∂<sup>i</sup> f <sup>k</sup>*(*x*)*∂<sup>j</sup> f <sup>l</sup>*

*tensor (for a counterexample, see [25], p 70).*

*rij*(*x*) = *<sup>∂</sup>iγ<sup>r</sup>*

*where* (Γ*<sup>k</sup>*

*The corresponding geometric object is thus made by a* 1*-form density ω* = (*ω*1, *ω*2, *ω*3) *that transforms like a* 1*-form up to the division by the square root of the Jacobian determinant. The unusual general Medolaghi form is:*

$$\Omega\_{\dot{l}} \equiv \omega\_{\dot{r}}(\mathfrak{x}) \partial\_{\dot{l}} \mathfrak{x}^{r} - (1/2) \omega\_{\dot{l}}(\mathfrak{x}) \partial\_{\dot{r}} \mathfrak{x}^{r} + \mathfrak{x}^{r} \partial\_{\dot{r}} \omega\_{\dot{l}}(\mathfrak{x}) = 0$$

*In a symbolic way ω* ∧ *dω is now a scalar and the only Vessiot structure equation is:*

$$
\omega\_1(\partial\_2\omega\_3 - \partial\_3\omega\_2) + \omega\_2(\partial\_3\omega\_1 - \partial\_1\omega\_3) + \omega\_3(\partial\_1\omega\_2 - \partial\_2\omega\_1) = \mathcal{L}
$$

*For the special section <sup>ω</sup>* = (1, <sup>−</sup>*x*3, 0) *we have c* <sup>=</sup> <sup>1</sup>*. If we choose <sup>ω</sup>*¯ = (1, 0, 0) *we may define* Γ¯ *by the system y*<sup>1</sup> <sup>2</sup> <sup>=</sup> 0, *<sup>y</sup>*<sup>1</sup> <sup>3</sup> <sup>=</sup> 0, *<sup>y</sup>*<sup>2</sup> 2*y*3 <sup>3</sup> <sup>−</sup> *<sup>y</sup>*<sup>2</sup> 3*y*3 <sup>2</sup> <sup>=</sup> *<sup>y</sup>*<sup>1</sup> <sup>1</sup> *but now c*¯ = 0 *and the equivalence problem j*1(*f*)−1(*ω*) = *ω*¯ *cannot even be solved formally. These results can be extended to an arbitrary odd dimension with much more work ([24], p 684).*

**Example 3.6.** *(Screw and complex structures) (n* = 2, *q* = 1*) In 1878 Clifford introduced abstract numbers of the form x*<sup>1</sup> + *�x*<sup>2</sup> *with �*<sup>2</sup> = 0 *in order to study helicoidal movements in the mechanics of rigid bodies. We may try to define functions of these numbers for which a derivative may have a meaning. Thus, if f*(*x*<sup>1</sup> + *�x*2) = *f* <sup>1</sup>(*x*1, *x*2) + *� f* <sup>2</sup>(*x*1, *x*2)*, then we should get:*

$$df = (A + \epsilon B)(d\mathbf{x}^1 + \epsilon d\mathbf{x}^2) = A d\mathbf{x}^1 + \epsilon (B d\mathbf{x}^1 + A d\mathbf{x}^2) = df^1 + \epsilon df^2$$

*Accordingly, we have to look for transformations y*<sup>1</sup> = *f* <sup>1</sup>(*x*1, *x*2), *y*<sup>2</sup> = *f* <sup>2</sup>(*x*1, *x*2) *satisfying the first order involutive system of finite Lie equations y*<sup>1</sup> <sup>2</sup> <sup>=</sup> 0, *<sup>y</sup>*<sup>2</sup> <sup>2</sup> <sup>−</sup> *<sup>y</sup>*<sup>1</sup> <sup>1</sup> = 0 *with no CC. As we have an algebraic Lie pseudogroup, a tricky computation ([24], p 467) allows to prove that* Γ *is made by the transformations preserving a mixed tensor with square equal to zero as follows:*

$$
\begin{pmatrix} y\_1^1 \ y\_2^1 \\ y\_1^2 \ y\_2^2 \end{pmatrix}^{-1} \begin{pmatrix} 0 \ 0 \\ 1 \ 0 \end{pmatrix} \begin{pmatrix} y\_1^1 \ y\_2^1 \\ y\_1^2 \ y\_2^2 \end{pmatrix} = \begin{pmatrix} 0 \ 0 \\ 1 \ 0 \end{pmatrix}.
$$

*We get the Lie form* <sup>Φ</sup><sup>1</sup> <sup>≡</sup> *<sup>y</sup>*<sup>1</sup> 2/*y*<sup>1</sup> <sup>1</sup> <sup>=</sup> 0, <sup>Φ</sup><sup>2</sup> <sup>≡</sup> (*y*<sup>1</sup> 1)2/(*y*<sup>1</sup> 1*y*2 <sup>2</sup> <sup>−</sup> *<sup>y</sup>*<sup>1</sup> 2*y*2 <sup>1</sup>) = 1 *and let the reader exhibit* F*. Finally, introducing similarly the abstract number i such that i*<sup>2</sup> <sup>=</sup> <sup>−</sup>1*, we get the Cauchy-Riemann system y*<sup>2</sup> <sup>2</sup> <sup>−</sup> *<sup>y</sup>*<sup>1</sup> <sup>1</sup> <sup>=</sup> 0, *<sup>y</sup>*<sup>1</sup> <sup>2</sup> <sup>+</sup> *<sup>y</sup>*<sup>2</sup> <sup>1</sup> = 0 *with no CC defining complex analytic transformations and the correponding geometric object or complex structure is a mixed tensor with square equal to minus the* 2 × 2 *identity matrix as we have now:*

$$
\begin{pmatrix} y\_1^1 \ y\_2^1 \\ y\_1^2 \ y\_2^2 \end{pmatrix}^{-1} \begin{pmatrix} 0 \\ 1 \ 0 \end{pmatrix} \begin{pmatrix} y\_1^1 \ y\_2^1 \\ y\_1^2 \ y\_2^2 \end{pmatrix} = \begin{pmatrix} 0 & -1 \\ 1 \ 0 \end{pmatrix}
$$

**Example 3.7.** *(Riemann structure) If ω is a section of* F = *S*2*T*<sup>∗</sup> *with det*(*ω*) �= 0 *we get: Lie form* <sup>Φ</sup>*ij*(*y*1) <sup>≡</sup> *<sup>ω</sup>kl*(*y*)*y<sup>k</sup> i yl <sup>j</sup>* = *ωij*(*x*) *Medolaghi form* <sup>Ω</sup>*ij* <sup>≡</sup> (L(*ξ*)*ω*)*ij* <sup>≡</sup> *<sup>ω</sup>rj*(*x*)*∂iξ<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>ir*(*x*)*∂jξ<sup>r</sup>* <sup>+</sup> *<sup>ξ</sup>r∂rωij*(*x*) = <sup>0</sup> *also called Killing system for historical reasons. A special section could be the Euclidean metric when n* = 1, 2, 3 *as in elasticity theory or the Minkowski metric when n* = 4 *as in special relativity. The main problem is that this system is not involutive unless we prolong the system to order two by differentiating once the equations. For such a purpose, introducing ω*−<sup>1</sup> = (*ωij*) *as usual, we may define: Christoffel symbols γ<sup>k</sup> ij*(*x*) = <sup>1</sup> <sup>2</sup>*ωkr*(*x*)(*∂iωrj*(*x*) + *<sup>∂</sup>jωri*(*x*) <sup>−</sup> *<sup>∂</sup>rωij*(*x*)) = *<sup>γ</sup><sup>k</sup> ji*(*x*)

*This is a new geometric object of order* 2 *allowing to obtain, as in Example 3.2, an isomorphism <sup>j</sup>*1(*ω*) � (*ω*, *<sup>γ</sup>*) *and the second order equations with f* <sup>−</sup><sup>1</sup> <sup>1</sup> = *g*1*:*

*Lie form g<sup>k</sup> <sup>l</sup>* (*y<sup>l</sup> ij* <sup>+</sup> *<sup>γ</sup><sup>l</sup> rs*(*y*)*y<sup>r</sup> i ys j* ) = *γ<sup>k</sup> ij*(*x*)

12 Will-be-set-by-IN-TECH

*The corresponding geometric object is thus made by a* 1*-form density ω* = (*ω*1, *ω*2, *ω*3) *that transforms like a* 1*-form up to the division by the square root of the Jacobian determinant. The unusual general*

*ω*1(*∂*2*ω*<sup>3</sup> − *∂*3*ω*2) + *ω*2(*∂*3*ω*<sup>1</sup> − *∂*1*ω*3) + *ω*3(*∂*1*ω*<sup>2</sup> − *∂*2*ω*1) = *c For the special section <sup>ω</sup>* = (1, <sup>−</sup>*x*3, 0) *we have c* <sup>=</sup> <sup>1</sup>*. If we choose <sup>ω</sup>*¯ = (1, 0, 0) *we may define*

*j*1(*f*)−1(*ω*) = *ω*¯ *cannot even be solved formally. These results can be extended to an arbitrary odd*

**Example 3.6.** *(Screw and complex structures) (n* = 2, *q* = 1*) In 1878 Clifford introduced abstract numbers of the form x*<sup>1</sup> + *�x*<sup>2</sup> *with �*<sup>2</sup> = 0 *in order to study helicoidal movements in the mechanics of rigid bodies. We may try to define functions of these numbers for which a derivative may have a*

*d f* = (*A* + *�B*)(*dx*<sup>1</sup> + *�dx*2) = *Adx*<sup>1</sup> + *�*(*Bdx*<sup>1</sup> + *Adx*2) = *d f* <sup>1</sup> + *�d f* <sup>2</sup> *Accordingly, we have to look for transformations y*<sup>1</sup> = *f* <sup>1</sup>(*x*1, *x*2), *y*<sup>2</sup> = *f* <sup>2</sup>(*x*1, *x*2) *satisfying the*

*an algebraic Lie pseudogroup, a tricky computation ([24], p 467) allows to prove that* Γ *is made by the*

 *y*<sup>1</sup> <sup>1</sup> *<sup>y</sup>*<sup>1</sup> 2 *y*2 <sup>1</sup> *<sup>y</sup>*<sup>2</sup> 2 = 0 0 1 0 

1)2/(*y*<sup>1</sup> 1*y*2 <sup>2</sup> <sup>−</sup> *<sup>y</sup>*<sup>1</sup> 2*y*2

*Finally, introducing similarly the abstract number i such that i*<sup>2</sup> <sup>=</sup> <sup>−</sup>1*, we get the Cauchy-Riemann*

*correponding geometric object or complex structure is a mixed tensor with square equal to minus the*

 *y*<sup>1</sup> <sup>1</sup> *<sup>y</sup>*<sup>1</sup> 2 *y*2 <sup>1</sup> *<sup>y</sup>*<sup>2</sup> 2 =

*i yl*

*Medolaghi form* <sup>Ω</sup>*ij* <sup>≡</sup> (L(*ξ*)*ω*)*ij* <sup>≡</sup> *<sup>ω</sup>rj*(*x*)*∂iξ<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>ir*(*x*)*∂jξ<sup>r</sup>* <sup>+</sup> *<sup>ξ</sup>r∂rωij*(*x*) = <sup>0</sup> *also called Killing system for historical reasons. A special section could be the Euclidean metric when n* = 1, 2, 3 *as in elasticity theory or the Minkowski metric when n* = 4 *as in special relativity. The main problem is that this system is not involutive unless we prolong the system to order two by differentiating*

*<sup>j</sup>* = *ωij*(*x*)

<sup>2</sup> <sup>=</sup> 0, *<sup>y</sup>*<sup>2</sup>

<sup>2</sup> <sup>−</sup> *<sup>y</sup>*<sup>1</sup>

<sup>1</sup> = 0 *with no CC defining complex analytic transformations and the*

 <sup>0</sup> <sup>−</sup><sup>1</sup> 1 0

<sup>2</sup>*ωkr*(*x*)(*∂iωrj*(*x*) + *<sup>∂</sup>jωri*(*x*) <sup>−</sup> *<sup>∂</sup>rωij*(*x*)) = *<sup>γ</sup><sup>k</sup>*

*∂rωi*(*x*) = 0

<sup>1</sup> *but now c*¯ = 0 *and the equivalence problem*

<sup>1</sup> = 0 *with no CC. As we have*

<sup>1</sup>) = 1 *and let the reader exhibit* F*.*

*ji*(*x*)

<sup>Ω</sup>*<sup>i</sup>* <sup>≡</sup> *<sup>ω</sup>r*(*x*)*∂iξ<sup>r</sup>* <sup>−</sup> (1/2)*ωi*(*x*)*∂rξ<sup>r</sup>* <sup>+</sup> *<sup>ξ</sup><sup>r</sup>*

*In a symbolic way ω* ∧ *dω is now a scalar and the only Vessiot structure equation is:*

*Medolaghi form is:*

Γ¯ *by the system y*<sup>1</sup>

*We get the Lie form* <sup>Φ</sup><sup>1</sup> <sup>≡</sup> *<sup>y</sup>*<sup>1</sup>

<sup>1</sup> <sup>=</sup> 0, *<sup>y</sup>*<sup>1</sup>

2 × 2 *identity matrix as we have now:*

*Christoffel symbols γ<sup>k</sup>*

<sup>2</sup> <sup>−</sup> *<sup>y</sup>*<sup>1</sup>

*system y*<sup>2</sup>

<sup>2</sup> <sup>=</sup> 0, *<sup>y</sup>*<sup>1</sup>

*dimension with much more work ([24], p 684).*

*first order involutive system of finite Lie equations y*<sup>1</sup>

 *y*<sup>1</sup> <sup>1</sup> *<sup>y</sup>*<sup>1</sup> 2 *y*2 <sup>1</sup> *<sup>y</sup>*<sup>2</sup> 2

2/*y*<sup>1</sup>

<sup>2</sup> <sup>+</sup> *<sup>y</sup>*<sup>2</sup>

 *y*<sup>1</sup> <sup>1</sup> *<sup>y</sup>*<sup>1</sup> 2 *y*2 <sup>1</sup> *<sup>y</sup>*<sup>2</sup> 2

*Lie form* <sup>Φ</sup>*ij*(*y*1) <sup>≡</sup> *<sup>ω</sup>kl*(*y*)*y<sup>k</sup>*

<sup>3</sup> <sup>=</sup> 0, *<sup>y</sup>*<sup>2</sup>

2*y*3 <sup>3</sup> <sup>−</sup> *<sup>y</sup>*<sup>2</sup> 3*y*3 <sup>2</sup> <sup>=</sup> *<sup>y</sup>*<sup>1</sup>

*meaning. Thus, if f*(*x*<sup>1</sup> + *�x*2) = *f* <sup>1</sup>(*x*1, *x*2) + *� f* <sup>2</sup>(*x*1, *x*2)*, then we should get:*

*transformations preserving a mixed tensor with square equal to zero as follows:*

<sup>−</sup><sup>1</sup> 0 0

<sup>1</sup> <sup>=</sup> 0, <sup>Φ</sup><sup>2</sup> <sup>≡</sup> (*y*<sup>1</sup>

<sup>−</sup><sup>1</sup> <sup>0</sup> <sup>−</sup><sup>1</sup> 1 0

**Example 3.7.** *(Riemann structure) If ω is a section of* F = *S*2*T*<sup>∗</sup> *with det*(*ω*) �= 0 *we get:*

*once the equations. For such a purpose, introducing ω*−<sup>1</sup> = (*ωij*) *as usual, we may define:*

*ij*(*x*) = <sup>1</sup>

1 0

*Medolaghi form* Γ*<sup>k</sup> ij* <sup>≡</sup> (L(*ξ*)*γ*)*<sup>k</sup> ij* <sup>≡</sup> *<sup>∂</sup>ijξ<sup>k</sup>* <sup>+</sup> *<sup>γ</sup><sup>k</sup> rj*(*x*)*∂iξ<sup>r</sup>* <sup>+</sup> *<sup>γ</sup><sup>k</sup> ir*(*x*)*∂jξ<sup>r</sup>* <sup>−</sup> *<sup>γ</sup><sup>r</sup> ij*(*x*)*∂rξ<sup>k</sup>* <sup>+</sup> *<sup>ξ</sup>r∂rγ<sup>k</sup> ij*(*x*) = 0

*where* (Γ*<sup>k</sup> ij*) *is a section of S*2*T*<sup>∗</sup> ⊗ *T. Surprisingly, the following expression:*

*Riemann tensor ρ<sup>k</sup> lij*(*x*) <sup>≡</sup> *<sup>∂</sup>iγ<sup>k</sup> lj*(*x*) <sup>−</sup> *<sup>∂</sup>jγ<sup>k</sup> li*(*x*) + *<sup>γ</sup><sup>r</sup> lj*(*x*)*γ<sup>k</sup> ri*(*x*) <sup>−</sup> *<sup>γ</sup><sup>r</sup> li*(*x*)*γ<sup>k</sup> rj*(*x*) *is still a first order geometric object and even a tensor as a section of* <sup>∧</sup>2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *T satisfying the purely algebraic relations :*

$$
\rho\_{\rm liij}^k + \rho\_{\rm ijl}^k + \rho\_{\rm jli}^k = 0, \qquad \omega\_{\rm rl} \rho\_{\rm kij}^l + \omega\_{\rm kr} \rho\_{\rm liij}^r = 0 \\
\Rightarrow \rho\_{\rm klij} = \omega\_{\rm kr} \rho\_{\rm liij}^r = \rho\_{\rm ijkl}.
$$

*Accordingly, the IC must express that the new first order equations* (L(*ξ*)*ρ*)*<sup>k</sup> lij* = 0 *are only linear combinations of the previous ones and we get the Vessiot structure equations:*

$$
\rho\_{lij}^k(\mathfrak{x}) = c(\delta\_i^k \omega\_{lj}(\mathfrak{x}) - \delta\_j^k \omega\_{li}(\mathfrak{x})) .
$$

*describing the constant Riemannian curvature condition of Eisenhart [10]. Finally, as we have ρr rij*(*x*) = *<sup>∂</sup>iγ<sup>r</sup> rj*(*x*) <sup>−</sup> *<sup>∂</sup>jγ<sup>r</sup> ri*(*x*) = <sup>0</sup>*, we can only introduce the Ricci tensor <sup>ρ</sup>ij*(*x*) = *<sup>ρ</sup><sup>r</sup> irj*(*x*) = *ρji*(*x*) *by contracting indices and the scalar curvature ρ*(*x*) = *ωij*(*x*)*ρij*(*x*) *in order to obtain ρ*(*x*) = *n*(*n* − 1)*c. It remains to obtain all these results in a purely formal way, for example to prove that the number of components of the Riemann tensor is equal to n*2(*n*<sup>2</sup> <sup>−</sup> <sup>1</sup>)/12 *without dealing with indices.*

**Remark 3.2.** *Comparing the various Vessiot structure equations containing structure constants, we discover at once that the many c appearing in the MC equations are absolutely on equal footing with the only c appearing in the other examples. As their factors are either constant, linear or quadratic, any identification of the quadratic terms appearing in the Riemann tensor with the quadratic terms appearing in the MC equations is definitively not correct or, in an equivalent but more abrupt way, the Cartan structure equations have nothing to do with the Vessiot structure equations. As we shall see, most of mathematical physics today is based on such a confusion.*

**Remark 3.3.** *Let us consider again Example 3.2 with ∂xx f*(*x*)/*∂<sup>x</sup> f*(*x*) = *ω*¯(*x*) *and introduce a variation η*(*f*(*x*)) = *δ f*(*x*) *as in analytical or continuum mechanics. We get similarly δ∂<sup>x</sup> f* = *<sup>∂</sup>x<sup>δ</sup> <sup>f</sup>* = *∂η <sup>∂</sup><sup>y</sup> ∂<sup>x</sup> f and so on, a result leading to δω*¯(*x*) = *∂<sup>x</sup> f*L(*η*)*ω*(*f*(*x*)) *where the Lie derivative involved is computed over the target. Let us now pass from the target to the source by introducing <sup>η</sup>* <sup>=</sup> *ξ∂<sup>x</sup> <sup>f</sup>* <sup>⇒</sup> *∂η <sup>∂</sup><sup>y</sup> ∂<sup>x</sup> f* = *∂xξ∂<sup>x</sup> f* + *ξ∂xx f and so on, a result leading to the particularly simple variation δω*¯ = L(*ξ*)*ω*¯ *over the soure. As another example of this general variational procedure, let us compare with the similar variations on which classical finite elasticity theory is based. Starting now with ωkl*(*f*(*x*))*∂<sup>i</sup> f <sup>k</sup>*(*x*)*∂<sup>j</sup> f <sup>l</sup>* (*x*) = *ω*¯*ij*(*x*)*, where ω is the Euclidean metric, we obtain* (*δω*¯)*ij*(*x*) = *∂<sup>i</sup> f <sup>k</sup>*(*x*)*∂<sup>j</sup> f <sup>l</sup>* (*x*)(L(*η*)*ω*)*kl*(*f*(*x*)) *where the Lie derivative involved is computed over the target. Passing now from the target to the source as before, we find the particularly simple variation δω*¯ = L(*ξ*)*ω*¯ *over the source. For "small" deformations, source and target are of course identified but it is not true that the infinitesimal deformation tensor is in general the limit of the finite deformation tensor (for a counterexample, see [25], p 70).*

*the characters α<sup>i</sup>*

*<sup>q</sup>* <sup>=</sup> *<sup>m</sup>* (*q*+*n*−*i*−1)!

*is obtained by only prolonging the β<sup>i</sup>*

*that case dim*(*gq*+1) = *α*<sup>1</sup>

*nonlinear systems as well.*

([27], VI,1.14, p 802):

*<sup>n</sup>*, ..., *z β*

*Janet sequence* ([4], p 144):

*coordinates z*<sup>1</sup>

believe.

*the other equations.*

(*q*−1)!((*n*−*i*)! <sup>−</sup> *<sup>β</sup><sup>i</sup>*

*<sup>q</sup>* + ... + *α<sup>n</sup>*

*<sup>n</sup> while zβ*+<sup>1</sup> *<sup>n</sup>* , ..., *<sup>z</sup>*

When *Rq* is involutive, the linear differential operator <sup>D</sup> : *<sup>E</sup> jq*

<sup>0</sup> −→ <sup>Θ</sup> −→ *<sup>T</sup>* <sup>D</sup>

canonical *linear Spencer sequence* ([4], p 150):

sequence for *Jq*+1(*E*) ⊂ *J*1(*Jq*(*E*)) ([25], p 152):

<sup>0</sup> −→ <sup>Θ</sup> *jq*

−→ *C*<sup>0</sup>

as the Janet sequence for the first order involutive system *Rq*+<sup>1</sup> ⊂ *J*1(*Rq*).

*<sup>D</sup>*<sup>1</sup> −→ *<sup>C</sup>*<sup>1</sup>

*<sup>q</sup> for i* = 1, ..., *n with α<sup>n</sup>*

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 15

*leading term* (*α*/*n*!)*r<sup>n</sup> when <sup>α</sup>* �<sup>=</sup> <sup>0</sup>*. Such a prolongation procedure allows to compute in a unique way the principal (pri) jets from the parametric (par) other ones. This definition may also be applied to*

We obtain the following theorem generalizing for PD control systems the well known first order Kalman form of OD control systems where the derivatives of the input do not appear

**Theorem 4.1.** *When Rq is involutive, its Spencer form is involutive and can be modified to a reduced Spencer form in such a way that β* = *dim*(*Rq*) − *α equations can be solved with respect to the jet*

order *q* with space of solutions Θ ⊂ *E* is said to be *involutive* and one has the canonical *linear*

<sup>D</sup><sup>1</sup> −→ *<sup>F</sup>*<sup>1</sup>

where each other operator is first order involutive and generates the *compatibility conditions* (CC) of the preceding one. As the Janet sequence can be cut at any place, *the numbering of the Janet bundles has nothing to do with that of the Poincaré sequence*, contrary to what many physicists

**Definition 4.3.** *The Janet sequence is said to be locally exact at Fr if any local section of Fr killed by* D*r*+<sup>1</sup> *is the image by* D*<sup>r</sup> of a local section of Fr*−1*. It is called locally exact if it is locally exact at each Fr for* 0 ≤ *r* ≤ *n. The Poincaré sequence is locally exact but counterexemples may exist ([23], p 202).*

*J*1(*Rq*) → *J*1(*Rq*)/*Rq*+<sup>1</sup> � *T*<sup>∗</sup> ⊗ *Rq*/*δ*(*gq*+1) = *C*<sup>1</sup> of order one induced by *<sup>D</sup>* : *Rq*+<sup>1</sup> <sup>→</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *Rq*. Introducing the *Spencer bundles Cr* <sup>=</sup> <sup>∧</sup>*rT*<sup>∗</sup> <sup>⊗</sup> *Rq*/*δ*(∧*r*−1*T*<sup>∗</sup> <sup>⊗</sup> *gq*+1), the first order involutive (*r* + 1)-*Spencer operator Dr*+<sup>1</sup> : *Cr* → *Cr*+<sup>1</sup> is induced by *<sup>D</sup>* : <sup>∧</sup>*rT*<sup>∗</sup> <sup>⊗</sup> *Rq*+<sup>1</sup> → ∧*r*+1*T*<sup>∗</sup> <sup>⊗</sup> *Rq* : *<sup>α</sup>* <sup>⊗</sup> *<sup>ξ</sup>q*+<sup>1</sup> <sup>→</sup> *<sup>d</sup><sup>α</sup>* <sup>⊗</sup> *<sup>ξ</sup><sup>q</sup>* + (−1)*r<sup>α</sup>* <sup>∧</sup> *<sup>D</sup>ξq*+<sup>1</sup> and we obtain the

*<sup>D</sup>*<sup>2</sup> −→ *<sup>C</sup>*<sup>2</sup>

The Janet sequence and the Spencer sequence are connected by the following *crucial* commutative *diagram* (1) where the Spencer sequence is induced by the locally exact central horizontal sequence which is at the same time the Janet sequence for *jq* and the Spencer

Equivalently, we have the involutive *first Spencer operator D*<sup>1</sup> : *C*<sup>0</sup> = *Rq*

−→ *F*<sup>0</sup>

*<sup>q</sup>* = *α. Then Rq is involutive if Rq*+<sup>1</sup>

*<sup>q</sup> equations of class i with respect to d*1, ..., *di for i* = 1, ..., *n. In*

*<sup>q</sup> and one can exhibit the Hilbert polynomial dim*(*Rq*+*r*) *in r with*

*<sup>β</sup>*+*<sup>α</sup> <sup>n</sup> do not appear. In this case zβ*<sup>+</sup>1, ..., *<sup>z</sup>β*+*<sup>α</sup> do not appear in*

<sup>D</sup><sup>2</sup> −→ ... <sup>D</sup>*<sup>n</sup>* −→ *Fn* −→ <sup>0</sup>

*<sup>D</sup>*<sup>3</sup> −→ ... *Dn* −→ *Cn* −→ <sup>0</sup>

<sup>→</sup> *Jq*(*E*) <sup>Φ</sup>

→ *Jq*(*E*)/*Rq* = *F*<sup>0</sup> of

*j*1 →

Introducing a copy *Y* of *X* in the general framework, (*f* , *δ f*) must be considered as a section of *V*(*X* × *Y*)=(*X* × *Y*)×*YT*(*Y*) = *X* × *T*(*Y*) over *X*. When *f* is invertible (care), then we may consider the map *f* : *X* → *Y* : (*x*) → (*y* = *f*(*x*)) and define *ξ* ∈ *T* by *η* = *T*(*f*)(*ξ*) or rather *<sup>η</sup>* <sup>=</sup> *<sup>j</sup>*1(*f*)(*ξ*) in the language of geometric object, as a way to identify *<sup>f</sup>* <sup>−</sup>1(*V*(*<sup>X</sup>* <sup>×</sup> *<sup>Y</sup>*)) with *T* = *T*(*X*). When *f* = *id*, this identification is canonical by considering vertical vectors along the diagonal <sup>Δ</sup> <sup>=</sup> {(*x*, *<sup>y</sup>*) <sup>∈</sup> *<sup>X</sup>* <sup>×</sup> *<sup>Y</sup>*|*<sup>y</sup>* <sup>=</sup> *<sup>x</sup>*} and we get *δω* <sup>=</sup> <sup>Ω</sup> <sup>∈</sup> *<sup>F</sup>*<sup>0</sup> <sup>=</sup> *<sup>ω</sup>*−1(*V*(F)). We point out that the above *vertical procedure* is a nice tool for studying nonlinear systems ([26], III, C and [27], III, 2).

#### **4. Janet versus Spencer : The linear sequences**

Let *<sup>μ</sup>* = (*μ*1, ..., *<sup>μ</sup>n*) be a multi-index with *length* |*μ*| = *<sup>μ</sup>*<sup>1</sup> + ... + *<sup>μ</sup>n*, *class i* if *<sup>μ</sup>*<sup>1</sup> = ... = *<sup>μ</sup>i*−<sup>1</sup> = 0, *<sup>μ</sup><sup>i</sup>* �<sup>=</sup> 0 and *<sup>μ</sup>* <sup>+</sup> <sup>1</sup>*<sup>i</sup>* = (*μ*1, ..., *<sup>μ</sup>i*−1, *<sup>μ</sup><sup>i</sup>* <sup>+</sup> 1, *<sup>μ</sup>i*<sup>+</sup>1, ..., *<sup>μ</sup>n*). We set *yq* <sup>=</sup> {*y<sup>k</sup> <sup>μ</sup>*|1 ≤ *k* ≤ *m*, 0 ≤ |*μ*| ≤ *<sup>q</sup>*} with *<sup>y</sup><sup>k</sup> <sup>μ</sup>* <sup>=</sup> *<sup>y</sup><sup>k</sup>* when <sup>|</sup>*μ*<sup>|</sup> <sup>=</sup> 0. If *<sup>E</sup>* is a vector bundle over *<sup>X</sup>* with local coordinates (*x<sup>i</sup>* , *yk*) for *i* = 1, ..., *n* and *k* = 1, ..., *m*, we denote by *Jq*(*E*) the *q*-*jet bundle* of *E* with local coordinates simply denoted by (*x*, *yq*) and *sections fq* : (*x*) <sup>→</sup> (*x*, *<sup>f</sup> <sup>k</sup>*(*x*), *<sup>f</sup> <sup>k</sup> <sup>i</sup>* (*x*), *<sup>f</sup> <sup>k</sup> ij*(*x*), ...) transforming like the section *jq*(*f*) : (*x*) <sup>→</sup> (*x*, *<sup>f</sup> <sup>k</sup>*(*x*), *<sup>∂</sup><sup>i</sup> <sup>f</sup> <sup>k</sup>*(*x*), *<sup>∂</sup>ij <sup>f</sup> <sup>k</sup>*(*x*), ...) when *<sup>f</sup>* is an arbitrary section of *<sup>E</sup>*. Then both *fq* ∈ *Jq*(*E*) and *jq*(*f*) ∈ *Jq*(*E*) are over *f* ∈ *E* and the *Spencer operator* just allows to distinguish them by introducing a kind of "*difference*" through the operator *D* : *Jq*+1(*E*) → *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *Jq*(*E*) : *fq*+<sup>1</sup> <sup>→</sup> *<sup>j</sup>*1(*fq*) <sup>−</sup> *fq*+<sup>1</sup> with local components (*∂<sup>i</sup> <sup>f</sup> <sup>k</sup>*(*x*) <sup>−</sup> *<sup>f</sup> <sup>k</sup> <sup>i</sup>* (*x*), *<sup>∂</sup><sup>i</sup> <sup>f</sup> <sup>k</sup> <sup>j</sup>* (*x*) <sup>−</sup> *<sup>f</sup> <sup>k</sup> ij*(*x*), ...) and more generally (*D fq*+1)*<sup>k</sup> μ*,*i* (*x*) = *∂<sup>i</sup> f <sup>k</sup> <sup>μ</sup>*(*x*) <sup>−</sup> *<sup>f</sup> <sup>k</sup> μ*+1*<sup>i</sup>* (*x*). In a symbolic way, *when changes of coordinates are not involved*, it is sometimes useful to write down the components of *D* in the form *di* = *∂<sup>i</sup>* − *δ<sup>i</sup>* and the restriction of *D* to the kernel *Sq*+1*T*<sup>∗</sup> ⊗ *E* of the canonical projection *<sup>π</sup>q*+<sup>1</sup> *<sup>q</sup>* : *Jq*+1(*E*) <sup>→</sup> *Jq*(*E*) is *minus* the *Spencer map <sup>δ</sup>* <sup>=</sup> *dx<sup>i</sup>* <sup>∧</sup> *<sup>δ</sup><sup>i</sup>* : *Sq*+1*T*<sup>∗</sup> <sup>⊗</sup> *<sup>E</sup>* <sup>→</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *SqT*<sup>∗</sup> <sup>⊗</sup> *<sup>E</sup>*. The kernel of *<sup>D</sup>* is made by sections such that *fq*+<sup>1</sup> = *<sup>j</sup>*1(*fq*) = *<sup>j</sup>*2(*fq*−1) = ... = *jq*+1(*f*). Finally, if *Rq* ⊂ *Jq*(*E*) is a *system* of order *q* on *E* locally defined by linear equations <sup>Φ</sup>*τ*(*x*, *yq*) <sup>≡</sup> *<sup>a</sup> τμ <sup>k</sup>* (*x*)*y<sup>k</sup> <sup>μ</sup>* = 0 and local coordinates (*x*, *z*) for the parametric jets up to order *q*, the *r*-*prolongation Rq*+*<sup>r</sup>* = *ρr*(*Rq*) = *Jr*(*Rq*) ∩ *Jq*+*r*(*E*) ⊂ *Jr*(*Jq*(*E*)) is locally defined when *<sup>r</sup>* <sup>=</sup> 1 by the linear equations <sup>Φ</sup>*τ*(*x*, *yq*) = 0, *di*Φ*τ*(*x*, *yq*+1) <sup>≡</sup> *<sup>a</sup> τμ <sup>k</sup>* (*x*)*y<sup>k</sup> <sup>μ</sup>*+1*<sup>i</sup>* + *∂ia τμ <sup>k</sup>* (*x*)*y<sup>k</sup> <sup>μ</sup>* = 0 and has *symbol gq*+*<sup>r</sup>* = *Rq*+*<sup>r</sup>* ∩ *Sq*+*rT*<sup>∗</sup> ⊗ *E* ⊂ *Jq*+*r*(*E*) if one looks at the *top order terms*. If *fq*+<sup>1</sup> ∈ *Rq*+<sup>1</sup> is over *fq* ∈ *Rq*, differentiating the identity *a τμ <sup>k</sup>* (*x*)*<sup>f</sup> <sup>k</sup> <sup>μ</sup>*(*x*) ≡ 0 with respect to *x<sup>i</sup>* and substracting the identity *a τμ <sup>k</sup>* (*x*)*<sup>f</sup> <sup>k</sup> μ*+1*<sup>i</sup>* (*x*) + *∂ia τμ <sup>k</sup>* (*x*)*<sup>f</sup> <sup>k</sup> <sup>μ</sup>*(*x*) ≡ 0, we obtain the identity *a τμ <sup>k</sup>* (*x*)(*∂<sup>i</sup> <sup>f</sup> <sup>k</sup> <sup>μ</sup>*(*x*) <sup>−</sup> *<sup>f</sup> <sup>k</sup> μ*+1*<sup>i</sup>* (*x*)) ≡ 0 and thus the restriction *D* : *Rq*+<sup>1</sup> → *T*<sup>∗</sup> ⊗ *Rq* ([23],[27],[33]).

**Definition 4.1.** *Rq is said to be formally integrable when the restriction <sup>π</sup>q*+<sup>1</sup> *<sup>q</sup>* : *Rq*+<sup>1</sup> → *Rq is an epimorphism* ∀*r* ≥ 0 *or, equivalently, when all the equations of order q* + *r are obtained by r prolongations only* ∀*r* ≥ 0*. In that case, Rq*+<sup>1</sup> ⊂ *J*1(*Rq*) *is a canonical equivalent formally integrable first order system on Rq with no zero order equations, called the Spencer form.*

**Definition 4.2.** *Rq is said to be involutive when it is formally integrable and all the sequences* ... *<sup>δ</sup>* → <sup>∧</sup>*sT*<sup>∗</sup> <sup>⊗</sup> *gq*+*<sup>r</sup> <sup>δ</sup>* → ... *are exact* ∀0 ≤ *s* ≤ *n*, ∀*r* ≥ 0*. Equivalently, using a linear change of local coordinates if necessary, we may successively solve the maximum number β<sup>n</sup> <sup>q</sup>* , *βn*−<sup>1</sup> *<sup>q</sup>* , ..., *β*<sup>1</sup> *<sup>q</sup> of equations with respect to the principal jet coordinates of strict order q and class n*, *n* − 1, ..., 1 *in order to introduce* 14 Will-be-set-by-IN-TECH

Introducing a copy *Y* of *X* in the general framework, (*f* , *δ f*) must be considered as a section of *V*(*X* × *Y*)=(*X* × *Y*)×*YT*(*Y*) = *X* × *T*(*Y*) over *X*. When *f* is invertible (care), then we may consider the map *f* : *X* → *Y* : (*x*) → (*y* = *f*(*x*)) and define *ξ* ∈ *T* by *η* = *T*(*f*)(*ξ*) or rather *<sup>η</sup>* <sup>=</sup> *<sup>j</sup>*1(*f*)(*ξ*) in the language of geometric object, as a way to identify *<sup>f</sup>* <sup>−</sup>1(*V*(*<sup>X</sup>* <sup>×</sup> *<sup>Y</sup>*)) with *T* = *T*(*X*). When *f* = *id*, this identification is canonical by considering vertical vectors along the diagonal <sup>Δ</sup> <sup>=</sup> {(*x*, *<sup>y</sup>*) <sup>∈</sup> *<sup>X</sup>* <sup>×</sup> *<sup>Y</sup>*|*<sup>y</sup>* <sup>=</sup> *<sup>x</sup>*} and we get *δω* <sup>=</sup> <sup>Ω</sup> <sup>∈</sup> *<sup>F</sup>*<sup>0</sup> <sup>=</sup> *<sup>ω</sup>*−1(*V*(F)). We point out that the above *vertical procedure* is a nice tool for studying nonlinear systems ([26], III, C

Let *<sup>μ</sup>* = (*μ*1, ..., *<sup>μ</sup>n*) be a multi-index with *length* |*μ*| = *<sup>μ</sup>*<sup>1</sup> + ... + *<sup>μ</sup>n*, *class i* if *<sup>μ</sup>*<sup>1</sup> = ... = *<sup>μ</sup>i*−<sup>1</sup> =

for *i* = 1, ..., *n* and *k* = 1, ..., *m*, we denote by *Jq*(*E*) the *q*-*jet bundle* of *E* with local coordinates

the section *jq*(*f*) : (*x*) <sup>→</sup> (*x*, *<sup>f</sup> <sup>k</sup>*(*x*), *<sup>∂</sup><sup>i</sup> <sup>f</sup> <sup>k</sup>*(*x*), *<sup>∂</sup>ij <sup>f</sup> <sup>k</sup>*(*x*), ...) when *<sup>f</sup>* is an arbitrary section of *<sup>E</sup>*. Then both *fq* ∈ *Jq*(*E*) and *jq*(*f*) ∈ *Jq*(*E*) are over *f* ∈ *E* and the *Spencer operator* just allows to distinguish them by introducing a kind of "*difference*" through the operator *D* : *Jq*+1(*E*) →

*<sup>μ</sup>*(*x*) <sup>−</sup> *<sup>f</sup> <sup>k</sup>*

*coordinates are not involved*, it is sometimes useful to write down the components of *D* in the form *di* = *∂<sup>i</sup>* − *δ<sup>i</sup>* and the restriction of *D* to the kernel *Sq*+1*T*<sup>∗</sup> ⊗ *E* of the canonical projection

*<sup>q</sup>* : *Jq*+1(*E*) <sup>→</sup> *Jq*(*E*) is *minus* the *Spencer map <sup>δ</sup>* <sup>=</sup> *dx<sup>i</sup>* <sup>∧</sup> *<sup>δ</sup><sup>i</sup>* : *Sq*+1*T*<sup>∗</sup> <sup>⊗</sup> *<sup>E</sup>* <sup>→</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *SqT*<sup>∗</sup> <sup>⊗</sup> *<sup>E</sup>*. The kernel of *<sup>D</sup>* is made by sections such that *fq*+<sup>1</sup> = *<sup>j</sup>*1(*fq*) = *<sup>j</sup>*2(*fq*−1) = ... = *jq*+1(*f*). Finally, if *Rq* ⊂ *Jq*(*E*) is a *system* of order *q* on *E* locally defined by linear equations

*q*, the *r*-*prolongation Rq*+*<sup>r</sup>* = *ρr*(*Rq*) = *Jr*(*Rq*) ∩ *Jq*+*r*(*E*) ⊂ *Jr*(*Jq*(*E*)) is locally defined when

and has *symbol gq*+*<sup>r</sup>* = *Rq*+*<sup>r</sup>* ∩ *Sq*+*rT*<sup>∗</sup> ⊗ *E* ⊂ *Jq*+*r*(*E*) if one looks at the *top order terms*. If

*an epimorphism* ∀*r* ≥ 0 *or, equivalently, when all the equations of order q* + *r are obtained by r prolongations only* ∀*r* ≥ 0*. In that case, Rq*+<sup>1</sup> ⊂ *J*1(*Rq*) *is a canonical equivalent formally integrable*

**Definition 4.2.** *Rq is said to be involutive when it is formally integrable and all the sequences* ... *<sup>δ</sup>*

*with respect to the principal jet coordinates of strict order q and class n*, *n* − 1, ..., 1 *in order to introduce*

(*x*) + *∂ia*

*μ*+1*<sup>i</sup>*

*<sup>μ</sup>* = 0 and local coordinates (*x*, *z*) for the parametric jets up to order

*τμ <sup>k</sup>* (*x*)*<sup>f</sup> <sup>k</sup>*

→ ... *are exact* ∀0 ≤ *s* ≤ *n*, ∀*r* ≥ 0*. Equivalently, using a linear change of local*

(*x*)) ≡ 0 and thus the restriction *D* : *Rq*+<sup>1</sup> → *T*<sup>∗</sup> ⊗ *Rq* ([23],[27],[33]).

*τμ <sup>k</sup>* (*x*)*y<sup>k</sup>*

*τμ <sup>k</sup>* (*x*)*<sup>f</sup> <sup>k</sup>*

*<sup>μ</sup>* <sup>=</sup> *<sup>y</sup><sup>k</sup>* when <sup>|</sup>*μ*<sup>|</sup> <sup>=</sup> 0. If *<sup>E</sup>* is a vector bundle over *<sup>X</sup>* with local coordinates (*x<sup>i</sup>*

*<sup>i</sup>* (*x*), *<sup>f</sup> <sup>k</sup>*

*<sup>μ</sup>*|1 ≤ *k* ≤ *m*, 0 ≤ |*μ*| ≤

*ij*(*x*), ...) transforming like

*<sup>j</sup>* (*x*) <sup>−</sup> *<sup>f</sup> <sup>k</sup>*

*τμ <sup>k</sup>* (*x*)*y<sup>k</sup>*

*<sup>μ</sup>*(*x*) ≡ 0 with respect to

*<sup>q</sup>* : *Rq*+<sup>1</sup> → *Rq is*

*<sup>i</sup>* (*x*), *<sup>∂</sup><sup>i</sup> <sup>f</sup> <sup>k</sup>*

(*x*). In a symbolic way, *when changes of*

*<sup>μ</sup>*+1*<sup>i</sup>* + *∂ia*

*<sup>μ</sup>*(*x*) ≡ 0, we obtain the identity

*<sup>q</sup>* , *βn*−<sup>1</sup> *<sup>q</sup>* , ..., *β*<sup>1</sup> , *yk*)

*ij*(*x*), ...)

*<sup>μ</sup>* = 0

→

*<sup>q</sup> of equations*

and [27], III, 2).

*<sup>q</sup>*} with *<sup>y</sup><sup>k</sup>*

*<sup>π</sup>q*+<sup>1</sup>

*a τμ <sup>k</sup>* (*x*)(*∂<sup>i</sup> <sup>f</sup> <sup>k</sup>*

<sup>Φ</sup>*τ*(*x*, *yq*) <sup>≡</sup> *<sup>a</sup>*

<sup>∧</sup>*sT*<sup>∗</sup> <sup>⊗</sup> *gq*+*<sup>r</sup> <sup>δ</sup>*

and more generally (*D fq*+1)*<sup>k</sup>*

*τμ <sup>k</sup>* (*x*)*y<sup>k</sup>*

*x<sup>i</sup>* and substracting the identity *a*

*μ*+1*<sup>i</sup>*

*<sup>μ</sup>*(*x*) <sup>−</sup> *<sup>f</sup> <sup>k</sup>*

**4. Janet versus Spencer : The linear sequences**

0, *<sup>μ</sup><sup>i</sup>* �<sup>=</sup> 0 and *<sup>μ</sup>* <sup>+</sup> <sup>1</sup>*<sup>i</sup>* = (*μ*1, ..., *<sup>μ</sup>i*−1, *<sup>μ</sup><sup>i</sup>* <sup>+</sup> 1, *<sup>μ</sup>i*<sup>+</sup>1, ..., *<sup>μ</sup>n*). We set *yq* <sup>=</sup> {*y<sup>k</sup>*

*<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *Jq*(*E*) : *fq*+<sup>1</sup> <sup>→</sup> *<sup>j</sup>*1(*fq*) <sup>−</sup> *fq*+<sup>1</sup> with local components (*∂<sup>i</sup> <sup>f</sup> <sup>k</sup>*(*x*) <sup>−</sup> *<sup>f</sup> <sup>k</sup>*

(*x*) = *∂<sup>i</sup> f <sup>k</sup>*

simply denoted by (*x*, *yq*) and *sections fq* : (*x*) <sup>→</sup> (*x*, *<sup>f</sup> <sup>k</sup>*(*x*), *<sup>f</sup> <sup>k</sup>*

*μ*,*i*

*<sup>r</sup>* <sup>=</sup> 1 by the linear equations <sup>Φ</sup>*τ*(*x*, *yq*) = 0, *di*Φ*τ*(*x*, *yq*+1) <sup>≡</sup> *<sup>a</sup>*

*τμ <sup>k</sup>* (*x*)*<sup>f</sup> <sup>k</sup> μ*+1*<sup>i</sup>*

**Definition 4.1.** *Rq is said to be formally integrable when the restriction <sup>π</sup>q*+<sup>1</sup>

*first order system on Rq with no zero order equations, called the Spencer form.*

*coordinates if necessary, we may successively solve the maximum number β<sup>n</sup>*

*fq*+<sup>1</sup> ∈ *Rq*+<sup>1</sup> is over *fq* ∈ *Rq*, differentiating the identity *a*

*the characters α<sup>i</sup> <sup>q</sup>* <sup>=</sup> *<sup>m</sup>* (*q*+*n*−*i*−1)! (*q*−1)!((*n*−*i*)! <sup>−</sup> *<sup>β</sup><sup>i</sup> <sup>q</sup> for i* = 1, ..., *n with α<sup>n</sup> <sup>q</sup>* = *α. Then Rq is involutive if Rq*+<sup>1</sup> *is obtained by only prolonging the β<sup>i</sup> <sup>q</sup> equations of class i with respect to d*1, ..., *di for i* = 1, ..., *n. In that case dim*(*gq*+1) = *α*<sup>1</sup> *<sup>q</sup>* + ... + *α<sup>n</sup> <sup>q</sup> and one can exhibit the Hilbert polynomial dim*(*Rq*+*r*) *in r with leading term* (*α*/*n*!)*r<sup>n</sup> when <sup>α</sup>* �<sup>=</sup> <sup>0</sup>*. Such a prolongation procedure allows to compute in a unique way the principal (pri) jets from the parametric (par) other ones. This definition may also be applied to nonlinear systems as well.*

We obtain the following theorem generalizing for PD control systems the well known first order Kalman form of OD control systems where the derivatives of the input do not appear ([27], VI,1.14, p 802):

**Theorem 4.1.** *When Rq is involutive, its Spencer form is involutive and can be modified to a reduced Spencer form in such a way that β* = *dim*(*Rq*) − *α equations can be solved with respect to the jet coordinates z*<sup>1</sup> *<sup>n</sup>*, ..., *z β <sup>n</sup> while zβ*+<sup>1</sup> *<sup>n</sup>* , ..., *<sup>z</sup> <sup>β</sup>*+*<sup>α</sup> <sup>n</sup> do not appear. In this case zβ*<sup>+</sup>1, ..., *<sup>z</sup>β*+*<sup>α</sup> do not appear in the other equations.*

When *Rq* is involutive, the linear differential operator <sup>D</sup> : *<sup>E</sup> jq* <sup>→</sup> *Jq*(*E*) <sup>Φ</sup> → *Jq*(*E*)/*Rq* = *F*<sup>0</sup> of order *q* with space of solutions Θ ⊂ *E* is said to be *involutive* and one has the canonical *linear Janet sequence* ([4], p 144):

$$0 \longrightarrow \Theta \longrightarrow T \stackrel{\mathcal{D}}{\longrightarrow} F\_0 \stackrel{\mathcal{D}\_1}{\longrightarrow} F\_1 \stackrel{\mathcal{D}\_2}{\longrightarrow} \dots \stackrel{\mathcal{D}\_n}{\longrightarrow} F\_n \longrightarrow 0$$

where each other operator is first order involutive and generates the *compatibility conditions* (CC) of the preceding one. As the Janet sequence can be cut at any place, *the numbering of the Janet bundles has nothing to do with that of the Poincaré sequence*, contrary to what many physicists believe.

**Definition 4.3.** *The Janet sequence is said to be locally exact at Fr if any local section of Fr killed by* D*r*+<sup>1</sup> *is the image by* D*<sup>r</sup> of a local section of Fr*−1*. It is called locally exact if it is locally exact at each Fr for* 0 ≤ *r* ≤ *n. The Poincaré sequence is locally exact but counterexemples may exist ([23], p 202).*

Equivalently, we have the involutive *first Spencer operator D*<sup>1</sup> : *C*<sup>0</sup> = *Rq j*1 → *J*1(*Rq*) → *J*1(*Rq*)/*Rq*+<sup>1</sup> � *T*<sup>∗</sup> ⊗ *Rq*/*δ*(*gq*+1) = *C*<sup>1</sup> of order one induced by *<sup>D</sup>* : *Rq*+<sup>1</sup> <sup>→</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *Rq*. Introducing the *Spencer bundles Cr* <sup>=</sup> <sup>∧</sup>*rT*<sup>∗</sup> <sup>⊗</sup> *Rq*/*δ*(∧*r*−1*T*<sup>∗</sup> <sup>⊗</sup> *gq*+1), the first order involutive (*r* + 1)-*Spencer operator Dr*+<sup>1</sup> : *Cr* → *Cr*+<sup>1</sup> is induced by *<sup>D</sup>* : <sup>∧</sup>*rT*<sup>∗</sup> <sup>⊗</sup> *Rq*+<sup>1</sup> → ∧*r*+1*T*<sup>∗</sup> <sup>⊗</sup> *Rq* : *<sup>α</sup>* <sup>⊗</sup> *<sup>ξ</sup>q*+<sup>1</sup> <sup>→</sup> *<sup>d</sup><sup>α</sup>* <sup>⊗</sup> *<sup>ξ</sup><sup>q</sup>* + (−1)*r<sup>α</sup>* <sup>∧</sup> *<sup>D</sup>ξq*+<sup>1</sup> and we obtain the canonical *linear Spencer sequence* ([4], p 150):

$$0 \longrightarrow \Theta \stackrel{j\_{\overline{\theta}}}{\longrightarrow} \mathbb{C}\_{0} \stackrel{D\_1}{\longrightarrow} \mathbb{C}\_{1} \stackrel{D\_2}{\longrightarrow} \mathbb{C}\_{2} \stackrel{D\_3}{\longrightarrow} \dots \stackrel{D\_n}{\longrightarrow} \mathbb{C}\_{n} \longrightarrow 0$$

as the Janet sequence for the first order involutive system *Rq*+<sup>1</sup> ⊂ *J*1(*Rq*).

The Janet sequence and the Spencer sequence are connected by the following *crucial* commutative *diagram* (1) where the Spencer sequence is induced by the locally exact central horizontal sequence which is at the same time the Janet sequence for *jq* and the Spencer sequence for *Jq*+1(*E*) ⊂ *J*1(*Jq*(*E*)) ([25], p 152):

*single OD equation yxxx* − *yx* = 0 *for the only y. Introducing the corresponding polynomial ideal* (*χ*<sup>3</sup> <sup>−</sup> *<sup>χ</sup>*)=(*χ*) <sup>∩</sup> (*<sup>χ</sup>* <sup>−</sup> <sup>1</sup>) <sup>∩</sup> (*<sup>χ</sup>* <sup>+</sup> <sup>1</sup>)*, we check that dx kills yxx* <sup>−</sup> *y, dx* <sup>−</sup> <sup>1</sup> *kills yxx* <sup>+</sup> *yx and dx* <sup>+</sup> <sup>1</sup>

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 17

More precisely, if *K* is a differential field containing **Q** with *n* commuting *derivations ∂i*, that is to say *∂i*(*a* + *b*) = *∂ia* + *∂ib* and *∂i*(*ab*)=(*∂ia*)*b* + *a∂ib*, ∀*a*, *b* ∈ *K* for *i* = 1, ..., *n*, we denote by *k* a subfield of constants. Let us introduce *m differential indeterminates y<sup>k</sup>* for *k* = 1, ..., *m* and

> *<sup>μ</sup>* = *y<sup>k</sup> μ*+1*<sup>i</sup>*

*of differential operators D* = *K*[*d*1, ..., *dn*] = *K*[*d*] with *dia* = *adi* + *∂ia*, ∀*a* ∈ *K* in the operator

elements in *Dy* indexed by *τ*, we may introduce the *differential module of equations I* = *D*Φ ⊂

In the algebraic framework considered, *only two possible formal constructions can be obtained from*

**Theorem 5.1.** *homD*(*M*, *D*) *is a right differential module that can be converted to a left differential module by introducing the right differential module structure of* <sup>∧</sup>*nT*∗*. As a differential geometric counterpart, we get the formal adjoint of* <sup>D</sup>*, namely ad*(D) : <sup>∧</sup>*nT*<sup>∗</sup> <sup>⊗</sup> *<sup>F</sup>*<sup>∗</sup> → ∧*nT*<sup>∗</sup> <sup>⊗</sup> *<sup>E</sup>*<sup>∗</sup> *usually constructed through an integration by parts and where E*∗ *is obtained from E by inverting the local*

**Remark 5.1.** *Such a result explains why dual objects in physics and engineering are no longer tensors but tensor densities, with no reference to any variational calculus. For example the EM potential is a section of T*<sup>∗</sup> *and the EM field is a section of* <sup>∧</sup>2*T*<sup>∗</sup> *while the EM induction is a section of* <sup>∧</sup>4*T*<sup>∗</sup> <sup>⊗</sup>

The filtration *D*<sup>0</sup> = *K* ⊆ *D*<sup>1</sup> = *K* ⊕ *T* ⊆ ... ⊆ *Dq* ⊆ ... ⊆ *D* of *D* by the order of operators induces a filtration/inductive limit 0 ⊆ *M*<sup>0</sup> ⊆ *M*<sup>1</sup> ⊆ ... ⊆ *Mq* ⊆ ... ⊆ *M* and provides by duality *over K* the projective limit *M*<sup>∗</sup> = *R* → ... → *Rq* → ... → *R*<sup>1</sup> → *R*<sup>0</sup> → 0 of formally integrable systems. As *D* is generated by *K* and *T* = *D*1/*D*0, we can define for any *f* ∈ *M*∗:

and check *dia* = *adi* + *∂ia*, *ξη* − *ηξ* = [*ξ*, *η*] in the operator sense by introducing the standard

**Remark 5.2.** *When m* = 1 *and D* = *k*[*d*] *is a commutative ring isomorphic to the polynomial ring A* = *k*[*χ*] *for the indeterminates χ*1, ..., *χn, this result exactly describes the inverse system of Macaulay*

**Definition 5.1.** *A simple module is a module having no other proper submodule than* 0*. A semi-simple module is a direct sum of simple modules. When A is a commutative integral domain and M a finitely generated module over A, the socle of M is the largest semi-simple submodule of M, that is soc*(*M*) = ⊕*soc*m(*M*) *where soc*m(*M*) *is the direct sum of all the isotypical simple submodules of M isomorphic to A*/m *for* m ∈ *max*(*A*) *the set of maximal proper ideals of A. The radical of a module is the intersection of all its maximum proper submodules. The quotient of a module by its radical is called the*

**Theorem 5.2.** *R* = *M*∗ *has a structure of differential module induced by the Spencer operator.*

*<sup>μ</sup>* = (*di f*)(*y<sup>k</sup>*

*<sup>μ</sup>*) = *∂<sup>i</sup> f <sup>k</sup>*

*<sup>μ</sup>* <sup>−</sup> *<sup>f</sup> <sup>k</sup>*

. We introduce the non-commutative *ring*

*<sup>μ</sup>*} is a finite number of

*di* ∈ *T*, ∀*m* ∈ *M*

*<sup>μ</sup>*+1*<sup>i</sup>* in a coherent

*τμ <sup>k</sup> <sup>y</sup><sup>k</sup>*

*kills yxx* − *yx, a result leading, as we shall see, to the only generator* {*ch*(*x*) − 1}*.*

sense and the *differential module Dy* <sup>=</sup> *Dy*<sup>1</sup> <sup>+</sup> ... <sup>+</sup> *Dym*. If {Φ*<sup>τ</sup>* <sup>=</sup> *<sup>a</sup>*

*Dy* and the finitely generated *residual differential module M* = *Dy*/*I*.

*M* when *D* = *K*[*d*], namely *homD*(*M*, *D*) and *M*<sup>∗</sup> = *homK*(*M*, *K*) ([3],[27],[32]).

*transition matrices, the simplest example being the way T*∗ *is obtained from T.*

<sup>∧</sup>2*<sup>T</sup>* � ∧2*T*<sup>∗</sup> *and the EM current is a section of* <sup>∧</sup>4*T*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>* � ∧3*T*<sup>∗</sup> *when n* <sup>=</sup> <sup>4</sup>*.*

(*a f*)(*m*) = *a f*(*m*) = *<sup>f</sup>*(*am*),(*<sup>ξ</sup> <sup>f</sup>*)(*m*) = *<sup>ξ</sup> <sup>f</sup>*(*m*) <sup>−</sup> *<sup>f</sup>*(*ξm*), <sup>∀</sup>*<sup>a</sup>* <sup>∈</sup> *<sup>K</sup>*, <sup>∀</sup>*<sup>ξ</sup>* <sup>=</sup> *<sup>a</sup><sup>i</sup>*

bracket of vector fields on *T*. Finally we get (*di f*)*<sup>k</sup>*

way.

*with* −*di* = *δ<sup>i</sup> ([M], §59,60).*

*top and is a semi-simple module ([3]).*

*n* commuting *formal derivatives di* with *diy<sup>k</sup>*

*SPENCER SEQUENCE* 000 0 ↓↓↓ ↓ <sup>0</sup> −→ <sup>Θ</sup> *jq* −→ *C*<sup>0</sup> *<sup>D</sup>*<sup>1</sup> −→ *<sup>C</sup>*<sup>1</sup> *<sup>D</sup>*<sup>2</sup> −→ *<sup>C</sup>*<sup>2</sup> *<sup>D</sup>*<sup>3</sup> −→ ... *Dn* −→ *Cn* −→ <sup>0</sup> ↓↓↓ ↓ <sup>0</sup> −→ *<sup>E</sup> jq* −→ *<sup>C</sup>*0(*E*) *<sup>D</sup>*<sup>1</sup> −→ *<sup>C</sup>*1(*E*) *<sup>D</sup>*<sup>2</sup> −→ *<sup>C</sup>*2(*E*) *<sup>D</sup>*<sup>3</sup> −→ ... *Dn* −→ *Cn*(*E*) −→ <sup>0</sup> � ↓ Φ<sup>0</sup> ↓ Φ<sup>1</sup> ↓ Φ<sup>2</sup> ↓ Φ*<sup>n</sup>* <sup>0</sup> −→ <sup>Θ</sup> −→ *<sup>E</sup>* <sup>D</sup> −→ *F*<sup>0</sup> <sup>D</sup><sup>1</sup> −→ *<sup>F</sup>*<sup>1</sup> <sup>D</sup><sup>2</sup> −→ *<sup>F</sup>*<sup>2</sup> <sup>D</sup><sup>3</sup> −→ ... <sup>D</sup>*<sup>n</sup>* −→ *Fn* −→ <sup>0</sup> ↓↓↓ ↓ 000 0 *JANET SEQUENCE*

In this diagram, *only depending on the left commutative square* D = Φ ◦ *jq*, the epimorhisms

Φ*<sup>r</sup>* : *Cr*(*E*) → *Fr* for 0 ≤ *r* ≤ *n* are successively induced by the canonical projection Φ = Φ<sup>0</sup> : *C*0(*E*) = *Jq*(*E*) → *Jq*(*E*)/*Rq* = *F*0.

**Example 4.1.** *(Screw structure): The system R*<sup>1</sup> <sup>⊂</sup> *<sup>J</sup>*1(*T*) *defined by <sup>ξ</sup>*<sup>1</sup> <sup>2</sup> <sup>=</sup> 0, *<sup>ξ</sup>*<sup>2</sup> <sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>1</sup> <sup>1</sup> = 0 *is involutive with par*(*R*2) = {*ξ*1, *<sup>ξ</sup>*2, *<sup>ξ</sup>*<sup>1</sup> <sup>1</sup>, *<sup>ξ</sup>*<sup>2</sup> <sup>1</sup>, *<sup>ξ</sup>*<sup>1</sup> <sup>11</sup>, *<sup>ξ</sup>*<sup>2</sup> <sup>11</sup>}*. The Spencer operator is not involutive as it is not even formally integrable because ∂*2*ξ*<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>1</sup> <sup>11</sup> <sup>=</sup> 0, *<sup>∂</sup>*1*ξ*<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>2</sup> <sup>11</sup> <sup>=</sup> <sup>0</sup> <sup>⇒</sup> *<sup>∂</sup>*1*ξ*<sup>1</sup> <sup>11</sup> <sup>−</sup> *<sup>∂</sup>*2*ξ*<sup>2</sup> <sup>11</sup> = 0*. We obtain dim*(*F*0) = 2, *dim*(*C*0(*T*)) = 6 ⇒ *dim*(*C*0) = *dim*(*R*1) = 4, *dim*(*F*1) = 0 ⇒ *dim*(*C*1(*T*)) = *dim*(*C*1) = 6, *dim*(*C*2(*T*)) = *dim*(*C*2) = 2 *and it is not evident at all that the first order involutive operator D*<sup>1</sup> : *C*<sup>0</sup> → *C*<sup>1</sup> *is defined by the* 6 *PD equations for* 4 *unknowns:*

$$
\partial\_2 \mathfrak{J}\_1^1 = 0, \partial\_2 \mathfrak{J}\_2^2 - \mathfrak{J}\_1^1 = 0, \partial\_2 \mathfrak{J}\_1^1 = 0, \partial\_2 \mathfrak{J}\_1^2 - \partial\_1 \mathfrak{J}\_1^1 = 0, \partial\_1 \mathfrak{J}\_1^1 - \mathfrak{J}\_1^1 = 0, \partial\_1 \mathfrak{J}\_1^2 - \mathfrak{J}\_1^2 = 0
$$

*The case of a complex structure is similar and left to the reader.*

#### **5. Differential modules and inverse systems**

An important but difficult problem in engineering physics is to study how the formal properties of a system of order *q* with *n* independent variables and *m* unknowns depend on the parameters involved in that system. This is particularly clear in classical control theory where the systems are classified into two categories, namely the "controllable" ones and the "uncontrollable" ones ([14],[27]). In order to understand the problem studied by Macaulay in [M], that is roughly to determine the minimum number of solutions of a system that must be known in order to determine all the others by using derivatives and linear combinations with constant coefficients in a field *k*, let us start with the following motivating example:

**Example 5.1.** *When n* = 1, *m* = 1, *q* = 3*, using a sub-index x for the derivatives with dxy* = *yx and so on, the general solution of yxxx* <sup>−</sup> *yx* <sup>=</sup> <sup>0</sup> *is y* <sup>=</sup> *ae<sup>x</sup>* <sup>+</sup> *be*−*<sup>x</sup>* <sup>+</sup> *<sup>c</sup>*<sup>1</sup> *with a*, *<sup>b</sup>*, *c constants and the derivative of e<sup>x</sup> is ex, the derivative of e*−*<sup>x</sup> is* <sup>−</sup>*e*−*<sup>x</sup> and the derivative of* <sup>1</sup> *is* <sup>0</sup>*. Hence we could believe that we need a basis* {1,*ex*,*e*−*x*} *with three generators for obtaining all the solutions through derivatives. Also, when n* = 1, *m* = 2, *k* = **R** *and a is a constant real parameter, the OD system y*<sup>1</sup> *xx* <sup>−</sup> *ay*<sup>1</sup> <sup>=</sup> 0, *<sup>y</sup>*<sup>2</sup> *<sup>x</sup>* = 0 *needs two generators* {(*x*, 0),(0, 1)} *when a* = 0 *with the only dx killing both y*<sup>1</sup> *<sup>x</sup> and y*<sup>2</sup> *but only one generator when a* �= 0*, namely* {(*ch*(*x*), 1)} *when a* = 1*. Indeed, setting y* <sup>=</sup> *<sup>y</sup>*<sup>1</sup> <sup>−</sup> *<sup>y</sup>*<sup>2</sup> *brings y*<sup>1</sup> <sup>=</sup> *yxx*, *<sup>y</sup>*<sup>2</sup> <sup>=</sup> *yxx* <sup>−</sup> *y and an equivalent system defined by the* 16 Will-be-set-by-IN-TECH

*SPENCER SEQUENCE*

*<sup>D</sup>*<sup>1</sup> −→ *<sup>C</sup>*<sup>1</sup>

<sup>D</sup><sup>1</sup> −→ *<sup>F</sup>*<sup>1</sup>

<sup>11</sup> <sup>=</sup> 0, *<sup>∂</sup>*1*ξ*<sup>2</sup>

<sup>1</sup> <sup>=</sup> 0, *<sup>∂</sup>*2*ξ*<sup>2</sup>

constant coefficients in a field *k*, let us start with the following motivating example:

<sup>0</sup> −→ <sup>Θ</sup> *jq*

<sup>0</sup> −→ *<sup>E</sup> jq*

*C*0(*E*) = *Jq*(*E*) → *Jq*(*E*)/*Rq* = *F*0.

*with par*(*R*2) = {*ξ*1, *<sup>ξ</sup>*2, *<sup>ξ</sup>*<sup>1</sup>

*system y*<sup>1</sup>

*dx killing both y*<sup>1</sup>

*formally integrable because ∂*2*ξ*<sup>2</sup>

*<sup>∂</sup>*2*ξ*<sup>1</sup> <sup>=</sup> 0, *<sup>∂</sup>*2*ξ*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>1</sup>

*xx* <sup>−</sup> *ay*<sup>1</sup> <sup>=</sup> 0, *<sup>y</sup>*<sup>2</sup>

<sup>0</sup> −→ <sup>Θ</sup> −→ *<sup>E</sup>* <sup>D</sup>

−→ *C*<sup>0</sup>

−→ *F*<sup>0</sup>

**Example 4.1.** *(Screw structure): The system R*<sup>1</sup> <sup>⊂</sup> *<sup>J</sup>*1(*T*) *defined by <sup>ξ</sup>*<sup>1</sup>

<sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>1</sup>

*operator D*<sup>1</sup> : *C*<sup>0</sup> → *C*<sup>1</sup> *is defined by the* 6 *PD equations for* 4 *unknowns:*

<sup>1</sup> <sup>=</sup> 0, *<sup>∂</sup>*2*ξ*<sup>1</sup>

*The case of a complex structure is similar and left to the reader.*

**5. Differential modules and inverse systems**

<sup>1</sup>, *<sup>ξ</sup>*<sup>2</sup> <sup>1</sup>, *<sup>ξ</sup>*<sup>1</sup> <sup>11</sup>, *<sup>ξ</sup>*<sup>2</sup>

000 0 ↓↓↓ ↓

*<sup>D</sup>*<sup>2</sup> −→ *<sup>C</sup>*<sup>2</sup>

↓↓↓ ↓

<sup>D</sup><sup>2</sup> −→ *<sup>F</sup>*<sup>2</sup>

↓↓↓ ↓ 000 0

� ↓ Φ<sup>0</sup> ↓ Φ<sup>1</sup> ↓ Φ<sup>2</sup> ↓ Φ*<sup>n</sup>*

*JANET SEQUENCE* In this diagram, *only depending on the left commutative square* D = Φ ◦ *jq*, the epimorhisms

Φ*<sup>r</sup>* : *Cr*(*E*) → *Fr* for 0 ≤ *r* ≤ *n* are successively induced by the canonical projection Φ = Φ<sup>0</sup> :

<sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>2</sup>

<sup>1</sup> <sup>−</sup> *<sup>∂</sup>*1*ξ*<sup>1</sup>

An important but difficult problem in engineering physics is to study how the formal properties of a system of order *q* with *n* independent variables and *m* unknowns depend on the parameters involved in that system. This is particularly clear in classical control theory where the systems are classified into two categories, namely the "controllable" ones and the "uncontrollable" ones ([14],[27]). In order to understand the problem studied by Macaulay in [M], that is roughly to determine the minimum number of solutions of a system that must be known in order to determine all the others by using derivatives and linear combinations with

**Example 5.1.** *When n* = 1, *m* = 1, *q* = 3*, using a sub-index x for the derivatives with dxy* = *yx and so on, the general solution of yxxx* <sup>−</sup> *yx* <sup>=</sup> <sup>0</sup> *is y* <sup>=</sup> *ae<sup>x</sup>* <sup>+</sup> *be*−*<sup>x</sup>* <sup>+</sup> *<sup>c</sup>*<sup>1</sup> *with a*, *<sup>b</sup>*, *c constants and the derivative of e<sup>x</sup> is ex, the derivative of e*−*<sup>x</sup> is* <sup>−</sup>*e*−*<sup>x</sup> and the derivative of* <sup>1</sup> *is* <sup>0</sup>*. Hence we could believe that we need a basis* {1,*ex*,*e*−*x*} *with three generators for obtaining all the solutions through derivatives. Also, when n* = 1, *m* = 2, *k* = **R** *and a is a constant real parameter, the OD*

*Indeed, setting y* <sup>=</sup> *<sup>y</sup>*<sup>1</sup> <sup>−</sup> *<sup>y</sup>*<sup>2</sup> *brings y*<sup>1</sup> <sup>=</sup> *yxx*, *<sup>y</sup>*<sup>2</sup> <sup>=</sup> *yxx* <sup>−</sup> *y and an equivalent system defined by the*

*dim*(*F*0) = 2, *dim*(*C*0(*T*)) = 6 ⇒ *dim*(*C*0) = *dim*(*R*1) = 4, *dim*(*F*1) = 0 ⇒ *dim*(*C*1(*T*)) = *dim*(*C*1) = 6, *dim*(*C*2(*T*)) = *dim*(*C*2) = 2 *and it is not evident at all that the first order involutive*

−→ *<sup>C</sup>*0(*E*) *<sup>D</sup>*<sup>1</sup> −→ *<sup>C</sup>*1(*E*) *<sup>D</sup>*<sup>2</sup> −→ *<sup>C</sup>*2(*E*) *<sup>D</sup>*<sup>3</sup> −→ ... *Dn* −→ *Cn*(*E*) −→ <sup>0</sup>

*<sup>D</sup>*<sup>3</sup> −→ ... *Dn* −→ *Cn* −→ <sup>0</sup>

<sup>D</sup><sup>3</sup> −→ ... <sup>D</sup>*<sup>n</sup>* −→ *Fn* −→ <sup>0</sup>

<sup>2</sup> <sup>=</sup> 0, *<sup>ξ</sup>*<sup>2</sup>

<sup>11</sup> <sup>−</sup> *<sup>∂</sup>*2*ξ*<sup>2</sup>

<sup>11</sup>}*. The Spencer operator is not involutive as it is not even*

<sup>11</sup> <sup>=</sup> <sup>0</sup> <sup>⇒</sup> *<sup>∂</sup>*1*ξ*<sup>1</sup>

<sup>1</sup> <sup>=</sup> 0, *<sup>∂</sup>*1*ξ*<sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>1</sup>

*<sup>x</sup>* = 0 *needs two generators* {(*x*, 0),(0, 1)} *when a* = 0 *with the only*

*<sup>x</sup> and y*<sup>2</sup> *but only one generator when a* �= 0*, namely* {(*ch*(*x*), 1)} *when a* = 1*.*

<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>1</sup>

<sup>1</sup> <sup>=</sup> 0, *<sup>∂</sup>*1*ξ*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>2</sup>

<sup>1</sup> = 0 *is involutive*

<sup>11</sup> = 0*. We obtain*

<sup>1</sup> = 0

*single OD equation yxxx* − *yx* = 0 *for the only y. Introducing the corresponding polynomial ideal* (*χ*<sup>3</sup> <sup>−</sup> *<sup>χ</sup>*)=(*χ*) <sup>∩</sup> (*<sup>χ</sup>* <sup>−</sup> <sup>1</sup>) <sup>∩</sup> (*<sup>χ</sup>* <sup>+</sup> <sup>1</sup>)*, we check that dx kills yxx* <sup>−</sup> *y, dx* <sup>−</sup> <sup>1</sup> *kills yxx* <sup>+</sup> *yx and dx* <sup>+</sup> <sup>1</sup> *kills yxx* − *yx, a result leading, as we shall see, to the only generator* {*ch*(*x*) − 1}*.*

More precisely, if *K* is a differential field containing **Q** with *n* commuting *derivations ∂i*, that is to say *∂i*(*a* + *b*) = *∂ia* + *∂ib* and *∂i*(*ab*)=(*∂ia*)*b* + *a∂ib*, ∀*a*, *b* ∈ *K* for *i* = 1, ..., *n*, we denote by *k* a subfield of constants. Let us introduce *m differential indeterminates y<sup>k</sup>* for *k* = 1, ..., *m* and *n* commuting *formal derivatives di* with *diy<sup>k</sup> <sup>μ</sup>* = *y<sup>k</sup> μ*+1*<sup>i</sup>* . We introduce the non-commutative *ring of differential operators D* = *K*[*d*1, ..., *dn*] = *K*[*d*] with *dia* = *adi* + *∂ia*, ∀*a* ∈ *K* in the operator sense and the *differential module Dy* <sup>=</sup> *Dy*<sup>1</sup> <sup>+</sup> ... <sup>+</sup> *Dym*. If {Φ*<sup>τ</sup>* <sup>=</sup> *<sup>a</sup> τμ <sup>k</sup> <sup>y</sup><sup>k</sup> <sup>μ</sup>*} is a finite number of elements in *Dy* indexed by *τ*, we may introduce the *differential module of equations I* = *D*Φ ⊂ *Dy* and the finitely generated *residual differential module M* = *Dy*/*I*.

In the algebraic framework considered, *only two possible formal constructions can be obtained from M* when *D* = *K*[*d*], namely *homD*(*M*, *D*) and *M*<sup>∗</sup> = *homK*(*M*, *K*) ([3],[27],[32]).

**Theorem 5.1.** *homD*(*M*, *D*) *is a right differential module that can be converted to a left differential module by introducing the right differential module structure of* <sup>∧</sup>*nT*∗*. As a differential geometric counterpart, we get the formal adjoint of* <sup>D</sup>*, namely ad*(D) : <sup>∧</sup>*nT*<sup>∗</sup> <sup>⊗</sup> *<sup>F</sup>*<sup>∗</sup> → ∧*nT*<sup>∗</sup> <sup>⊗</sup> *<sup>E</sup>*<sup>∗</sup> *usually constructed through an integration by parts and where E*∗ *is obtained from E by inverting the local transition matrices, the simplest example being the way T*∗ *is obtained from T.*

**Remark 5.1.** *Such a result explains why dual objects in physics and engineering are no longer tensors but tensor densities, with no reference to any variational calculus. For example the EM potential is a section of T*<sup>∗</sup> *and the EM field is a section of* <sup>∧</sup>2*T*<sup>∗</sup> *while the EM induction is a section of* <sup>∧</sup>4*T*<sup>∗</sup> <sup>⊗</sup> <sup>∧</sup>2*<sup>T</sup>* � ∧2*T*<sup>∗</sup> *and the EM current is a section of* <sup>∧</sup>4*T*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>* � ∧3*T*<sup>∗</sup> *when n* <sup>=</sup> <sup>4</sup>*.*

The filtration *D*<sup>0</sup> = *K* ⊆ *D*<sup>1</sup> = *K* ⊕ *T* ⊆ ... ⊆ *Dq* ⊆ ... ⊆ *D* of *D* by the order of operators induces a filtration/inductive limit 0 ⊆ *M*<sup>0</sup> ⊆ *M*<sup>1</sup> ⊆ ... ⊆ *Mq* ⊆ ... ⊆ *M* and provides by duality *over K* the projective limit *M*<sup>∗</sup> = *R* → ... → *Rq* → ... → *R*<sup>1</sup> → *R*<sup>0</sup> → 0 of formally integrable systems. As *D* is generated by *K* and *T* = *D*1/*D*0, we can define for any *f* ∈ *M*∗:

$$(af)(m) = af(m) = f(am), \\ (\sharp f)(m) = \sharp f(m) - f(\sharp m), \forall a \in \mathcal{K}, \forall \xi = a^l d\_l \in \mathcal{T}, \forall m \in \mathcal{M}$$

and check *dia* = *adi* + *∂ia*, *ξη* − *ηξ* = [*ξ*, *η*] in the operator sense by introducing the standard bracket of vector fields on *T*. Finally we get (*di f*)*<sup>k</sup> <sup>μ</sup>* = (*di f*)(*y<sup>k</sup> <sup>μ</sup>*) = *∂<sup>i</sup> f <sup>k</sup> <sup>μ</sup>* <sup>−</sup> *<sup>f</sup> <sup>k</sup> <sup>μ</sup>*+1*<sup>i</sup>* in a coherent way.

**Theorem 5.2.** *R* = *M*∗ *has a structure of differential module induced by the Spencer operator.*

**Remark 5.2.** *When m* = 1 *and D* = *k*[*d*] *is a commutative ring isomorphic to the polynomial ring A* = *k*[*χ*] *for the indeterminates χ*1, ..., *χn, this result exactly describes the inverse system of Macaulay with* −*di* = *δ<sup>i</sup> ([M], §59,60).*

**Definition 5.1.** *A simple module is a module having no other proper submodule than* 0*. A semi-simple module is a direct sum of simple modules. When A is a commutative integral domain and M a finitely generated module over A, the socle of M is the largest semi-simple submodule of M, that is soc*(*M*) = ⊕*soc*m(*M*) *where soc*m(*M*) *is the direct sum of all the isotypical simple submodules of M isomorphic to A*/m *for* m ∈ *max*(*A*) *the set of maximal proper ideals of A. The radical of a module is the intersection of all its maximum proper submodules. The quotient of a module by its radical is called the top and is a semi-simple module ([3]).*

**Theorem 5.4.** *When M is n-pure, one may use the chinese remainder theorem ([19], p 41) in order to prove that the minimum number of generators of R is equal to the maximum number of isotypical components that can be found among the various components of soc*(*M*) *or top*(*R*)*. When M is r-pure but r* ≤ *n* − 1*, the minimum number of generators of R is smaller or equal to the smallest non-zero*

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 19

Nonlinear operators do not in general admit CC as can be seen by considering the involutive

happens in the study of Lie pseudogroups. However, the kernel of a linear operator D : *E* → *F* is always taken with respet to the zero section of *F*, while it must be taken with respect to a prescribed section by a *double arrow* for a nonlinear operator. Keeping in mind the linear Janet sequence and the examples of Vessiot structure equations already presented, one obtains: **Theorem 6.1.** *There exists a nonlinear Janet sequence associated with the Lie form of an involutive*

0 → Γ → *aut*(*X*) ⇒ F ⇒ F<sup>1</sup>

*where the kernel of the first operator f* <sup>→</sup> <sup>Φ</sup>*<sup>ω</sup>* ◦ *jq*(*f*) = <sup>Φ</sup>*ω*(*jq*(*f*)) = *jq*(*f*)−1(*ω*) *is taken with respect to the section ω of* F *while the kernel of the second operator ω* → *I*(*j*1(*ω*)) ≡ *A*(*ω*)*∂xω* +

**Corollary 6.1.** *By linearization at the identity, one obtains the involutive Lie operator* D = D*<sup>ω</sup>* : *T* → *F*<sup>0</sup> : *ξ* → L(*ξ*)*ω with kernel* Θ = {*ξ* ∈ *T*|L(*ξ*)*ω* = 0} ⊂ *T satisfying* [Θ, Θ] ⊂ Θ *and the*

Now we notice that *T* is a natural vector bundle of order 1 and *Jq*(*T*) is thus a natural vector bundle of order *q* + 1. Looking at the way a vector field and its derivatives are transformed

*<sup>u</sup>*(*f*(*x*))*<sup>f</sup> <sup>u</sup>*

*η<sup>q</sup>* = *fq*+1(*ξq*) *from any section ξ<sup>q</sup>* ∈ *Jq*(*T*) *and any section fq*+<sup>1</sup> ∈ Π*q*+<sup>1</sup> *by the formula:*

*r ξr*

*where the left member belongs to V*(Π*q*)*. Similarly Rq* ⊂ *Jq*(*T*) *is associated with* R*q*+<sup>1</sup> ⊂ Π*q*+1*.*

In order to construct another nonlinear sequence, we need a few basic definitions on *Lie groupoids* and *Lie algebroids* that will become substitutes for Lie groups and Lie algebras. As in the beginning of section 3, the first idea is to use the chain rule for derivatives *jq*(*g* ◦ *f*) = *jq*(*g*) ◦ *jq*(*f*) whenever *f* , *g* ∈ *aut*(*X*) can be composed and to replace both *jq*(*f*) and *jq*(*g*) respectively by *fq* and *gq* in order to obtain the new section *gq* ◦ *fq*. This kind of "composition" law can be written in a pointwise symbolic way by introducing another copy

**Lemma 6.1.** *Jq*(*T*) *is associated with* Π*q*+<sup>1</sup> = Π*q*+1(*X*, *X*) *that is we can obtain a new section*

*<sup>μ</sup>* + ... + *<sup>f</sup> <sup>k</sup>*

*B*(*ω*) *is taken with respect to the zero section of the vector bundle* F<sup>1</sup> *over* F*.*

under any *f* ∈ *aut*(*X*) while replacing *jq*(*f*) by *fq*, we obtain:

*r fr*

*<sup>r</sup>* (*x*)*ξ<sup>r</sup>*

*<sup>d</sup>μη<sup>k</sup>* <sup>≡</sup> *<sup>η</sup><sup>k</sup>*

*ηk*(*f*(*x*)) = *f <sup>k</sup>*

and so on, a result leading to:

*corresponding linear Janet sequence where F*<sup>0</sup> <sup>=</sup> *<sup>ω</sup>*−1(*V*(F)) *and F*<sup>1</sup> <sup>=</sup> *<sup>ω</sup>*−1(F1)*.*

(*x*) <sup>⇒</sup> *<sup>η</sup><sup>k</sup>*

*<sup>μ</sup>* + ... = *<sup>f</sup> <sup>k</sup>*

Φ*<sup>ω</sup>* ◦ *jq I* ◦ *j*<sup>1</sup>

*ω* ◦ *α* 0

*<sup>i</sup>* (*x*) = *<sup>f</sup> <sup>k</sup>*

*μ*+1*<sup>r</sup> ξr*

*<sup>r</sup>* (*x*)*ξ<sup>r</sup>*

*<sup>i</sup>*(*x*) + *<sup>f</sup> <sup>k</sup>*

, ∀0 ≤ |*μ*| ≤ *q*

*ri*(*x*)*ξ<sup>r</sup>*

(*x*)

<sup>2</sup> (*y*11)<sup>2</sup> <sup>=</sup> *<sup>v</sup>* with *<sup>m</sup>* <sup>=</sup> 1, *<sup>n</sup>* <sup>=</sup> 2, *<sup>q</sup>* <sup>=</sup> 2, contrary to what

**6. Janet versus Spencer : The nonlinear sequences**

<sup>3</sup> (*y*11)<sup>3</sup> <sup>=</sup> *<sup>u</sup>*, *<sup>y</sup>*<sup>12</sup> <sup>−</sup> <sup>1</sup>

*character.*

example *<sup>y</sup>*<sup>22</sup> <sup>−</sup> <sup>1</sup>

*system of finite Lie equations:*

The "*secret* " of Macaulay is expressed by the next theorem:

**Theorem 5.3.** *Instead of using the socle of M over A, one may use duality over k in order to deal with the short exact sequence* 0 → *rad*(*R*) → *R* → *top*(*R*) → 0 *where top*(*R*) *is the dual of soc*(*M*)*.*

However, Nakayama's lemma ([3],[19],[32]) cannot be used in general unless *R* is finitely generated over *k* and thus over *D*. The main idea of Macaulay has been to overcome this difficulty by dealing only with *unmixed* ideals when *m* = 1. As a generalization, one can state ([27]):

**Definition 5.2.** *One has the purity filtration* 0 = *tn*(*M*) ⊆ ... ⊆ *t*0(*M*) = *t*(*M*) ⊆ *M where any involutive system of order p defining Dm is such that αn*−*<sup>r</sup> <sup>p</sup>* = 0, ..., *α<sup>n</sup> <sup>p</sup>* = 0 *when m* ∈ *tr*(*M*) *and M is said to be r-pure if tr*(*M*) = 0, *tr*−1(*M*) = *M. With t*(*M*) = {*<sup>m</sup>* ∈ *<sup>M</sup>* | ∃<sup>0</sup> �= *<sup>a</sup>* ∈ *<sup>A</sup>*, *am* = <sup>0</sup>} *we say that M is a* 0*-pure or torsion-free module if t*(*M*) = 0 *and a torsion module if t*(*M*) = *M.*

**Example 5.2.** *With n* = 2, *q* = 2*, let us consider the involutive system y*(0,2) ≡ *y*<sup>22</sup> = 0, *y*(1,1) ≡ *y*<sup>12</sup> = 0*. Then z*� = *y*<sup>1</sup> *satisfies z*� <sup>2</sup> = 0 *while z*�� = *y*<sup>2</sup> *satisfies z*�� <sup>2</sup> = 0, *z*�� <sup>1</sup> = 0 *and we have the filtration* 0 = *t*2(*M*) ⊂ *t*1(*M*) ⊂ *t*0(*M*) = *t*(*M*) = *M with z*�� ∈ *t*1(*M*), *z*� ∈ *t*0(*M*) *but z*� ∈/ *t*1(*M*)*. This classification of observables has never been applied to engineering systems like the ones to be found in magnetohydrodynamics (MHD) because the mathematics involved are not known.*

**Remark 5.3.** *A standard result in commutative algebra allows to embed any torsion-free module into a free module ([32]). Such a property provides the possibility to parametrize the solution space of the corresponding system of OD/PD equations by a finite number of potential like arbitrary functions. For this, in order to test the possibility to parametrize a given operator* D1*, one may construct the adjoint operator ad*(D1) *and look for generating CC in the form of an operator ad*(D)*. As ad*(D) ◦ *ad*(D1) = *ad*(D<sup>1</sup> ◦ D) = 0 ⇒ D<sup>1</sup> ◦ D = 0*, it only remains to check that the CC of* D *are generated by* D1*. When n* = 1 *this result amounts to Kalman test and the fact that a classical OD control system is controllable if and only if it is parametrizable, a result showing that controllability is an intrinsic structural property of a control system, not depending on the choice of inputs and outputs contrary to a well established engineering tradition ([14],[27]). When n* = 2*, the formal adjoint of the only CC for the deformation tensor has been used in the Introduction in order to parametrize the stress equation by means of the Airy function. This result is also valid for the non-commutative ring D* = *K*[*d*]*.*

**Example 5.3.** *With K* = **Q**(*x*1, *x*2, *x*3)*, infinitesimal contact transformations are defined by the system <sup>∂</sup>*2*ξ*<sup>1</sup> <sup>−</sup> *<sup>x</sup>*3*∂*2*ξ*<sup>2</sup> <sup>+</sup> *<sup>x</sup>*3*∂*1*ξ*<sup>1</sup> <sup>−</sup> (*x*3)2*∂*1*ξ*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>3</sup> <sup>=</sup> 0, *<sup>∂</sup>*3*ξ*<sup>1</sup> <sup>−</sup> *<sup>x</sup>*3*∂*3*ξ*<sup>2</sup> <sup>=</sup> <sup>0</sup>*. Multiplying by test functions* (*λ*1, *λ*2) *and integrating by parts, we obtain the adjoint operator (up to sign):*

$$
\partial\_2 \lambda^1 + \mathbf{x}^3 \partial\_1 \lambda^1 + \partial\_3 \lambda^2 = \mu^1, \quad -\mathbf{x}^3 \partial\_2 \lambda^1 - (\mathbf{x}^3)^2 \partial\_1 \lambda^1 - \mathbf{x}^3 \partial\_3 \lambda^2 - \lambda^2 = \mu^2, \quad \lambda^1 = \mu^3
$$

*It follows that <sup>λ</sup>*<sup>1</sup> <sup>=</sup> *<sup>μ</sup>*3, *<sup>λ</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>*μ*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*3*μ*<sup>1</sup> <sup>⇒</sup> *<sup>∂</sup>*2*μ*<sup>3</sup> <sup>+</sup> *<sup>x</sup>*3*∂*1*μ*<sup>3</sup> <sup>−</sup> *<sup>∂</sup>*3*μ*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*3*∂*3*μ*<sup>1</sup> <sup>−</sup> <sup>2</sup>*μ*<sup>1</sup> <sup>=</sup> <sup>0</sup>*. Multiplying again by a test function <sup>φ</sup>, we discover the parametrization <sup>ξ</sup>*<sup>1</sup> <sup>=</sup> *<sup>x</sup>*3*∂*3*<sup>φ</sup>* <sup>−</sup> *<sup>φ</sup>*, *<sup>ξ</sup>*<sup>2</sup> <sup>=</sup> *<sup>∂</sup>*3*φ*, *<sup>ξ</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>*∂*2*<sup>φ</sup>* <sup>−</sup> *<sup>x</sup>*3*∂*1*<sup>φ</sup> which is not evident at first sight.*

When *M* is *r*-pure, Theorem 4.3 provides the exact sequence 0 → *M* → *k*(*χ*1, ..., *χn*−*r*) ⊗ *M*, also discovered by Macaulay ([M], §77, 82), and one obtains the following key result for studying the *identifiability* of OD/PD control systems (see *localization* in ([19],[27],32[29],[30],[32]).

**Theorem 5.4.** *When M is n-pure, one may use the chinese remainder theorem ([19], p 41) in order to prove that the minimum number of generators of R is equal to the maximum number of isotypical components that can be found among the various components of soc*(*M*) *or top*(*R*)*. When M is r-pure but r* ≤ *n* − 1*, the minimum number of generators of R is smaller or equal to the smallest non-zero character.*

#### **6. Janet versus Spencer : The nonlinear sequences**

18 Will-be-set-by-IN-TECH

**Theorem 5.3.** *Instead of using the socle of M over A, one may use duality over k in order to deal with the short exact sequence* 0 → *rad*(*R*) → *R* → *top*(*R*) → 0 *where top*(*R*) *is the dual of soc*(*M*)*.*

However, Nakayama's lemma ([3],[19],[32]) cannot be used in general unless *R* is finitely generated over *k* and thus over *D*. The main idea of Macaulay has been to overcome this difficulty by dealing only with *unmixed* ideals when *m* = 1. As a generalization, one can state

**Definition 5.2.** *One has the purity filtration* 0 = *tn*(*M*) ⊆ ... ⊆ *t*0(*M*) = *t*(*M*) ⊆ *M where any*

*is said to be r-pure if tr*(*M*) = 0, *tr*−1(*M*) = *M. With t*(*M*) = {*<sup>m</sup>* ∈ *<sup>M</sup>* | ∃<sup>0</sup> �= *<sup>a</sup>* ∈ *<sup>A</sup>*, *am* = <sup>0</sup>} *we say that M is a* 0*-pure or torsion-free module if t*(*M*) = 0 *and a torsion module if t*(*M*) = *M.*

**Example 5.2.** *With n* = 2, *q* = 2*, let us consider the involutive system y*(0,2) ≡ *y*<sup>22</sup> = 0, *y*(1,1) ≡

*the filtration* 0 = *t*2(*M*) ⊂ *t*1(*M*) ⊂ *t*0(*M*) = *t*(*M*) = *M with z*�� ∈ *t*1(*M*), *z*� ∈ *t*0(*M*) *but z*� ∈/ *t*1(*M*)*. This classification of observables has never been applied to engineering systems like the ones to be found in magnetohydrodynamics (MHD) because the mathematics involved are not known.*

**Remark 5.3.** *A standard result in commutative algebra allows to embed any torsion-free module into a free module ([32]). Such a property provides the possibility to parametrize the solution space of the corresponding system of OD/PD equations by a finite number of potential like arbitrary functions. For this, in order to test the possibility to parametrize a given operator* D1*, one may construct the adjoint operator ad*(D1) *and look for generating CC in the form of an operator ad*(D)*. As ad*(D) ◦ *ad*(D1) = *ad*(D<sup>1</sup> ◦ D) = 0 ⇒ D<sup>1</sup> ◦ D = 0*, it only remains to check that the CC of* D *are generated by* D1*. When n* = 1 *this result amounts to Kalman test and the fact that a classical OD control system is controllable if and only if it is parametrizable, a result showing that controllability is an intrinsic structural property of a control system, not depending on the choice of inputs and outputs contrary to a well established engineering tradition ([14],[27]). When n* = 2*, the formal adjoint of the only CC for the deformation tensor has been used in the Introduction in order to parametrize the stress equation by means of the*

**Example 5.3.** *With K* = **Q**(*x*1, *x*2, *x*3)*, infinitesimal contact transformations are defined by the system <sup>∂</sup>*2*ξ*<sup>1</sup> <sup>−</sup> *<sup>x</sup>*3*∂*2*ξ*<sup>2</sup> <sup>+</sup> *<sup>x</sup>*3*∂*1*ξ*<sup>1</sup> <sup>−</sup> (*x*3)2*∂*1*ξ*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>3</sup> <sup>=</sup> 0, *<sup>∂</sup>*3*ξ*<sup>1</sup> <sup>−</sup> *<sup>x</sup>*3*∂*3*ξ*<sup>2</sup> <sup>=</sup> <sup>0</sup>*. Multiplying by*

*<sup>∂</sup>*2*λ*<sup>1</sup> <sup>+</sup> *<sup>x</sup>*3*∂*1*λ*<sup>1</sup> <sup>+</sup> *<sup>∂</sup>*3*λ*<sup>2</sup> <sup>=</sup> *<sup>μ</sup>*1, <sup>−</sup>*x*3*∂*2*λ*<sup>1</sup> <sup>−</sup> (*x*3)2*∂*1*λ*<sup>1</sup> <sup>−</sup> *<sup>x</sup>*3*∂*3*λ*<sup>2</sup> <sup>−</sup> *<sup>λ</sup>*<sup>2</sup> <sup>=</sup> *<sup>μ</sup>*2, *<sup>λ</sup>*<sup>1</sup> <sup>=</sup> *<sup>μ</sup>*<sup>3</sup>

*It follows that <sup>λ</sup>*<sup>1</sup> <sup>=</sup> *<sup>μ</sup>*3, *<sup>λ</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>*μ*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*3*μ*<sup>1</sup> <sup>⇒</sup> *<sup>∂</sup>*2*μ*<sup>3</sup> <sup>+</sup> *<sup>x</sup>*3*∂*1*μ*<sup>3</sup> <sup>−</sup> *<sup>∂</sup>*3*μ*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*3*∂*3*μ*<sup>1</sup> <sup>−</sup> <sup>2</sup>*μ*<sup>1</sup> <sup>=</sup> <sup>0</sup>*. Multiplying again by a test function <sup>φ</sup>, we discover the parametrization <sup>ξ</sup>*<sup>1</sup> <sup>=</sup> *<sup>x</sup>*3*∂*3*<sup>φ</sup>* <sup>−</sup> *<sup>φ</sup>*, *<sup>ξ</sup>*<sup>2</sup> <sup>=</sup>

When *M* is *r*-pure, Theorem 4.3 provides the exact sequence 0 → *M* → *k*(*χ*1, ..., *χn*−*r*) ⊗ *M*, also discovered by Macaulay ([M], §77, 82), and one obtains the following key result for studying the *identifiability* of OD/PD control systems (see *localization* in

*test functions* (*λ*1, *λ*2) *and integrating by parts, we obtain the adjoint operator (up to sign):*

<sup>2</sup> = 0 *while z*�� = *y*<sup>2</sup> *satisfies z*��

*<sup>p</sup>* = 0 *when m* ∈ *tr*(*M*) *and M*

<sup>1</sup> = 0 *and we have*

<sup>2</sup> = 0, *z*��

The "*secret* " of Macaulay is expressed by the next theorem:

*involutive system of order p defining Dm is such that αn*−*<sup>r</sup> <sup>p</sup>* = 0, ..., *α<sup>n</sup>*

*Airy function. This result is also valid for the non-commutative ring D* = *K*[*d*]*.*

*<sup>∂</sup>*3*φ*, *<sup>ξ</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>*∂*2*<sup>φ</sup>* <sup>−</sup> *<sup>x</sup>*3*∂*1*<sup>φ</sup> which is not evident at first sight.*

([19],[27],32[29],[30],[32]).

*y*<sup>12</sup> = 0*. Then z*� = *y*<sup>1</sup> *satisfies z*�

([27]):

Nonlinear operators do not in general admit CC as can be seen by considering the involutive example *<sup>y</sup>*<sup>22</sup> <sup>−</sup> <sup>1</sup> <sup>3</sup> (*y*11)<sup>3</sup> <sup>=</sup> *<sup>u</sup>*, *<sup>y</sup>*<sup>12</sup> <sup>−</sup> <sup>1</sup> <sup>2</sup> (*y*11)<sup>2</sup> <sup>=</sup> *<sup>v</sup>* with *<sup>m</sup>* <sup>=</sup> 1, *<sup>n</sup>* <sup>=</sup> 2, *<sup>q</sup>* <sup>=</sup> 2, contrary to what happens in the study of Lie pseudogroups. However, the kernel of a linear operator D : *E* → *F* is always taken with respet to the zero section of *F*, while it must be taken with respect to a prescribed section by a *double arrow* for a nonlinear operator. Keeping in mind the linear Janet sequence and the examples of Vessiot structure equations already presented, one obtains:

**Theorem 6.1.** *There exists a nonlinear Janet sequence associated with the Lie form of an involutive system of finite Lie equations:*

$$\begin{array}{ccc} 0 \to \Gamma \to \operatorname{aut}(X) & \stackrel{\Phi\_{\omega} \circ j\_{\boldsymbol{q}}}{\Longrightarrow} & \mathcal{F} & \stackrel{I \circ j\_{1}}{\Longrightarrow} & \mathcal{F}\_{1} \\ & & \omega \circ \mathfrak{a} & 0 \end{array}$$

*where the kernel of the first operator f* <sup>→</sup> <sup>Φ</sup>*<sup>ω</sup>* ◦ *jq*(*f*) = <sup>Φ</sup>*ω*(*jq*(*f*)) = *jq*(*f*)−1(*ω*) *is taken with respect to the section ω of* F *while the kernel of the second operator ω* → *I*(*j*1(*ω*)) ≡ *A*(*ω*)*∂xω* + *B*(*ω*) *is taken with respect to the zero section of the vector bundle* F<sup>1</sup> *over* F*.*

**Corollary 6.1.** *By linearization at the identity, one obtains the involutive Lie operator* D = D*<sup>ω</sup>* : *T* → *F*<sup>0</sup> : *ξ* → L(*ξ*)*ω with kernel* Θ = {*ξ* ∈ *T*|L(*ξ*)*ω* = 0} ⊂ *T satisfying* [Θ, Θ] ⊂ Θ *and the corresponding linear Janet sequence where F*<sup>0</sup> <sup>=</sup> *<sup>ω</sup>*−1(*V*(F)) *and F*<sup>1</sup> <sup>=</sup> *<sup>ω</sup>*−1(F1)*.*

Now we notice that *T* is a natural vector bundle of order 1 and *Jq*(*T*) is thus a natural vector bundle of order *q* + 1. Looking at the way a vector field and its derivatives are transformed under any *f* ∈ *aut*(*X*) while replacing *jq*(*f*) by *fq*, we obtain:

$$\eta^k(f(\mathbf{x})) = f\_r^k(\mathbf{x})\mathfrak{z}^r(\mathbf{x}) \Rightarrow \eta\_u^k(f(\mathbf{x}))f\_i^u(\mathbf{x}) = f\_r^k(\mathbf{x})\mathfrak{z}\_i^r(\mathbf{x}) + f\_{ri}^k(\mathbf{x})\mathfrak{z}^r(\mathbf{x})$$

and so on, a result leading to:

**Lemma 6.1.** *Jq*(*T*) *is associated with* Π*q*+<sup>1</sup> = Π*q*+1(*X*, *X*) *that is we can obtain a new section η<sup>q</sup>* = *fq*+1(*ξq*) *from any section ξ<sup>q</sup>* ∈ *Jq*(*T*) *and any section fq*+<sup>1</sup> ∈ Π*q*+<sup>1</sup> *by the formula:*

$$d\_{\mu}\eta^{k} \equiv \eta\_{r}^{k}f^{r}\_{\mu} + \dots = f^{k}\_{r}\mathfrak{T}^{r}\_{\mu} + \dots + f^{k}\_{\mu+1,r}\mathfrak{T}^{r} \; \forall 0 \le |\mu| \le q$$

*where the left member belongs to V*(Π*q*)*. Similarly Rq* ⊂ *Jq*(*T*) *is associated with* R*q*+<sup>1</sup> ⊂ Π*q*+1*.*

In order to construct another nonlinear sequence, we need a few basic definitions on *Lie groupoids* and *Lie algebroids* that will become substitutes for Lie groups and Lie algebras. As in the beginning of section 3, the first idea is to use the chain rule for derivatives *jq*(*g* ◦ *f*) = *jq*(*g*) ◦ *jq*(*f*) whenever *f* , *g* ∈ *aut*(*X*) can be composed and to replace both *jq*(*f*) and *jq*(*g*) respectively by *fq* and *gq* in order to obtain the new section *gq* ◦ *fq*. This kind of "composition" law can be written in a pointwise symbolic way by introducing another copy

We refer to ([26], p 215) for the inductive proof of the local exactness, providing the only

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 21

There is no need for double-arrows in this framework as the kernels are taken with respect to the zero section of the vector bundles involved. We finally notice that the main difference with the gauge sequence is that *all the indices range from* 1 *to n* and that the condition *det*(*A*) �= 0

*<sup>j</sup>*,*<sup>i</sup>* <sup>−</sup> (*χ<sup>r</sup>* ,*i χk <sup>r</sup>*,*<sup>j</sup>* <sup>−</sup> *<sup>χ</sup><sup>r</sup>* ,*j χk r*,*i* ) = 0

*<sup>i</sup>* ) �= 0 by assumption.

*D*¯ �

*<sup>q</sup>* = *idT or equivalently a section of T*<sup>∗</sup> ⊗ *Rq over idT* ∈ *T*<sup>∗</sup> ⊗ *T and is called a*

*<sup>D</sup>*¯ <sup>1</sup> −→ *<sup>C</sup>*1(*T*) *<sup>D</sup>*¯ <sup>2</sup> −→ *<sup>C</sup>*2(*T*)

*<sup>D</sup>*¯ <sup>2</sup> −→ *<sup>C</sup>*<sup>2</sup>

*<sup>μ</sup>*+1*<sup>r</sup>* <sup>+</sup> *terms*(*order* <sup>≤</sup> *<sup>q</sup>*) and *<sup>D</sup>*¯ : <sup>Π</sup>*q*+<sup>1</sup> <sup>→</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *Jq*(*T*) restricts to

*<sup>μ</sup>*+1*<sup>r</sup>* <sup>+</sup> *terms*(*order* <sup>≤</sup> *<sup>q</sup>*). Setting *<sup>χ</sup><sup>k</sup>*

*<sup>q</sup> is defined by κ*�

*χ*� *<sup>q</sup>* = *κ*�

−→ ∧2*T*<sup>∗</sup> <sup>⊗</sup> *Jq*−1(*T*)

*πq*

<sup>→</sup><sup>0</sup> *<sup>T</sup>* <sup>→</sup> <sup>0</sup> *is a map <sup>χ</sup>*�

*<sup>q</sup>*(*η*)] − *χ*�

*<sup>q</sup>*(*ξ*), *χ*�

*<sup>q</sup> if and only if A* = 0*. In particular*

*<sup>q</sup>* : *T* →

*<sup>q</sup>*([*ξ*, *η*])*.*

*<sup>μ</sup>*,*<sup>i</sup>* <sup>=</sup> *<sup>A</sup><sup>r</sup>*

*i τk <sup>μ</sup>*,*r*,

*<sup>q</sup>* → *Rq*

*<sup>q</sup>*(*ξ*, *η*)=[*χ*�

formulas that will be used later on and can be checked directly by the reader:

*lj*,*<sup>i</sup>* <sup>−</sup> (*χ<sup>r</sup>* ,*i χk lr*,*<sup>j</sup>* <sup>+</sup> *<sup>χ</sup><sup>r</sup> l*,*i χk <sup>r</sup>*,*<sup>j</sup>* <sup>−</sup> *<sup>χ</sup><sup>r</sup> l*,*j χk <sup>r</sup>*,*<sup>i</sup>* <sup>−</sup> *<sup>χ</sup><sup>r</sup>* ,*j χk lr*,*i* ) = 0

*D*¯ −→ *T*<sup>∗</sup> ⊗ *Rq*

*<sup>i</sup>*,*<sup>j</sup>* <sup>+</sup> *<sup>χ</sup><sup>k</sup>*

,*<sup>i</sup>* <sup>−</sup> *<sup>χ</sup><sup>k</sup>*

*∂iχ<sup>k</sup>*

*<sup>l</sup>*,*<sup>i</sup>* <sup>−</sup> *<sup>χ</sup><sup>k</sup>*

<sup>0</sup> −→ <sup>Γ</sup> *jq*<sup>+</sup><sup>1</sup>

*∂iχ<sup>k</sup>*

*Rq such that <sup>π</sup><sup>q</sup>*

*We notice that χ*�

(*δ<sup>k</sup> <sup>i</sup>* , <sup>−</sup>*γ<sup>k</sup>* <sup>0</sup> ◦ *χ*�

*∂iχ<sup>k</sup>*

*Proof.* : With <sup>|</sup>*μ*<sup>|</sup> <sup>=</sup> *<sup>q</sup>* we have *<sup>χ</sup><sup>k</sup>*

*<sup>μ</sup>*,*<sup>r</sup>* <sup>=</sup> <sup>−</sup>*g<sup>k</sup>*

*l f l*

we obtain *τ<sup>k</sup>*

*<sup>D</sup>*¯ <sup>1</sup> : <sup>Π</sup>*<sup>q</sup>* <sup>→</sup> *<sup>C</sup>*1(*T*).

*<sup>l</sup>*,*<sup>j</sup>* <sup>−</sup> *<sup>∂</sup>jχ<sup>k</sup>*

*Rq-connection. Its curvature κ*�

*<sup>l</sup>*,*<sup>j</sup>* <sup>−</sup> *<sup>∂</sup>jχ<sup>k</sup>*

amounts to <sup>Δ</sup> <sup>=</sup> *det*(*∂<sup>i</sup> <sup>f</sup> <sup>k</sup>*) �<sup>=</sup> 0 because *det*(*<sup>f</sup> <sup>k</sup>*

,*<sup>j</sup>* <sup>−</sup> *<sup>∂</sup>jχ<sup>k</sup>*

**Corollary 6.2.** *There is a restricted nonlinear differential sequence:*

**Definition 6.3.** *A splitting of the short exact sequence* <sup>0</sup> <sup>→</sup> *<sup>R</sup>*<sup>0</sup>

−→ R*q*+1

*<sup>q</sup>* ∈ ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>R</sup>*<sup>0</sup>

**Remark 6.1.** *Rewriting the previous formulas with A instead of χ*<sup>0</sup> *we get:*

*<sup>l</sup>*,*<sup>i</sup>* <sup>−</sup> *<sup>χ</sup><sup>r</sup> l*,*i χk <sup>r</sup>*,*<sup>j</sup>* <sup>+</sup> *<sup>χ</sup><sup>r</sup> l*,*j χk <sup>r</sup>*,*<sup>i</sup>* <sup>−</sup> *<sup>A</sup><sup>r</sup> i χk lr*,*<sup>j</sup>* <sup>+</sup> *<sup>A</sup><sup>r</sup> jχk lr*,*<sup>i</sup>* = 0

<sup>0</sup> −→ *aut*(*X*) *jq*

*is such that <sup>D</sup>*¯ <sup>1</sup> *and <sup>D</sup>*¯ <sup>2</sup> *are involutive whenever* <sup>R</sup>*<sup>q</sup> is involutive.*

*ij*) *is the only existing symmetric connection for the Killing system.*

*<sup>j</sup>* <sup>−</sup> *<sup>∂</sup>jA<sup>k</sup>*

*complex structures, the quadratic terms disappear but the last two terms are left.*

**Corollary 6.3.** *When det*(*A*) �= 0 *there is a nonlinear stabilized sequence at order q:*

*called nonlinear Spencer sequence where D*¯ <sup>1</sup> *and D*¯ <sup>2</sup> *are involutive and its restriction:*

<sup>0</sup> −→ <sup>Γ</sup> *jq*

*<sup>μ</sup>*,*<sup>i</sup>* <sup>=</sup> <sup>−</sup>*g<sup>k</sup>*

−→ Π*<sup>q</sup>*

−→ R*q*

*l Ar i f l* *<sup>D</sup>*¯ <sup>1</sup> −→ *<sup>C</sup>*<sup>1</sup>

*<sup>i</sup>* <sup>−</sup> *<sup>A</sup><sup>r</sup> i χk <sup>r</sup>*,*<sup>j</sup>* <sup>+</sup> *<sup>A</sup><sup>r</sup> jχk <sup>r</sup>*,*<sup>i</sup>* = 0

*When q* = 1, *g*<sup>2</sup> = 0 *and though surprising it may look like, we find back exactly all the formulas presented by E. and F. Cosserat in ([C], p 123 and [16]). Even more strikingly, in the case of a Riemann structure, the last two terms disappear but the quadratic terms are left while, in the case of screw and*

*<sup>q</sup>* <sup>=</sup> <sup>−</sup>*χ<sup>q</sup> is a connection with <sup>D</sup>*¯ �

*∂iA<sup>k</sup>*

*li*,*<sup>j</sup>* <sup>+</sup> *<sup>χ</sup><sup>k</sup>*

*Z* of *X* with local coordinates (*z*) as follows:

$$(\gamma\_{\mathfrak{q}} : \Pi\_{\mathfrak{q}}(Y, Z) \times\_Y \Pi\_{\mathfrak{q}}(X, Y) \to \Pi\_{\mathfrak{q}}(X, Z) : ((y, z, \frac{\partial z}{\partial y'} \dots), (\mathbf{x}, y, \frac{\partial y}{\partial \mathbf{x'}} \dots) \to (\mathbf{x}, z, \frac{\partial z}{\partial y} \frac{\partial y}{\partial \mathbf{x'}} \dots))$$

We may also define *jq*(*f*)−<sup>1</sup> = *jq*(*f* <sup>−</sup>1) and obtain similarly an "inversion" law.

**Definition 6.1.** *A fibered submanifold* R*<sup>q</sup>* ⊂ Π*<sup>q</sup> is called a system of finite Lie equations or a Lie groupoid of order q if we have an induced source projection α<sup>q</sup>* : R*<sup>q</sup>* → *X, target projection β<sup>q</sup>* : R*<sup>q</sup>* → *X, composition γ<sup>q</sup>* : R*q*×*X*R*<sup>q</sup>* → R*q, inversion ι<sup>q</sup>* : R*<sup>q</sup>* → R*<sup>q</sup> and identity idq* : *X* → R*q. In the sequel we shall only consider transitive Lie groupoids such that the map* (*αq*, *βq*) : R*<sup>q</sup>* → *X* × *X is an epimorphism and we shall denote by* <sup>R</sup><sup>0</sup> *<sup>q</sup>* <sup>=</sup> *id*−1(R*q*) *the isotropy Lie group bundle of* <sup>R</sup>*q. Also, one can prove that the new system ρr*(R*q*) = R*q*+*<sup>r</sup> obtained by differentiating r times all the defining equations of* R*<sup>q</sup> is a Lie groupoid of order q* + *r. Finally, one can write down the Lie form and obtain* <sup>R</sup>*<sup>q</sup>* <sup>=</sup> { *fq* <sup>∈</sup> <sup>Π</sup>*q*<sup>|</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup> *<sup>q</sup>* (*ω*) = *ω*}*.*

Now, using the *algebraic bracket* {*jq*+1(*ξ*), *jq*+1(*η*)} = *jq*([*ξ*, *η*]), ∀*ξ*, *η* ∈ *T*, we may obtain by bilinearity a *differential bracket* on *Jq*(*T*) extending the bracket on *T*:

$$[\xi\_{q\prime}\eta\_q] = \{\xi\_{q+1\prime}\eta\_{q+1}\} + i(\xi)D\eta\_{q+1} - i(\eta)D\xi\_{q+1\prime}\psi\xi\_{q\prime}\eta\_q \in I\_q(T)$$

which does not depend on the respective lifts *ξq*+<sup>1</sup> and *ηq*+<sup>1</sup> of *ξ<sup>q</sup>* and *η<sup>q</sup>* in *Jq*+1(*T*). This bracket on sections satisfies the Jacobi identity and we set:

**Definition 6.2.** *We say that a vector subbundle Rq* ⊂ *Jq*(*T*) *is a system of infinitesimal Lie equations or a Lie algebroid if* [*Rq*, *Rq*] <sup>⊂</sup> *Rq, that is to say* [*ξq*, *<sup>η</sup>q*] <sup>∈</sup> *Rq*, <sup>∀</sup>*ξq*, *<sup>η</sup><sup>q</sup>* <sup>∈</sup> *Rq. The kernel R*<sup>0</sup> *<sup>q</sup> of the projection <sup>π</sup><sup>q</sup>* <sup>0</sup> : *Rq* <sup>→</sup> *T is the isotropy Lie algebra bundle of* <sup>R</sup><sup>0</sup> *<sup>q</sup> and* [*R*<sup>0</sup> *<sup>q</sup>*, *R*<sup>0</sup> *<sup>q</sup>*] <sup>⊂</sup> *<sup>R</sup>*<sup>0</sup> *<sup>q</sup> does not contain derivatives. Such a definition can be checked by means of computer algebra.*

**Proposition 6.1.** *There is a nonlinear differential sequence:*

$$0 \longrightarrow \operatorname{aut}(X) \stackrel{j\_{q+1}}{\longrightarrow} \Pi\_{q+1}(X,X) \stackrel{\tilde{D}}{\longrightarrow} T^\* \otimes J\_q(T) \stackrel{\tilde{D}'}{\longrightarrow} \wedge^2 T^\* \otimes J\_{q-1}(T)$$

*with D f* ¯ *<sup>q</sup>*+<sup>1</sup> <sup>≡</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup> *<sup>q</sup>*+<sup>1</sup> ◦ *<sup>j</sup>*1(*fq*) <sup>−</sup> *idq*+<sup>1</sup> <sup>=</sup> *<sup>χ</sup><sup>q</sup>* <sup>⇒</sup> *<sup>D</sup>*¯ � *χq*(*ξ*, *η*) ≡ *Dχq*(*ξ*, *η*) − {*χq*(*ξ*), *χq*(*η*)} = 0*. Moreover, setting χ*<sup>0</sup> = *A* − *id* ∈ *T*<sup>∗</sup> ⊗ *T, this sequence is locally exact if det*(*A*) �= 0*.*

*Proof.* There is a canonical inclusion <sup>Π</sup>*q*+<sup>1</sup> <sup>⊂</sup> *<sup>J</sup>*1(Π*q*) defined by *<sup>y</sup><sup>k</sup> <sup>μ</sup>*,*<sup>i</sup>* <sup>=</sup> *<sup>y</sup><sup>k</sup> <sup>μ</sup>*+1*<sup>i</sup>* and the composition *f* <sup>−</sup><sup>1</sup> *<sup>q</sup>*+<sup>1</sup> ◦ *<sup>j</sup>*1(*fq*) is a well defined section of *<sup>J</sup>*1(Π*q*) over the section *<sup>f</sup>* <sup>−</sup><sup>1</sup> *<sup>q</sup>* ◦ *fq* = *idq* of Π*<sup>q</sup>* like *idq*+1. The difference *χ<sup>q</sup>* = *f* <sup>−</sup><sup>1</sup> *<sup>q</sup>*+<sup>1</sup> ◦ *j*1(*fq*) − *idq*+<sup>1</sup> is thus a section of *T*<sup>∗</sup> ⊗ *V*(Π*q*) over *idq* and we have already noticed that *id*−<sup>1</sup> *<sup>q</sup>* (*V*(Π*q*)) = *Jq*(*T*). For *<sup>q</sup>* = 1 we get with *<sup>g</sup>*<sup>1</sup> = *<sup>f</sup>* <sup>−</sup><sup>1</sup> <sup>1</sup> :

$$\chi^k\_{,i} = \mathbf{g}^k\_{l}\partial\_{\mathbf{i}}f^l - \delta^k\_{\mathbf{i}} = A^k\_{\mathbf{i}} - \delta^k\_{\mathbf{i}}\prime \qquad \chi^k\_{\mathbf{j},\mathbf{i}} = \mathbf{g}^k\_l(\partial\_{\mathbf{i}}f^l\_{\mathbf{j}} - A^r\_{\mathbf{i}}f^l\_{r\mathbf{j}})$$

We also obtain from Lemma 6.1 the useful formula *f <sup>k</sup> <sup>r</sup> χ<sup>r</sup> <sup>μ</sup>*,*<sup>i</sup>* <sup>+</sup> ... <sup>+</sup> *<sup>f</sup> <sup>k</sup> μ*+1*<sup>r</sup> χr* ,*<sup>i</sup>* <sup>=</sup> *<sup>∂</sup><sup>i</sup> <sup>f</sup> <sup>k</sup> <sup>μ</sup>* <sup>−</sup> *<sup>f</sup> <sup>k</sup> μ*+1*<sup>i</sup>* allowing to determine *χ<sup>q</sup>* inductively.

We refer to ([26], p 215) for the inductive proof of the local exactness, providing the only formulas that will be used later on and can be checked directly by the reader:

$$
\partial\_i \chi^k\_{,j} - \partial\_j \chi^k\_{,i} - \chi^k\_{i,j} + \chi^k\_{j,i} - (\chi^r\_{,i} \chi^k\_{r,j} - \chi^r\_{,j} \chi^k\_{r,i}) = 0
$$

$$
\partial\_i \chi^k\_{l,j} - \partial\_j \chi^k\_{l,i} - \chi^k\_{li,j} + \chi^k\_{lj,i} - (\chi^r\_{,i} \chi^k\_{lr,j} + \chi^r\_{l,i} \chi^k\_{r,j} - \chi^r\_{l,j} \chi^k\_{r,i} - \chi^r\_{,j} \chi^k\_{lr,i}) = 0
$$

There is no need for double-arrows in this framework as the kernels are taken with respect to the zero section of the vector bundles involved. We finally notice that the main difference with the gauge sequence is that *all the indices range from* 1 *to n* and that the condition *det*(*A*) �= 0 amounts to <sup>Δ</sup> <sup>=</sup> *det*(*∂<sup>i</sup> <sup>f</sup> <sup>k</sup>*) �<sup>=</sup> 0 because *det*(*<sup>f</sup> <sup>k</sup> <sup>i</sup>* ) �= 0 by assumption.

**Corollary 6.2.** *There is a restricted nonlinear differential sequence:*

20 Will-be-set-by-IN-TECH

**Definition 6.1.** *A fibered submanifold* R*<sup>q</sup>* ⊂ Π*<sup>q</sup> is called a system of finite Lie equations or a Lie groupoid of order q if we have an induced source projection α<sup>q</sup>* : R*<sup>q</sup>* → *X, target projection β<sup>q</sup>* : R*<sup>q</sup>* → *X, composition γ<sup>q</sup>* : R*q*×*X*R*<sup>q</sup>* → R*q, inversion ι<sup>q</sup>* : R*<sup>q</sup>* → R*<sup>q</sup> and identity idq* : *X* → R*q. In the sequel we shall only consider transitive Lie groupoids such that the map* (*αq*, *βq*) : R*<sup>q</sup>* → *X* × *X is*

*one can prove that the new system ρr*(R*q*) = R*q*+*<sup>r</sup> obtained by differentiating r times all the defining equations of* R*<sup>q</sup> is a Lie groupoid of order q* + *r. Finally, one can write down the Lie form and obtain*

Now, using the *algebraic bracket* {*jq*+1(*ξ*), *jq*+1(*η*)} = *jq*([*ξ*, *η*]), ∀*ξ*, *η* ∈ *T*, we may obtain by

[*ξq*, *ηq*] = {*ξq*+1, *ηq*+1} + *i*(*ξ*)*Dηq*+<sup>1</sup> − *i*(*η*)*Dξq*+1, ∀*ξq*, *η<sup>q</sup>* ∈ *Jq*(*T*) which does not depend on the respective lifts *ξq*+<sup>1</sup> and *ηq*+<sup>1</sup> of *ξ<sup>q</sup>* and *η<sup>q</sup>* in *Jq*+1(*T*). This

**Definition 6.2.** *We say that a vector subbundle Rq* ⊂ *Jq*(*T*) *is a system of infinitesimal Lie equations or a Lie algebroid if* [*Rq*, *Rq*] <sup>⊂</sup> *Rq, that is to say* [*ξq*, *<sup>η</sup>q*] <sup>∈</sup> *Rq*, <sup>∀</sup>*ξq*, *<sup>η</sup><sup>q</sup>* <sup>∈</sup> *Rq. The kernel R*<sup>0</sup>

*<sup>q</sup>*+<sup>1</sup> ◦ *<sup>j</sup>*1(*fq*) is a well defined section of *<sup>J</sup>*1(Π*q*) over the section *<sup>f</sup>* <sup>−</sup><sup>1</sup>

*<sup>i</sup>* , *<sup>χ</sup><sup>k</sup>*

*<sup>j</sup>*,*<sup>i</sup>* <sup>=</sup> *<sup>g</sup><sup>k</sup>*

*<sup>r</sup> χ<sup>r</sup>*

−→ *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *Jq*(*T*) *<sup>D</sup>*¯ �

We may also define *jq*(*f*)−<sup>1</sup> = *jq*(*f* <sup>−</sup>1) and obtain similarly an "inversion" law.

*∂z ∂y*

, ...),(*x*, *y*,

*∂y ∂x*

*<sup>q</sup>* <sup>=</sup> *id*−1(R*q*) *the isotropy Lie group bundle of* <sup>R</sup>*q. Also,*

*<sup>q</sup> and* [*R*<sup>0</sup>

*<sup>q</sup>*, *R*<sup>0</sup>

−→ ∧2*T*<sup>∗</sup> <sup>⊗</sup> *Jq*−1(*T*)

*<sup>q</sup>*+<sup>1</sup> ◦ *j*1(*fq*) − *idq*+<sup>1</sup> is thus a section of *T*<sup>∗</sup> ⊗ *V*(Π*q*) over

*<sup>q</sup>* (*V*(Π*q*)) = *Jq*(*T*). For *<sup>q</sup>* = 1 we get with *<sup>g</sup>*<sup>1</sup> = *<sup>f</sup>* <sup>−</sup><sup>1</sup>

*<sup>μ</sup>*,*<sup>i</sup>* <sup>+</sup> ... <sup>+</sup> *<sup>f</sup> <sup>k</sup>*

*<sup>j</sup>* <sup>−</sup> *<sup>A</sup><sup>r</sup> i f l rj*)

> *μ*+1*<sup>r</sup> χr*

*<sup>l</sup>* (*∂<sup>i</sup> <sup>f</sup> <sup>l</sup>*

*χq*(*ξ*, *η*) ≡ *Dχq*(*ξ*, *η*) − {*χq*(*ξ*), *χq*(*η*)} = 0*.*

*<sup>μ</sup>*,*<sup>i</sup>* <sup>=</sup> *<sup>y</sup><sup>k</sup>*

,*<sup>i</sup>* <sup>=</sup> *<sup>∂</sup><sup>i</sup> <sup>f</sup> <sup>k</sup>*

*<sup>q</sup>*] <sup>⊂</sup> *<sup>R</sup>*<sup>0</sup>

, ...) → (*x*, *z*,

*∂z ∂y ∂y ∂x* , ...)

*<sup>q</sup> of the*

*<sup>q</sup> does not contain*

*<sup>μ</sup>*+1*<sup>i</sup>* and the

*<sup>μ</sup>* <sup>−</sup> *<sup>f</sup> <sup>k</sup> μ*+1*<sup>i</sup>*

<sup>1</sup> :

*<sup>q</sup>* ◦ *fq* = *idq* of

*Z* of *X* with local coordinates (*z*) as follows:

*an epimorphism and we shall denote by* <sup>R</sup><sup>0</sup>

*<sup>q</sup>* (*ω*) = *ω*}*.*

bilinearity a *differential bracket* on *Jq*(*T*) extending the bracket on *T*:

<sup>0</sup> : *Rq* <sup>→</sup> *T is the isotropy Lie algebra bundle of* <sup>R</sup><sup>0</sup>

−→ <sup>Π</sup>*q*+1(*X*, *<sup>X</sup>*) *<sup>D</sup>*¯

*Proof.* There is a canonical inclusion <sup>Π</sup>*q*+<sup>1</sup> <sup>⊂</sup> *<sup>J</sup>*1(Π*q*) defined by *<sup>y</sup><sup>k</sup>*

*<sup>i</sup>* <sup>=</sup> *<sup>A</sup><sup>k</sup>*

*<sup>i</sup>* <sup>−</sup> *<sup>δ</sup><sup>k</sup>*

*Moreover, setting χ*<sup>0</sup> = *A* − *id* ∈ *T*<sup>∗</sup> ⊗ *T, this sequence is locally exact if det*(*A*) �= 0*.*

*derivatives. Such a definition can be checked by means of computer algebra.*

*<sup>q</sup>*+<sup>1</sup> ◦ *<sup>j</sup>*1(*fq*) <sup>−</sup> *idq*+<sup>1</sup> <sup>=</sup> *<sup>χ</sup><sup>q</sup>* <sup>⇒</sup> *<sup>D</sup>*¯ �

*<sup>l</sup> <sup>∂</sup><sup>i</sup> <sup>f</sup> <sup>l</sup>* <sup>−</sup> *<sup>δ</sup><sup>k</sup>*

We also obtain from Lemma 6.1 the useful formula *f <sup>k</sup>*

bracket on sections satisfies the Jacobi identity and we set:

**Proposition 6.1.** *There is a nonlinear differential sequence:*

<sup>0</sup> −→ *aut*(*X*) *jq*<sup>+</sup><sup>1</sup>

Π*<sup>q</sup>* like *idq*+1. The difference *χ<sup>q</sup>* = *f* <sup>−</sup><sup>1</sup>

*idq* and we have already noticed that *id*−<sup>1</sup>

*χk* ,*<sup>i</sup>* <sup>=</sup> *<sup>g</sup><sup>k</sup>*

allowing to determine *χ<sup>q</sup>* inductively.

<sup>R</sup>*<sup>q</sup>* <sup>=</sup> { *fq* <sup>∈</sup> <sup>Π</sup>*q*<sup>|</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup>

*projection <sup>π</sup><sup>q</sup>*

*with D f* ¯ *<sup>q</sup>*+<sup>1</sup> <sup>≡</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup>

composition *f* <sup>−</sup><sup>1</sup>

*γ<sup>q</sup>* : Π*q*(*Y*, *Z*)×*Y*Π*q*(*X*,*Y*) → Π*q*(*X*, *Z*) : ((*y*, *z*,

$$0 \longrightarrow \Gamma \stackrel{l\_{q+1}}{\longrightarrow} \mathcal{R}\_{q+1} \stackrel{\bar{D}}{\longrightarrow} T^\* \otimes \mathcal{R}\_q \stackrel{\mathcal{D}'}{\longrightarrow} \wedge^2 T^\* \otimes I\_{q-1}(T)$$

**Definition 6.3.** *A splitting of the short exact sequence* <sup>0</sup> <sup>→</sup> *<sup>R</sup>*<sup>0</sup> *<sup>q</sup>* → *Rq πq* <sup>→</sup><sup>0</sup> *<sup>T</sup>* <sup>→</sup> <sup>0</sup> *is a map <sup>χ</sup>*� *<sup>q</sup>* : *T* → *Rq such that <sup>π</sup><sup>q</sup>* <sup>0</sup> ◦ *χ*� *<sup>q</sup>* = *idT or equivalently a section of T*<sup>∗</sup> ⊗ *Rq over idT* ∈ *T*<sup>∗</sup> ⊗ *T and is called a Rq-connection. Its curvature κ*� *<sup>q</sup>* ∈ ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>R</sup>*<sup>0</sup> *<sup>q</sup> is defined by κ*� *<sup>q</sup>*(*ξ*, *η*)=[*χ*� *<sup>q</sup>*(*ξ*), *χ*� *<sup>q</sup>*(*η*)] − *χ*� *<sup>q</sup>*([*ξ*, *η*])*. We notice that χ*� *<sup>q</sup>* <sup>=</sup> <sup>−</sup>*χ<sup>q</sup> is a connection with <sup>D</sup>*¯ � *χ*� *<sup>q</sup>* = *κ*� *<sup>q</sup> if and only if A* = 0*. In particular* (*δ<sup>k</sup> <sup>i</sup>* , <sup>−</sup>*γ<sup>k</sup> ij*) *is the only existing symmetric connection for the Killing system.*

**Remark 6.1.** *Rewriting the previous formulas with A instead of χ*<sup>0</sup> *we get:*

$$
\partial\_i A^k\_j - \partial\_j A^k\_i - A^r\_i \chi^k\_{r,j} + A^r\_j \chi^k\_{r,i} = 0
$$

$$
\partial\_i \chi^k\_{l,j} - \partial\_j \chi^k\_{l,i} - \chi^r\_{l,i} \chi^k\_{r,j} + \chi^r\_{l,j} \chi^k\_{r,i} - A^r\_i \chi^k\_{lr,j} + A^r\_j \chi^k\_{lr,i} = 0
$$

*When q* = 1, *g*<sup>2</sup> = 0 *and though surprising it may look like, we find back exactly all the formulas presented by E. and F. Cosserat in ([C], p 123 and [16]). Even more strikingly, in the case of a Riemann structure, the last two terms disappear but the quadratic terms are left while, in the case of screw and complex structures, the quadratic terms disappear but the last two terms are left.*

**Corollary 6.3.** *When det*(*A*) �= 0 *there is a nonlinear stabilized sequence at order q:*

$$0 \longrightarrow aut(X) \stackrel{j\_q}{\longrightarrow} \Pi\_q \stackrel{\mathcal{D}\_1}{\longrightarrow} \mathcal{C}\_1(T) \stackrel{\mathcal{D}\_2}{\longrightarrow} \mathcal{C}\_2(T)$$

*called nonlinear Spencer sequence where D*¯ <sup>1</sup> *and D*¯ <sup>2</sup> *are involutive and its restriction:*

$$0 \longrightarrow \Gamma \stackrel{j\_q}{\longrightarrow} \mathcal{R}\_q \stackrel{D\_1}{\longrightarrow} \mathcal{C}\_1 \stackrel{D\_2}{\longrightarrow} \mathcal{C}\_2$$

*is such that <sup>D</sup>*¯ <sup>1</sup> *and <sup>D</sup>*¯ <sup>2</sup> *are involutive whenever* <sup>R</sup>*<sup>q</sup> is involutive.*

*Proof.* : With <sup>|</sup>*μ*<sup>|</sup> <sup>=</sup> *<sup>q</sup>* we have *<sup>χ</sup><sup>k</sup> <sup>μ</sup>*,*<sup>i</sup>* <sup>=</sup> <sup>−</sup>*g<sup>k</sup> l Ar i f l <sup>μ</sup>*+1*<sup>r</sup>* <sup>+</sup> *terms*(*order* <sup>≤</sup> *<sup>q</sup>*). Setting *<sup>χ</sup><sup>k</sup> <sup>μ</sup>*,*<sup>i</sup>* <sup>=</sup> *<sup>A</sup><sup>r</sup> i τk <sup>μ</sup>*,*r*, we obtain *τ<sup>k</sup> <sup>μ</sup>*,*<sup>r</sup>* <sup>=</sup> <sup>−</sup>*g<sup>k</sup> l f l <sup>μ</sup>*+1*<sup>r</sup>* <sup>+</sup> *terms*(*order* <sup>≤</sup> *<sup>q</sup>*) and *<sup>D</sup>*¯ : <sup>Π</sup>*q*+<sup>1</sup> <sup>→</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *Jq*(*T*) restricts to *<sup>D</sup>*¯ <sup>1</sup> : <sup>Π</sup>*<sup>q</sup>* <sup>→</sup> *<sup>C</sup>*1(*T*).

*Proof.* : Choosing *fq*+1, *gq*+1, *hq*+<sup>1</sup> ∈ R*q*+<sup>1</sup> such that *gq*+<sup>1</sup> ◦ *fq*+<sup>1</sup> = *fq*+<sup>1</sup> ◦ *hq*+<sup>1</sup> and passing to the limits *gq*+<sup>1</sup> = *idq*+<sup>1</sup> + *tηq*+<sup>1</sup> + ... and *hq*+<sup>1</sup> = *idq*+<sup>1</sup> + *tξq*+<sup>1</sup> + ... when *t* → 0, we obtain the

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 23

) + *f <sup>k</sup> μ*+1*<sup>r</sup>*

*<sup>ξ</sup>q*+<sup>1</sup> − {*χq*+1, ¯

,*i ξk r*)

> *r*,*i ξr <sup>j</sup>* <sup>−</sup> *<sup>χ</sup><sup>r</sup> j*,*i ξk <sup>r</sup>* <sup>−</sup> *<sup>χ</sup><sup>r</sup>* ,*i ξk jr*)

*∂rα<sup>i</sup>* + *αr∂iξ<sup>r</sup>* + *χ<sup>s</sup>*

,*i ξr rs*)

*ri is the variation*

*<sup>q</sup>*+1(*j*1(*ω*)) = *j*1(*ω*¯). We obtain

*<sup>q</sup>* ))−1(*j*1(*ω*)) <sup>−</sup> *<sup>j</sup>*1(*ω*) = *<sup>L</sup>*(*σq*)*<sup>ω</sup>* <sup>=</sup> <sup>0</sup>

*<sup>q</sup>*+1(*j*1(*ω*)) = *j*1(*ω*¯), it follows that

*<sup>q</sup>*+<sup>1</sup> = *gq*+<sup>1</sup> ◦ *fq*+<sup>1</sup> and the new

,*r∂iξ<sup>r</sup>* <sup>−</sup> *<sup>χ</sup><sup>r</sup>*

*<sup>j</sup>*,*r∂iξ<sup>r</sup>* <sup>+</sup> *<sup>χ</sup><sup>k</sup>*

*ξ<sup>r</sup>* + ... + *f <sup>k</sup>*

*ξq*+<sup>1</sup> = *ξq*+<sup>1</sup> + *χq*+1(*ξ*). This transformation is invertible if

*ξq*+1}*:*

*r ξr μ*

(*∂<sup>i</sup> f <sup>k</sup> <sup>μ</sup>* <sup>−</sup> *<sup>f</sup> <sup>k</sup> μ*+1*<sup>i</sup>*

,*<sup>i</sup>* <sup>+</sup> *<sup>χ</sup><sup>k</sup>*

*<sup>j</sup>*,*<sup>i</sup>* <sup>+</sup> *<sup>χ</sup><sup>k</sup>*

*For the Killing system R*<sup>1</sup> ⊂ *J*1(*T*) *with g*<sup>2</sup> = 0*, these variations are exactly the ones that can be found in ([C], (50)+(49), p 124 with a printing mistake corrected on p 128) when replacing a* 3 × 3 *skewsymmetric matrix by the corresponding vector. The last unavoidable Proposition is thus essential in order to bring back the nonlinear framework of finite elasticity to the linear framewok of infinitesimal*

*<sup>r</sup>* <sup>−</sup> *<sup>ξ</sup><sup>r</sup>*

*ri*)+(*ξ<sup>r</sup>*

*<sup>r</sup>* <sup>=</sup> <sup>−</sup>*αiξ<sup>i</sup> by introducing a connection. Accordingly, <sup>ξ</sup><sup>r</sup>*

*<sup>q</sup>* (*ω*)) = *<sup>j</sup>*1(*ω*¯) and *<sup>f</sup>* <sup>−</sup><sup>1</sup>

*<sup>q</sup>*+1(*j*1(*ω*)) = *<sup>f</sup>* �−<sup>1</sup>

*<sup>q</sup>*+<sup>1</sup> ∈ *T*<sup>∗</sup> ⊗ *Rq over the target*, even if *fq*+<sup>1</sup> may not be a section of R*q*+1.

, we have related cocycles at F in the Janet sequence *over the source* with

*<sup>q</sup>*+<sup>1</sup> by a gauge transformation.

*This is exactly the variation obtained by Weyl ([W], (76), p 289) who was assuming implicitly A* = 0

*of the EM potential itself, that is the δAi of engineers used in order to exhibit the Maxwell equations from a variational principle ([W],* § *26) but the introduction of the Spencer operator is new in this*

Finally, chasing in diagram (1) , the Spencer sequence is locally exact at *C*<sup>1</sup> if and only if the Janet sequence is locally exact at *F*<sup>0</sup> because the central sequence is locally exact. *The situation is much more complicate in the nonlinear framewok*. Let *ω*¯ be a section of F satisfying the same CC as *ω*, namely *I*(*j*1(*ω*)) = 0. It follows that we can find a section *fq*+<sup>1</sup> ∈ Π*q*+<sup>1</sup> such that

*<sup>q</sup>*+1(*j*1(*ω*)) <sup>⇒</sup> (*fq*+<sup>1</sup> ◦ *<sup>j</sup>*1(*<sup>f</sup>* <sup>−</sup><sup>1</sup>

*ξ* = *ξ* + *χ*0(*ξ*) = *A*(*ξ*) is an isomorphism of *T*.

*<sup>i</sup>* )+(*ξr∂rχ<sup>k</sup>*

*ij*)+(*ξr∂rχ<sup>k</sup>*

*For the conformal Killing system <sup>R</sup>*<sup>ˆ</sup> <sup>1</sup> <sup>⊂</sup> *<sup>J</sup>*1(*T*) *(see next section) we obtain:*

*<sup>r</sup>*,*<sup>i</sup>* <sup>⇒</sup> *δα<sup>i</sup>* = (*∂iξ<sup>r</sup>*

*<sup>q</sup>* )(*j*1(*ω*)) = *j*1(*f* <sup>−</sup><sup>1</sup>

*<sup>q</sup>*+<sup>1</sup> <sup>∈</sup> <sup>Π</sup>*q*+<sup>1</sup> are such that *<sup>f</sup>* <sup>−</sup><sup>1</sup>

*<sup>q</sup>*+<sup>1</sup> differs from the initial *<sup>σ</sup><sup>q</sup>* <sup>=</sup> *D f* ¯ <sup>−</sup><sup>1</sup>

*<sup>q</sup>*+1)(*j*1(*ω*)) = *j*1(*ω*) ⇒ ∃*gq*+<sup>1</sup> ∈ R*q*+<sup>1</sup> such that *f* �

local formula:

and thus *ηq*+<sup>1</sup> = *fq*+2( ¯

*δχ<sup>k</sup>*

*δχ<sup>k</sup>*

and only if *<sup>ξ</sup>* <sup>→</sup> ¯

*when setting* ¯

*framework.*

*f* <sup>−</sup><sup>1</sup>

(*f* �

*σ*�

*ξr*

*<sup>q</sup>* (*ω*) = *<sup>ω</sup>*¯ <sup>⇒</sup> *<sup>j</sup>*1(*<sup>f</sup>* <sup>−</sup><sup>1</sup>

and thus *σ<sup>q</sup>* = *D f* ¯ <sup>−</sup><sup>1</sup>

As *σ<sup>q</sup>* is killed by *D*¯ �

therefore *j*1(*f* <sup>−</sup><sup>1</sup>

Now, if *fq*+1, *f* �

*<sup>q</sup>*+<sup>1</sup> ◦ *<sup>f</sup>* <sup>−</sup><sup>1</sup>

*<sup>q</sup>* = *D f* ¯ �−<sup>1</sup>

*dμη<sup>k</sup>* = *η<sup>k</sup>*

**Example 6.1.** *For q* = 1*, we obtain from δχ<sup>q</sup>* = *D* ¯

,*<sup>i</sup>* = (*∂iξ<sup>k</sup>* <sup>−</sup> *<sup>ξ</sup><sup>k</sup>*

*<sup>ξ</sup><sup>k</sup>* <sup>−</sup> ¯ *ξk <sup>i</sup>* )+(*χ<sup>k</sup> r*,*i* ¯ *<sup>ξ</sup><sup>r</sup>* <sup>−</sup> *<sup>χ</sup><sup>r</sup>* ,*i* ¯ *ξk r*)

*elasticity that only depends on the linear Spencer operator.*

*<sup>j</sup>* <sup>−</sup> *<sup>ξ</sup><sup>k</sup>*

= (*∂<sup>i</sup>* ¯

= (*∂<sup>i</sup>* ¯ *ξk <sup>j</sup>* <sup>−</sup> ¯ *ξk ij*)+(*χ<sup>k</sup> rj*,*i* ¯ *ξ<sup>r</sup>* + *χ<sup>k</sup> r*,*i* ¯ *ξr <sup>j</sup>* <sup>−</sup> *<sup>χ</sup><sup>r</sup> j*,*i* ¯ *ξk <sup>r</sup>* <sup>−</sup> *<sup>χ</sup><sup>r</sup>* ,*i* ¯ *ξk jr*)

*<sup>j</sup>*,*<sup>i</sup>* = (*∂iξ<sup>k</sup>*

*α<sup>i</sup>* = *χ<sup>r</sup>*

*<sup>q</sup>* )(*j*1(*ω*)) = *<sup>f</sup>* <sup>−</sup><sup>1</sup>

cocycles at *T*<sup>∗</sup> ⊗ *Rq* or *C*<sup>1</sup> *over the target*.

*<sup>r</sup>* <sup>=</sup> <sup>0</sup> <sup>⇔</sup> *<sup>ξ</sup><sup>r</sup>*

*r fr*

*ξq*+1) with ¯

*<sup>μ</sup>* + ... = *<sup>ξ</sup><sup>i</sup>*

Finally, setting *<sup>A</sup>*−<sup>1</sup> <sup>=</sup> *<sup>B</sup>* <sup>=</sup> *id* <sup>−</sup> *<sup>τ</sup>*0, we obtain successively:

$$\begin{aligned} \partial\_i \chi^k\_{\mu,j} - \partial\_j \chi^k\_{\mu,i} + terms(\chi\_q) - (A^r\_i \chi^k\_{\mu+1\_{r,j}} - A^r\_j \chi^k\_{\mu+1\_{r,i}}) &= 0 \\ B^i\_r B^j\_s (\partial\_i \chi^k\_{\mu,j} - \partial\_j \chi^k\_{\mu,i}) + terms(\chi\_q) - (\tau^k\_{\mu+1\_{r,s}} - \tau^k\_{\mu+1\_{s,r}}) &= 0 \end{aligned}$$

We obtain therefore *<sup>D</sup>τq*+<sup>1</sup> <sup>+</sup> *terms*(*τq*) = 0 and *<sup>D</sup>*¯ � : *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *Jq*(*T*) → ∧2*T*<sup>∗</sup> <sup>⊗</sup> *Jq*−1(*T*) restricts to *<sup>D</sup>*¯ <sup>2</sup> : *<sup>C</sup>*1(*T*) <sup>→</sup> *<sup>C</sup>*2(*T*).

In the case of Lie groups of transformations, the symbol of the involutive system *Rq must* be *gq* = 0 providing an isomorphism R*q*+<sup>1</sup> � R*<sup>q</sup>* ⇒ *Rq*+<sup>1</sup> � *Rq* and we have therefore *Cr* <sup>=</sup> <sup>∧</sup>*rT*<sup>∗</sup> <sup>⊗</sup> *Rq* for *<sup>r</sup>* <sup>=</sup> 1, ..., *<sup>n</sup>* in the linear Spencer sequence.

**Remark 6.2.** *The passage from χq to τq is exactly the one done by E. and F. Cosserat in ([C], p 190). However, even if is a good idea to pass from the source to the target, the way they realize it is based on a subtle misunderstanding that we shall correct later on in Proposition 6.3.*

$$\begin{aligned} \text{If } f\_{q+1}, g\_{q+1} \in \Pi\_{q+1} \text{ and } f'\_{q+1} = \mathbf{g}\_{q+1} \circ f\_{q+1}, \text{we get:}\\ \bar{D}f'\_{q+1} = f\_{q+1}^{-1} \circ \mathbf{g}\_{q+1}^{-1} \circ \mathbf{j}\_1(\mathbf{g}\_q) \circ \mathbf{j}\_1(f\_q) - \mathrm{id}\_{q+1} = f\_{q+1}^{-1} \circ \bar{D}\mathbf{g}\_{q+1} \circ \mathbf{j}\_1(f\_q) + \bar{D}f\_{q+1} \end{aligned}$$

**Definition 6.4.** *For any section fq*+<sup>1</sup> ∈ R*q*+1*, the transformation:*

$$\chi\_{\eta} \longrightarrow \chi'\_{\eta} = f\_{\eta+1}^{-1} \circ \chi\_{\eta} \circ j\_1(f\_{\eta}) + \vec{D}f\_{\eta+1}$$

*is called a gauge transformation and exchanges the solutions of the field equations D*¯ � *χ<sup>q</sup>* = 0*.*

Introducing the *formal Lie derivative* on *Jq*(*T*) by the formulas:

$$\begin{aligned} L(\mathfrak{f}\_{q+1})\eta\_{\mathfrak{q}} &= \{\mathfrak{f}\_{q+1}, \eta\_{\mathfrak{q}+1}\} + i(\mathfrak{f})D\eta\_{\mathfrak{q}+1} = [\mathfrak{f}\_{\mathfrak{q}\prime}\eta\_{\mathfrak{q}}] + i(\mathfrak{q})D\mathfrak{f}\_{\mathfrak{q}+1} \\ (L(j\_1(\mathfrak{f}\_{q+1}))\chi\_{\mathfrak{q}})(\xi) &= L(\mathfrak{f}\_{q+1})(\chi\_{\mathfrak{q}}(\xi)) - \chi\_{\mathfrak{q}}([\mathfrak{f},\xi]) \end{aligned}$$

and passing to the limit with *fq*+<sup>1</sup> = *idq*+<sup>1</sup> + *tξq*+<sup>1</sup> + ... for *t* → 0 *over the source*, we get:

**Lemma 6.2.** *An infinitesimal gauge transformation has the form:*

$$
\delta \chi\_{\mathfrak{q}} = D \mathfrak{f}\_{\mathfrak{q}+1} + L(j\_1(\mathfrak{f}\_{\mathfrak{q}+1})) \chi\_{\mathfrak{q}},
$$

*Passing again to the limit but now over the target with χ<sup>q</sup>* = *D f* ¯ *<sup>q</sup>*+<sup>1</sup> *and gq*+<sup>1</sup> = *idq*+<sup>1</sup> + *tηq*+<sup>1</sup> + ...*, we obtain the variation:*

$$\delta \chi\_{\emptyset} = f\_{q+1}^{-1} \circ D \eta\_{q+1} \circ j\_1(f\_{\emptyset})$$

**Proposition 6.2.** *The same variation is obtained whenever ηq*+<sup>1</sup> = *fq*+2(*ξq*+<sup>1</sup> + *χq*+1(*ξ*)) *with χq*+<sup>1</sup> = *D f* ¯ *<sup>q</sup>*+2*, a transformation which only depends on j*1(*fq*+1) *and is invertible if and only if det*(*A*) �= 0*.*

*Proof.* : Choosing *fq*+1, *gq*+1, *hq*+<sup>1</sup> ∈ R*q*+<sup>1</sup> such that *gq*+<sup>1</sup> ◦ *fq*+<sup>1</sup> = *fq*+<sup>1</sup> ◦ *hq*+<sup>1</sup> and passing to the limits *gq*+<sup>1</sup> = *idq*+<sup>1</sup> + *tηq*+<sup>1</sup> + ... and *hq*+<sup>1</sup> = *idq*+<sup>1</sup> + *tξq*+<sup>1</sup> + ... when *t* → 0, we obtain the local formula:

$$d\_{\mu}\eta^{k} = \eta^{k}\_{r}f^{r}\_{\mu} + \dots = \mathfrak{J}^{i}(\partial\_{i}f^{k}\_{\mu} - f^{k}\_{\mu+1\_{l}}) + f^{k}\_{\mu+1\_{l}}\mathfrak{J}^{r} + \dots + f^{k}\_{r}\mathfrak{J}^{r}\_{\mu}$$

and thus *ηq*+<sup>1</sup> = *fq*+2( ¯ *ξq*+1) with ¯ *ξq*+<sup>1</sup> = *ξq*+<sup>1</sup> + *χq*+1(*ξ*). This transformation is invertible if and only if *<sup>ξ</sup>* <sup>→</sup> ¯ *ξ* = *ξ* + *χ*0(*ξ*) = *A*(*ξ*) is an isomorphism of *T*.

**Example 6.1.** *For q* = 1*, we obtain from δχ<sup>q</sup>* = *D* ¯ *<sup>ξ</sup>q*+<sup>1</sup> − {*χq*+1, ¯ *ξq*+1}*:*

22 Will-be-set-by-IN-TECH

) + *terms*(*χq*) <sup>−</sup> (*τ<sup>k</sup>*

We obtain therefore *<sup>D</sup>τq*+<sup>1</sup> <sup>+</sup> *terms*(*τq*) = 0 and *<sup>D</sup>*¯ � : *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *Jq*(*T*) → ∧2*T*<sup>∗</sup> <sup>⊗</sup> *Jq*−1(*T*) restricts to

In the case of Lie groups of transformations, the symbol of the involutive system *Rq must* be *gq* = 0 providing an isomorphism R*q*+<sup>1</sup> � R*<sup>q</sup>* ⇒ *Rq*+<sup>1</sup> � *Rq* and we have therefore

**Remark 6.2.** *The passage from χq to τq is exactly the one done by E. and F. Cosserat in ([C], p 190). However, even if is a good idea to pass from the source to the target, the way they realize it is based on*

> *L*(*ξq*+1)*η<sup>q</sup>* = {*ξq*+1, *ηq*+1} + *i*(*ξ*)*Dηq*+<sup>1</sup> = [*ξq*, *ηq*] + *i*(*η*)*Dξq*+<sup>1</sup> (*L*(*j*1(*ξq*+1))*χq*)(*ζ*) = *L*(*ξq*+1)(*χq*(*ζ*)) − *χq*([*ξ*, *ζ*])

> > *δχ<sup>q</sup>* = *Dξq*+<sup>1</sup> + *L*(*j*1(*ξq*+1))*χ<sup>q</sup>*

*Passing again to the limit but now over the target with χ<sup>q</sup>* = *D f* ¯ *<sup>q</sup>*+<sup>1</sup> *and gq*+<sup>1</sup> = *idq*+<sup>1</sup> + *tηq*+<sup>1</sup> + ...*,*

**Proposition 6.2.** *The same variation is obtained whenever ηq*+<sup>1</sup> = *fq*+2(*ξq*+<sup>1</sup> + *χq*+1(*ξ*)) *with χq*+<sup>1</sup> = *D f* ¯ *<sup>q</sup>*+2*, a transformation which only depends on j*1(*fq*+1) *and is invertible if and only if*

*<sup>q</sup>*+<sup>1</sup> ◦ *Dηq*+<sup>1</sup> ◦ *j*1(*fq*)

and passing to the limit with *fq*+<sup>1</sup> = *idq*+<sup>1</sup> + *tξq*+<sup>1</sup> + ... for *t* → 0 *over the source*, we get:

*<sup>q</sup>*+<sup>1</sup> ◦ *<sup>χ</sup><sup>q</sup>* ◦ *<sup>j</sup>*1(*fq*) + *D f* ¯ *<sup>q</sup>*+<sup>1</sup>

*i χk*

*<sup>μ</sup>*+1*r*,*<sup>j</sup>* <sup>−</sup> *<sup>A</sup><sup>r</sup>*

*<sup>μ</sup>*+1*r*,*<sup>s</sup>* <sup>−</sup> *<sup>τ</sup><sup>k</sup>*

*jχk μ*+1*r*,*i*

) = 0

*<sup>μ</sup>*+1*s*,*r*) = 0

*<sup>q</sup>*+<sup>1</sup> ◦ *Dg*¯ *<sup>q</sup>*+<sup>1</sup> ◦ *<sup>j</sup>*1(*fq*) + *D f* ¯ *<sup>q</sup>*+<sup>1</sup>

*χ<sup>q</sup>* = 0*.*

*<sup>μ</sup>*,*<sup>i</sup>* <sup>+</sup> *terms*(*χq*) <sup>−</sup> (*A<sup>r</sup>*

Finally, setting *<sup>A</sup>*−<sup>1</sup> <sup>=</sup> *<sup>B</sup>* <sup>=</sup> *id* <sup>−</sup> *<sup>τ</sup>*0, we obtain successively:

*<sup>μ</sup>*,*<sup>j</sup>* <sup>−</sup> *<sup>∂</sup>jχ<sup>k</sup>*

*Cr* <sup>=</sup> <sup>∧</sup>*rT*<sup>∗</sup> <sup>⊗</sup> *Rq* for *<sup>r</sup>* <sup>=</sup> 1, ..., *<sup>n</sup>* in the linear Spencer sequence.

**Definition 6.4.** *For any section fq*+<sup>1</sup> ∈ R*q*+1*, the transformation:*

*χ<sup>q</sup>* −→ *χ*�

Introducing the *formal Lie derivative* on *Jq*(*T*) by the formulas:

**Lemma 6.2.** *An infinitesimal gauge transformation has the form:*

*a subtle misunderstanding that we shall correct later on in Proposition 6.3.*

*<sup>q</sup>*+<sup>1</sup> = *gq*+<sup>1</sup> ◦ *fq*+1*, we get:*

*<sup>q</sup>*+<sup>1</sup> ◦ *<sup>j</sup>*1(*gq*) ◦ *<sup>j</sup>*1(*fq*) <sup>−</sup> *idq*+<sup>1</sup> <sup>=</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup>

*<sup>q</sup>* = *<sup>f</sup>* <sup>−</sup><sup>1</sup>

*is called a gauge transformation and exchanges the solutions of the field equations D*¯ �

*δχ<sup>q</sup>* = *f* <sup>−</sup><sup>1</sup>

*μ*,*i*

*<sup>μ</sup>*,*<sup>j</sup>* <sup>−</sup> *<sup>∂</sup>jχ<sup>k</sup>*

*∂iχ<sup>k</sup>*

*Bi rBj s*(*∂iχ<sup>k</sup>*

*<sup>D</sup>*¯ <sup>2</sup> : *<sup>C</sup>*1(*T*) <sup>→</sup> *<sup>C</sup>*2(*T*).

*If fq*+1, *gq*+<sup>1</sup> ∈ Π*q*+<sup>1</sup> *and f* �

*<sup>q</sup>*+<sup>1</sup> <sup>=</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup>

*<sup>q</sup>*+<sup>1</sup> ◦ *<sup>g</sup>*−<sup>1</sup>

*D f* ¯ �

*we obtain the variation:*

*det*(*A*) �= 0*.*

$$
\begin{array}{l}
\delta\chi^{k}\_{,i} = (\partial\_{i}\widetilde{\mathfrak{z}}^{k} - \widetilde{\mathfrak{z}}^{k}\_{i}) + (\widetilde{\mathfrak{z}}^{r}\partial\_{r}\chi^{k}\_{,i} + \chi^{k}\_{,r}\partial\_{i}\widetilde{\mathfrak{z}}^{r} - \chi^{r}\_{,i}\widetilde{\mathfrak{z}}^{k}\_{r}) \\
= (\partial\_{i}\widetilde{\mathfrak{z}}^{k} - \widetilde{\mathfrak{z}}^{k}\_{i}) + (\chi^{k}\_{r,i}\widetilde{\mathfrak{z}}^{r} - \chi^{r}\_{,i}\widetilde{\mathfrak{z}}^{k}\_{r}) \\
\delta\chi^{k}\_{j,i} = (\partial\_{i}\widetilde{\mathfrak{z}}^{k}\_{j} - \widetilde{\mathfrak{z}}^{k}\_{ij}) + (\widetilde{\mathfrak{z}}^{r}\partial\_{r}\chi^{k}\_{j,i} + \chi^{k}\_{j,r}\partial\_{i}\widetilde{\mathfrak{z}}^{r} + \chi^{k}\_{r,i}\widetilde{\mathfrak{z}}^{r}\_{j} - \chi^{r}\_{,j}\widetilde{\mathfrak{z}}^{k}\_{r} - \chi^{r}\_{,i}\widetilde{\mathfrak{z}}^{k}\_{j}) \\
= (\partial\_{i}\widetilde{\mathfrak{z}}^{k}\_{j} - \widetilde{\mathfrak{z}}^{k}\_{ij}) + (\chi^{k}\_{rj,i}\widetilde{\mathfrak{z}}^{r} + \chi^{k}\_{r,i}\widetilde{\mathfrak{z}}^{r}\_{j} - \chi^{r}\_{,j}\widetilde{\mathfrak{z}}^{k}\_{r} - \chi^{r}\_{,i}\widetilde{\mathfrak{z}}^{k}\_{j}) \\
\end{array}
$$

*For the Killing system R*<sup>1</sup> ⊂ *J*1(*T*) *with g*<sup>2</sup> = 0*, these variations are exactly the ones that can be found in ([C], (50)+(49), p 124 with a printing mistake corrected on p 128) when replacing a* 3 × 3 *skewsymmetric matrix by the corresponding vector. The last unavoidable Proposition is thus essential in order to bring back the nonlinear framework of finite elasticity to the linear framewok of infinitesimal elasticity that only depends on the linear Spencer operator.*

*For the conformal Killing system <sup>R</sup>*<sup>ˆ</sup> <sup>1</sup> <sup>⊂</sup> *<sup>J</sup>*1(*T*) *(see next section) we obtain:*

$$\mathfrak{a}\_{i} = \chi^{r}\_{r,i} \Rightarrow \delta\mathfrak{a}\_{i} = (\partial\_{i}\mathfrak{\zeta}^{r}\_{r} - \mathfrak{\zeta}^{r}\_{ri}) + (\mathfrak{\zeta}^{r}\partial\_{r}\mathfrak{a}\_{i} + \mathfrak{a}\_{r}\partial\_{i}\mathfrak{\zeta}^{r} + \chi^{s}\_{,i}\mathfrak{\zeta}^{r}\_{rs})$$

*This is exactly the variation obtained by Weyl ([W], (76), p 289) who was assuming implicitly A* = 0 *when setting* ¯ *ξr <sup>r</sup>* <sup>=</sup> <sup>0</sup> <sup>⇔</sup> *<sup>ξ</sup><sup>r</sup> <sup>r</sup>* <sup>=</sup> <sup>−</sup>*αiξ<sup>i</sup> by introducing a connection. Accordingly, <sup>ξ</sup><sup>r</sup> ri is the variation of the EM potential itself, that is the δAi of engineers used in order to exhibit the Maxwell equations from a variational principle ([W],* § *26) but the introduction of the Spencer operator is new in this framework.*

Finally, chasing in diagram (1) , the Spencer sequence is locally exact at *C*<sup>1</sup> if and only if the Janet sequence is locally exact at *F*<sup>0</sup> because the central sequence is locally exact. *The situation is much more complicate in the nonlinear framewok*. Let *ω*¯ be a section of F satisfying the same CC as *ω*, namely *I*(*j*1(*ω*)) = 0. It follows that we can find a section *fq*+<sup>1</sup> ∈ Π*q*+<sup>1</sup> such that *f* <sup>−</sup><sup>1</sup> *<sup>q</sup>* (*ω*) = *<sup>ω</sup>*¯ <sup>⇒</sup> *<sup>j</sup>*1(*<sup>f</sup>* <sup>−</sup><sup>1</sup> *<sup>q</sup>* )(*j*1(*ω*)) = *j*1(*f* <sup>−</sup><sup>1</sup> *<sup>q</sup>* (*ω*)) = *<sup>j</sup>*1(*ω*¯) and *<sup>f</sup>* <sup>−</sup><sup>1</sup> *<sup>q</sup>*+1(*j*1(*ω*)) = *j*1(*ω*¯). We obtain therefore *j*1(*f* <sup>−</sup><sup>1</sup> *<sup>q</sup>* )(*j*1(*ω*)) = *<sup>f</sup>* <sup>−</sup><sup>1</sup> *<sup>q</sup>*+1(*j*1(*ω*)) <sup>⇒</sup> (*fq*+<sup>1</sup> ◦ *<sup>j</sup>*1(*<sup>f</sup>* <sup>−</sup><sup>1</sup> *<sup>q</sup>* ))−1(*j*1(*ω*)) <sup>−</sup> *<sup>j</sup>*1(*ω*) = *<sup>L</sup>*(*σq*)*<sup>ω</sup>* <sup>=</sup> <sup>0</sup> and thus *σ<sup>q</sup>* = *D f* ¯ <sup>−</sup><sup>1</sup> *<sup>q</sup>*+<sup>1</sup> ∈ *T*<sup>∗</sup> ⊗ *Rq over the target*, even if *fq*+<sup>1</sup> may not be a section of R*q*+1. As *σ<sup>q</sup>* is killed by *D*¯ � , we have related cocycles at F in the Janet sequence *over the source* with cocycles at *T*<sup>∗</sup> ⊗ *Rq* or *C*<sup>1</sup> *over the target*.

Now, if  $f\_{q+1}, f\_{q+1}' \in \Pi\_{q+1}$  are such that  $f\_{q+1}^{-1}(j\_1(\omega)) = f\_{q+1}'^{-1}(j\_1(\omega)) = j\_1(\bar{\omega})$ , it follows that  $(f\_{q+1}' \circ f\_{q+1}^{-1})(j\_1(\omega)) = j\_1(\omega) \Rightarrow \exists g\_{q+1} \in \mathcal{R}\_{q+1}$  such that  $f\_{q+1}' = g\_{q+1} \circ f\_{q+1}$  and the new  $\sigma\_q' = \overline{D} f\_{q+1}^{\ell-1}$  differs from the initial  $\sigma\_q = \overline{D} f\_{q+1}^{-1}$  by a gauge transformation.

*adjoint of the second Spencer operator D*<sup>2</sup> : *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>R</sup>*<sup>1</sup> → ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>R</sup>*1*:*

*follows: d dt* <sup>=</sup> *dx dt ∂ ∂x*

<sup>+</sup> *dy dt ∂ ∂y* + *dz dt ∂ ∂z* + *∂ ∂t* ⇒

+ ... + *pyz* − *pzy* +

+ *∂ ∂t* ( *P*

*∂qxx ∂x*

*presented for the first time.*

*first Spencer operator, one has ∂iξ<sup>r</sup>*

*because of the inclusion R*<sup>1</sup> � *<sup>R</sup>*<sup>2</sup> <sup>⊂</sup> *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup>*.*

*EM field with EM potential ξ<sup>r</sup>*

*look for the factors of ξ*1, *ξ*<sup>2</sup> *and ξ*1,2 *in the integration by parts of the sum:*

*<sup>σ</sup>*1,1(*∂*1*ξ*<sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*1,1) + *<sup>σ</sup>*1,2(*∂*2*ξ*<sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*1,2) + *<sup>σ</sup>*2,1(*∂*1*ξ*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*2,1) + *<sup>σ</sup>*2,2(*∂*2*ξ*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*2,2) + *<sup>μ</sup>*12,*<sup>r</sup>*

*coherent way with the Airy parametrization obtained when <sup>φ</sup>*<sup>1</sup> <sup>=</sup> *<sup>∂</sup>*2*φ*, *<sup>φ</sup>*<sup>2</sup> <sup>=</sup> *<sup>∂</sup>*1*φ*, *<sup>φ</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>*φ.*

*∂pxx ∂x*

> *C* Δ *dy*

*ijr* <sup>=</sup> *<sup>∂</sup>iξ<sup>r</sup>*

*parameters) obtained by adding the only dilatation with infinitesimal generators x<sup>i</sup>*

1 Δ *dP dt* <sup>+</sup> *C* Δ *dy dt* <sup>−</sup> *<sup>B</sup>* Δ *dz dt* <sup>=</sup> *<sup>∂</sup> ∂x*

<sup>Δ</sup>)+(*pyz* <sup>+</sup>

*rj* <sup>−</sup> *<sup>ξ</sup><sup>r</sup>*

*<sup>σ</sup>i*,*<sup>j</sup>* <sup>=</sup> *<sup>∂</sup>rφi*,*jr* , *<sup>μ</sup>ij*,*<sup>r</sup>* <sup>=</sup> *<sup>∂</sup>sψij*,*rs* <sup>+</sup> *<sup>φ</sup>j*,*ir* <sup>−</sup> *<sup>φ</sup>i*,*jr* **Example 7.1.** *When n* = 2*, lowering the indices by means of the constant metric ω, we just need to*

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 25

*Finally, setting φ*1,12 = *φ*1, *φ*2,12 = *φ*2, *ψ*12,12 = *φ*3*, we obtain the nontrivial parametrization σ*1,1 = *<sup>∂</sup>*2*φ*1, *<sup>σ</sup>*1,2 <sup>=</sup> <sup>−</sup>*∂*1*φ*1, *<sup>σ</sup>*2,1 <sup>=</sup> <sup>−</sup>*∂*2*φ*2, *<sup>σ</sup>*2,2 <sup>=</sup> *<sup>∂</sup>*1*φ*2, *<sup>μ</sup>*12,1 <sup>=</sup> *<sup>∂</sup>*2*φ*<sup>3</sup> <sup>+</sup> *<sup>φ</sup>*1, *<sup>μ</sup>*12,2 <sup>=</sup> <sup>−</sup>*∂*1*φ*<sup>3</sup> <sup>−</sup> *<sup>φ</sup>*<sup>2</sup> *in a*

**Remark 7.1.** *First of all, it is clear that [C] (p 13,14 for n* = 1*, p 75,76 for n* = 2*) still deals with m* = 3 *for the "ambient space", that is with the construction of the nonlinear gauge sequence, in particular for the dynamical study of a line with arc length s and time t considered as a surface, hence with no way to pass from the source to the target, only possible, as we have seen, when m* = *n* = 3 *by using the nonlinear Spencer sequence. For n* = 4*, the group of rigid motions of space is extended to space-time by using only a translation on time and we can rewrite the formulas in ([C], p 167) as*

+ ... +

*dt* ) <sup>−</sup> (*pzy* <sup>+</sup>

*It is essential to notice that the Cosserat equations for n* = 3 *are still introduced today in a phenomenological way ([35] is a good example), contrary to the "deductive" way used in ([C], p 1-6) and that "intuition" will never allow to provide the relativistic Cosserat equations for n* = 4 *which are*

**Theorem 7.2.** *The Weyl equations ([W], §35) are exactly described by the formal adjoint of the first Spencer operator D*<sup>1</sup> : *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup> <sup>→</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup> *when n* <sup>=</sup> <sup>4</sup> *and can be parametrized by the formal adjoint of the second Spencer operator D*<sup>2</sup> : *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup> <sup>→</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup>*. In particular, among the components of the*

*projects onto d* : *<sup>T</sup>*<sup>∗</sup> → ∧2*T*<sup>∗</sup> *and thus D*<sup>2</sup> *projects onto the first set of Maxwell equations described by d* : <sup>∧</sup>2*T*<sup>∗</sup> → ∧3*T*∗*. Indeed, the Spencer sequence projects onto the Poincaré sequence with a shift by* +1 *in the degree of the exterior forms involved because both sequences are made with first order involutive operators and the comment after diagram (1) can thus be used. By duality, the second set of Maxwell equations thus appears among the Weyl equations which project onto the Cosserat equations*

**Remark 7.2.** *When n* = 4*, the Poincaré group (*10 *parameters) is a subgroup of the conformal group (*15 *parameters) which is not a maximal subgroup because it is a subgroup of the Weyl group (*11

*optical group is another subgroup with* 10 *parameters which is maximal and the same procedure may be applied to all these subgroups in order to study coupling phenomena. It is also important to notice that*

1 Δ *dA dt* <sup>=</sup> *<sup>∂</sup> ∂x*

> *B* Δ *dz dt* )

*rj and thus the components <sup>∂</sup>iξ<sup>r</sup>*

*ri* = *Ai coming from the second order jets (elations). It follows that D*<sup>1</sup>

(*pxx* +

(*qxx* +

*A* Δ *dx dt* ) + ... <sup>+</sup>

*P* Δ *dx dt* ) + ...

*rj* <sup>−</sup> *<sup>∂</sup>jξ<sup>r</sup>*

(*∂rξ*1,2 − *ξ*1,2*r*)

*∂ ∂t* ( *A* Δ )

*ri* = *Fij of the*

*∂i. However, the*

Conversely, let *fq*+1, *f* � *<sup>q</sup>*+<sup>1</sup> <sup>∈</sup> <sup>Π</sup>*q*+<sup>1</sup> be such that *<sup>σ</sup><sup>q</sup>* <sup>=</sup> *D f* ¯ <sup>−</sup><sup>1</sup> *<sup>q</sup>*+<sup>1</sup> <sup>=</sup> *D f* ¯ �−<sup>1</sup> *<sup>q</sup>*+<sup>1</sup> = *σ*� *<sup>q</sup>*. It follows that *D*¯ (*f* <sup>−</sup><sup>1</sup> *<sup>q</sup>*+<sup>1</sup> ◦ *f* � *<sup>q</sup>*+1) = 0 and one can find *g* ∈ *aut*(*X*) such that *f* � *<sup>q</sup>*+<sup>1</sup> = *fq*+<sup>1</sup> ◦ *jq*+1(*g*) providing *ω*¯ � = *f* �−<sup>1</sup> *<sup>q</sup>* (*ω*)=(*fq* ◦ *jq*(*g*))−1(*ω*) = *jq*(*g*)−1(*<sup>f</sup>* <sup>−</sup><sup>1</sup> *<sup>q</sup>* (*ω*)) = *jq*(*g*)−1(*ω*¯).

**Proposition 6.3.** *Natural transformations of* F *over the source in the nonlinear Janet sequence correspond to gauge transformations of T*<sup>∗</sup> ⊗ *Rq or C*<sup>1</sup> *over the target in the nonlinear Spencer sequence. Similarly, the Lie derivative* D*ξ* = L(*ξ*)*ω* ∈ *F*<sup>0</sup> *in the linear Janet sequence corresponds to the Spencer operator Dξq*+<sup>1</sup> ∈ *T*<sup>∗</sup> ⊗ *Rq or D*1*ξ<sup>q</sup>* ∈ *C*<sup>1</sup> *in the linear Spencer sequence.*

With a slight abuse of language *<sup>δ</sup> <sup>f</sup>* <sup>=</sup> *<sup>η</sup>* ◦ *<sup>f</sup>* <sup>⇔</sup> *<sup>δ</sup> <sup>f</sup>* ◦ *<sup>f</sup>* <sup>−</sup><sup>1</sup> <sup>=</sup> *<sup>η</sup>* <sup>⇔</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup> ◦ *<sup>δ</sup> <sup>f</sup>* <sup>=</sup> *<sup>ξ</sup>* when *<sup>η</sup>* <sup>=</sup> *<sup>T</sup>*(*f*)(*ξ*) and we get *jq*(*f*)−1(*ω*) = *<sup>ω</sup>*¯ <sup>⇒</sup> *jq*(*<sup>f</sup>* <sup>+</sup> *<sup>δ</sup> <sup>f</sup>*)−1(*ω*) = *<sup>ω</sup>*¯ <sup>+</sup> *δω*¯ that is *jq*(*<sup>f</sup>* <sup>−</sup><sup>1</sup> ◦ (*<sup>f</sup>* <sup>+</sup> *<sup>δ</sup> <sup>f</sup>*))−1(*ω*¯) = *<sup>ω</sup>*¯ <sup>+</sup> *δω*¯ <sup>⇒</sup> *δω*¯ <sup>=</sup> <sup>L</sup>(*ξ*)*ω*¯ and *jq*((*<sup>f</sup>* <sup>+</sup> *<sup>δ</sup> <sup>f</sup>*) ◦ *<sup>f</sup>* <sup>−</sup><sup>1</sup> ◦ *<sup>f</sup>*)−1(*ω*) = *jq*(*f*)−1(*jq*((*<sup>f</sup>* <sup>+</sup> *<sup>δ</sup> <sup>f</sup>*) ◦ *<sup>f</sup>* <sup>−</sup>1)−1(*ω*)) <sup>⇒</sup> *δω*¯ <sup>=</sup> *jq*(*f*)−1(L(*η*)*ω*).

Passing to the infinitesimal point of view, we obtain the following generalization of Remark 3.3 which is important for applications ([2], AJSE-mathematics):

**Corollary 6.4.** *δω*¯ <sup>=</sup> <sup>L</sup>(*ξ*)*ω*¯ <sup>=</sup> *jq*(*f*)−1(L(*η*)*ω*)*.*

**Example 6.2.** *In Example 3.1 with n* = 1, *q* = 1*, we have ω*(*f*(*x*))*fx*(*x*) = *ω*¯(*x*), *ω*(*f*(*x*))*fxx*(*x*) + *∂yω*(*f*(*x*))*f* <sup>2</sup> *<sup>x</sup>* (*x*) = *∂xω*¯(*x*) *and obtain therefore ωσy*,*<sup>y</sup>* + *σ*,*y∂yω* ≡ −*ω*(1/ *fx*)(*∂<sup>x</sup> fx* − *fxx*)(1/*∂<sup>x</sup> f*) + ((*fx*/*∂<sup>x</sup> f*) − 1)*∂yω* = 0 *whenever y* = *f*(*x*)*. The case of an affine stucture needs more work.*

#### **7. Cosserat versus Weyl: New perspectives for physics**

As an application of the previous mehods, let us now consider the *conformal Killing system*:

$$\hat{\mathcal{R}}\_1 \subset J\_1(T) \,\,\omega\_{r\dot{j}}\xi^r\_{\dot{i}} + \omega\_{ir}\xi^r\_{\dot{j}} + \xi^r \partial\_r \omega\_{i\dot{j}} = A(\boldsymbol{\chi})\omega\_{i\dot{j}}$$

with symbols:

$$\hat{\mathfrak{g}}\_2 \subset \mathbb{S}\_2 T^\* \otimes T \qquad \qquad \qquad n\_{\vec{\imath}\vec{\jmath}}^{\vec{\varkappa}} = \delta\_{\vec{\imath}}^k \mathfrak{f}\_{r\vec{\jmath}}^r + \delta\_{\vec{\jmath}}^k \mathfrak{f}\_{r\vec{\imath}}^r - \omega\_{\vec{\imath}\vec{\jmath}} \omega^{\vec{\imath}\vec{\varkappa}} \mathfrak{f}\_{r\vec{\imath}}^r \qquad \Rightarrow \qquad \hat{\mathfrak{g}}\_3 = 0, \forall n \ge 3$$

obtained by eliminating the arbitrary function *A*(*x*), where *ω* is the Euclidean metric when *n* = 1 (line), *n* = 2 (plane) or *n* = 3 (space) and the Minskowskian metric when *n* = 4 (space-time).

The brothers Cosserat were only dealing with the *Killing subsystem*:

$$\mathcal{R}\_1 \subset \mathcal{R}\_1 \tag{10.15} \qquad \qquad \qquad \omega\_{r\{}^\sharp \!\!\!/} + \omega\_{\text{ir}} \sharp^r\_{\text{j}} + \xi^r \partial\_{\text{r}} \omega\_{\text{ij}} = 0$$

that is with {*ξk*, *<sup>ξ</sup><sup>k</sup> <sup>i</sup>* <sup>|</sup> *<sup>ξ</sup><sup>r</sup> <sup>r</sup>* = 0, *ξ<sup>k</sup> ij* = 0} = {*translations*,*rotations*} when *A*(*x*) = 0, while, *in a somehow complementary way*, Weyl was mainly dealing with {*ξ<sup>r</sup> <sup>r</sup>*, *ξ<sup>r</sup> ri*} = {*dilatation*,*elations*}. Accordingly, one has ([7]):

**Theorem 7.1.** *The Cosserat equations ([C], p 137 for n* = 3*, p 167 for n* = 4*):*

$$
\partial\_r \sigma^{i,r} = f^i \qquad , \qquad \partial\_r \mu^{ij,r} + \sigma^{i,j} - \sigma^{j,i} = m^{ij}
$$

*are exactly described by the formal adjoint of the first Spencer operator D*<sup>1</sup> : *R*<sup>1</sup> → *T*<sup>∗</sup> ⊗ *R*1*. Introducing <sup>φ</sup>r*,*ij* <sup>=</sup> <sup>−</sup>*φr*,*ji and <sup>ψ</sup>rs*,*ij* <sup>=</sup> <sup>−</sup>*ψrs*,*ji* <sup>=</sup> <sup>−</sup>*ψsr*,*ij, they can be parametrized by the formal* *adjoint of the second Spencer operator D*<sup>2</sup> : *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>R</sup>*<sup>1</sup> → ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>R</sup>*1*:*

24 Will-be-set-by-IN-TECH

**Proposition 6.3.** *Natural transformations of* F *over the source in the nonlinear Janet sequence correspond to gauge transformations of T*<sup>∗</sup> ⊗ *Rq or C*<sup>1</sup> *over the target in the nonlinear Spencer sequence. Similarly, the Lie derivative* D*ξ* = L(*ξ*)*ω* ∈ *F*<sup>0</sup> *in the linear Janet sequence corresponds to*

With a slight abuse of language *<sup>δ</sup> <sup>f</sup>* <sup>=</sup> *<sup>η</sup>* ◦ *<sup>f</sup>* <sup>⇔</sup> *<sup>δ</sup> <sup>f</sup>* ◦ *<sup>f</sup>* <sup>−</sup><sup>1</sup> <sup>=</sup> *<sup>η</sup>* <sup>⇔</sup> *<sup>f</sup>* <sup>−</sup><sup>1</sup> ◦ *<sup>δ</sup> <sup>f</sup>* <sup>=</sup> *<sup>ξ</sup>* when *<sup>η</sup>* <sup>=</sup> *<sup>T</sup>*(*f*)(*ξ*) and we get *jq*(*f*)−1(*ω*) = *<sup>ω</sup>*¯ <sup>⇒</sup> *jq*(*<sup>f</sup>* <sup>+</sup> *<sup>δ</sup> <sup>f</sup>*)−1(*ω*) = *<sup>ω</sup>*¯ <sup>+</sup> *δω*¯ that is *jq*(*<sup>f</sup>* <sup>−</sup><sup>1</sup> ◦ (*<sup>f</sup>* <sup>+</sup> *<sup>δ</sup> <sup>f</sup>*))−1(*ω*¯) = *<sup>ω</sup>*¯ <sup>+</sup> *δω*¯ <sup>⇒</sup> *δω*¯ <sup>=</sup> <sup>L</sup>(*ξ*)*ω*¯ and *jq*((*<sup>f</sup>* <sup>+</sup> *<sup>δ</sup> <sup>f</sup>*) ◦ *<sup>f</sup>* <sup>−</sup><sup>1</sup> ◦ *<sup>f</sup>*)−1(*ω*) = *jq*(*f*)−1(*jq*((*<sup>f</sup>* <sup>+</sup>

Passing to the infinitesimal point of view, we obtain the following generalization of Remark

**Example 6.2.** *In Example 3.1 with n* = 1, *q* = 1*, we have ω*(*f*(*x*))*fx*(*x*) = *ω*¯(*x*), *ω*(*f*(*x*))*fxx*(*x*) +

*fxx*)(1/*∂<sup>x</sup> f*) + ((*fx*/*∂<sup>x</sup> f*) − 1)*∂yω* = 0 *whenever y* = *f*(*x*)*. The case of an affine stucture needs*

As an application of the previous mehods, let us now consider the *conformal Killing system*:

*<sup>i</sup>* <sup>+</sup> *<sup>ω</sup>irξ<sup>r</sup>*

obtained by eliminating the arbitrary function *A*(*x*), where *ω* is the Euclidean metric when *n* = 1 (line), *n* = 2 (plane) or *n* = 3 (space) and the Minskowskian metric when *n* = 4

*<sup>∂</sup>rσi*,*<sup>r</sup>* <sup>=</sup> *<sup>f</sup> <sup>i</sup>* , *<sup>∂</sup>rμij*,*<sup>r</sup>* <sup>+</sup> *<sup>σ</sup>i*,*<sup>j</sup>* <sup>−</sup> *<sup>σ</sup>j*,*<sup>i</sup>* <sup>=</sup> *<sup>m</sup>ij*

*are exactly described by the formal adjoint of the first Spencer operator D*<sup>1</sup> : *R*<sup>1</sup> → *T*<sup>∗</sup> ⊗ *R*1*. Introducing <sup>φ</sup>r*,*ij* <sup>=</sup> <sup>−</sup>*φr*,*ji and <sup>ψ</sup>rs*,*ij* <sup>=</sup> <sup>−</sup>*ψrs*,*ji* <sup>=</sup> <sup>−</sup>*ψsr*,*ij, they can be parametrized by the formal*

*ij* <sup>=</sup> *<sup>δ</sup><sup>k</sup> i ξr rj* <sup>+</sup> *<sup>δ</sup><sup>k</sup> j ξr*

*<sup>x</sup>* (*x*) = *∂xω*¯(*x*) *and obtain therefore ωσy*,*<sup>y</sup>* + *σ*,*y∂yω* ≡ −*ω*(1/ *fx*)(*∂<sup>x</sup> fx* −

*<sup>j</sup>* <sup>+</sup> *<sup>ξ</sup>r∂rωij* <sup>=</sup> *<sup>A</sup>*(*x*)*ωij*

*ri* <sup>−</sup> *<sup>ω</sup>ijωksξ<sup>r</sup>*

*<sup>i</sup>* <sup>+</sup> *<sup>ω</sup>irξ<sup>r</sup>*

*<sup>j</sup>* <sup>+</sup> *<sup>ξ</sup><sup>r</sup>*

*ij* = 0} = {*translations*,*rotations*} when *A*(*x*) = 0, while, *in a*

*<sup>r</sup>*, *ξ<sup>r</sup>*

*∂rωij* = 0

*<sup>q</sup>*+<sup>1</sup> <sup>=</sup> *D f* ¯ �−<sup>1</sup>

*<sup>q</sup>* (*ω*)) = *jq*(*g*)−1(*ω*¯).

*<sup>q</sup>*+<sup>1</sup> = *σ*�

*<sup>q</sup>*+<sup>1</sup> = *fq*+<sup>1</sup> ◦ *jq*+1(*g*) providing

*rs* ⇒ *g*ˆ3 = 0, ∀*n* ≥ 3

*ri*} = {*dilatation*,*elations*}.

*<sup>q</sup>*. It follows that

*<sup>q</sup>*+<sup>1</sup> <sup>∈</sup> <sup>Π</sup>*q*+<sup>1</sup> be such that *<sup>σ</sup><sup>q</sup>* <sup>=</sup> *D f* ¯ <sup>−</sup><sup>1</sup>

*the Spencer operator Dξq*+<sup>1</sup> ∈ *T*<sup>∗</sup> ⊗ *Rq or D*1*ξ<sup>q</sup>* ∈ *C*<sup>1</sup> *in the linear Spencer sequence.*

*<sup>q</sup>*+1) = 0 and one can find *g* ∈ *aut*(*X*) such that *f* �

*<sup>q</sup>* (*ω*)=(*fq* ◦ *jq*(*g*))−1(*ω*) = *jq*(*g*)−1(*<sup>f</sup>* <sup>−</sup><sup>1</sup>

3.3 which is important for applications ([2], AJSE-mathematics):

**7. Cosserat versus Weyl: New perspectives for physics**

*<sup>R</sup>*<sup>ˆ</sup> <sup>1</sup> <sup>⊂</sup> *<sup>J</sup>*1(*T*) *<sup>ω</sup>rjξ<sup>r</sup>*

The brothers Cosserat were only dealing with the *Killing subsystem*:

*somehow complementary way*, Weyl was mainly dealing with {*ξ<sup>r</sup>*

**Theorem 7.1.** *The Cosserat equations ([C], p 137 for n* = 3*, p 167 for n* = 4*):*

*<sup>δ</sup> <sup>f</sup>*) ◦ *<sup>f</sup>* <sup>−</sup>1)−1(*ω*)) <sup>⇒</sup> *δω*¯ <sup>=</sup> *jq*(*f*)−1(L(*η*)*ω*).

**Corollary 6.4.** *δω*¯ <sup>=</sup> <sup>L</sup>(*ξ*)*ω*¯ <sup>=</sup> *jq*(*f*)−1(L(*η*)*ω*)*.*

*<sup>g</sup>*ˆ2 <sup>⊂</sup> *<sup>S</sup>*2*T*<sup>∗</sup> <sup>⊗</sup> *T nξ<sup>k</sup>*

*<sup>i</sup>* <sup>|</sup> *<sup>ξ</sup><sup>r</sup>*

*<sup>R</sup>*<sup>1</sup> <sup>⊂</sup> *<sup>R</sup>*<sup>ˆ</sup> <sup>1</sup> *<sup>ω</sup>rjξ<sup>r</sup>*

*<sup>r</sup>* = 0, *ξ<sup>k</sup>*

Conversely, let *fq*+1, *f* �

*D*¯ (*f* <sup>−</sup><sup>1</sup> *<sup>q</sup>*+<sup>1</sup> ◦ *f* �

*ω*¯ � = *f* �−<sup>1</sup>

*∂yω*(*f*(*x*))*f* <sup>2</sup>

with symbols:

(space-time).

that is with {*ξk*, *<sup>ξ</sup><sup>k</sup>*

Accordingly, one has ([7]):

*more work.*

$$
\sigma^{i,j} = \partial\_r \phi^{i,jr} \qquad \qquad \qquad \mu^{ij,r} = \partial\_s \psi^{ij,rs} + \phi^{j,ir} - \phi^{i,jr}
$$

**Example 7.1.** *When n* = 2*, lowering the indices by means of the constant metric ω, we just need to look for the factors of ξ*1, *ξ*<sup>2</sup> *and ξ*1,2 *in the integration by parts of the sum:*

*<sup>σ</sup>*1,1(*∂*1*ξ*<sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*1,1) + *<sup>σ</sup>*1,2(*∂*2*ξ*<sup>1</sup> <sup>−</sup> *<sup>ξ</sup>*1,2) + *<sup>σ</sup>*2,1(*∂*1*ξ*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*2,1) + *<sup>σ</sup>*2,2(*∂*2*ξ*<sup>2</sup> <sup>−</sup> *<sup>ξ</sup>*2,2) + *<sup>μ</sup>*12,*<sup>r</sup>* (*∂rξ*1,2 − *ξ*1,2*r*) *Finally, setting φ*1,12 = *φ*1, *φ*2,12 = *φ*2, *ψ*12,12 = *φ*3*, we obtain the nontrivial parametrization σ*1,1 = *<sup>∂</sup>*2*φ*1, *<sup>σ</sup>*1,2 <sup>=</sup> <sup>−</sup>*∂*1*φ*1, *<sup>σ</sup>*2,1 <sup>=</sup> <sup>−</sup>*∂*2*φ*2, *<sup>σ</sup>*2,2 <sup>=</sup> *<sup>∂</sup>*1*φ*2, *<sup>μ</sup>*12,1 <sup>=</sup> *<sup>∂</sup>*2*φ*<sup>3</sup> <sup>+</sup> *<sup>φ</sup>*1, *<sup>μ</sup>*12,2 <sup>=</sup> <sup>−</sup>*∂*1*φ*<sup>3</sup> <sup>−</sup> *<sup>φ</sup>*<sup>2</sup> *in a coherent way with the Airy parametrization obtained when <sup>φ</sup>*<sup>1</sup> <sup>=</sup> *<sup>∂</sup>*2*φ*, *<sup>φ</sup>*<sup>2</sup> <sup>=</sup> *<sup>∂</sup>*1*φ*, *<sup>φ</sup>*<sup>3</sup> <sup>=</sup> <sup>−</sup>*φ.*

**Remark 7.1.** *First of all, it is clear that [C] (p 13,14 for n* = 1*, p 75,76 for n* = 2*) still deals with m* = 3 *for the "ambient space", that is with the construction of the nonlinear gauge sequence, in particular for the dynamical study of a line with arc length s and time t considered as a surface, hence with no way to pass from the source to the target, only possible, as we have seen, when m* = *n* = 3 *by using the nonlinear Spencer sequence. For n* = 4*, the group of rigid motions of space is extended to space-time by using only a translation on time and we can rewrite the formulas in ([C], p 167) as follows:*

$$\frac{d}{dt} = \frac{d\mathbf{x}}{dt}\frac{\partial}{\partial\mathbf{x}} + \frac{d\mathbf{y}}{dt}\frac{\partial}{\partial\mathbf{y}} + \frac{d\mathbf{z}}{dt}\frac{\partial}{\partial\mathbf{z}} + \frac{\partial}{\partial t} \Rightarrow \frac{\partial p\_{\mathbf{xx}}}{\partial\mathbf{x}} + \dots + \frac{1}{\Delta}\frac{dA}{dt} = \frac{\partial}{\partial\mathbf{x}}(p\_{\mathbf{xx}} + \frac{A}{\Delta}\frac{d\mathbf{x}}{dt}) + \dots + \frac{\partial}{\partial t}(\frac{A}{\Delta})$$

$$\begin{split} \frac{\partial q\_{\text{xx}}}{\partial \mathbf{x}} + \dots &+ p\_{\text{yz}} - p\_{\text{zy}} + \frac{1}{\Delta} \frac{dP}{dt} + \frac{\mathbb{C}}{\Delta} \frac{dy}{dt} - \frac{\mathbb{B}}{\Delta} \frac{dz}{dt} = \frac{\partial}{\partial \mathbf{x}} (q\_{\text{xx}} + \frac{P}{\Delta} \frac{dx}{dt}) + \dots \\ &+ \frac{\partial}{\partial t} (\frac{P}{\Delta}) + (p\_{\text{yz}} + \frac{\mathbb{C}}{\Delta} \frac{dy}{dt}) - (p\_{\text{zy}} + \frac{\mathbb{B}}{\Delta} \frac{dz}{dt}) \end{split}$$

*It is essential to notice that the Cosserat equations for n* = 3 *are still introduced today in a phenomenological way ([35] is a good example), contrary to the "deductive" way used in ([C], p 1-6) and that "intuition" will never allow to provide the relativistic Cosserat equations for n* = 4 *which are presented for the first time.*

**Theorem 7.2.** *The Weyl equations ([W], §35) are exactly described by the formal adjoint of the first Spencer operator D*<sup>1</sup> : *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup> <sup>→</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup> *when n* <sup>=</sup> <sup>4</sup> *and can be parametrized by the formal adjoint of the second Spencer operator D*<sup>2</sup> : *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup> <sup>→</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup>*. In particular, among the components of the first Spencer operator, one has ∂iξ<sup>r</sup> rj* <sup>−</sup> *<sup>ξ</sup><sup>r</sup> ijr* <sup>=</sup> *<sup>∂</sup>iξ<sup>r</sup> rj and thus the components <sup>∂</sup>iξ<sup>r</sup> rj* <sup>−</sup> *<sup>∂</sup>jξ<sup>r</sup> ri* = *Fij of the EM field with EM potential ξ<sup>r</sup> ri* = *Ai coming from the second order jets (elations). It follows that D*<sup>1</sup> *projects onto d* : *<sup>T</sup>*<sup>∗</sup> → ∧2*T*<sup>∗</sup> *and thus D*<sup>2</sup> *projects onto the first set of Maxwell equations described by d* : <sup>∧</sup>2*T*<sup>∗</sup> → ∧3*T*∗*. Indeed, the Spencer sequence projects onto the Poincaré sequence with a shift by* +1 *in the degree of the exterior forms involved because both sequences are made with first order involutive operators and the comment after diagram (1) can thus be used. By duality, the second set of Maxwell equations thus appears among the Weyl equations which project onto the Cosserat equations because of the inclusion R*<sup>1</sup> � *<sup>R</sup>*<sup>2</sup> <sup>⊂</sup> *<sup>R</sup>*<sup>ˆ</sup> <sup>2</sup>*.*

**Remark 7.2.** *When n* = 4*, the Poincaré group (*10 *parameters) is a subgroup of the conformal group (*15 *parameters) which is not a maximal subgroup because it is a subgroup of the Weyl group (*11 *parameters) obtained by adding the only dilatation with infinitesimal generators x<sup>i</sup> ∂i. However, the optical group is another subgroup with* 10 *parameters which is maximal and the same procedure may be applied to all these subgroups in order to study coupling phenomena. It is also important to notice that*

this action by differentiating *q* times the action law in order to eliminate the parameters in the following commutative and exact diagram where R*<sup>q</sup>* is a Lie groupoid with local coordinates (*x*, *yq*), *source* projection *α<sup>q</sup>* : (*x*, *yq*) → (*x*) and *target* projection *β<sup>q</sup>* : (*x*, *yq*) → (*y*) when *q* is

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 27

0 → *X* × *G* −→ R*<sup>q</sup>* → 0 � *α<sup>q</sup>* � � *β<sup>q</sup> X* × *G* → *X* × *X* The link between the various sections of the trivial principal bundle on the left (*gauging procedure*) and the various corresponding sections of the Lie groupoid on the right with respect

> 0 → *X* × *G* = R*<sup>q</sup>* → 0 *a* = *cst* ↑↓↑ *a*(*x*) *jq*(*f*) ↑↓↑ *fq X* = *X*

**Theorem 7.3.** *In the above situation, the nonlinear Spencer sequence is isomorphic to the nonlinear*

*Proof.* If we consider the action *y* = *f*(*x*, *a*) and start with a section (*x*) → (*x*, *a*(*x*)) of *X* × *G*,

*<sup>∂</sup><sup>b</sup>* <sup>=</sup> *<sup>θ</sup>*(*x*)*ω*(*b*) from the first fundamental theorem of Lie. With <sup>−</sup>*ω*(*b*)*db* <sup>=</sup> <sup>−</sup>*dbb*−<sup>1</sup> <sup>=</sup> *<sup>a</sup>*−1*da*,

*<sup>∂</sup>a<sup>τ</sup>* )*∂ia<sup>τ</sup>*

*<sup>∂</sup>xr <sup>θ</sup><sup>r</sup>*

*<sup>μ</sup>*+1*i*,*<sup>j</sup>* <sup>+</sup> *<sup>χ</sup><sup>k</sup>*

*<sup>μ</sup>* <sup>=</sup> *<sup>A</sup><sup>ρ</sup> <sup>i</sup> <sup>A</sup><sup>σ</sup>*

Introducing now the Lie algebra G = *Te*(*G*) and the Lie algebroid *Rq* ⊂ *Jq*(*T*), namely the linearization of R*<sup>q</sup>* at the *q*-jet of the identity *y* = *x*, we get the commutative and exact

> 0 → *X* × G = *Rq* → 0 *λ* = *cst* ↑↓↑ *λ*(*x*) *jq*(*ξ*) ↑↓↑ *ξ<sup>q</sup> X* = *X*

*<sup>μ</sup>*+1*<sup>i</sup>* <sup>=</sup> *di*(*∂μ <sup>f</sup> <sup>k</sup>*(*x*, *<sup>a</sup>*(*x*)) <sup>−</sup> *∂μ*+1*<sup>i</sup> <sup>f</sup> <sup>k</sup>*(*x*, *<sup>a</sup>*(*x*))

*τ*(*x*))*ω<sup>τ</sup>*

*<sup>μ</sup>*+1*j*,*<sup>i</sup>* = (*∂iA<sup>τ</sup>*

*<sup>j</sup> ∂μ*([*θρ*, *θσ*])*<sup>k</sup>* <sup>=</sup> *<sup>c</sup><sup>τ</sup>*

*<sup>σ</sup>*(*b*) *<sup>∂</sup>b<sup>σ</sup> <sup>∂</sup>a<sup>τ</sup> <sup>∂</sup>ia<sup>τ</sup>*

*<sup>τ</sup>*(*x*) from the inductive formula allowing to define *<sup>χ</sup><sup>q</sup>* = *D f* ¯ *<sup>q</sup>*+1.

*<sup>j</sup>* <sup>−</sup> *<sup>∂</sup>jA<sup>τ</sup>*

*ρσA<sup>ρ</sup> <sup>i</sup> <sup>A</sup><sup>σ</sup> <sup>j</sup> ∂μθ<sup>k</sup> τ*.

*<sup>i</sup>* )*∂μθ<sup>k</sup> τ*

*D*¯ → *T*<sup>∗</sup> ⊗ *Rq*

*<sup>X</sup>* <sup>×</sup> *<sup>G</sup>* <sup>→</sup> *<sup>T</sup>*<sup>∗</sup> ⊗ G *MC*→ ∧2*T*<sup>∗</sup> ⊗ G ↓↓ ↓

*D*¯ �

→ ∧2*T*<sup>∗</sup> <sup>⊗</sup> *Rq*

*<sup>μ</sup>*(*x*) = *∂μ <sup>f</sup> <sup>k</sup>*(*x*, *<sup>a</sup>*(*x*))) of <sup>R</sup>*q*. Setting *<sup>b</sup>* <sup>=</sup> *<sup>a</sup>*−<sup>1</sup> <sup>=</sup> *<sup>b</sup>*(*a*),

*∂x ∂ f ∂b ∂b <sup>∂</sup><sup>a</sup>* <sup>+</sup> *<sup>∂</sup><sup>y</sup>*

*<sup>∂</sup><sup>a</sup>* = 0 with

to the source projection is expressed by the next commutative and exact diagram:

*gauge sequence and we have the following commutative and locally exact diagram:*

*The action is essential in the Spencer sequence but disappears in the gauge sequence.*

we get *<sup>y</sup>* <sup>=</sup> *<sup>f</sup>*(*x*, *<sup>a</sup>*) <sup>⇒</sup> *<sup>x</sup>* <sup>=</sup> *<sup>f</sup>*(*y*, *<sup>b</sup>*) <sup>⇒</sup> *<sup>y</sup>* <sup>=</sup> *<sup>f</sup>*(*f*(*y*, *<sup>b</sup>*(*a*), *<sup>a</sup>*) and thus *<sup>∂</sup><sup>y</sup>*

<sup>=</sup> *∂μ*( *<sup>∂</sup> <sup>f</sup> <sup>k</sup>*

<sup>=</sup> <sup>−</sup>*∂μ*( *<sup>∂</sup> <sup>f</sup> <sup>k</sup>*

0 → Γ → R*<sup>q</sup>*

we obtain the section (*x*) <sup>→</sup> (*x*, *<sup>f</sup> <sup>k</sup>*

(*x*) = *A<sup>τ</sup>*

*∂<sup>i</sup> f <sup>k</sup> <sup>μ</sup>* <sup>−</sup> *<sup>f</sup> <sup>k</sup>*

*<sup>i</sup>* (*x*)*∂μθ<sup>k</sup>*

*∂iχ<sup>k</sup>*

As for the commutatitvity of the right square, we have:

*<sup>μ</sup>*,*<sup>j</sup>* <sup>−</sup> *<sup>∂</sup>jχ<sup>k</sup>*

({*χq*+1(*∂i*), *<sup>χ</sup>q*+1(*∂j*)})*<sup>k</sup>*

*<sup>μ</sup>*,*<sup>i</sup>* <sup>−</sup> *<sup>χ</sup><sup>k</sup>*

*∂ f*

we obtain:

and thus *χ<sup>k</sup>*

diagram:

*μ*,*i*

large enough:

*the first and second sets of Maxwell equations are invariant by any diffeomorphism and the conformal group is only the group of invariance of the Minkowski constitutive laws in vacuum ([20])([27], p 492).*

**Remark 7.3.** *Though striking it may look like, there is no conceptual difference between the Cosserat and Maxwell equations on space-time. As a byproduct, separating space from time, there is no conceptual difference between the Lamé constants (mass per unit volume) of elasticity and the magnetic (dielectric) constants of EM appearing in the respective wave speeds. For example, the speed of longitudinal free vibrations of a thin elastic bar with Young modulus E and mass per unit volume ρ is v* = *E <sup>ρ</sup> while the speed of light in a medium with magnetic constant μ and dielectric constant � is v* = 1/*<sup>μ</sup> � . In the first case, we have the* 1*-dimensional dynamical equations:*

$$\delta \int (\frac{1}{2}E(\frac{\partial \tilde{\xi}}{\partial \mathbf{x}})^2 - \frac{1}{2}\rho(\frac{\partial \tilde{\xi}}{\partial t})^2)d\mathbf{x}dt = 0 \Rightarrow E\frac{\partial^2 \tilde{\xi}}{\partial \mathbf{x}^2} - \rho \frac{\partial^2 \tilde{\xi}}{\partial t^2} = 0$$

*In the second case, studying the propagation in vacuum for simplicity, one uses to set H* = (1/*μ*0) *B*, *D* = *�*<sup>0</sup> *E with �*0*μ*0*c*<sup>2</sup> = 1 *in the induction equations and to substitute the space-time parametrization dA* = *F of the field equations dF* = 0 *in the variational condition δ* ( 1 <sup>2</sup> *�*<sup>0</sup> *<sup>E</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup> <sup>2</sup> (1/*μ*0) *B*2)*dxdt* = 0*. However, the second order PD equations thus obtained become wave equations only if one assumes the Lorentz condition div*(*A*) = *ωij∂iAj* = 0 *([20]). This is not correct because the Lagrangian of the corresponding variational problem with constraint must contain the additional term λdiv*(*A*) *where λ is a Lagrange multiplier providing the equations* ✷*A* = *dλ as a* 1*-form and thus* ✷*F* = 0 *as a* 2*-form when* ✷ *is the Dalembertian ([27], p 885).*

**Remark 7.4.** *When studying static phenomena, �* = (*�ij*) *and E* = (*E<sup>i</sup>* ) *are now on equal footing in the Lagrangian, exactly like in the technique of finite elements. Starting with a homogeneous medium at rest with no stress and electric induction, we may consider a quadratic Lagrangian Aijkl�ij�kl* + *BijEiEj* + *Cijk�ijEk obtained by moving the indices by means of the Euclidean metric. The two first terms describe (pure) linear elasticity and electrostatic while only the last quadratic coupling term may be used in order to describe coupling phenomena. For an isotropic medium, the* 3*-tensor C must vanish and such a coupling phenomenon, called piezzoelectricity, can only appear in non-isotropic media like crystals, providing the additional stress σij* = *CijkEk and/or an additional electric induction D<sup>k</sup>* = *Cijk�ij. Accordingly, if the medium is fixed, for example between the plates of a condenser, an electric field may provide stress inside while, if the medium is deformed as in the piezzo-lighters, an electric induction may appear and produce a spark. Finally, for an isotropic medium, we can only add a cubic coupling term Cijkl�ijEkEl responsible for photoelasticity as it provides the additional electric induction D<sup>l</sup>* = (*Cijkl�ij*)*Ek, modifying therefore the dielectric constant by a term depending linearly on the deformation and thus modifying the index of refraction n because �μ*0*c*<sup>2</sup> = *n*<sup>2</sup> *with �*0*μ*0*c*<sup>2</sup> = 1 *in vacuum leads to �* <sup>=</sup> *<sup>n</sup>*2*�*0*. We may also identify the dimensionless "speed" vk*/*<sup>c</sup>* � 1, <sup>∀</sup>*<sup>k</sup>* <sup>=</sup> 1, 2, 3 *(time derivative of position) with a first jet (Lorentz rotation) by setting <sup>∂</sup>*4*ξ<sup>k</sup>* <sup>−</sup> *<sup>ξ</sup><sup>k</sup>* <sup>4</sup> = 0 *and introduce the speed of deformation by the formula* 2*νij* = *ωrj*(*∂iξ<sup>r</sup>* <sup>4</sup> <sup>−</sup> *<sup>ξ</sup><sup>r</sup> <sup>i</sup>*4) + *<sup>ω</sup>ir*(*∂jξ<sup>r</sup>* <sup>4</sup> <sup>−</sup> *<sup>ξ</sup><sup>r</sup> <sup>j</sup>*4) = *<sup>ω</sup>rj∂iξ<sup>r</sup>* <sup>4</sup> + *ωir∂jξ<sup>r</sup>* <sup>4</sup> <sup>=</sup> *<sup>∂</sup>*4(*ωrj∂iξ<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>ir∂jξr*) = *<sup>ω</sup>rj∂iv<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>ir∂jv<sup>r</sup>* <sup>=</sup> *<sup>∂</sup>*4*�ij*, <sup>∀</sup><sup>1</sup> <sup>≤</sup> *<sup>i</sup>*, *<sup>j</sup>* <sup>≤</sup> <sup>3</sup> *in order to obtain streaming birefringence in a similar way. These results perfectly agree with most of the field-matter couplings known in engineering sciences ([28]) but contradict gauge theory ([15],[26]) and general relativity ([W],[21]).*

In order to justify the last remark, let *G* be a Lie group with identity *e* and parameters *a* acting on *X* through the group action *X* × *G* → *X* : (*x*, *a*) → *y* = *f*(*x*, *a*) and (local) infinitesimal generators *θτ* satisfying [*θρ*, *θσ*] = *c<sup>τ</sup> ρσθτ* for *ρ*, *σ*, *τ* = 1, ..., *dim*(*G*). We may prolong the *graph* of 26 Will-be-set-by-IN-TECH

*the first and second sets of Maxwell equations are invariant by any diffeomorphism and the conformal group is only the group of invariance of the Minkowski constitutive laws in vacuum ([20])([27], p 492).* **Remark 7.3.** *Though striking it may look like, there is no conceptual difference between the Cosserat and Maxwell equations on space-time. As a byproduct, separating space from time, there is no conceptual difference between the Lamé constants (mass per unit volume) of elasticity and the magnetic (dielectric) constants of EM appearing in the respective wave speeds. For example, the speed of longitudinal free vibrations of a thin elastic bar with Young modulus E and mass per unit volume ρ is*

*<sup>ρ</sup> while the speed of light in a medium with magnetic constant μ and dielectric constant � is*

)2)*dxdt* <sup>=</sup> <sup>0</sup> <sup>⇒</sup> *<sup>E</sup> <sup>∂</sup>*2*<sup>ξ</sup>*

*B*2)*dxdt* = 0*. However, the second order PD equations thus obtained become wave equations*

*E with �*0*μ*0*c*<sup>2</sup> = 1 *in the induction equations and to substitute the space-time*

<sup>4</sup> <sup>−</sup> *<sup>ξ</sup><sup>r</sup>*

<sup>4</sup> <sup>=</sup> *<sup>∂</sup>*4(*ωrj∂iξ<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>ir∂jξr*) = *<sup>ω</sup>rj∂iv<sup>r</sup>* <sup>+</sup> *<sup>ω</sup>ir∂jv<sup>r</sup>* <sup>=</sup> *<sup>∂</sup>*4*�ij*, <sup>∀</sup><sup>1</sup> <sup>≤</sup> *<sup>i</sup>*, *<sup>j</sup>* <sup>≤</sup> <sup>3</sup> *in order to obtain streaming birefringence in a similar way. These results perfectly agree with most of the field-matter couplings known in engineering sciences ([28]) but contradict gauge theory ([15],[26]) and general*

In order to justify the last remark, let *G* be a Lie group with identity *e* and parameters *a* acting on *X* through the group action *X* × *G* → *X* : (*x*, *a*) → *y* = *f*(*x*, *a*) and (local) infinitesimal

*<sup>i</sup>*4) + *<sup>ω</sup>ir*(*∂jξ<sup>r</sup>*

*ρσθτ* for *ρ*, *σ*, *τ* = 1, ..., *dim*(*G*). We may prolong the *graph* of

<sup>4</sup> <sup>−</sup> *<sup>ξ</sup><sup>r</sup>*

*In the second case, studying the propagation in vacuum for simplicity, one uses to set H* =

*<sup>E</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup>

*only if one assumes the Lorentz condition div*(*A*) = *ωij∂iAj* = 0 *([20]). This is not correct because the Lagrangian of the corresponding variational problem with constraint must contain the additional term λdiv*(*A*) *where λ is a Lagrange multiplier providing the equations* ✷*A* = *dλ as a* 1*-form and*

*the Lagrangian, exactly like in the technique of finite elements. Starting with a homogeneous medium at rest with no stress and electric induction, we may consider a quadratic Lagrangian Aijkl�ij�kl* + *BijEiEj* + *Cijk�ijEk obtained by moving the indices by means of the Euclidean metric. The two first terms describe (pure) linear elasticity and electrostatic while only the last quadratic coupling term may be used in order to describe coupling phenomena. For an isotropic medium, the* 3*-tensor C must vanish and such a coupling phenomenon, called piezzoelectricity, can only appear in non-isotropic media like crystals, providing the additional stress σij* = *CijkEk and/or an additional electric induction D<sup>k</sup>* = *Cijk�ij. Accordingly, if the medium is fixed, for example between the plates of a condenser, an electric field may provide stress inside while, if the medium is deformed as in the piezzo-lighters, an electric induction may appear and produce a spark. Finally, for an isotropic medium, we can only add a cubic coupling term Cijkl�ijEkEl responsible for photoelasticity as it provides the additional electric induction D<sup>l</sup>* = (*Cijkl�ij*)*Ek, modifying therefore the dielectric constant by a term depending linearly on the deformation and thus modifying the index of refraction n because �μ*0*c*<sup>2</sup> = *n*<sup>2</sup> *with �*0*μ*0*c*<sup>2</sup> = 1 *in vacuum leads to �* <sup>=</sup> *<sup>n</sup>*2*�*0*. We may also identify the dimensionless "speed" vk*/*<sup>c</sup>* � 1, <sup>∀</sup>*<sup>k</sup>* <sup>=</sup> 1, 2, 3

*<sup>∂</sup>x*<sup>2</sup> <sup>−</sup> *<sup>ρ</sup>*

*E* = (*E<sup>i</sup>*

*∂*2*ξ <sup>∂</sup>t*<sup>2</sup> <sup>=</sup> <sup>0</sup>

> ( 1 <sup>2</sup> *�*<sup>0</sup>

) *are now on equal footing in*

<sup>4</sup> = 0 *and introduce*

*<sup>j</sup>*4) = *<sup>ω</sup>rj∂iξ<sup>r</sup>*

<sup>4</sup> +

*� . In the first case, we have the* 1*-dimensional dynamical equations:*

*parametrization dA* = *F of the field equations dF* = 0 *in the variational condition δ*

*(time derivative of position) with a first jet (Lorentz rotation) by setting <sup>∂</sup>*4*ξ<sup>k</sup>* <sup>−</sup> *<sup>ξ</sup><sup>k</sup>*

*the speed of deformation by the formula* 2*νij* = *ωrj*(*∂iξ<sup>r</sup>*

)<sup>2</sup> <sup>−</sup> <sup>1</sup> 2 *ρ*( *∂ξ ∂t*

*thus* ✷*F* = 0 *as a* 2*-form when* ✷ *is the Dalembertian ([27], p 885).* **Remark 7.4.** *When studying static phenomena, �* = (*�ij*) *and*

*v* =

*v* =

(1/*μ*0)

<sup>2</sup> (1/*μ*0)

*ωir∂jξ<sup>r</sup>*

*relativity ([W],[21]).*

generators *θτ* satisfying [*θρ*, *θσ*] = *c<sup>τ</sup>*

*E*

1/*<sup>μ</sup>*

*δ* ( 1 2 *E*( *∂ξ ∂x*

*B*, *D* = *�*<sup>0</sup>

this action by differentiating *q* times the action law in order to eliminate the parameters in the following commutative and exact diagram where R*<sup>q</sup>* is a Lie groupoid with local coordinates (*x*, *yq*), *source* projection *α<sup>q</sup>* : (*x*, *yq*) → (*x*) and *target* projection *β<sup>q</sup>* : (*x*, *yq*) → (*y*) when *q* is large enough:

$$\begin{array}{ccccc} 0 \to X \times G \longrightarrow & \mathcal{R}\_{\emptyset} & \to 0 \\ & \parallel & \mathfrak{a}\_{\emptyset} \not\subset \\ X \times G \to & X & \times & X \end{array} \to \begin{array}{c} \mathcal{R}\_{\emptyset} & \to 0 \\ \searrow \mathcal{B}\_{\emptyset} & \\ X & \times & X \end{array}$$

The link between the various sections of the trivial principal bundle on the left (*gauging procedure*) and the various corresponding sections of the Lie groupoid on the right with respect to the source projection is expressed by the next commutative and exact diagram:

$$\begin{array}{rcl} 0 \to & X \\ & a = cst \uparrow\downarrow\uparrow a(x) \\ & X \end{array} \begin{array}{rcl} \mathcal{G} = & \mathcal{R}\_q \\ & j\_q(f) \uparrow\downarrow\uparrow f\_q \\ & X \end{array} \xrightarrow{\to} 0$$

**Theorem 7.3.** *In the above situation, the nonlinear Spencer sequence is isomorphic to the nonlinear gauge sequence and we have the following commutative and locally exact diagram:*

$$\begin{array}{ccccc} & \begin{array}{c} X \times G \to & T^\* \otimes \mathcal{G} & \xrightarrow{M\!\!\!\! \!\! } & \wedge^2 T^\* \otimes \mathcal{G} \\ \downarrow & & \downarrow & & \downarrow \\ 0 \to \Gamma \to & \mathcal{R}\_q & \xrightarrow{D} T^\* \otimes \mathcal{R}\_q & \xrightarrow{D'} \wedge^2 T^\* \otimes \mathcal{R}\_q \end{array} \end{array}$$

*The action is essential in the Spencer sequence but disappears in the gauge sequence.*

*Proof.* If we consider the action *y* = *f*(*x*, *a*) and start with a section (*x*) → (*x*, *a*(*x*)) of *X* × *G*, we obtain the section (*x*) <sup>→</sup> (*x*, *<sup>f</sup> <sup>k</sup> <sup>μ</sup>*(*x*) = *∂μ <sup>f</sup> <sup>k</sup>*(*x*, *<sup>a</sup>*(*x*))) of <sup>R</sup>*q*. Setting *<sup>b</sup>* <sup>=</sup> *<sup>a</sup>*−<sup>1</sup> <sup>=</sup> *<sup>b</sup>*(*a*), we get *<sup>y</sup>* <sup>=</sup> *<sup>f</sup>*(*x*, *<sup>a</sup>*) <sup>⇒</sup> *<sup>x</sup>* <sup>=</sup> *<sup>f</sup>*(*y*, *<sup>b</sup>*) <sup>⇒</sup> *<sup>y</sup>* <sup>=</sup> *<sup>f</sup>*(*f*(*y*, *<sup>b</sup>*(*a*), *<sup>a</sup>*) and thus *<sup>∂</sup><sup>y</sup> ∂x ∂ f ∂b ∂b <sup>∂</sup><sup>a</sup>* <sup>+</sup> *<sup>∂</sup><sup>y</sup> <sup>∂</sup><sup>a</sup>* = 0 with *∂ f <sup>∂</sup><sup>b</sup>* <sup>=</sup> *<sup>θ</sup>*(*x*)*ω*(*b*) from the first fundamental theorem of Lie. With <sup>−</sup>*ω*(*b*)*db* <sup>=</sup> <sup>−</sup>*dbb*−<sup>1</sup> <sup>=</sup> *<sup>a</sup>*−1*da*, we obtain:

$$\begin{aligned} \partial\_i f^k\_{\mu} - f^k\_{\mu+1\_i} &= d\_i(\partial\_{\mu} f^k(\mathfrak{x}, a(\mathfrak{x})) - \partial\_{\mu+1\_i} f^k(\mathfrak{x}, a(\mathfrak{x})) \\ &= \partial\_{\mu} (\frac{\partial f^k}{\partial a^\top}) \partial\_i a^\tau \\ &= -\partial\_{\mu} (\frac{\partial f^k}{\partial \mathfrak{x}^\tau} \theta^r\_\tau(\mathfrak{x})) \omega^\tau\_\sigma(b) \frac{\partial b^\sigma}{\partial a^\tau} \partial\_i a^\tau \end{aligned}$$

and thus *χ<sup>k</sup> μ*,*i* (*x*) = *A<sup>τ</sup> <sup>i</sup>* (*x*)*∂μθ<sup>k</sup> <sup>τ</sup>*(*x*) from the inductive formula allowing to define *<sup>χ</sup><sup>q</sup>* = *D f* ¯ *<sup>q</sup>*+1. As for the commutatitvity of the right square, we have:

$$\begin{split} \partial\_{i} \chi^{k}\_{\mu,j} - \partial\_{j} \chi^{k}\_{\mu,i} - \chi^{k}\_{\mu+1,j} + \chi^{k}\_{\mu+1,i} &= (\partial\_{i} A^{\tau}\_{j} - \partial\_{j} A^{\tau}\_{i}) \partial\_{\mu} \theta^{k}\_{\tau} \\ (\{\chi\_{q+1}(\partial\_{i}), \chi\_{q+1}(\partial\_{j})\})^{k}\_{\mu} &= A^{\rho}\_{i} A^{\sigma}\_{j} \partial\_{\mu} ([\theta\_{\rho}, \theta\_{\sigma}])^{k} = c^{\tau}\_{\rho\sigma} A^{\rho}\_{i} A^{\sigma}\_{j} \partial\_{\mu} \theta^{k}\_{\tau}. \end{split}$$

Introducing now the Lie algebra G = *Te*(*G*) and the Lie algebroid *Rq* ⊂ *Jq*(*T*), namely the linearization of R*<sup>q</sup>* at the *q*-jet of the identity *y* = *x*, we get the commutative and exact diagram:

$$\begin{array}{rcl} 0 \to & X \\ & \lambda = cst \,\uparrow\downarrow\uparrow\lambda \,(x) \\ & X \end{array} = \begin{array}{rcl} \mathcal{R}\_q & \to \, 0 \\ \mathcal{j}\_{\emptyset}(\xi) \,\uparrow\downarrow\uparrow\xi\_{\emptyset} \\ & X \end{array}$$

As *q* = 1 and *g*<sup>2</sup> = 0 ⇒ *g*<sup>3</sup> = 0 we have *s* = 1 and no CC of order 1. The generating CC of order 2 only depend on *<sup>F</sup>*<sup>1</sup> <sup>=</sup> *<sup>ω</sup>*−1(F1) according to section 2 where *<sup>F</sup>*<sup>1</sup> is now defined by the following commutative diagram with exact columns but the first on the left and exact rows:

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 29

It follows from a chase([26], p 55)([27], p 192)([32], p 171) that there is a short exact *connecting sequence* <sup>0</sup> <sup>→</sup> *<sup>B</sup>*2(*g*1) <sup>→</sup> *<sup>Z</sup>*2(*g*1) <sup>→</sup> *<sup>F</sup>*<sup>1</sup> <sup>→</sup> 0 leading to an isomorphism *<sup>F</sup>*<sup>1</sup> � *<sup>H</sup>*2(*g*1). The Riemann tensor is thus a section of *Riemann* = *F*<sup>1</sup> = *H*2(*g*1) = *Z*2(*g*1) in the Killing case with *dim*(*Riemann*)=(*n*2(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)2/4) <sup>−</sup> (*n*2(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>n</sup>* <sup>+</sup> <sup>2</sup>)/6)=(*n*2(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)2/4) <sup>−</sup> (*n*2(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)(*<sup>n</sup>* <sup>−</sup> <sup>2</sup>)/6) = *<sup>n</sup>*2(*n*<sup>2</sup> <sup>−</sup> <sup>1</sup>)/12 by using either the upper row or the left column and *we find back the*

However, for the conformal Killing system, we still have *q* = 1 but the situation is much more delicate because *<sup>g</sup>*<sup>3</sup> <sup>=</sup> 0 for *<sup>n</sup>* <sup>≥</sup> 3 and *<sup>H</sup>*2(*g*ˆ2) = 0 only for *<sup>n</sup>* <sup>≥</sup> 4 ([26], p 435). Hence, setting

The inclusion *g*<sup>1</sup> ⊂ *g*ˆ1 and the relations *g*<sup>2</sup> = 0, *g*ˆ3 = 0 finally induce the following *crucial*

↓ ↓↓

A diagonal chase allows to identify *Ricci* with *S*2*T*∗ *without contracting indices* and provides the splitting of *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>*<sup>∗</sup> into *<sup>S</sup>*2*T*<sup>∗</sup> (*gravitation*) and <sup>∧</sup>2*T*<sup>∗</sup> (*electromagnetism*) in the lower horizontal sequence *obtained by using the Spencer sequence*, solving thus an old conjecture. However, *T*<sup>∗</sup> ⊗ *T*<sup>∗</sup> � *T*<sup>∗</sup> ⊗ *g*ˆ2 has only to do with second order jets (*elations*) and not a word is left from the standard approach to GR. In addition, we obtain the following important theorem explaining

0 ↓

0 *Ricci* ↓ ↓

↓ ↓↓ *JANET*

<sup>→</sup> *<sup>Z</sup>*2(*g*ˆ1) <sup>→</sup> *Weyl* <sup>→</sup> <sup>0</sup>

<sup>0</sup> <sup>→</sup> *<sup>Z</sup>*2(*g*1) <sup>→</sup> *Riemann* <sup>→</sup> <sup>0</sup>

→ ∧2*T*<sup>∗</sup> <sup>→</sup> <sup>0</sup>

<sup>1</sup> <sup>=</sup> *<sup>H</sup>*2(*g*ˆ1) �<sup>=</sup> *<sup>Z</sup>*2(*g*ˆ1).

<sup>0</sup> <sup>=</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>*/*g*ˆ1, the Weyl tensor is a section of *Weyl* <sup>=</sup> *<sup>F</sup>*<sup>ˆ</sup>

*δ*

↓ ↓ 0 0 *SPENCER*

00 0 ↓↓ ↓ 0 → *g*<sup>3</sup> → *S*3*T*<sup>∗</sup> ⊗ *T* → *S*2*T*<sup>∗</sup> ⊗ *F*<sup>0</sup> → *F*<sup>1</sup> → 0 ↓ *δ* ↓ *δ* ↓ *δ* 0 → *T*<sup>∗</sup> ⊗ *g*<sup>2</sup> → *T*<sup>∗</sup> ⊗ *S*2*T*<sup>∗</sup> ⊗ *T* → *T*<sup>∗</sup> ⊗ *T*<sup>∗</sup> ⊗ *F*<sup>0</sup> → 0 ↓ *δ* ↓ *δ* ↓ *δ* <sup>0</sup> → ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>g</sup>*<sup>1</sup> → ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>* → ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>F</sup>*<sup>0</sup> <sup>→</sup> <sup>0</sup> ↓ *δ* ↓ *δ* ↓

<sup>0</sup> → ∧3*T*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>* <sup>=</sup> <sup>∧</sup>3*T*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>* <sup>→</sup> <sup>0</sup>

↓ ↓ 0 0

*two algebraic properties of the Riemann tensor without using indices*.

commutative and exact *diagram* (2) ([25], p 430):

0 → *T*<sup>∗</sup> ⊗ *g*ˆ2

for the first time classical results in an intrinsic way:

<sup>→</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>*<sup>∗</sup> *<sup>δ</sup>*

<sup>0</sup> <sup>→</sup> *<sup>S</sup>*2*T*<sup>∗</sup> *<sup>δ</sup>*

similarly *F*ˆ

where the upper isomorphism is described by *<sup>λ</sup>τ*(*x*) <sup>→</sup> *<sup>ξ</sup><sup>k</sup> <sup>μ</sup>*(*x*) = *λτ*(*x*)*∂μθ<sup>k</sup> <sup>τ</sup>*(*x*) for *q* large enough. The unusual Lie algebroid structure on *X* × G is described by the formula: ([*λ*, *λ*� ])*<sup>τ</sup>* = *c<sup>τ</sup> ρσλρλ*�*<sup>σ</sup>* + (*λρθρ*).*λ*�*<sup>τ</sup>* <sup>−</sup> (*λ*�*σθσ*).*λ<sup>τ</sup>* which is induced by the ordinary bracket [*ξ*, *ξ*� ] on *T* and thus depends on the action. Applying the Spencer operator, we finally obtain *∂iξ<sup>k</sup> <sup>μ</sup>*(*x*) <sup>−</sup> *<sup>ξ</sup><sup>k</sup> μ*+1*<sup>i</sup>* (*x*) = *∂iλτ*(*x*)*∂μθ<sup>k</sup> <sup>τ</sup>*(*x*) and the linear Spencer sequence is isomorphic to the linear gauge sequence already introduced which is no longer depending on the action as it is only the tensor product of the Poincaré sequence by G.

**Example 7.2.** *Let us consider the group of affine transformations of the real line y* = *a*1*x* + *a*<sup>2</sup> *with n* = 1, *dim*(*G*) = 2, *q* = 2*,* R<sup>2</sup> *defined by the system yxx* = 0*, R*<sup>2</sup> *defined by ξxx* = 0 *and the two infinitesimal generators θ*<sup>1</sup> = *x <sup>∂</sup> <sup>∂</sup><sup>x</sup>* , *<sup>θ</sup>*<sup>2</sup> <sup>=</sup> *<sup>∂</sup> <sup>∂</sup><sup>x</sup> . We get f*(*x*) = *<sup>a</sup>*1(*x*)*<sup>x</sup>* <sup>+</sup> *<sup>a</sup>*2(*x*), *fx*(*x*) = *<sup>a</sup>*1(*x*), *fxx*(*x*) = <sup>0</sup> *and thus <sup>χ</sup>*,*x*(*x*)=(1/ *fx*(*x*))*∂<sup>x</sup> <sup>f</sup>*(*x*) <sup>−</sup> <sup>1</sup> = (1/*a*1(*x*))(*x∂<sup>x</sup> <sup>a</sup>*1(*x*) + *∂<sup>x</sup> a*2(*x*)) = *xA*<sup>1</sup> *<sup>x</sup>*(*x*) + *A*<sup>2</sup> *<sup>x</sup>*(*x*), *χx*,*x*(*x*)=(1/ *fx*(*x*))(*∂<sup>x</sup> fx*(*x*) − (1/ *fx*(*x*))*∂<sup>x</sup> f*(*x*)*fxx*(*x*)) = (1/*a*1(*x*))*∂<sup>x</sup> a*1(*x*) = *A*<sup>1</sup> *<sup>x</sup>*(*x*), *χxx*,*x*(*x*) = 0*. Similarly, we get ξ*(*x*) = *λ*1(*x*)*x* + *λ*2(*x*), *ξx*(*x*) = *<sup>λ</sup>*1(*x*), *<sup>ξ</sup>xx*(*x*) = <sup>0</sup>*. Finally, integrating by part the sum <sup>σ</sup>*(*∂x<sup>ξ</sup>* <sup>−</sup> *<sup>ξ</sup>x*) + *<sup>μ</sup>*(*∂xξ<sup>x</sup>* <sup>−</sup> *<sup>ξ</sup>xx*) *we obtain the dual of the Spencer operator as ∂xσ* = *f* , *∂xμ* + *σ* = *m that is to say the Cosserat equations for the affine group of the real line.*

It finally remains to study GR within this framework, as it is only "*added*" by Weyl in an independent way and, for simplicity, we shall restrict to the linearized aspect. First of all, it becomes clear from diagram (1) that the mathematical foundation of GR is based on a confusion between the operator D<sup>1</sup> (*classical curvature alone*) in the Janet sequence when D is the Killing operator brought to involution and the operator *D*<sup>2</sup> (*gauge curvature=curvature+torsion*) in the corresponding Spencer sequence. It must also be noticed that, according to the same diagram, the bigger is the underlying group, the bigger are the Spencer bundles while, on the contrary, the smaller are the Janet bundles depending on the invariants of the group action (deformation tensor in classical elasticity is a good example). Precisely, as already noticed in Theorem 7.2, if *<sup>G</sup>* <sup>⊂</sup> *<sup>G</sup>*ˆ, the Spencer sequence for *<sup>G</sup>* is *contained into* the Spencer sequence for *G*ˆ while the Janet sequence for *G projects onto* the Janet sequence for *G*ˆ, *the best picture for understanding such a phenomenon is that of two children sitting on the ends of a beam and playing at see-saw*.

Such a confusion is also combined with another one well described in ([40], p 631) by the chinese saying "*To put Chang's cap on Li's head*", namely to relate the Ricci tensor (usually obtained from the Riemann tensor by contraction of indices) to the energy-momentum tensor (space-time stress), without taking into account the previous confusion relating the gauge curvature to *rotations* only while the (classical and Cosserat) stress has only to do with *translations*. In addition, it must be noticed that *the Cosserat and Maxwell equations can be parametrized while the Einstein equations cannot be parametrized* ([29]).

In order to escape from this dilemna, let us denote by *B*2(*gq*), *Z*2(*gq*) and *H*2(*gq*) = *Z*2(*gq*)/*B*2(*gq*) the coboundary (image of the left *δ*), cocycle (kernel of the right *δ*) and cohomology bundles of the *δ*-sequence *T*<sup>∗</sup> ⊗ *gq*+<sup>1</sup> *δ* → ∧2*T*<sup>∗</sup> <sup>⊗</sup> *gq δ* → ∧3*T*<sup>∗</sup> <sup>⊗</sup> *Sq*−1*T*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>*. It can be proved that the order of the generating CC of a formally integrable operator of order *<sup>q</sup>* is equal to *<sup>s</sup>* <sup>+</sup> 1 when *<sup>s</sup>* is the smallest integer such that *<sup>H</sup>*2(*gq*+*r*) = 0, <sup>∀</sup>*<sup>r</sup>* <sup>≥</sup> *<sup>s</sup>* ([26]). As an example with *n* = 3, we let the reader prove that the second order systems *y*<sup>33</sup> = 0, *y*<sup>23</sup> − *y*<sup>11</sup> = 0, *y*<sup>22</sup> = 0 and *y*<sup>33</sup> − *y*<sup>11</sup> = 0, *y*<sup>23</sup> = 0, *y*<sup>22</sup> − *y*<sup>11</sup> = 0 have both three second order generating CC ([30]). For the Killing system *R*<sup>1</sup> ⊂ *J*1(*T*) with symbol *g*1, we have *F*<sup>0</sup> = *J*1(*T*)/*R*<sup>1</sup> = *T*<sup>∗</sup> ⊗ *T*/*g*<sup>1</sup> and the short exact sequence 0 → *g*<sup>1</sup> → *T*<sup>∗</sup> ⊗ *T* → *F*<sup>0</sup> → 0. 28 Will-be-set-by-IN-TECH

large enough. The unusual Lie algebroid structure on *X* × G is described by the formula:

linear gauge sequence already introduced which is no longer depending on the action as it is

**Example 7.2.** *Let us consider the group of affine transformations of the real line y* = *a*1*x* + *a*<sup>2</sup> *with n* = 1, *dim*(*G*) = 2, *q* = 2*,* R<sup>2</sup> *defined by the system yxx* = 0*, R*<sup>2</sup> *defined by ξxx* = 0 *and*

*<sup>a</sup>*1(*x*), *fxx*(*x*) = <sup>0</sup> *and thus <sup>χ</sup>*,*x*(*x*)=(1/ *fx*(*x*))*∂<sup>x</sup> <sup>f</sup>*(*x*) <sup>−</sup> <sup>1</sup> = (1/*a*1(*x*))(*x∂<sup>x</sup> <sup>a</sup>*1(*x*) +

*<sup>λ</sup>*1(*x*), *<sup>ξ</sup>xx*(*x*) = <sup>0</sup>*. Finally, integrating by part the sum <sup>σ</sup>*(*∂x<sup>ξ</sup>* <sup>−</sup> *<sup>ξ</sup>x*) + *<sup>μ</sup>*(*∂xξ<sup>x</sup>* <sup>−</sup> *<sup>ξ</sup>xx*) *we obtain the dual of the Spencer operator as ∂xσ* = *f* , *∂xμ* + *σ* = *m that is to say the Cosserat equations for the*

It finally remains to study GR within this framework, as it is only "*added*" by Weyl in an independent way and, for simplicity, we shall restrict to the linearized aspect. First of all, it becomes clear from diagram (1) that the mathematical foundation of GR is based on a confusion between the operator D<sup>1</sup> (*classical curvature alone*) in the Janet sequence when D is the Killing operator brought to involution and the operator *D*<sup>2</sup> (*gauge curvature=curvature+torsion*) in the corresponding Spencer sequence. It must also be noticed that, according to the same diagram, the bigger is the underlying group, the bigger are the Spencer bundles while, on the contrary, the smaller are the Janet bundles depending on the invariants of the group action (deformation tensor in classical elasticity is a good example). Precisely, as already noticed in Theorem 7.2, if *<sup>G</sup>* <sup>⊂</sup> *<sup>G</sup>*ˆ, the Spencer sequence for *<sup>G</sup>* is *contained into* the Spencer sequence for *G*ˆ while the Janet sequence for *G projects onto* the Janet sequence for *G*ˆ, *the best picture for understanding such a phenomenon is that of two children sitting on the ends*

Such a confusion is also combined with another one well described in ([40], p 631) by the chinese saying "*To put Chang's cap on Li's head*", namely to relate the Ricci tensor (usually obtained from the Riemann tensor by contraction of indices) to the energy-momentum tensor (space-time stress), without taking into account the previous confusion relating the gauge curvature to *rotations* only while the (classical and Cosserat) stress has only to do with *translations*. In addition, it must be noticed that *the Cosserat and Maxwell equations can be*

In order to escape from this dilemna, let us denote by *B*2(*gq*), *Z*2(*gq*) and *H*2(*gq*) = *Z*2(*gq*)/*B*2(*gq*) the coboundary (image of the left *δ*), cocycle (kernel of the right *δ*) and

It can be proved that the order of the generating CC of a formally integrable operator of order *<sup>q</sup>* is equal to *<sup>s</sup>* <sup>+</sup> 1 when *<sup>s</sup>* is the smallest integer such that *<sup>H</sup>*2(*gq*+*r*) = 0, <sup>∀</sup>*<sup>r</sup>* <sup>≥</sup> *<sup>s</sup>* ([26]). As an example with *n* = 3, we let the reader prove that the second order systems *y*<sup>33</sup> = 0, *y*<sup>23</sup> − *y*<sup>11</sup> = 0, *y*<sup>22</sup> = 0 and *y*<sup>33</sup> − *y*<sup>11</sup> = 0, *y*<sup>23</sup> = 0, *y*<sup>22</sup> − *y*<sup>11</sup> = 0 have both three second order generating CC ([30]). For the Killing system *R*<sup>1</sup> ⊂ *J*1(*T*) with symbol *g*1, we have *F*<sup>0</sup> = *J*1(*T*)/*R*<sup>1</sup> = *T*<sup>∗</sup> ⊗ *T*/*g*<sup>1</sup> and the short exact sequence 0 → *g*<sup>1</sup> → *T*<sup>∗</sup> ⊗ *T* → *F*<sup>0</sup> → 0.

*δ*

→ ∧2*T*<sup>∗</sup> <sup>⊗</sup> *gq*

*δ*

→ ∧3*T*<sup>∗</sup> <sup>⊗</sup> *Sq*−1*T*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>*.

*parametrized while the Einstein equations cannot be parametrized* ([29]).

cohomology bundles of the *δ*-sequence *T*<sup>∗</sup> ⊗ *gq*+<sup>1</sup>

*<sup>∂</sup><sup>x</sup>* , *<sup>θ</sup>*<sup>2</sup> <sup>=</sup> *<sup>∂</sup>*

] on *T* and thus depends on the action. Applying the Spencer operator, we finally obtain

*ρσλρλ*�*<sup>σ</sup>* + (*λρθρ*).*λ*�*<sup>τ</sup>* <sup>−</sup> (*λ*�*σθσ*).*λ<sup>τ</sup>* which is induced by the ordinary bracket

*<sup>τ</sup>*(*x*) and the linear Spencer sequence is isomorphic to the

*<sup>x</sup>*(*x*), *χx*,*x*(*x*)=(1/ *fx*(*x*))(*∂<sup>x</sup> fx*(*x*) − (1/ *fx*(*x*))*∂<sup>x</sup> f*(*x*)*fxx*(*x*)) =

*<sup>x</sup>*(*x*), *χxx*,*x*(*x*) = 0*. Similarly, we get ξ*(*x*) = *λ*1(*x*)*x* + *λ*2(*x*), *ξx*(*x*) =

*<sup>μ</sup>*(*x*) = *λτ*(*x*)*∂μθ<sup>k</sup>*

*<sup>∂</sup><sup>x</sup> . We get f*(*x*) = *<sup>a</sup>*1(*x*)*<sup>x</sup>* <sup>+</sup> *<sup>a</sup>*2(*x*), *fx*(*x*) =

*<sup>τ</sup>*(*x*) for *q*

where the upper isomorphism is described by *<sup>λ</sup>τ*(*x*) <sup>→</sup> *<sup>ξ</sup><sup>k</sup>*

(*x*) = *∂iλτ*(*x*)*∂μθ<sup>k</sup>*

*the two infinitesimal generators θ*<sup>1</sup> = *x <sup>∂</sup>*

*<sup>x</sup>*(*x*) + *A*<sup>2</sup>

only the tensor product of the Poincaré sequence by G.

([*λ*, *λ*�

[*ξ*, *ξ*�

*∂iξ<sup>k</sup>*

])*<sup>τ</sup>* = *c<sup>τ</sup>*

*μ*+1*<sup>i</sup>*

*<sup>μ</sup>*(*x*) <sup>−</sup> *<sup>ξ</sup><sup>k</sup>*

*∂<sup>x</sup> a*2(*x*)) = *xA*<sup>1</sup>

(1/*a*1(*x*))*∂<sup>x</sup> a*1(*x*) = *A*<sup>1</sup>

*affine group of the real line.*

*of a beam and playing at see-saw*.

As *q* = 1 and *g*<sup>2</sup> = 0 ⇒ *g*<sup>3</sup> = 0 we have *s* = 1 and no CC of order 1. The generating CC of order 2 only depend on *<sup>F</sup>*<sup>1</sup> <sup>=</sup> *<sup>ω</sup>*−1(F1) according to section 2 where *<sup>F</sup>*<sup>1</sup> is now defined by the following commutative diagram with exact columns but the first on the left and exact rows:

00 0 ↓↓ ↓ 0 → *g*<sup>3</sup> → *S*3*T*<sup>∗</sup> ⊗ *T* → *S*2*T*<sup>∗</sup> ⊗ *F*<sup>0</sup> → *F*<sup>1</sup> → 0 ↓ *δ* ↓ *δ* ↓ *δ* 0 → *T*<sup>∗</sup> ⊗ *g*<sup>2</sup> → *T*<sup>∗</sup> ⊗ *S*2*T*<sup>∗</sup> ⊗ *T* → *T*<sup>∗</sup> ⊗ *T*<sup>∗</sup> ⊗ *F*<sup>0</sup> → 0 ↓ *δ* ↓ *δ* ↓ *δ* <sup>0</sup> → ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>g</sup>*<sup>1</sup> → ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>* → ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>F</sup>*<sup>0</sup> <sup>→</sup> <sup>0</sup> ↓ *δ* ↓ *δ* ↓ <sup>0</sup> → ∧3*T*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>* <sup>=</sup> <sup>∧</sup>3*T*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>* <sup>→</sup> <sup>0</sup> ↓ ↓ 0 0

It follows from a chase([26], p 55)([27], p 192)([32], p 171) that there is a short exact *connecting sequence* <sup>0</sup> <sup>→</sup> *<sup>B</sup>*2(*g*1) <sup>→</sup> *<sup>Z</sup>*2(*g*1) <sup>→</sup> *<sup>F</sup>*<sup>1</sup> <sup>→</sup> 0 leading to an isomorphism *<sup>F</sup>*<sup>1</sup> � *<sup>H</sup>*2(*g*1). The Riemann tensor is thus a section of *Riemann* = *F*<sup>1</sup> = *H*2(*g*1) = *Z*2(*g*1) in the Killing case with *dim*(*Riemann*)=(*n*2(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)2/4) <sup>−</sup> (*n*2(*<sup>n</sup>* <sup>+</sup> <sup>1</sup>)(*<sup>n</sup>* <sup>+</sup> <sup>2</sup>)/6)=(*n*2(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)2/4) <sup>−</sup> (*n*2(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)(*<sup>n</sup>* <sup>−</sup> <sup>2</sup>)/6) = *<sup>n</sup>*2(*n*<sup>2</sup> <sup>−</sup> <sup>1</sup>)/12 by using either the upper row or the left column and *we find back the two algebraic properties of the Riemann tensor without using indices*.

However, for the conformal Killing system, we still have *q* = 1 but the situation is much more delicate because *<sup>g</sup>*<sup>3</sup> <sup>=</sup> 0 for *<sup>n</sup>* <sup>≥</sup> 3 and *<sup>H</sup>*2(*g*ˆ2) = 0 only for *<sup>n</sup>* <sup>≥</sup> 4 ([26], p 435). Hence, setting similarly *F*ˆ <sup>0</sup> <sup>=</sup> *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>*/*g*ˆ1, the Weyl tensor is a section of *Weyl* <sup>=</sup> *<sup>F</sup>*<sup>ˆ</sup> <sup>1</sup> <sup>=</sup> *<sup>H</sup>*2(*g*ˆ1) �<sup>=</sup> *<sup>Z</sup>*2(*g*ˆ1). The inclusion *g*<sup>1</sup> ⊂ *g*ˆ1 and the relations *g*<sup>2</sup> = 0, *g*ˆ3 = 0 finally induce the following *crucial* commutative and exact *diagram* (2) ([25], p 430):

$$\begin{array}{ccccc} & & 0 & & 0 & & \\ & & \downarrow & & & \\ & & 0 & & \text{Ricci} & & \\ 0 & \rightarrow & Z^2(\mathfrak{g}\_1) \rightarrow & \text{Riemann} & \rightarrow 0 & \\ & \downarrow & & \downarrow & & \downarrow & \\ 0 & \rightarrow T^\* \otimes \mathfrak{g}\_2 & \stackrel{\delta}{\rightarrow} & Z^2(\mathfrak{g}\_1) \rightarrow & \text{Weyl} & \rightarrow 0 \\ & \downarrow & & \downarrow & & \downarrow & \\ 0 \rightarrow S\_2 T^\* \stackrel{\delta}{\rightarrow} T^\* \otimes T^\* & \stackrel{\delta}{\rightarrow} & \wedge & \wedge^2 T^\* & \rightarrow & 0 \\ & \downarrow & & \downarrow & & \\ & 0 & & 0 & & \\ & & & \text{SPENCER} & & \\ \end{array}$$

A diagonal chase allows to identify *Ricci* with *S*2*T*∗ *without contracting indices* and provides the splitting of *<sup>T</sup>*<sup>∗</sup> <sup>⊗</sup> *<sup>T</sup>*<sup>∗</sup> into *<sup>S</sup>*2*T*<sup>∗</sup> (*gravitation*) and <sup>∧</sup>2*T*<sup>∗</sup> (*electromagnetism*) in the lower horizontal sequence *obtained by using the Spencer sequence*, solving thus an old conjecture. However, *T*<sup>∗</sup> ⊗ *T*<sup>∗</sup> � *T*<sup>∗</sup> ⊗ *g*ˆ2 has only to do with second order jets (*elations*) and not a word is left from the standard approach to GR. In addition, we obtain the following important theorem explaining for the first time classical results in an intrinsic way:

*known today*, wich explain in a unique way all the above results and the resulting field-matter

Spencer Operator and Applications: From Continuum Mechanics to Mathematical Physics 31

sections of *C*<sup>1</sup> � *T*<sup>∗</sup> ⊗ G parametrized by *D*<sup>1</sup> and thus killed by *D*2.

Accordingly, paraphrasing W. Shakespeare, we may say:

sur la théorie de l'action Euclidienne", 557-629.

[3] I. Assem: Algèbres et Modules, Masson, Paris, 1997.

translation: Hydrodynamique, Dunod, Paris, 1955.

and hope future will fast give an answer !.

In gauge theory, the structure of electromagnetism is coming from the unitary group *U*(1), the unit circle in the complex plane, which is *not* acting on space-time, as the *only* possibility to obtain a pure 2-form from <sup>∧</sup>2*T*<sup>∗</sup> ⊗ G is to have *dim*(*G*) = 1. However, we have explained the structure of electromagnetism from that of the conformal group of space-time, with a *shift by one step* in the interpretation of the Spencer sequence involved because the "*fields*" are now

In general relativity, we have similarly proved that the standard way of introducing the Ricci tensor was based on a *double confusion* between the Janet and Spencer sequences described by *diagrams* (1) *and* (2). In particular we have explained why the intrinsic structure of this tensor *necessarily* depends on the difference existing between the Weyl group and the conformal group which is coming from second order jets, relating for the first time on equal footing electromagnetism and gravitation to the Spencer *δ*-cohomology of various symbols.

" TO ACT OR NOT TO ACT, THAT IS THE QUESTION "

[1] P. Appell: Traité de Mécanique Rationnelle, Gauthier-Villars, Paris, 1909. Particularly t II concerned with analytical mechanics and t III with a Note by E. and F. Cosserat "Note

[2] V. Arnold: Méthodes mathématiques de la mécanique classique, Appendice 2 (Géodésiques des métriques invariantes à gauche sur des groupes de Lie et hydrodynamique des fluides parfaits), MIR, moscow, 1974,1976. (For more details, see also: J.-F. POMMARET: Arnold's hydrodynamics revisited, AJSE-mathematics, 1, 1,

[4] G. Birkhoff: Hydrodynamics, Princeton University Press, Princeton, 1954. French

[5] E. Cartan: Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion, C. R. Académie des Sciences Paris, 174, 1922, 437-439, 593-595, 734-737, 857-860. [6] E. Cartan: Sur les variétés à connexion affine et la théorie de la relativité généralisée,

[7] O. Chwolson: Traité de Physique (In particular III, 2, 537 + III, 3, 994 + V, 209), Hermann,

[9] J. Drach: Thèse de Doctorat: Essai sur une théorie générale de l'intégration et sur la

Ann. Ec. Norm. Sup., 40, 1923, 325-412; 41, 1924, 1-25; 42, 1925, 17-88.

[8] E. Cosserat, F. Cosserat: Théorie des Corps Déformables, Hermann, Paris, 1909.

classification des transcendantes, in Ann. Ec. Norm. Sup., 15, 1898, 243-384. [10] L.P. Eisenhart: Riemannian Geometry, Princeton University Press, Princeton, 1926. [11] H. Goldschmidt: Sur la structure des équations de Lie, J. Differential Geometry, 6, 1972,

couplings.

**9. References**

2009, 157-174).

Paris, 1914.

357-373 and 7, 1972, 67-95.

**Theorem 7.4.** *There exist canonical splittings of the various δ-maps appearing in the above diagram which allow to split the vertical short exact sequence on the right.*

*Proof.* We recall first that a short exact sequence 0 <sup>→</sup> *<sup>M</sup>*� *<sup>f</sup>* <sup>→</sup> *<sup>M</sup> <sup>g</sup>* → *M*" → 0 of modules *splits*, that is *M* � *M*� ⊕ *M*", if and only if there exists a map *u* : *M* → *M*� with *u* ◦ *f* = *idM*� or a map *<sup>v</sup>* : *<sup>M</sup>*" <sup>→</sup> *<sup>M</sup>* with *<sup>g</sup>* ◦ *<sup>v</sup>* <sup>=</sup> *idM*" ([3], p 73)([32], p 33). Hence, starting with (*τ<sup>k</sup> li*,*j* ) ∈ *T*<sup>∗</sup> ⊗ *g*ˆ2, we may introduce (*ρ<sup>k</sup> <sup>l</sup>*,*ij* <sup>=</sup> *<sup>τ</sup><sup>k</sup> li*,*<sup>j</sup>* <sup>−</sup> *<sup>τ</sup><sup>k</sup> lj*,*i* ) <sup>∈</sup> *<sup>B</sup>*2(*g*ˆ1) <sup>⊂</sup> *<sup>Z</sup>*2(*g*ˆ1) ⊂ ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>g</sup>*ˆ1 but now *<sup>ϕ</sup>ij* <sup>=</sup> *<sup>ρ</sup><sup>r</sup> <sup>r</sup>*,*ij* = *τr ri*,*<sup>j</sup>* <sup>−</sup> *<sup>τ</sup><sup>r</sup> rj*,*<sup>i</sup>* <sup>=</sup> *<sup>ρ</sup>ij* <sup>−</sup> *<sup>ρ</sup>ji* �<sup>=</sup> 0 with *<sup>ρ</sup>ij* <sup>=</sup> *<sup>ρ</sup><sup>r</sup> <sup>i</sup>*,*rj* because we have *<sup>ρ</sup><sup>k</sup> <sup>l</sup>*,*ij* <sup>+</sup> *<sup>ρ</sup><sup>k</sup> <sup>i</sup>*,*jl* <sup>+</sup> *<sup>ρ</sup><sup>k</sup> <sup>j</sup>*,*li* = 0. With *τ* = *ωijτ<sup>r</sup> ri*,*<sup>j</sup>* and *<sup>ρ</sup>* <sup>=</sup> *<sup>ω</sup>ijρij*, we obtain (*<sup>n</sup>* <sup>−</sup>2)*τ<sup>r</sup> ri*,*<sup>j</sup>* = (*n* −1)*ρij* + *ρji* − (*n*/2(*n* −1))*ωijρ* and thus *<sup>n</sup><sup>ρ</sup>* <sup>=</sup> <sup>2</sup>(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)*τ*. The lower sequence splits with *<sup>ϕ</sup>ij* <sup>→</sup> *<sup>τ</sup>ij* <sup>=</sup> *<sup>τ</sup><sup>r</sup> ri*,*<sup>j</sup>* = (1/2)*ϕij* → *τij* − *τji* = *ϕij* and *<sup>ρ</sup>ij* <sup>=</sup> *<sup>ρ</sup>ji* <sup>⇔</sup> *<sup>ϕ</sup>ij* <sup>=</sup> 0 in *<sup>Z</sup>*2(*g*1) ⊂ ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>g</sup>*1. It follows from a chase that the kernel of the canonical projection *Riemann* <sup>→</sup> *Weyl* is defined by *<sup>ρ</sup><sup>k</sup> <sup>l</sup>*,*ij* <sup>=</sup> *<sup>τ</sup><sup>k</sup> li*,*<sup>j</sup>* <sup>−</sup> *<sup>τ</sup><sup>k</sup> lj*,*<sup>i</sup>* with (*ρ<sup>k</sup> <sup>l</sup>*,*ij*) <sup>∈</sup> *<sup>Z</sup>*2(*g*1) <sup>⊂</sup> *Z*2(*g*ˆ1) and (*τ<sup>k</sup> li*,*j* ) ∈ *T*<sup>∗</sup> ⊗ *g*ˆ2. Accordingly (*n* − 2)*τij* = *nρij* − (*n*/2(*n* − 1))*ωijρ* provides the isomorphism *Ricci* � *<sup>S</sup>*2*T*<sup>∗</sup> and we get *<sup>n</sup>ρ<sup>k</sup> <sup>l</sup>*,*ij* <sup>=</sup> *<sup>δ</sup><sup>k</sup> <sup>i</sup> <sup>τ</sup>lj* <sup>−</sup> *<sup>δ</sup><sup>k</sup> <sup>j</sup> <sup>τ</sup>li* <sup>+</sup> *<sup>ω</sup>ljωksτsi* <sup>−</sup> *<sup>ω</sup>liωksτsj*, that is:

$$\rho\_{l, \mathbf{j}\rangle}^k = \frac{1}{(n-2)} (\delta\_{\mathbf{i}}^k \rho\_{l\mathbf{j}} - \delta\_{\mathbf{j}}^k \rho\_{l\mathbf{i}} + \omega\_{\mathbf{l}\mathbf{j}} \omega^{\mathbf{k}s} \rho\_{\mathbf{si}} - \omega\_{\mathbf{l}\mathbf{i}} \omega^{\mathbf{k}s} \rho\_{\mathbf{sj}}) - \frac{1}{(n-1)(n-2)} (\delta\_{\mathbf{i}}^k \omega\_{\mathbf{l}\mathbf{j}} - \delta\_{\mathbf{j}}^k \omega\_{\mathbf{l}\mathbf{i}}) \rho\_{\mathbf{i}\mathbf{j}}$$

We check that *ρ<sup>r</sup> <sup>i</sup>*,*rj* = *ρij*, obtaining therefore a splitting of the right vertical sequence in the last diagram that allows to define the Weyl tensor by difference. These purely algebraic results only depend on *ω* independently of any conformal factor.

**Example 7.3.** *The free movement of a body in a constant static gravitational fieldg is described by <sup>d</sup><sup>x</sup> dt* − *v* = 0, *<sup>d</sup><sup>v</sup> dt* <sup>−</sup>*<sup>g</sup>* <sup>=</sup> 0, *<sup>∂</sup><sup>g</sup> <sup>∂</sup>xi* − <sup>0</sup> = <sup>0</sup> *where the "speed" is considered as a first order jet (Lorentz rotation) and the "gravity" as a second order jet (elation). Hence an accelerometer merely helps measuring the part of the Spencer operator dealing with second order jets (equivalence principle). As a byproduct, the difference ∂*<sup>4</sup> *f <sup>k</sup>* <sup>4</sup> <sup>−</sup> *<sup>f</sup> <sup>k</sup>* <sup>44</sup> *under the constraint <sup>∂</sup>*<sup>4</sup> *<sup>f</sup> <sup>k</sup>* <sup>−</sup> *<sup>f</sup> <sup>k</sup>* <sup>4</sup> *identifying the "speed" with a first order jet allows to provide a modern version of the Gauss principle of least constraint where the extremum is now obtained with respect to the second order jets and not with respect to the "acceleration" as usual ([1], p 470). The corresponding infinitesimal variational principle δ* (*ρ*(*∂*4*ξ*<sup>4</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>4</sup> <sup>4</sup>) + *<sup>g</sup><sup>i</sup>* (*∂iξ<sup>r</sup> <sup>r</sup>* <sup>−</sup> *<sup>ξ</sup><sup>r</sup> ri*) + *gij*(*∂iξ<sup>r</sup> rj* <sup>−</sup> <sup>0</sup>))*dx* <sup>=</sup> <sup>0</sup> *provides the Poisson law of gravitation with <sup>ρ</sup>* <sup>=</sup> *cst and <sup>g</sup>* = (*g<sup>i</sup>* ) *when <sup>g</sup>ij* <sup>=</sup> *λωij* <sup>⇒</sup> *gi* <sup>=</sup> <sup>−</sup>*∂iλ. The last term of this gravitational action in vacuum is thus of the form λdiv*(*A*)*, that is exactly the term responsible for the Lorentz constraint in Remark 7.6.*

#### **8. Conclusion**

In continuum mechanics, the classical approach is based on differential invariants and only involves derivatives of finite transformations. Accordingly, the corresponding variational calculus can only describe forces as it only involves translations. It has been the idea of E. and F. Cosserat to change drastically this point of view by considering a new differential geometric tool, now called Spencer sequence, and a corresponding variational calculus involving *both* translations and rotations in order to describe torsors, that is *both* forces and couples.

About at the same time, H. Weyl tried to describe electromagnetism and gravitation by using, *in a similar but complementary way*, the dilatation and elations of the conformal group of space-time. We have shown that the underlying Spencer sequence has additional terms, *not* *known today*, wich explain in a unique way all the above results and the resulting field-matter couplings.

In gauge theory, the structure of electromagnetism is coming from the unitary group *U*(1), the unit circle in the complex plane, which is *not* acting on space-time, as the *only* possibility to obtain a pure 2-form from <sup>∧</sup>2*T*<sup>∗</sup> ⊗ G is to have *dim*(*G*) = 1. However, we have explained the structure of electromagnetism from that of the conformal group of space-time, with a *shift by one step* in the interpretation of the Spencer sequence involved because the "*fields*" are now sections of *C*<sup>1</sup> � *T*<sup>∗</sup> ⊗ G parametrized by *D*<sup>1</sup> and thus killed by *D*2.

In general relativity, we have similarly proved that the standard way of introducing the Ricci tensor was based on a *double confusion* between the Janet and Spencer sequences described by *diagrams* (1) *and* (2). In particular we have explained why the intrinsic structure of this tensor *necessarily* depends on the difference existing between the Weyl group and the conformal group which is coming from second order jets, relating for the first time on equal footing electromagnetism and gravitation to the Spencer *δ*-cohomology of various symbols.

Accordingly, paraphrasing W. Shakespeare, we may say:

#### " TO ACT OR NOT TO ACT, THAT IS THE QUESTION "

and hope future will fast give an answer !.

#### **9. References**

30 Will-be-set-by-IN-TECH

**Theorem 7.4.** *There exist canonical splittings of the various δ-maps appearing in the above diagram*

that is *M* � *M*� ⊕ *M*", if and only if there exists a map *u* : *M* → *M*� with *u* ◦ *f* = *idM*� or a map

and *<sup>ρ</sup>ij* <sup>=</sup> *<sup>ρ</sup>ji* <sup>⇔</sup> *<sup>ϕ</sup>ij* <sup>=</sup> 0 in *<sup>Z</sup>*2(*g*1) ⊂ ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>g</sup>*1. It follows from a chase that the kernel of the

*<sup>l</sup>*,*ij* <sup>=</sup> *<sup>δ</sup><sup>k</sup>*

*<sup>j</sup> <sup>ρ</sup>li* <sup>+</sup> *<sup>ω</sup>ljωksρsi* <sup>−</sup> *<sup>ω</sup>liωksρsj*) <sup>−</sup> <sup>1</sup>

last diagram that allows to define the Weyl tensor by difference. These purely algebraic results

**Example 7.3.** *The free movement of a body in a constant static gravitational fieldg is described by <sup>d</sup><sup>x</sup>*

*and the "gravity" as a second order jet (elation). Hence an accelerometer merely helps measuring the part of the Spencer operator dealing with second order jets (equivalence principle). As a byproduct,*

*allows to provide a modern version of the Gauss principle of least constraint where the extremum is now obtained with respect to the second order jets and not with respect to the "acceleration" as usual ([1],*

*rj* <sup>−</sup> <sup>0</sup>))*dx* <sup>=</sup> <sup>0</sup> *provides the Poisson law of gravitation with <sup>ρ</sup>* <sup>=</sup> *cst and <sup>g</sup>* = (*g<sup>i</sup>*

*<sup>g</sup>ij* <sup>=</sup> *λωij* <sup>⇒</sup> *gi* <sup>=</sup> <sup>−</sup>*∂iλ. The last term of this gravitational action in vacuum is thus of the form*

In continuum mechanics, the classical approach is based on differential invariants and only involves derivatives of finite transformations. Accordingly, the corresponding variational calculus can only describe forces as it only involves translations. It has been the idea of E. and F. Cosserat to change drastically this point of view by considering a new differential geometric tool, now called Spencer sequence, and a corresponding variational calculus involving *both*

About at the same time, H. Weyl tried to describe electromagnetism and gravitation by using, *in a similar but complementary way*, the dilatation and elations of the conformal group of space-time. We have shown that the underlying Spencer sequence has additional terms, *not*

translations and rotations in order to describe torsors, that is *both* forces and couples.

*<sup>v</sup>* : *<sup>M</sup>*" <sup>→</sup> *<sup>M</sup>* with *<sup>g</sup>* ◦ *<sup>v</sup>* <sup>=</sup> *idM*" ([3], p 73)([32], p 33). Hence, starting with (*τ<sup>k</sup>*

<sup>→</sup> *<sup>M</sup> <sup>g</sup>*

) <sup>∈</sup> *<sup>B</sup>*2(*g*ˆ1) <sup>⊂</sup> *<sup>Z</sup>*2(*g*ˆ1) ⊂ ∧2*T*<sup>∗</sup> <sup>⊗</sup> *<sup>g</sup>*ˆ1 but now *<sup>ϕ</sup>ij* <sup>=</sup> *<sup>ρ</sup><sup>r</sup>*

*li*,*<sup>j</sup>* <sup>−</sup> *<sup>τ</sup><sup>k</sup>*

(*n* − 1)(*n* − 2)

*<sup>l</sup>*,*ij* <sup>+</sup> *<sup>ρ</sup><sup>k</sup>*

*ri*,*<sup>j</sup>* = (*n* −1)*ρij* + *ρji* − (*n*/2(*n* −1))*ωijρ* and thus

*<sup>i</sup>*,*rj* because we have *<sup>ρ</sup><sup>k</sup>*

*<sup>l</sup>*,*ij* <sup>=</sup> *<sup>τ</sup><sup>k</sup>*

) ∈ *T*<sup>∗</sup> ⊗ *g*ˆ2. Accordingly (*n* − 2)*τij* = *nρij* − (*n*/2(*n* − 1))*ωijρ* provides the

*<sup>i</sup>*,*rj* = *ρij*, obtaining therefore a splitting of the right vertical sequence in the

*<sup>∂</sup>xi* − <sup>0</sup> = <sup>0</sup> *where the "speed" is considered as a first order jet (Lorentz rotation)*

*<sup>i</sup> <sup>τ</sup>lj* <sup>−</sup> *<sup>δ</sup><sup>k</sup>*

→ *M*" → 0 of modules *splits*,

*li*,*j*

*<sup>i</sup>*,*jl* <sup>+</sup> *<sup>ρ</sup><sup>k</sup>*

*lj*,*<sup>i</sup>* with (*ρ<sup>k</sup>*

*<sup>j</sup> <sup>τ</sup>li* <sup>+</sup> *<sup>ω</sup>ljωksτsi* <sup>−</sup> *<sup>ω</sup>liωksτsj*, that is:

<sup>4</sup> *identifying the "speed" with a first order jet*

<sup>4</sup>) + *<sup>g</sup><sup>i</sup>*

(*∂iξ<sup>r</sup> <sup>r</sup>* <sup>−</sup> *<sup>ξ</sup><sup>r</sup> ri*) +

(*ρ*(*∂*4*ξ*<sup>4</sup> <sup>−</sup> *<sup>ξ</sup>*<sup>4</sup>

(*δk*

*<sup>i</sup> <sup>ω</sup>lj* <sup>−</sup> *<sup>δ</sup><sup>k</sup>*

*ri*,*<sup>j</sup>* = (1/2)*ϕij* → *τij* − *τji* = *ϕij*

) ∈ *T*<sup>∗</sup> ⊗ *g*ˆ2,

*<sup>j</sup>*,*li* = 0. With

*<sup>l</sup>*,*ij*) <sup>∈</sup> *<sup>Z</sup>*2(*g*1) <sup>⊂</sup>

*<sup>j</sup> ωli*)*ρ*

*dt* −

) *when*

*<sup>r</sup>*,*ij* =

*which allow to split the vertical short exact sequence on the right.*

*Proof.* We recall first that a short exact sequence 0 <sup>→</sup> *<sup>M</sup>*� *<sup>f</sup>*

*li*,*<sup>j</sup>* <sup>−</sup> *<sup>τ</sup><sup>k</sup> lj*,*i*

*<sup>n</sup><sup>ρ</sup>* <sup>=</sup> <sup>2</sup>(*<sup>n</sup>* <sup>−</sup> <sup>1</sup>)*τ*. The lower sequence splits with *<sup>ϕ</sup>ij* <sup>→</sup> *<sup>τ</sup>ij* <sup>=</sup> *<sup>τ</sup><sup>r</sup>*

*<sup>l</sup>*,*ij* <sup>=</sup> *<sup>τ</sup><sup>k</sup>*

*rj*,*<sup>i</sup>* <sup>=</sup> *<sup>ρ</sup>ij* <sup>−</sup> *<sup>ρ</sup>ji* �<sup>=</sup> 0 with *<sup>ρ</sup>ij* <sup>=</sup> *<sup>ρ</sup><sup>r</sup>*

*ri*,*<sup>j</sup>* and *<sup>ρ</sup>* <sup>=</sup> *<sup>ω</sup>ijρij*, we obtain (*<sup>n</sup>* <sup>−</sup>2)*τ<sup>r</sup>*

canonical projection *Riemann* <sup>→</sup> *Weyl* is defined by *<sup>ρ</sup><sup>k</sup>*

only depend on *ω* independently of any conformal factor.

*p 470). The corresponding infinitesimal variational principle δ*

<sup>44</sup> *under the constraint <sup>∂</sup>*<sup>4</sup> *<sup>f</sup> <sup>k</sup>* <sup>−</sup> *<sup>f</sup> <sup>k</sup>*

*λdiv*(*A*)*, that is exactly the term responsible for the Lorentz constraint in Remark 7.6.*

we may introduce (*ρ<sup>k</sup>*

*τr ri*,*<sup>j</sup>* <sup>−</sup> *<sup>τ</sup><sup>r</sup>*

*τ* = *ωijτ<sup>r</sup>*

*Z*2(*g*ˆ1) and (*τ<sup>k</sup>*

*<sup>l</sup>*,*ij* <sup>=</sup> <sup>1</sup>

We check that *ρ<sup>r</sup>*

*the difference ∂*<sup>4</sup> *f <sup>k</sup>*

**8. Conclusion**

*ρk*

*v* = 0, *<sup>d</sup><sup>v</sup>*

*gij*(*∂iξ<sup>r</sup>*

*li*,*j*

(*n* − 2)

*dt* <sup>−</sup>*<sup>g</sup>* <sup>=</sup> 0, *<sup>∂</sup><sup>g</sup>*

isomorphism *Ricci* � *<sup>S</sup>*2*T*<sup>∗</sup> and we get *<sup>n</sup>ρ<sup>k</sup>*

(*δk <sup>i</sup> <sup>ρ</sup>lj* <sup>−</sup> *<sup>δ</sup><sup>k</sup>*

<sup>4</sup> <sup>−</sup> *<sup>f</sup> <sup>k</sup>*


**2** 

Jianlin Liu

*China* 

**Transversality Condition** 

 **in Continuum Mechanics** 

*Department of Engineering Mechanics, China University of Petroleum,* 

Nature creates all kinds of miraculous similar phenomena in the real world. For example, the spiral morphologies exist in nebula, sunflower seed array, grapevine, and DNA. There are also a lot of similarities in physical theories and principles, such as the analogy between a harmonic vibration system and an RLC oscillation circuit, between a membrane and a sand-heap in elasticity and plasticity, and between fluid mechanics and electricity or magnetism. The great scientist Maxwell pointed out that the form of the capillary surface is identical with that of the elastic curve, which was later tested by the experiment of Clanet and Quere (2002), and then was analyzed by Liu in detail (2009). Exploring these similarities and analogies can help us understand the underlining secret of nature, and pave the way to

For this study, we mainly focus on the similarity in the adhesion of materials and devices at micro and nano scales, which may be caused by van der Waals force, Casimir force, capillary force or other interaction forces. Among others, the adhesion of a slender structure as micro-beam or carbon nanotube (CNT) is of great value for both theoretical and practical aspects. In these systems, due to considerable surface to volume ratio in low-dimensional micro/nano-systems, surface tension or interfacial energy will dominate over the volume force as their dimensions shrink to micro/nano-meters, which presents a lot of novel behaviors distinct with those of the macroscopic systems (Poncharal, et al., 1999). The typical phenomenon is stiction of the micro-beams, such as the micro/nano-wires and micro/nano-belts which are widely used as building blocks of micro-sensors, resonators, probes, transistors and actuators in M/NMES (micro/nano-electro-mechanical systems). In micro-contact printing technology, adhesion associated with van der Waals force leads to stamp deformation (Hui, et al., 2002), and the micro-machined MEMS structures will spontaneously adhere on the substrate under the influence of solid surface energy or liquid surface tension (Zhao, et al., 2003). This failure due to stiction has become a major limitation to push the better application of these novel engineering devices, and the problem has been highlighted as a hot topic in the past decades. The main reason of stiction is that in the small spacings, the slender structures with high compliance are easily brought into contact with

Another related issue is the self-collapse of a single wall carbon nanotube (SWCNT), in which process its initially circular cross-section will jump to a flat ribbon like shape. The

incorporate several similar phenomena into a unified analysis frame.

the substrate of strong surface energy.

**1. Introduction** 

