Airy, Beltrami, Maxwell, Einstein and Lanczos Potentials Revisited ()
Received 19 February 2016; accepted 25 April 2016; published 28 April 2016
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1. Introduction
The language of differential modules has been recently introduced in control theory as a way to understand in an intrinsic way the structural properties of systems of ordinary differential (OD) or partial differential (PD) equations (controllability, observability, identifiability, ...) [1] - [10] . A similar comment can be done for optimal control that is for variational calculus with differential constraints, and the author thanks Prof. Lars Andersson (Einstein Institute, Potsdam) for having suggested him to study the Lanczos potential within this new framework.
We start providing a few explicit examples in order to convince the reader that the corresponding computations are often becoming so tricky that nobody could achieve them or even imagine any underlying general algorithm, for example in the study of the mathematical foundations of control theory, elasticity theory or general relativity.
EXAMPLE 1.1: OD Control Theory
With one independent variable x, for example the time t in control theory or the curvilinear abcissa s in the study of a beam, and three unknowns
. Setting formally
for
and so on, let us consider the system made by the two first order OD equations depending on a variable coefficient
:
![](//html.scirp.org/file/8-7502655x13.png)
In control theory, if
is a constant parameter, one could bring the system to any first order Kalman form and check that the corresponding control system is controllable if and only if
, that is
and
(exercise), independently of the choice of 1 input and 2 outputs among the 3 control variables [11] . In addition to that, using the second OD equation in the form
and substituting in the first, we get the only second order OD equation:
![](//html.scirp.org/file/8-7502655x19.png)
a result leading to a kind of “vicious circle” because the only way to test controllability is ... to bring this second order equation back to a first order system and there are a lot of possibilities. Again, in any case, the only critical values are
and
. Of course, one could dream about a direct approach providing the same result in an intrinsic way. Introducing the operator
as the (formal) derivative with respect to x, we may rewrite the last equation in the form:
![]()
Replacing the operators
and
by the polynomials
and
, the two poly- nomials have a common root
or
and we find back the desired critical values
but such a result is not intrinsic at all. However, we notice that, for example
.
Introducing
, we get
while
that is, setting
, we get now
. Calling “torsion element” any scalar quantity made from the unknowns and their derivatives but satisfying at least one OD equation, we discover that such quantities do exist ... if and only if
or
(exercise). Of course, the existence of any torsion element breaks at once the controllability of the system but the converse is not evident at all, a result leading nevertheless to the feeling that a control system is controllable if and only if no torsion element can be found and such an idea can be extended “mutatis mutandis” to any system of PD equations [6] . However, this result could be useful if and only if there is a test for checking such a property of the system.
Now, using a variable parameter
, not a word of the preceding approach is left but the concept of a torsion element still exists. We shall prove, at the end of the paper, that the condition
becomes
and that the computations needed are quite far from the previous ones. We ask the reader familiar with classical control theory to make his mind a few minutes (or hours!) to agree with us by trying to recover himself such a differential condition.
EXAMPLE 1.2: OD Optimal Control Theory
OD optimal control is the study of OD variational calculus with OD constraints described by OD control systems. However, while studying optimal control, the author of this paper has been surprised to discover that, in all cases, the OD constraints were defined by means of controllable control systems. It is only at the end of this paper that the importance of such an assumption will be explained. For the moment, we shall provide an example allowing to exhibit all the difficulties involved. For this, let
be a solution of the following single input/single output (SISO) OD control system where a is a constant parameter:
![]()
Proceeding as before, the two polynomials replacing the respective operators are
,
and can only have the common root
. Accordingly, the system is controllable if and only if
for any choice of input and output. Now, let us introduce the so-called “cost function” and let us look at the extremum of the
integral
under the previous OD constraint. It is well known that the proper way to study
such a problem is to introduce a Lagrange multiplier
and to vary the new integral:
![]()
The corresponding Euler-Lagrange (EL) equations are:
![]()
to which we must add the OD constraint when varying
. Summing the two EL equations, we get
and two possibilities:
1)
compatible with the constraint.
2)
.
Substituting, we get:
![]()
This system may not be formally integrable. Indeed, by substraction, we get
and must consider the following two possibilities:
![]()
Summarising the results so far obtained, we discover that the Lagrange multiplier is known if and only if the system is controllable. Also, if
, we may exhibit the parametrization
and the cost function becomes
. Equivalently, when the system is controllable it can be parametrized and the variational problem with constraint becomes a variational problem without any constraint which, some- times, does not provide EL equations. We finally understand that extending such a situation to PD variational calculus with PD constraints needs new techniques.
EXAMPLE 1.3: Elasticity Theory
In classical elasticity, the stress tensor density
existing inside an elastic body is a symmetric 2-tensor density introduced by A. Cauchy in 1822. The corresponding Cauchy stress equations can be written as
where the right member describes the local density of forces applied to the body, for example gravitation. With zero second member, we study the possibility to “parametrize” the system of PD equations
, namely to express its general solution by means of a certain number of arbitrary functions or potentials, called stress functions. Of course, the problem is to know about the number of such functions and the order of the parametrizing operator. In what follows, the space has n local coordinates
. For
one may introduce the Euclidean metric
while, for
, one may consider the Minkowski metric. A few definitions used thereafter will be provided later on.
・
: There is no possible parametrization of
.
・
: The stress equations become
. Their second order parametrization
has been provided by George Biddell Airy (1801-1892) in 1863 [12] . It can be simply recovered in the following manner:
![]()
![]()
We get the second order system:
![]()
which is involutive with one equation of class 2, 2 equations of class 1 and it is easy to check that the 2 corresponding first order CC are just the stress equations. As we have a system with constant coefficients, we may use localization [6] [13] in order to transform the 2 PD equations into the 2 linear equations
and get
![]()
Setting
, we finally get
and obtain the previous parametrization by delocalizing, that is replacing now
by
.
・
: Things become quite more delicate when we try to parametrize the 3 PD equations:
![]()
Of course, localization could be used similarly by dealing with the 3 linear equations:
![]()
having rank 3 for 6 unknowns but, even if we succeed bringing all the fractions to the same denominator as before after easy but painful calculus, there is an additional difficulty which is well hidden. Indeed, coming back to the previous Example when
, say
, we should get ![]()
. Hence, setting
, we only get a
parametrization of the first order OD equation
leading to
. Accordingly, localization does indeed provide a parametrization, ... if we already know there exists a possibility to parametrize the given system or if we are able to check that we have obtained such a parametrization by using involution, a way to supersede the use of Janet or Gröbner bases as was proved for the case
[14] . Also, if we proceed along such a way, we should surely loose any geometric argument that could exist.
A direct computational approach has been provided by Eugenio Beltrami (1835-1900) in 1892 [15] , James Clerk Maxwell (1831-1879) in 1870 [16] and Giacinto Morera (1856-1909) in 1892 [17] by introducing the 6 stress functions
through the parametrization obtained by considering:
![]()
![]()
and the additional 4 relations obtained by using a cyclic permutation of
. The system:
![]()
is involutive with 3 equations of class 3, 3 equations of class 2 and no equation of class 1. The 3 CC are describing the stress equations which admit therefore a parametrization ... justifying the localization approach “a posteriori” but without any geometric framework [18] .
Surprisingly, the Maxwell parametrization is obtained by keeping
while setting
in order to obtain the system:
![]()
However, this system may not be involutive and no CC can be found “a priori” because the coordinate system is surely not d-regular. Indeed, effecting the linear change of coordinates
, we obtain the involutive system:
![]()
and it is easy to check that the 3 CC obtained just amount to the desired 3 stress equations when coming back to the original system of coordinates. Again, if there is a geometrical background, this change of local coordinates is hidding it totally. Moreover, we notice that the stress functions kept in the procedure are just the ones on which
is acting. The reason for such an apparently technical choice is related to very general deep arguments in the theory of differential modules that will only be explained at the end of the paper. The Morera parametrization is obtained similarly by keeping now
while setting
.
・
: As already explained, localization cannot be applied directly as we don't know if a parametrization may exist and in any case no analogy with the previous situations
could be used. Moreover, no known differential geometric background could be used at first sight in order to provide a hint towards the solution. Now, if
is the Minkowski metric and
is the gravitational potential, then
and a perturbation
of
may satisfy in vacuum the 10 second order Einstein equations for the 10 W:
![]()
by introducing the corresponding second order Einstein operator
when
[19] . Though it is well known that the corresponding second order Einstein operator is parametrizing the stress equations, the challenge of parametrizing Einstein equations has been proposed in 1970 by J. Wheeler for 1000 $ and solved negatively in 1995 by the author who only received 1 $. We shall see that, exactly as before and though it is quite striking, the key ingredient will be to use the linearized Riemann tensor considered as a second order operator [6] [20] . As an even more striking fact, we shall discover that the condition
has only to do with Spencer cohomology for the symbol of the conformal Killing equations.
EXAMPLE 1.4: PD Control Theory
The aim of this last example is to prove that the possibility to exhibit two different parametrizations of the stress equations which has been presented in the previous example has surely nothing to do with the proper mathematical background of elasticity theory!
For this, let us consider the (trivially involutive) inhomogeneous PD equations with two independent variables
, two unknown functions
and a second member
:
![]()
Multiplying on the left by a test function
and integrating by parts, the corresponding inhomogeneous adjoint system of PD equations is:
![]()
Using crossed derivatives, we get
and substituting, we get the two CC:
![]()
The corresponding generating CC for the second member
is:
![]()
Therefore
is differentially dependent on
but
is also differentially dependent on
.
Multiplying the first equation by the test function
, the second equation by the test function
, adding and integrating by parts, we get the canonical parametrization
:
![]()
of the initial system with zero second member. The system (up to sign) is involutive and the kernel of this parametrization has differential rank equal to 1.
Keeping
while setting
, we get the first minimal parametrization
:
![]()
The system is again involutive (up to sign) and the parametrization is minimal because the kernel of this parametrization has differential rank equal to 0. With a similar comment, setting now
while keeping
, we get the second minimal parametrization
:
![]()
EXAMPLE 1.5: PD Optimal Control Theory
Let us revisit briefly the foundation of n-dimensional elasticity theory as it can be found today in any textbook, restricting our study to
for simplicity. If
is a point in the plane and
is the displacement vector, lowering the indices by means of the Euclidean metric, we may introduce the “small” deformation tensor
with
(independent) components
. 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
and vary the total free energy
with
by introducing
for
in order
to obtain
. Accordingly, the “decision” to define the stress tensor
by
a symmetric matrix with
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
with a linear dependence
on the outward normal unit vector
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
. However, even if we assume this symmetry, we now need the different summation
.
An integration by parts and a change of sign produce the integral
leading to the stress equations
already considered. This 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. It is however widely used through the technique of “finite elements” where it can also be applied to electromagnetism (EM) with similar quadratic (piezoelectricity) or cubic (photoelasticity) Lagrangian integrals. In this situation, the 4-potential A of EM is used in place of
while the EM field
is used in place of
and the Maxwell equations
are used in place of the Riemann CC for
.
However, there exists another equivalent procedure dealing with a variational calculus with constraint. Indeed, as we shall see later on, the deformation tensor is not any symmetric tensor as it must satisfy
compatibility conditions (CC), that is only
when
. In this case, introducing
the Lagrange multiplier
, we have to vary the new integral
for an
arbitrary
. Setting
, a double integration by parts now provides the parametrization
of the stress equations by means of the Airy function
and the formal adjoint of the Riemann CC, on the condition to observe that we have in fact
as another way to under- stand the deep meaning of the factor “2” in the summation. The same variational calculus with constraint may thus also be used in order to “shortcut” the introduction of the EM potential.
Finally, using the constitutive relations of the material establishing an isomorphism
, one can also introduce a local free energy
in a variational problem having now for constraint the stress equations, with the same comment as above (see [6] , p. 915, for more details). The well known Minkowski constitutive relations
can be similarly used for EM.
Lagrangian that should be quadratic in the Riemann tensor
while con-
sidering it independently of its expression through the second order derivatives of a metric
with inverse
or the first order derivatives of the corresponding Christoffel symbols
. According to the previous paragraph, the corresponding variational calculus must involve PD constraints made by the Bianchi identities and the new Lagrangian to vary must therefore contain as many Lagrange multipliers as the number of Bianchi identities (care!) that can be written under the form:
![]()
Meanwhile, Lanczos and followers have been looking for a kind of parametrization of the Bianchi identities, exactly like the Lagrange multiplier has been used as an Airy potential for the stress equations. However, we shall prove that the definition of a Riemann candidate and the answer to this question cannot be done without the knowledge of the Spencer cohomology. Moreover, we have pointed out the existence of well known couplings between elasticity and electromagnetism, namely piezoelectricity and photoelasticity, which are showing that, in the respective Lagrangians, the EM field is on equal footing with the deformation tensor and not with the Riemann tensor. This fact is showing the shift by one step that must be used in the physical inter-pretation of the differential sequences involved and cannot be avoided. Meanwhile, the ordinary derivatives
can be used in place of the covariant derivatives
when dealing with the linearized framework as the Christoffel symbols vanish when Euclidean or Minkowskian metrics are used.
The next tentative of Lanczos has been to extend his approach to the Weyl tensor:
![]()
The main problem is now that the Spencer cohomology of the symbols of the conformal Killing equations, in particular the 2-acyclicity, will be absolutely needed in order to study the Vessiot structure equations providing the Weyl tensor and its relation with the Riemann tensor. It will follow that the CC for the Weyl tensor are not first order contrary to the CC for the Riemann tensor made by the Bianchi identities, another reason for justifying the above shift by one step.
Finally, comparing the various parametrizations already obtained in the previous examples, it seems that the procedures are similar, even when dealing with systems having variable coefficients. The purpose of the paper is to prove that, in order to obtain a general algorithm, we shall need a lot of new tools involving at the same time commutative algebra, homological algebra, differential algebra and differential geometry that will be recalled in the next sections. Finally, like in any good crime story, it is only at the real end of the paper that we shall be able to revisit and compare all these examples in a unique framework.
2) MODULE THEORY
Before entering the heart of the next section dealing with extension modules, we need a few technical definitions and results from commutative algebra [13] [32] .
DEFINITION 2.1: A ring A is said to be unitary if it has a (unique) element
such that
and commutative if
. A non-zero element
is called a zero-divisor if one can find a non-zero
such that
and a ring is called an integral domain if it has no zero-divisor. From now on, all rings considered will be unitary integral domains as we shall deal mainly with rings of partial diffe- rential operators.
DEFINITION 2.2: A ring K is called a field if every non-zero element
is a unit, that is one can find an element
such that
.
DEFINITION 2.3: A module M over a ring A or simply an A-module is a set of elements
which is an abelian group for an addition
with an action
satisfying:
・ ![]()
・ ![]()
・ ![]()
・ ![]()
The set of modules over a ring A will be denoted by
. A module over a field is called a vector space.
DEFINITION 2.4: A map
between two A-modules is called a homomorphism over A if
and
. We successively define:
・ ![]()
・ ![]()
・ ![]()
・ ![]()
with an isomorphism
induced by f.
DEFINITION 2.5: We say that a chain of modules and homomorphisms is a sequence if the composition of two successive such homomorphisms is zero. A sequence is said to be exact if the kernel of each map is equal to the image of the map preceding it. An injective homomorphism is called a monomorphism, a surjective homo- morphism is called an epimorphism and a bijective homomorphism is called an isomorphism. A short exact sequence is an exact sequence made by a monomorphism followed by an epimorphism.
PROPOSITION 2.6: If one has a short exact sequence:
![]()
then the following conditions are equivalent:
・ There exists an epimorphism
such that
(left inverse of f).
・ There exists a monomorphism
such that
(right inverse of g).
DEFINITION 2.7: In the above situation, we say that the short exact sequence splits. The relation
provides an isomorphism
with inverse
. The short exact sequence
cannot split over
.
For the sake of clarity, as a few results will also be valid for modules over non-commutative rings, we shall denote by
a bimodule M which is a left module for A with operation
and a right module for B with operation
. In the commutative case, lower indices are not needed. If
and
are two left A-modules, the set of A-linear maps
will be denoted by
or simply
when there will be no confusion and there is a canonical isomorphism
with inverse
. When A is commutative,
is again an A-module for the law
. In the non-commutative case, things are much more complicate and we have:
LEMMA 2.8: Given
and
, then
becomes a right module over B for the law
. A similar result can be obtained with
and
, where
now becomes a left module over B for the law
.
THEOREM 2.9: If
are A-modules, the sequence:
![]()
is exact if and only if the sequence:
![]()
is exact for any A-module N.
COROLLARY 2.10: The short exact sequence:
![]()
splits if and only if the short exact sequence:
![]()
is exact for any module N.
DEFINITION 2.11: If M is a module over a ring A, a system of generators of M over A is a family
of elements of M such that any element of M can be written
with only a finite number of nonzero
. An A-module is called noetherian if every submodule of M (and thus M itself) is finitely generated.
One has the following standard technical result:
PROPOSITION 2.12: In a short exact sequence of modules, the central module is noetherian if and only if the two other modules are noetherian. As a byproduct, if A is a noetherian ring and M is a finitely generated module over A, then M is noetherian.
Accordingly, if M is generated by
, there is an epimorphism
. The kernel of this epimorphism is thus also finitely generated, say by
and we therefore obtain the exact sequence
that can be extended inductively to the left. Such a property will always be assumed in the sequel.
DEFINITION 2.13: In this case, we say that M is finitely presented.
We now turn to the definition and brief study of tensor products of modules over rings that will not be necessarily commutative unless stated explicitly.
Let
be a right A-module and
be a left A-module. We may introduce the free
-module made by finite formal linear combinations of elements of
with coefficients in
.
DEFINITION 2.14: The tensor product of M and N over A is the
-module
obtained by quotienting the above
-module by the submodule generated by the elements of the form:
![]()
and the image of
will be denoted by
.
It follows from the definition that we have the relations:
![]()
and there is a canonical isomorphism
. When A is commutative, we may use left modules only and
becomes a left A-module.
EXAMPLE 2.15: If
and
, we have
because
.
We present the technique of localization in order to introduce rings and modules of fractions.
Definition 2.16: A subset S of a ring A is said to be multiplicatively closed if
and
. By a left ring of fractions or left localization of a noncommutative ring A with respect to a multiplicatively closed subset S of A, we mean a ring denoted by
with a monomorphism
or simply a such that:
1) s is invertible in
, with inverse
or simply
.
2) Each element of
or fraction has the form
for some
.
We have to distinguish carefully
from
and we recover the standard notation
of the com- mutative case when two fractions
and
can be reduced to the same denominator
. The follow- ing proposition is essential for constructing localizations.
Proposition 2.17: If there exists a left localization of A with respect to S, then we must have
. A set S satisfying this condition is called a left Ore set.
Proof: As
must be a ring, the element
in
must be of the form
for some
. Accordingly,
with
.
Q.E.D.
Lemma 2.18: If S is a left Ore set in a ring A, then
and two fractions can be brought to the same denominator.
Proof: From the left Ore condition, we can find
and
such that
. More generally, we can find
such that
and we successively get:
![]()
so that the two fractions
and
can be brought to the same denominator
.
Q.E.D.
Let us now define an equivalence relation on
by saying that
if one can find
such that
and
. Such a relation is clearly reflexive and symmetric, thus we only need to prove that it is transitive. So let
and
. Then we can find
such that
and
. Also we can find
such that
and
. Now, from the Ore condition, one can find
such that
and thus
, that is to say
. As A is an integral domain, we have
as wished. We finally define
to be the quotient of
by the above equivalence relation with
. The sum
will be defined to be
and the product
will be defined to be
whenever
.
A similar approach can be used in order to define and construct modules of fractions whenever S satisfies the two conditions of the last proposition. For this we need a preliminary lemma:
LEMMA 2.19: If S is a left Ore set in a ring A and M is a left module over A, the set:
![]()
is a submodule of M called the S-torsion submodule of M.
Proof: If
, we may find
such that
. Now, we can find
such that
and we successively get
. Also,
, using the Ore condition for S, we can find
such that
and we get
.
Q.E.D.
DEFINITION 2.20: By a left module of fractions or left localization of M with respect to S, we mean a left module
over
both with a homomorphism
such that:
1) Each element of
has the form
for
.
2)
.
In order to construct
, we shall define an equivalence relation on
by saying that
if there exists
such that
and
. The main property of localization is ex- pressed by the following theorem:
Theorem 2.21: If one has an exact sequence:
![]()
then one also has the exact sequence:
![]()
where
.
As a link between tensor product and localization, we notice that the multiplication map
given by
induces an isomorphism
of modules over
when
is considered as a right module over A with
and M as a left module over A. In particular, when A is a commutative integral domain and
, the field
is called the field of fractions of A and we have the short exact sequence:
![]()
If now M is a left A-module, we may tensor this sequence by M on the right with
but we do not get in general an exact sequence. The defect of exactness on the left is nothing else but the torsion submodule
and we have the long exact sequence:
![]()
as we may describe the central map as follows:
![]()
As we saw in the Introduction, such a result allows to understand why controllability has to do with localization which is introduced implicitly through the transfer matrix in control theory. In particular, a module M is said to be a torsion module if
and a torsion-free module if
.
DEFINITION 2.22: A module in
is called a free module if it has a basis, that is a system of generators linearly independent over A. When a module F is free, the number of generators in a basis, and thus in any basis, is called the rank of F over A and is denoted by
. In particular, if F is free of finite rank r, then
. More generally, a module P is said to be projective if there exists another (projective) module Q such that
and any short exact sequence splits if it ends with a projective module (see [6] , p. 638-644) for a formal test).
If M is any module over a ring A and F is a maximum free submodule of M, then
is a torsion module. Indeed, if
, then one can find
such that
because, otherwise,
should be free submodules of M with a strict inclusion. In that case, the rank of M is by definition the rank of F over A. When A is commutative, one has:
LEMMA 2.23:
.
Proof: Taking the tensor product by K over A of the short exact sequence
, we get an isomorphism
because
(exercise) and the lemma follows from the definition of the rank.
Q.E.D.
PROPOSITION 2.24: (additivity property of the rank) If
is a short exact sequence of modules over a ring A, then we have
.
Proof : Let us consider the following diagram with exact left/right columns and central row:
![]()
where
is a maximum free submodule of
and
is a torsion module. Pulling back by g the image under
of a basis of
, we may obtain by linearity a map
and we define
where
and
are the canonical projections on each factor of the direct sum. We have
and
. Hence, the diagram is commutative and thus exact with
trivially. Finally, if
and
are torsion modules, it is easy to check that T is a torsion module too and
is thus a maximum free submodule of M.
Q.E.D.
DEFINITION 2.25: If
is any morphism, the rank of f will be defined to be
.
We provide a few additional properties of the rank that will be used in the sequel. For this we shall set
and, for any morphism
we shall denote by
the corresponding morphism which is such that
.
PROPOSITION 2.26: When A is a commutative integral domain and M is a finitely presented module over A, then
.
Proof: Applying
to the short exact sequence in the proof of the preceding lemma while taking into account
, we get a monomorphism
and obtain therefore
. However, as
with
because M is finitely generated, we get
too because
. It
follows that
and thus
.
Now, if
is a finite presentation of M, applying
to this presentation, we get the ker/coker exact sequence:
![]()
Applying
to this sequence while taking into account the isomorphisms
, we get the ker/coker exact sequence:
![]()
Counting the ranks, we obtain:
![]()
and thus:
![]()
As both two numbers in this sum are non-negative, they must be zero and we finally get
.
Q.E.D.
COROLLARY 2.27: Under the condition of the proposition, we have
.
Proof: Introducing the
exact sequence:
![]()
we have:
. Applying
and taking into account Theorem 2.9, we have the exact sequence:
![]()
and thus:
. Using the preceding proposition, we get
and
, that is to say
.
Q.E.D.
3) HOMOLOGICAL ALGEBRA
We need a few definitions and results from homological algebra [13] [33] [34] and start recalling the well known Cramer’s rule for linear systems through the exactness of the ker/coker sequence for modules when
is a linear map (homomorphism):
![]()
In the case of vector spaces over a field K, we successively have
, ![]()
,
of compatibility conditions, and obtain by sub- straction:
![]()
In the case of modules, we may replace the dimension by the rank and obtain the same relations because of the additive property of the rank. We may also define cohomology theory as follows:
DEFINITION 3.1: If one has a sequence
, that is if
, then one may introduce the submodules
and define the cohomology at M to be the quotient
.
We now introduce the extension modules in an elementary manner, using the standard notation
. Using a free resolution of an A-module M, that is to say a long exact sequence:
![]()
where
are free modules, namely modules isomorphic to powers of A and
. We may take out M and obtain the deleted sequence:
![]()
which is of course no longer exact. We may apply the functor
and obtain the sequence:
![]()
in order to state:
DEFINITION 3.2: We set:
![]()
The extension modules have the following three main properties [6] [13] [33] [34] :
PROPOSITION 3.3: The extension modules do not depend on the resolution of M chosen.
PROPOSITION 3.4: If
is a short exact sequence of A-modules, then we have the following connecting long exact sequence:
![]()
of extension modules. Moreover
whenever P is a projective module.
PROPOSITION 3.5:
is a torsion module,
.
Proof: Having in mind that
and
, we obtain
and
. However, we started from a resolution, that is an exact se-
quence in which
. It follows that
and thus
, that is to say
is a torsion module for
,
.
Q.E.D.
The next theorem and its corollary constitute the main results that will be used for applications through a classification of modules [1] [2] [6] - [8] [34] [35] :
THEOREM 3.6: The following long exact sequence:
![]()
is isomorphic to the ker/coker long exact sequence for the central morphism
which is defined by
.
Proof: Introducing
, we may obtain two short exact sequences, a left one starting with K and a right one finishing with K as follows:
![]()
Using the two corresponding long exact connecting sequences, we get
from the one starting with
which is also providing the left exact column of the next diagram and the exact central row of this diagram from the one starting with
. The Theorem is finally obtained by a chase proving that the full diagram is commutative and exact:
![]()
Q.E.D.
COROLLARY 3.7:
.
Proof: As
is a torsion module, we have therefore
. Now, if
, we may find
such that
and
because A is an integral domain, that is
and thus
.
Q.E.D.
DEFINITION 3.8: A module M will be called torsion-free if
and reflexive if
.
Despite all these results, a major difficulty still remains. Indeed, we have
as a left module over A but, using the bimodule structure of
and Lemma 2.13, it follows that
is a right module over A and thus
is also a right module over A. However, as we shall see, all the differential modules used through applications will be left modules over the ring of differential operators and it will therefore not be possible to use dual sequences as we did without being able to “pass from left to right and vice-versa”. For this purpose we now need many delicate results from differential geometry, in particular a way to deal with the formal adjoint of an operator as we did many times in the Introduction.
4) SYSTEM THEORY
If E is a vector bundle over the base manifold X with projection
and local coordinates
projecting onto
for
and
, identifying a map with its graph, a (local) section
is such that
on U and we write
or simply
. For any
change of local coordinates
on E, the change of section is ![]()
such that
. The new vector bundle
obtained by changing the transi- tion matrix A to its inverse
is called the dual vector bundle of E. Differentiating with respect to
and using new coordinates
in place of
, we obtain
. Introducing a multi-index
with length
and prolonging the procedure up to order q, we may construct in this way, by patching coordinates, a vector bundle
over X, called the jet bundle of
order q with local coordinates
with
and
. We have therefore epimor-
phisms
. For a later use, we shall set
and define the operator
on sections by the local formula ![]()
. Moreover, a jet coordinate
is said to be of class i if ![]()
. We finally introduce the Spencer operator
with
.
DEFINITION 4.1: A system of PD equations of order q on E is a vector subbundle
locally defined by a constant rank system of linear equations for the jets of order q of the form
. Its first
prolongation
will be defined by the equations ![]()
which may not provide a system of constant rank as can easily be seen for
where the rank drops at
.
The next definition of formal integrability (FI) will be crucial for our purpose.
DEFINITION 4.2: A system
is said to be formally integrable if the
are vector bundles
(regularity condition) and no new equation of order
can be obtained by prolonging the given PD equations more than r times,
or, equivalently, we have induced epimorphisms
allowing to compute “step by step” formal power series solutions.
A formal test has been first sketched by C. Riquier in 1910 [36] , then improved by M. Janet in 1920 [20] [37] and by E. Cartan in 1945 [38] , finally rediscovered in 1965, totally independently, by B. Buchberger who introduced Gröbner bases, using the name of his thesis advisor. However all these tentatives have been largely superseded and achieved in an intrinsic way, again totally independently of the previous approaches, by D.C. Spencer in 1965 [20] [39] [40] .
DEFINITION 4.3: The family
of vector spaces over X defined by the purely linear equations
for
is called the symbol at order
and only depends on
.
The following procedure, where one may have to change linearly the independent variables if necessary, is the key towards the next definition which is intrinsic even though it must be checked in a particular coordinate system called d-regular (see [6] [20] and [39] for more details):
・ Equations of class n: Solve the maximum number
of equations with respect to the jets of order q and class n. Then call
multiplicative variables.
![]()
・ Equations of class i: Solve the maximum number of remaining equations with respect to the jets of order q and class i. Then call
multiplicative variables and
non-multiplicative variables.
![]()
・ Remaining equations of order
: Call
non-multiplicative variables.
DEFINITION 4.4: The above multiplicative and non-multiplicative variables can be visualized respectively by integers and dots in the corresponding Janet board. A system of PD equations is said to be involutive if its first prolongation can be achieved by prolonging its equations only with respect to the corresponding multiplicative variables. The following numbers are called characters:
![]()
For an involutive system,
can be given arbitrarily.
For an involutive system of order q in the above solved form, we shall use to denote by
the principal jet coordinates, namely the leading terms of the solved equations in the sense of involution. Accordingly, any formal derivative of a principal jet coordinate is again a principal jet coordinate. The remaining jet coordinates will be called parametric jet coordinates and denoted by
. Now, the symbol of
is the zero symbol and is thus trivially involutive at any order q. Accordingly, if we introduce the multiplicative variables
for the parametric jets of order q and class i, the formal derivative or a parametric jet of strict order q and class i by one of its multiplicative variables is uniquely obtained and cannot be a principal jet of order
which is coming from a uniquely defined principal jet of order q and class i.
PROPOSITION 4.5: Using the Janet board and the definition of involutivity, we get:
![]()
Let T be the tangent vector bundle of vector fields on X,
be the cotangent vector bundle of 1-forms on X and
be the vector bundle of s-forms on X with usual bases
where we have set
. Also, let
be the vector bundle of symmetric q-covariant tensors. Moreover, if
are two vector fields on X, we may define their bracket
by the local formula
leading to the Jacobi identity
. We may finally introduce the exterior derivative
with
in the Poincaré sequence:
![]()
In a purely algebraic setting, one has [20] [39] [40] :
PROPOSITION 4.6: There exists a map
which restricts to
and
.
Proof: Let us introduce the family of s-forms
and set
. We obtain at once
.
Q.E.D.
The kernel of each
in the first case is equal to the image of the preceding
but this may no longer be true in the restricted case and we set (see [39] , p. 85-88 for more details):
DEFINITION 4.7: We denote by
and
respectively
the coboundary space, cocycle space and cohomology space at
of the restricted d-sequence which only depend on gq and may not be vector bundles. The symbol gq is said to be s-acyclic if
, involutive if it is n-acyclic and finite type if
becomes trivially involutive for r large enough. For a later use, we notice that a symbol
is involutive and of finite type if and only if
. Finally,
is involutive
if we set
.
FI CRITERION 4.8: If
is an epimorphism of vector bundles and
is 2-acyclic (involutive), then
is formally integrable (involutive).
EXAMPLE 4.9: The system
defined by the three PD equations
![]()
is homogeneous and thus automatically formally integrable but
is not involutive though finite type because
. Elementary computations of ranks of matrices show that the d-map:
![]()
is a
isomorphism and thus
is 2-acyclic with
, a crucial intrinsic property totally absent from any “old” work and quite more easy to handle than its Koszul dual.
The main use of involution is to construct differential sequences that are made up by successive compatibility conditions (CC) of order one. In particular, when
is involutive, the differential operator
of order q with space of solutions
is said to be involutive and one has the canonical linear Janet sequence ( [31] , p. 144):
![]()
where each other operator is first order involutive and generates the CC of the preceding one with the Janet bundles
. As the Janet sequence can be “cut at any place”, that is can also be constructed anew from any intermediate operator, the numbering of the Janet bundles has nothing to do with that of the Poincaré sequence for the exterior derivative, contrary to what many physicists still believe (
with
provides the simplest example). Moreover, the fiber dimension of the Janet bundles can be computed at once inductively from the board of multiplicative and non-multiplicative variables that can be exhibited for
by working out the board for
and so on. For this, the number of rows of this new board is the number of dots appearing in the initial board while the number
of dots in the column i just indicates the number of CC of class i for
with
. When
is not involutive but formally integrable and the r-prolongation of its symbol
becomes 2-acyclic, it is known that the generating CC are of order
(see [39] , Example 6, p. 120 and previous Example).
DEFINITION 4.10: More generally, a differential sequence is said to be formally exact if each operator generates the CC of the operator preceding it.
EXAMPLE 4.11: ( [41] , §38, p 40, is providing the first intuition of formal integrability) The second order system
is neither formally integrable nor involutive. Indeed, we get
and
, that is to say each first and second prolongation does bring a new second order PD equation. Considering the new system
, the question is to decide whether this system is involutive or not. In such a simple situation, as there is no PD equation of class 3, the evident permutation of coordinates
provides the following involutive second order system with one equation of class 3, 2 equations of class 2 and 1 equation of class 1:
![]()
We have
and the corresponding CC system is easily seen to be the following involutive first order system:
![]()
The final CC system is the involutive first order system:
![]()
We get therefore the (formally exact) Janet sequence:
![]()
However, keeping only
and
while using the fact that
commutes with
, we get the formally exact sequence
which is not a Janet sequence. We finally check that each
is separately differentially dependent on
because we have successively
.
Finally, we may extend the restriction
of the Spencer operator to:
![]()
in order to construct the first Spencer sequence which is another resolution of
because the kernel of the first D is such that
when q is large enough.
5) DIFFERENTIAL MODULES
Let K be a differential field, that is a field containing
with n commuting derivations
with
such that
,
and
. Using an implicit summation on multi-indices, we may introduce the (noncom- mutative) ring of differential operators
with elements
such that
and
. The highest value of
with
is called the order of the operator P and the ring D with multiplication
is filtred by the order q of the operators. We have the filtration
. Moreover, it is clear that D, as an algebra, is generated by
and
with
if we identify an element
with the vector field
of differential geometry, but with
now. It follows that
is a bimodule over itself, being at the same time a left D-module
by the composition
and a right D-module
by the composition
with
.
If we introduce differential indeterminates
, we may extend
to
for
. Therefore, setting
and calling
the differential module of equations, we obtain by residue the differential module or D-module
, denoting the residue of
by
when there can be a confusion. Introducing the two free differential modules
, we obtain equivalently the free presentation
of order q when
and
. We shall moreover assume that
provides a strict morphism, namely that the corresponding system
is formally integrable. It follows that M can be endowed with a quotient filtration obtained from that of
which is defined by the order of the jet coordinates yq in
. We have therefore the inductive limit
with
but it is important to notice that
in this particular case. It also follows from Noetherian arguments and involution that
though we have in general only
. As
, we may introduce the forgetful functor
.
More generally, introducing the successive CC as in the preceding section while changing slightly the numbering of the respective operators, we may finally obtain the free resolution of M, namely the exact sequence
. In actual practice, one must never forget that
acts on the left on column vectors in the operator case and on the right on row vectors in the module case. Also, with a slight abuse of language, when
is involutive as in Section 2 and thus
is involutive, one should say that M has an involutive presentation of order q or that
is involutive.
DEFINITION 5.1: Setting
, we have
and
. Such a definition can be extended to any matrix of operators by using the transposed matrix of adjoint operators and we get:
![]()
from integration by part, where
is a row vector of test functions and
the usual contraction. We quote
the useful formulas
(care about the
minus sign) and
as in ( [32] , p. 610-612).
REMARK 5.2: As can be seen from the examples of the Introduction, when
is involutive, then
may not be involutive. In the differential framework, we may set
. Comparing to similar concepts used in differential algebra, this number is just the maximum number of differentially independent equations to be found in the differential module I of equations. Indeed, pointing out that differential indeter- minates in differential algebra are nothing else than jet coordinates in differential geometry and using standard
notations, we have
. In that case, the differential ideal I automatically generates a prime
differential ideal
providing a differential extension
with
and differential transcendence degree
, a result explaining the notations [39] . Now, from the dimension formulas of
, we obtain at once
and thus
in a coherent way with any free presentation of M starting with
. However,
acts on the left in differential geometry but on the right in the theory of differential modules. For an operator of order zero, we recognize the fact that the rank of a matrix is equal to the rank of the transposed matrix.
PROPOSITION 5.3: If
is a local diffeomorphisms on X, we may set
and we have the identity:
![]()
and the adjoint of the well defined intrinsic operator
is ( minus) the well defined intrinsic operator
. Accordingly, if we have an operator
, we obtain the formal adjoint operator
.
Now, with operational notations, let us consider the two differential sequences:
![]()
![]()
where
generates all the CC of
. Then
but
may not gene- rate all the CC of
. Passing to the module framework, we just recognize the definition of
. Now, exactly like we defined the differential module M from
, let us define the differential module N from
. Then
does not depend on the presentation of M.
Having in mind that D is a K-algebra, that K is a left D-module with the standard action
and that D is a bimodule over itself, we have only two possible constructions leading to the following two definitions:
DEFINITION 5.4: We may define the inverse system
of M and introduce
as the inverse system of order q.
DEFINITION 5.5: We may define the right differential module
.
The first definition is leading to the inverse systems introduced by Macaulay in [41] (see [43] for details). As for the second, we have (see [1] , p. 21, [6] , p. 483-495, [42] , [43] for details).
THEOREM 5.6: When M and N are left D-modules, then
and
are left D-modules. In particular
is also a left D-module for the Spencer operator. Moreover, the structures of left D-modules existing therefore on
and
are now coherent with the adjoint isomor- phism for
:
![]()
Proof: For any
, let us define:
![]()
![]()
It is easy to check that
in the operator sense and that
is the standard bracket of vector fields. We have in particular with d in place of any
:
![]()
For any
with arbitrary
and
, we may then define:
![]()
![]()
and conclude similarly with:
![]()
Using K in place of N, we finally get
that is we recognize exactly the Spencer operator with now
and thus:
![]()
In fact, R is the projective limit of
in a coherent way with jet theory [18] [19] .
The next result is entrelacing the two left structures that we have just provided through the formula
defining the map
whenever
is given and
. Using any
, we get successively in L:
![]()
and thus
or simply
.
Q.E.D.
COROLLARY 5.7: If M and N are right D-modules, then
is a left D-module. Moreover, if M is a left D-module and N is a right D-module, then
is a right D-module.
Proof: If M and N are right D-modules, we just need to set
and conclude as before. Similarly, if M is a left D-module and N is a right D-module, we just need to set
.
Q.E.D.
REMARK 5.8: When
and
, , then
cannot be endowed with any left or right differential structure. When
and
, then
cannot be endowed with any left or right differential structure (see [1] , p. 24 for more details).
As
is a bimodule, then
is a right D-module according to Lemma 2.13 and the module N defined by the ker/coker sequence
is thus a right module
.
COROLLARY 5.9: We have the side changing procedure
with inverse
whenever
.
Proof: According to the above Theorem, we just need to prove that
has a natural right module structure over D. For this, if
is a volume form with coefficient
, we may set
when
. As D is generated by K and T, we just need to check that the above formula has an intrinsic meaning for any
. In that case, we check at once:
![]()
by introducing the Lie derivative of
with respect to
, along the intrinsic formula
where
is the interior multiplication and d is the exterior derivative of exterior forms. According to well known properties of the Lie derivative, we get:
![]()
Q.E.D.
Collecting all the results so far obtained, if a differential operator
is given in the framework of differential geometry, we may keep the same notation
in the framework of differential modules which are left modules over the ring D of linear differential operators and apply duality, provided we use the notation
and deal with right differential modules or use the notation
and deal again with left differential modules by using the
conversion procedure.
DEFINITION 5.10: If a differential operator
is given, a direct problem is to find (generating) compatibility conditions (CC) as an operator
such that
. Conversely, given
, the inverse problem will be to look for
such that
generates the CC of
and we shall say that
is parametrized by
if such an operator
is existing.
REMARK 5.11: Of course, solving the direct problem (Janet, Spencer) is necessary for solving the inverse problem. However, though the direct problem always has a solution, the inverse problem may not have a solution at all and the case of the Einstein operator is one of the best non-trivial PD counterexamples (compare [10] to [34] ). It is rather striking to discover that, in the case of OD operators, it took almost 50 years to understand that the possibility to solve the inverse problem was equivalent to the controllability of the corresponding control system (compare [11] to [34] ).
As
, any operator is the adjoint of a certain operator and we get:
FORMAL TEST 5.12: The double duality test needed in order to check whether
or not and to find out a parametrization if
has 5 steps which are drawn in the following diagram where
generates the CC of
and
generates the CC of
:
![]()
THEOREM 5.13:
parametrized by
.
REMARK 5.14: When an operator
can be parametrized by an operator
, we may ask whether or not
can be parametrized again by an operator
and so on. A good comparison can be made with hunting rifles as a few among them, called double rifles, are equipped with a double trigger mechanism, allowing to shoot again once one has already shot. In a mathematical manner, the question is to know whether the diffe- rential module defined by
is torsion-free or reflexive. The main difficulty is that these intrinsic properties highly depend on the choice of the parametrizing operator. The simplest example is provided by the Poincaré sequence for n = 3 made by the successive
operators. Indeed, any student knows that
is parametrizing div and that grad is parametrizing curl. However, we may parametrize
by
choosing
with 2 potentials
only instead of the usual 3 poten-
tials
and cannot proceed ahead as before. Other important examples will be provided in the next section dealing with applications, in particular the one involving Einstein equations when
. This comment points out the reason for using the extension modules.
It remains to study a delicate question on which all the examples of the Introduction were focussing. Indeed, if a parametrization of a given system of OD or PD equations is possible, that is, equivalently, if the corresponding differential module is torsion-free, it appears that different parametrizations may exist with quite different numbers of potentials needed. Accordingly, it should be useful to know about the possibility to have upper and lower bounds for these numbers when
, particularly in elasticity theory, because when
, an OD module M with
being automatically isomorphic to a free module E, the number of potentials needed is equal to
. We shall use the language of differential modules in order to improve and apply a few results already presented in ( [7] , Theorem 7+ Appendix).
THEOREM 5.15: Let
be a finite free presentation of the differential module
and assume we already know that
by using the formal test. Accordingly, we have obtained the exact sequence
of free differential modules where
is the parametrizing operator. Then, there exists other parametrizations
called minimal parametrizations and such that
is a torsion module.
Proof: We first explain the reason for using the word “minimal”. Indeed, we have
but also
and thus
as a way to get a lower bound for the number of potentials but not to get a differential geometric framework.
While applying the formal test in the operator language,
is describing the (generating) CC of
and we shall denote by
the (generating) CC of
as we did in Example 1.3. In the module framework, going on with left differential modules, when F is a free left module, we shall denote by
the corresponding converted left differential module of the right differential module
. The reader not familiar with duality may look at the situations met in electromagnetism and elasticity in ( [6] , p. 492-495). If
and
is the largest free differential submodule of L (
in Example 1.3, D in Example 1.4), then
is a torsion module and we have the following commutative and exact diagram:
![]()
where the central vertical monomorphism
is obtained by pulling a basis of
back to
as we did in the diagram of Proposition 2.24. Coming back to the operators
and
, we get the following commutative and exact diagram allowing to define
by composition:
![]()
We have:
![]()
and obtain by duality the following commutative and exact diagram where
:
![]()
However, though the upper sequence
is exact by definition because
, the lower induced sequence
may not be exact. With
for simplicity,
and the induced epimorphism
, we obtain:
![]()
![]()
![]()
![]()
Accordingly,
is a minimal parametrization of
contrary to
in general and we invite the reader to repeat the proof by using operators and their adjoints as in the formal test.
Q.E.D.
6) APPLICATIONS
EXAMPLE 6.1: OD Control Theory Revisited
The following result is well known and can be found in any textbook of algebra [6] [13] [32] :
PROPOSITION 6.2: If A is a principal ideal domain, that is if any ideal in A is generated by a single element, then any torsion-free module over A is free and thus projective.
As this is the case of the ring
when
, we obtain the following corollary of the preceding parametrizing Theorem, allowing to extend the Kalman test of controllability to PD systems with variable coefficients as we did in the Introduction (see [6] - [9] [11] for details).
COROLLARY 6.3: If
is surjective, then
is injective if and only if M is projective.
Proof: As
is surjective, replacing M by P, we have the following short exact sequence:
![]()
As P is projective, this short exact sequence splits with
[6] [13] [32] . Using Proposition 2.6, we can construct a right inverse operator
of
with now
for the corresponding morphisms. Applying duality and Corollary 2.10, we get the short exact sequence:
![]()
It follows that
is surjective and the adjoint operator
is injective because
.
Conversely, if
is injective, there exists a left inverse
of
providing a right inverse
of
(care). We may thus use again Corollary 2.10 because
and
. Meanwhile, we have proved that, if
and
, it is always possible to find an injective parametrization but Example 1.4 is showing that this result is no longer true when
.
Q.E.D.
Multiplying the control system of Example 1.1 by a test function
and integrating by parts, the kernel of the operator thus obtained is defined by the OD equations:
![]()
The formal adjoint of the operator defining the control system is thus injective if and only if we have
, a result absolutely not evident at first sight but explaining why we used the same notation for a test function and for a Lagrange multiplier.
EXAMPLE 6.4: Elasticity Theory Revisited
The Killing operator
is a defined by
with
where
is the displacement vector,
is the Lie derivative of
with respect to
and
is the infinitesimal deformation tensor of textbooks. It is a Lie operator because its solutions
satisfy
. The corresponding first order Killing system
is not involutive because its symbol
is finite type with first prolongation
and thus
. Accordingly, as
is a flat constant metric, the second order CC are described by an operator
coming from the linearization of the Riemann tensor obtained in a standard way by setting
with a small parameter
, dividing by t and taking the limit when
. Finally, raising the index by means of the metric, the adjoint operator
is defined by the intrinsic stress equations
where
is the covariant derivative and
the Christoffel symbols ( [6] , p. 494, [44] , p. 236).
・ Airy parametrization of the stress equations when
gives
and we have thus 1 potential only. By duality, working out the corresponding adjoint operators, we obtain the two formally exact sequences:
![]()
Accordingly, the canonical and the minimal parametrizations coincide when
. We discover that the Airy parametrization is nothing else than the formal adjoint of the Riemann CC for the deformation tensor:
![]()
where the indices of the displacement vector
are lowered by means of the euclidean metric of
. We do not believe this result is known in such a general framework.
・ Beltrami parametrization of the stress equations when
gives
and we have thus 6 potentials. However, Maxwell/Morera parametrizations of the stress equations when
both give
and we have thus 3 potentials only.
![]()
Accordingly, the canonical parametrization has 6 potentials while any minimal parametrization has 3 potentials. We finally notice that the Cauchy operator is parametrized by the Beltrami operator which is again para- metrized by the adjoint of the Bianchi operator obtained by linearizing the Bianchi identities existing for the Riemann tensor, a property not held by any minimal parametrization as we already noticed.
・ For
, we shall prove below that the Einstein parametrization of the stress equations is neither canonical nor minimal in the following diagram:
![]()
obtained by using the fact that the Einstein operator, linearization of the Einstein tensor at the Minkowski metric, is self-adjoint, the 6 terms being exchanged between themselves [10] [45] . The upper div induced by Bianchi has nothing to do with the lower Cauchy stress equations, contrary to what is still believed today. It also follows that the Einstein equations in vacuum cannot be parametrized as we have the following diagram of operators (see [6] and [34] for more details or [10] for a computer algebra exhibition of this result):
![]()
・ It remains therefore to compute all these numbers for an arbitrary dimension
. For this, we notice that the successive prolongations
defined by
for
have kernel
. The symbol morphism
with kernel
is induced by the projection of
onto
(see [39] , p. 256 or [41] , p. 233 for details). If we use such a procedure for a first order system with no zero or first order CC, we have
. The Killing system
is formally integrable (
involutive) if and only if
has constant Riemannian curvature:
![]()
with
when
is the flat Minkowski metric [12] [20] [39] . In general, we may apply the Spencer d-map to the top row obtained with
in order to get the commutative diagram:
![]()
with exact rows and exact columns but the first that may not be exact at
. We shall denote by
the coboundary as the image of the central
, by
the cocycle as the kernel of the lower
and by
the Spencer d- cohomology at
.
In the classical Killing system,
is defined by
.
Applying the previous diagram, we discover that the Riemann tensor
is a section of the bundle
with ![]()
by using the top row or the left column. We obtain at once the two properties of the (linearized) Riemann tensor through the chase involved, namely
is killed by both
and
. However, we have no indices for
and cannot therefore exhibit the Ricci tensor or the Einstein tensor of general relativity by means of the usual contraction or
trace. We recall briefly their standard definitions by stating
.
Similarly, going one step further, we get the (linearized) Bianchi identities with
![]()
as in ( [46] , p. 168-171). This approach is relating for the first time the concept of Riemann tensor candidate, introduced by Lanczos and others, to the Spencer d-cohomology of the Killing symbols.
Counting the differential ranks is now easy because
is formally integrable with finite type symbol and thus
is involutive with symbol
. We get:
![]()
![]()
that is
when
and
when
. Collecting all the results, we obtain that the canonical parametrization needs
potentials while any minimal parametrization only needs
potentials. The Einstein parametrization is thus “in between” because
.
The conformal Killing system
is defined by eliminating the function
in the system
. It is also a Lie operator
with solutions
satisfying
. Its symbol ![]()
is defined by the linear equations
which do not depend on any conformal factor and
is finite type when
because
but
is now 2-acyclic only when
and 3-acyclic only when
[20] [46] - [48] . It is known that
and thus
too (by a chase) are formally integrable if and only if
has zero Weyl tensor:
![]()
We may use the formula
of Proposition 2.6 in the split short exact sequence induced by the inclusions
:
![]()
according to the Vessiot structure equations, in particular if
has constant Riemannian curvature and thus
[20] [39] [45] - [47] . Using the same diagrams as before, we get
for defining any Weyl tensor candidate. As a byproduct, the linearized Weyl operator is of order 2 with a symbol
which is not 2-acyclic by applying the d-map to the short exact sequence:
![]()
and chasing through the commutative diagram thus obtained with
. As
becomes 2-acyclic after one prolongation of
only, it follows that the generating CC for the Weyl operator are of order 2, a result showing that the so-called Bianchi identities for the Weyl tensor are not CC in the strict sense of the definition as they do not involve only the Weyl tensor. Of course, these results could not have been discovered by Lanczos and followers because the formal theory of Lie pseudogroups and the Vessiot structure equations are still not acknowledged today.
For this reason, we provide a few hints in order to explain why the Vessiot structure equations sometimes contain a few constants, sometimes none at all as we just saw (see [39] [49] and [50] for more details). Indeed, isometries preserve the metric
while conformal isometries preserve the symmetric
tensor density
. The respective variations are related by the similitude formula
which only depends on
and not on a conformal factor. It follows that ![]()
and that
may be identified with the sub-bundle
with the above well defined epi- morphism
induced by the inclusion
. We set [39] [49] [50] :
DEFINITION 6.5: We say that a vector bundle F is associated with a Lie operator
if, for any solution
of
, there exists a first order operator
called Lie derivative with respect to
and such that:
1) ![]()
2) ![]()
3) ![]()
4) If E and F are two such associated vector bundles, then:
![]()
In such a case, we may introduce
.
PROPOSITION 6.6: Using capital letters for linearized objects, we have:
1)
of
in T.
2)
.
3)
.
4)
.
5) The Lie derivative commutes with the Janet operators
.
We have in particular
(care to sign).
Proof: Two (nondegenerate) metrics
give the same Killing system
if and only if
with the multiplicative group parameter
. Therefore, if
is FI, then the two metrics have respective constant curvatures c and
. Setting
while linearizing these finite transformations with
gives
when
.
Q.E.D.
However, we have yet not proved the most difficult result that could not be obtained without homological algebra and the next example will explain this additional difficulty.
EXAMPLE 6.7: With
for
, we get
for
. Then
is defined by
while
is defined by
but the CC of
are generated by
. Using operators, we have the differential sequences:
![]()
where the upper sequence is formally exact at
but the lower sequence is not formally exact at
.
Passing to the module framework, we obtain the sequences:
![]()
where the lower sequence is not exact at
.
Therefore, we have to prove that the extension modules vanish, that is
generates the CC of
and, conversely, that
generates the CC of
. We also remind the reader that it has not been easy to exhibit the CC of the Maxwell or Morera parametrizations when
and that a direct checking for
should be strictly impossible. It has been proved by L. P. Eisenhart in 1926 [49] that the solution space
of the Killing system has
infinitesimal generators
linearly independent over the constants if and only if
had constant Riemannian curvature, namely zero in our case. As we have a Lie group of transformations preserving the metric, the three theorems of Sophus Lie assert than
where the structure constants c define a Lie algebra
. We have therefore
with
. Hence, we may replace locally the Killing system by the system
, getting therefore the differential sequence:
![]()
which is the tensor product of the Poincaré sequence by
. Finally, it follows from Proposition 3.3 that the extension modules do not depend on the resolution used and thus vanish because the Poincaré sequence is self adjoint (up to sign), that is
generates the CC of
at any position, exactly like d generates the CC of d at any position. This (difficult) result explains why the differential modules we have met were torsion-free or even reflexive. We invite the reader to compare with the situation of the Maxwell equations in electro-mag- netisme (see [6] , p. 492-494 for more details). However, we have explained in [6] [45] - [47] [51] why neither the Janet sequence nor the Poincaré sequence can be used in physics and must be replaced by the Spencer sequence which is another resolution of
[39] [40] [46] .
EXAMPLE 6.8: PD Control Theory Revisited
Comparing with the Theorem allowing to construct a minimal parametrization, we started with
and computed
with generating CC
, obtaining therefore finally the generating CC
, that is
. In that case,
in the diagram providing the minimal parametrization. This result explains why we had two potentials
in the canonical parametrization and only one, namely
or
, in the minimal parametrizations but it is not possible to imagine the underlying procedure.
EXAMPLE 6.9: OD/PD Optimal Control Revisited
Using the notations of the Formal Test 5.12, let us assume that the two differential sequences:
![]()
are formally exact, that is
generates the CC of
and
generates the CC of
, namely
is a potential for
and
is a potential for
. We may consider a variational problem for a cost function
under the linear OD or PD constraint described by
.
・ Introducing convenient Lagrange multipliers
while setting
for simplicity, we must vary the integral:
![]()
Integrating by parts, we obtain the EL equations:
![]()
to which we have to add the constraint
obtained by varying
. If
is an injective operator, in particular if
is formally surjective (no CC) while
and M is torsion-free or
and M is projective, then one can obtain λ explicitly and eliminate it by substitution ( [7] ). Otherwise, using the CC
of
in order to eliminate
, we have to study the formal integrability of the combined system:
![]()
which may be a difficult task as we already saw through the examples of the Introduction.
・ We may also transform the given variational problem with constraint into a variational problem without any constraint if and only if the differential constraint can be parametrized. Using the parametrization of
by
, we may vary the integral:
![]()
whenever
and integrate by parts for arbitrary
in order to obtain the EL equations:
![]()
in a coherent way with the previous approach and the Poincaré duality
.
As a byproduct, if the field equations
can be parametrized by a potential
through the formula
, then the induction equations
can be obtained by duality in a coherent way with the double duality test, ... on the condition to know what sequence must be used.
However, we have already proved in [45] - [47] [51] [52] that the Cauchy stress equations must be replaced by the Cosserat couple-stress equations and that the Janet sequence (only used in this paper) must be thus re- placed by the Spencer sequence. Accordingly, it becomes clear that the work of Lanczos and followers has been based on a double confusion between fields and inductions on one side, but also between the Janet sequence and the Spencer sequence on the other side.
FUNDAMENTAL RESULT 6.10: The Janet and Spencer sequences for any Lie operator of finite type are formally exact by construction, both with their corresponding adjoint sequences. Lanczos has been trying to parametrize
by
when
parametrizes
. On the contrary, we have proved that one must parametrize
by
when
parametrizes
as in the famous infinitesimal equivalence problem ( [20] , p. 332-336), with a shift by one step. This is also the only way which is coherent with the corresponding non-linear sequences and the finite equivalence problem [39] [46] [47] [50] [52] [53] .
2. Conclusion
The effective usefulness of the double duality test seems absolutely magical in actual practice but the reader may not forget about the amount of mathematics needed from different domains. Unhappily, in our opinion based on a long experience in dealing with applications, the most difficult part is concerned with formal integrability and involution needed in order to compute the various differential ranks involved. However, the above methods, though largely superseding the pioneering approaches of Janet and Cartan, are still not known in mechanics and in mathematical physics, particularly in general relativity or even in control theory despite many tentatives done twenty years ago. We hope that this paper will help to improve this situation in a near future, in particular when dealing with partial differential optimal control, which is with variational calculus with OD or PD constraints along the way that has been initiated by Lanczos for eliminating the corresponding Lagrange multipliers or using them as potentials while studying the mathematical foundations of general relativity.