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Airy, Beltrami, Maxwell, Einstein and Lanczos Potentials Revisited ()

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*stress functions*, needed in order to study the parametrizations of the stress equations, respectively provided by G.B. Airy (1863) for 2-dimensional elasticity, then by E. Beltrami (1892), J.C. Maxwell (1870) for 3-dimensional elasticity, finally by A. Einstein (1915) for 4-dimensional elasticity, both with a variational procedure introduced by C. Lanczos (1949, 1962) in order to relate potentials to Lagrange multipliers. Using the methods of Algebraic Analysis, namely mixing differential geometry with homological algebra and combining the

*double duality test*involved with the

*Spencer cohomology*, we shall be able to extend these results to an arbitrary situation with an arbitrary dimension n. We shall also explain why double duality is perfectly adapted to variational calculus with differential constraints as a way to eliminate the corresponding Lagrange multipliers. For example, the

*canonical parametrization*of the stress equations is just described by the formal adjoint of the components of the linearized Riemann tensor considered as a linear second order differential operator but the minimum number of potentials needed is equal to for any

*minimal parametrization*, the Einstein parametrization being “

*in between*” with potentials. We provide all the above results without even using indices for writing down explicit formulas in the way it is done in any textbook today, but it could be

*strictly impossible*to obtain them without using the above methods. We also revisit the

*possibility*(Maxwell equations of electromagnetism) or the

*impossibility*(Einstein equations of gravitation) to obtain canonical or minimal parametrizations for various equations of physics. It is nevertheless important to notice that, when n and the algorithms presented are known, most of the calculations can be achieved by using computers for the corresponding symbolic computations. Finally, though the paper is mathematically oriented as it aims providing new insights towards the mathematical foundations of general relativity, it is written in a rather self-contained way.

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Cite this paper

*Journal of Modern Physics*,

**7**, 699-728. doi: 10.4236/jmp.2016.77068.

Received 19 February 2016; accepted 25 April 2016; published 28 April 2016

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:

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:

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 lass="lazy 100" data-original="//html.scirp.org/file/8-7502655x918.png" />, 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.

Conflicts of Interest

The authors declare no conflicts of interest.

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