Abstract and Applied Analysis

Volume 2014 (2014), Article ID 482963, 32 pages

http://dx.doi.org/10.1155/2014/482963

Research Article

Automorphisms of Ordinary Differential Equations

Václav Tryhuk and Veronika Chrastinová

Department of Mathematics, Faculty of Civil Engineering, Brno University of Technology, Veveří 331/95, 602 00 Brno, Czech Republic

Received 26 July 2013; Accepted 4 October 2013; Published 28 January 2014

Academic Editor: Josef Diblik

Copyright © 2014 Václav Tryhuk and Veronika Chrastinová. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The paper deals with the local theory of internal symmetries of underdetermined systems of ordinary differential equations in full generality. The symmetries need not preserve the choice of the independent variable, the hierarchy of dependent variables, and the order of derivatives. Internal approach to the symmetries of one-dimensional constrained variational integrals is moreover proposed without the use of multipliers.

1. Preface

The theory of symmetries of determined systems (the solution depends on constants) of ordinary differential equations was ultimately established in Lie’s and Cartan’s era in the most possible generality and the technical tools (infinitesimal transformations and moving frames) are well known. Recall that the calculations are performed in finite-dimensional spaces given in advance and the results are expressed in terms of Lie groups or Lie-Cartan pseudogroups.

We deal with underdetermined systems (more unknown functions than the number of equations) of ordinary differential equations here. Then the symmetry problem is rather involved. Even the system of three first-order quasilinear equations with four unknown functions (equivalently, three Pfaffian equations with five variables) treated in the famous Cartan’s article [1] and repeatedly referred to in actual literature was not yet clearly explained in detail. Paradoxically, the common tools (the calculations in given finite-order jet space) are quite sufficient for this particular example. We will later see that they are insufficient to analyze the seemingly easier symmetry problem of one first-order equation with three unknown functions (alternatively, two Pfaffian equations with five variables) in full generality since the order of derivatives need not be preserved in this case and the finite-order jet spaces may be destroyed. Recall that even the higher-order symmetries (automorphisms) of empty systems of differential equations (i.e., of the infinite order jet spaces without any additional differential constraints) are nontrivial [2–4] and cannot be included into the classical Lie-Cartan theory of transformation groups. Such symmetries need not preserve any finite-dimensional space and therefore the invariant differential forms (the Maurer-Cartan forms, the moving coframes) need not exist.

Let us outline the very core of the subject for better clarity by using the common jet terminology. We start with the higher-order transformations of curves lying in the space with coordinates . The transformations are defined by certain formulae

where the -smooth real-valued functions depend on a finite number of the familiar jet variables

The resulting curve again lying in appears as follows. We put

and assuming

there exists the inverse function which provides the desired result

One can also easily obtain the well-known prolongation formula

for the derivatives by using the Pfaffian equations

Functions satisfying (4) and may be arbitrary here.

At this place, in order to obtain coherent theory, introduction of the familiar infinite-order jet space of -parametrized curves briefly designated as with coordinates is necessary. Then formulae ((1), (6)) determine a mapping , a morphism of the jet space . If the inverse given by certain formulae

exists, we speak of an automorphism (in alternative common terms, symmetry) of the jet space . It should be noted that we tacitly deal with the local theory in the sense that all formulae and identities, all mappings, and transformation groups to follow are in fact considered only on certain open subsets of the relevant underlying spaces which is not formally declared by the notation. Expressively saying, in order to avoid the clumsy purism, we follow the reasonable 19th century practice and do not rigorously indicate the true definition domains.

After this preparation, a system of differential equations is traditionally identified with the subspace given by certain equations

(We tacitly suppose that is a “reasonable subspace” and omit the technical details.) This is the infinitely prolonged system. The total derivative vector field defined on is tangent to the subspace and may be regarded as a vector field on , as well. The morphism transforms into the subspace given by the equations

This is again a system of differential equations. In our paper, we are interested only in the particular case when . Then, if the inverse locally exists on a neighbourhood of the subspace in the total jet space, we speak of the external symmetry   of the system of differential equations (9). Let us, however, deal with the natural restriction of the mapping to the subspace . If there exists the inverse of the restriction, we speak of the internal symmetry. Internal symmetries do not depend on the localizations of in . More precisely, differential equations can be introduced without any reference to jet spaces and the internal symmetries can be defined without the use of localizations. On this occasion, we are also interested in groups of internal symmetries. They are generated by special vector fields, the infinitesimal symmetries.

In the actual literature, differential equations are as a rule considered in finite-dimensional jet spaces. Then the internal and external symmetries become rather delicate and differ from our concepts since the higher-order symmetries are not taken into account. We will not discuss such conceptual confusion in this paper with the belief that the following two remarks (and Remark 5) should be quite sufficient in this respect.

Remark 1 (on the symmetries). The true structure of the jet space is determined by the contact module   which involves all contact forms

Then the above morphisms   given in ((1), (6)) are characterized by the property . Recall that invertible morphisms are automorphisms. Let us introduce the subspace of all points (9). This is equipped with the restriction of the contact module. Recall that we are interested only in the case (abbreviation of ). Let be the restriction of . If is a morphism then is a morphism in the sense that . Recall that we have the internal symmetry, if is moreover invertible. If also is invertible, we have the external symmetry  . The internal symmetries can be defined without any reference to and as follows. Let be any invertible mapping such that . This can be always extended to a morphism   of the ambient jet space. (Hint, recurrence (6) holds true both in and in .) So we may conclude that such is just the internal symmetry. Moreover, if there exists invertible extension of , then is even the external symmetry but the latter concept already depends on the localization of in .

Remark 2 (on infinitesimal symmetries). Let us consider a vector field

on the jet space . Let us moreover suppose from now on (where denotes the Lie derivative see also Definition 8). In common terminology, such vector fields are called generalized (higher-order, Lie-Bäcklund) infinitesimal symmetries of the jet space . However need not in general generate any true group of transformations and we therefore prefer the “unorthodox” term a variation   here. (See Section 7 and especially Remark 35 where the reasons for this term are clarified.) The common term infinitesimal symmetry is retained only for the favourable case when generates a local one-parameter Lie group [5]. Let us consider the above subspace . If is tangent to , then there exists the natural restriction of to . Clearly and we speak of the (internal) variation  . If moreover generates a group in , we have the (internal) infinitesimal symmetry  . The internal concepts on can be easily introduced without any reference to the ambient space . This is not the case for the concept of the external infinitesimal symmetry   which supposes that appropriate extension   of on the ambient space   generates a Lie group.

We deal only with the internal symmetries and infinitesimal symmetries in this paper. It is to be noted once more that infinite-dimensional underlying spaces are necessary if we wish to obtain a coherent theory. The common technical tools invented in the finite-dimensional spaces will be only slightly adapted; alas, the ingenious methods proposed, for example, in [6–8] seem to be not suitable for this aim and so we undertake the elementary approach [9] here.

2. Technical Tools

We introduce infinite-dimensional manifold modelled on the space with local coordinates in full accordance with [9]. The manifold is equipped with the structural algebra   of -smooth functions expressed as in terms of coordinates. Transformations (mappings) are (locally) given by certain formulae

and analogous (invertible) formulae describe the change of coordinates at the overlapping coordinate systems.

Let be the -module of differential 1-forms

The familiar rules of exterior calculus can be applied without any change, in particular for the above transformation .

Let be the -module of vector fields . In terms of coordinates we have

where the coefficients may be quite arbitrary. We identify with the linear functional on determined by the familiar duality pairing

With this principle in mind, if certain forms generate the -module, then the values

uniquely determine the vector field and (17) can be very expressively (and unorthodoxly) recorded by

This is a mere symbolical record, not the true infinite series. However, if is a basis of the module in the sense that every admits a unique representation , finite sum) then the coefficients can be quite arbitrary and (18) may be regarded as a true infinite series. The arising vector fields provide a weak basis (infinite expansions, see [9]) of dual to the basis of . In this transcription, (15) is alternatively expressed as

We recall the Lie derivative   acting on exterior differential forms. The image of a vector field defined by the property

need not exist. It is defined if is invertible.

We consider various submodules of differential forms together with the relevant orthogonal submodules consisting of all vector fields such that . The existence of (local) -bases in all submodules of to appear in our reasonings is tacitly postulated. Dimension of an -module is the number of elements of an -basis. Omitting some “exceptional points,” it may be confused with the dimension of the corresponding -module (the localization) at a fixed place . On this occasion, it should be noted that the image

of a tangent vector at exists as a vector at the place .

Let us also remark with regret that any rigorous exposition of classical analysis in the infinite-dimensional space is not yet available; however, certain adjustments of finite-dimensional results are not difficult. For instance, the following invertibility theorem will latently occur in the proof of Theorem 20.

Theorem 3. A mapping is invertible if and only if any of the following equivalent conditions is satisfied: the pull-back is invertible, the pull-back is invertible, and if is a (fixed, equivalently: arbitrary) basis of module , then again is a basis.

Hint. A nonlinear version of the familiar Gauss elimination procedure for infinite dimension [9] provides a direct proof with difficulties concerning the definition domain of the resulting inverse mapping. Nevertheless if is moreover a morphism of a diffiety (see Definition 8) then the prolongation procedure ensures the local existence of in the common sense.

3. Fundamental Concepts

We introduce a somewhat unusual intrinsical approach to underdetermined systems of ordinary differential equations in terms of the above underlying space , a submodule of differential 1-forms, and its orthogonal submodule of vector fields.

Definition 4. A codimension one submodule is called a diffiety if there exists a good filtration

by finite-dimensional submodules such that

To every subset , let denote the submodule with generators . Since (easy), the second requirement (23) can be a little formally simplified as .

Remark 5. This is a global coordinate-free definition; however, we again deal only with the local theory from now on in the sense that the definition domains (of filtrations (22), of independent variable to follow, and so on) are not specified. It should be noted on this occasion that the common geometrical approach [6–8] to differential equations rests on the use of the rigid structure of finite-order jets. Many classical concepts then become incorrect, if the higher-order mappings are allowed but we cannot adequately discuss this important topic here. Rather subtle difficulties are also passed over already in the common approach to the fundamental jet theory. For instance, smooth curves in the plane with coordinates are parametrized either by (i.e., ) or by coordinate (i.e., ) in the common so-called “geometrical” approach [6–8]. However, then already the Lie’s classical achievements concerning contact transformations [10, 11] with curves parametrized either by or by cannot be involved. Quite analogously, the “higher-order” parameterizations and mappings [2–5] are in fact rejected in the common “rigid” jet theory with a mere point symmetries.

Definition 6. Let a differential generate together with the total module of all differential 1-forms. Then is called the independent variable to diffiety . The vector field (abbreviation) such that

is called total (or formal) derivative of with respect to the independent variable . This vector field is a basis of the one-dimensional module for every fixed particular choice of the independent variable .

Remark 7. Let us state some simple properties of diffieties. The proofs are quite easy and may be omitted. A form is lying in if and only if . In particular in accordance with the identities

(This trivial property clarifies the more restrictive condition (23).) Moreover clearly

and in particular

for all coordinates. We have very useful -generators of diffiety . The independent variable and the filtrations (22) can be capriciously modified. In particular the -lift [9]

with large enough ensures that for all . We will be, however, interested just in the reverse concept “” latently involved in the “standard adaptation” of filtrations to appear later on.

Definition 8. A transformation is called a morphism of the diffiety if . Invertible morphisms are automorphisms (or symmetries) of . A vector field satisfying is called the variation of . If moreover (locally) generates a one-parameter group of transformations, we speak of the infinitesimal symmetry   of diffiety .

Remark 9. Let us mention the transformation groups in more detail. A local one-parameter group of transformations is given by certain formulae

in terms of local coordinates, where is supposed. Then the special vector field (15) defined by

is called the infinitesimal transformation of the group (29). In the opposite direction, we recall that a general vector field (15) generates the local group (29) if and only if the Lie system

is satisfied. Alas, a given vector field (19) need not in general generate any transformation group since the Lie system need not admit any solution (29).

With all fundamental concepts available, let us eventually recall the familiar and thoroughly discussed in [9] interrelation between the diffieties and the corresponding classical concept of differential equations for the convenience of reader. In brief terms, the idea is quite simple. A given system of differential equations is represented by a system of Pfaffian equations and the module generated by such 1-forms is just the diffiety. More precisely, we deal with the infinite prolongations as follows.

In one direction, let a system of underdetermined ordinary differential equations be given. We may deal with the first-order system

without any true loss of generality. Then (32) completed with

provides the infinite prolongation. The corresponding diffiety is generated by the forms

in the space with coordinates

Clearly

is the total derivative and the submodules of all forms (34) with determine a quite simple filtration (22) with respect to the order of contact forms. (Hint: use the formulae

and .) However, there exist many other and more useful filtrations; see the examples to follow later on.

The particular case of the empty system (32) can be easily related to the case of the jet space of all -parametrized curves in of the Section 1. The relevant diffiety is identified with the module of all contact forms (11), of course.

In the reverse direction, let a diffiety be given on the space . In accordance with (27), the forms generate . So we have the Pfaffian system and therefore the system of differential equations

of rather unpleasant kind. Then, due to the existence of a filtration (22) and (23), one can obtain also the above classical system of differential equations (32) together with the prolongation (33) by means of appropriate change of coordinates [9]. This is, however, a lengthy procedure and a shorter approach can be described as follows. Let the second requirement (23) be satisfied, if . Suppose that the forms generate module . Then all forms

generate the diffiety . The corresponding Pfaffian system is equivalent to certain infinite prolongation of differential equations, namely,

(direct verification), and in general

We have the infinite prolongation of the classical system and this is just the system that corresponds to diffiety .

Altogether taken, differential equations uniquely determine the corresponding diffieties; however, a given diffiety leads to many rather dissimilar but equivalent systems of differential equations with regard to the additional choice of dependent and independent variables.

Remark 10. Definitions 4–8 make good sense even if is a finite-dimensional manifold and then provide the well-known intrinsical approach to determined systems of differential equations. They are identified with vector fields (better, fields of directions) in the finite-dimensional space . Choosing a certain independent variable , the equations are represented by the vector field or, more visually, by the corresponding -flow. The general theory becomes trivial; we may, for example, choose for all in filtration (22).

4. On the Structure of Diffieties

Definition 11. To every submodule of a diffiety , let be the submodule of all such that . Filtration (22) and (23) is called a standard one, if

For every , the first condition ensures that the inclusions are equivalent and the second condition ensures that implies .

Theorem 12. Appropriate adaptation of some lower-order terms of a given filtration (22) and (23) provides a standard filtration in a unique manner [9]. Equivalently and in more detail, there exists unique standard filtration such that for appropriate and all large enough. Equivalently and briefly, there exists unique standard filtration such that for appropriate .

Proof. The mapping naturally induces certain -homomorphism

of factor modules denoted by for better clarity. Homomorphisms are surjective and therefore even bijective for all large enough, say for . However, the injectivity of implies . It follows that we have strongly decreasing sequence

which necessarily terminates with the stationarity . Denoting

we have the sought strongly increasing standard filtration

of diffiety . In particular .

Proof of Theorem 12 was of the algorithmical nature and provides a useful standard basis of diffiety as follows. Assume that the forms

(recall that whence ) and moreover the classes of forms

(recall that is injective mapping), the classes of forms

(recall that is injective mapping), and in general the classes of forms

Alternatively saying, the following forms constitute a basis:

and so on. Let us denote

In terms of this notation

we have the standard basis of .

Clearly and it follows that there is only a finite number of initial forms

with the lower zero indice. The following forms satisfy the recurrence and the (equivalent) congruence

In this sense, the linearly independent forms    are generalizations of the classical contact forms    of the jet theory.

Theorem 13. Let be a standard filtration of diffiety . Then the submodule is generated by all differentials .

Proof. First assume . Then whence . Clearly for appropriate . This implies , if therefore . It follows that contains all differentials .

Conversely let . Due to the equality , we have whence ), consequently

It follows that identically by using and (55). (Hint: look at assumed top order product where . Then involves only one summand with which is impossible since .) Therefore and the Frobenius theorem can be applied. Module has a basis consisting of total differentials.

Definition 14. We may denote since this module does not depend on the choice of the filtration (22). Together with the original basis occurring in (53), there exists alternative basis with differentials. In the particular case , hence, , we speak of a controllable diffiety .

Remark 15. The controllability is a familiar concept of the theory of underdetermined ordinary differential equations or Pfaffian systems in finite-dimensional spaces [12]; however, some aspects due to diffieties are worth mentioning here. If is a nontrivial module, the underlying space is fibered by the leaves depending on parameters. A curve is called a solution of diffiety , if . Since , we have

therefore every solution of diffiety is contained in a certain leaf (the Figure 2(a)).

In the controllable case, such foliation of the space does not exist. However, the construction of the standard filtration need not be of the “universal nature.” There may exist some “exceptional points” where the terms of the standard basis are not independent. We may even obtain a solution consisting of such exceptional points and then there appears the “infinitesimal leaf” of the noncontrollability along which means that is a Mayer extremal (the Figure 2(b)). We refer to article [13] inspired by the beautiful paper [14]. In the present paper, such exceptional points are tacitly excluded. They produce singularities of the symmetry groups and deserve a special, not yet available approach. It should be noted that the noncontrollable case also causes some technical difficulties. We may however suppose without much loss of generality since the noncontrollable diffiety can be restricted to a leaf and regarded as a diffiety depending on parameters .

Theorem 16. The total number of initial forms does not depend on the choice of the good filtration (22).

Proof. Filtration (22) differs from the standard filtration only in lower terms whence

Let another filtration of diffiety provide (corresponding standard filtration and therefore) certain number of (other) initial forms. Then

However for appropriate and whence

by using (23) and the equality easily follows.

5. On the Morphisms and Variations

A huge literature on the point symmetries (scheme (a) of Figure 3, the order of derivatives is preserved) of differential equations is available. On the contrary, we can mention only a few fundamental principles for the generalized (or higher-order) symmetries (scheme (c) of Figure 3) since the general theory deserves quite another paper. Our modest aim is to clarify a little the mechanisms of the particular examples to follow. We will also deal with generalized (or higher-order) groups of symmetries and the relevant generalized infinitesimal symmetries (scheme (b) Figure 3) with ambiguous higher-order invariant subspaces (the dotted lines). Figure 3 should be therefore regarded as a rough description of the topics to follow and we also refer to Section 9 for more transparent details. The main difficulty of the higher-order theory lies in the fact that the dotted domains are not known in advance. Modules represent the “natural” filtration with respect to the primary order of contact forms in the ambient jet space, see the examples. They depend on the accidental inclusion mentioned in Section 1 and do not have any true geometrical sense in the internal approach. It is to be therefore surprisingly observed that the seemingly “exotic” at the first glance concept of higher-order transformations of Section 1 should be regarded for reasonable and the only possible in the coordinate free theory. On the other hand, an important distinction between the group-like morphisms with large number of finite-dimensional invariant subspaces (scheme (a) and (b)) and the genuine order-destroying morphisms without such subspaces (scheme (c)) is of the highest importance.

We are passing to rigorous exposition. Let us recall the diffiety on the space , the independent variable with the corresponding vector field , the controllability submodule with the basis , and a standard basis of diffiety .

Let us begin with morphisms.

Lemma 17. If is a morphism of then and the recurrence

modulo holds true.

Proof. If is a morphism then therefore (use Theorem 13) and . It follows that

modulo and . This implies (61) by comparing both factors of .

Remark 18. On this occasion, the following useful principles of calculation are worth mentioning:

and in general

In terms of notation (21), we conclude that and therefore

if the morphism of diffiety is invertible.

Let us turn to invertible morphisms.

Lemma 19. The inverse of a morphism again is a morphism.

Proof. Assume ). Then

where . Hence and therefore .

We have if is a morphism and moreover hence in the invertible case. The converse and rather useful assertion is as follows.

Theorem 20. A morphism of diffiety is invertible if and only if .

This may be obtained easily from the following result.

Lemma 21. Let and . Then is invertible.

Proof. Proof of the Lemma 21 is analogous as in [2, Theorem 2] and we briefly recall only the main principles here. It is sufficient to prove the invertibility of .

Assuming then by virtue of recurrence (61). It follows that and is surjective. We prove that   is even injectivity by using the well-known algebraical interrelation between filtrations and gradations.

Let us introduce filtrations (, resp.) as follows: the submodule is generated by and all forms where . We also introduce the gradations

(formally ). It follows that the naturally induced mapping is surjective and it is sufficient to prove that this induced is also injective.

We are passing to the most delicate part of the proof. The surjectivity of implies that for large enough. Therefore by applying the recursion (61) which implies

On the other hand, assume the noninjectivity therefore the existence of a nontrivial identity

Then by applying operator and recurrence (61). Due to the existence of such identities, it follows that

and this is a contradiction.

Remark 22. Recall that if is a mapping and a submodule, then denotes the submodule with generators in accordance with the common practice in the algebraical module theory. Let in particular be a diffiety and assume for simplicity. Then module is generated by all forms and therefore by all forms , see Lemma 17. It follows that the invertibility of the morphism depends only on the properties of the forms , see Lemma 21. In this sense, the invertibility problem is reduced to the finite-dimensional reasonings.

We turn to the variations.

Lemma 23. A vector field is a variation of diffiety if and only if

and all are functions only of variables .

Proof. We suppose which is equivalent to the congruences

by using ((26) and (55)). So we have obtained (72) and moreover identities .

It is sufficient to prove that the latter identities imply . However, every differential can be represented as

in terms of the standard basis. Assuming in particular , we have already obtained the equation and then identities easily follow by applying the common rule together with (26). This concludes the proof.

Theorem 24. A variation of diffiety is infinitesimal symmetry of if and only if all forms are contained in a finite-dimensional module.

We omit lengthy proof and refer to more general results [5, Lemma 5.4, Theorem 5.6, and especially Theorem 11.1]. In future examples, we apply other and quite elementary arguments in order to avoid the nontrivial Theorem 24.

Remark 25. It follows from Lemma 23 that variations of diffiety can be represented by the universal series

where are arbitrary composed functions and are arbitrary functions in . We have explicit formulae for all variations (in common terms, for all Lie-Bäcklund infinitesimal symmetries) of a given system of ordinary differential equations. Recall that these variations need not generate any true group, and though the criterion in Theorem 24 is formally simple, it is not easy to be applied. Lemma 17 can be regarded as a counterpart to Lemma 23 since it ensures quite analogous result for the morphism or, better saying, for the pullback of a morphism. In more detail, the quite arbitrary choice of the initial terms of recurrence (61) is in principle possible but provides a mere formal result (corresponding to the formal nature of variations ) and does not ensure the existence of true morphism . We may refer to articles [2, 3] where the formal part (the algebra) is distinguished from the nonformal part (the analysis) in the higher-order algorithms.

We conclude this Section with the only gratifying result [9, point on page 40].

Theorem 26. The standard filtration is unique in the case .

Proof. Let us take a fixed filtration (22) and the corresponding standard filtration (46). Since , we have only one initial form and therefore is a basis of ; see (53). Let us take another standard filtration . Then the module has certain basis

These forms together with all generate the module and this is possible only if . We conclude that which implies hence for all .

Remark 27. It follows that in the particular case , every symmetry and infinitesimal symmetry preserves all terms of the (unique) standard filtration. So we have a large family of finite-dimensional subspaces of the underlying space which are preserved too. The classical methods acting in finite-dimensional spaces uniquely determined in advance can be applied and are quite sufficient in this case  .

Remark 28. In more generality, one could also consider two diffieties and on the underlying spaces and , respectively. Though we do not deal with the isomorphism problems of two diffieties and here, let us mention that such isomorphism is defined as invertible mapping of underlying spaces satisfying . Quite equivalent “absolute equivalence” problem was introduced in [15] and resolved just for the case (in our terminology) by using finite-dimensional methods. We have discovered alternative approach here: the isomorphism identifies the unique standard filtrations of and of . On this occasion, it is worth mentioning Cartan’s pessimistic notice (rather unusual in his work) to the case “Je dois ajourter que la géneralization de la théorie de l’equivalence absolu aux systémes differentiels dont la solution générale dépend de deux functions arbitraires d’un argument n’est pas immédiate et souléve d’asses grosses difficultiés.” The same notice can be literally repeated also for the theory of the higher-order symmetries treated in this paper.

6. The Order-Preserving Case of Infinitesimal Symmetries

We are passing to the first example which intentionally concerns the well-known “towering” problem in order to examine our method reliably. Let us deal with infinitesimal symmetries of differential equation

involving two unknown functions and . In external theory, (77) is identified with the subspace defined by the conditions

in the jet space . We use simplified notation of coordinates and contact forms

here. We are, however, interested in internal theory, that is, in the diffiety corresponding to (77). Diffiety appears if the contact forms

are restricted to the subspace . In accordance with the common practice, let us again simplify as

the notation of the restrictions to and moreover will be regarded as a vector field on from now on.

Let us outline the lengthy path of future reasonings for the convenience of reader. We begin with preparatory points (ι)–(ι ιι). The underlying space together with the diffiety is introduced and the standard basis , abbreviation of diffiety is determined. The standard basis is related to the “common” basis of by means of formulae (93). We obtain explicit representation (99) for the variations   with two arbitrary functions   and as the final result. Variations generating the true group (i.e., the infinitesimal symmetries   of ) satisfy certain strong conditions discovered in points and . The conditions are expressed by the resolving system (107) and (108) or, alternatively, by (112)–(114) only in terms of the functions , , , and . This rather complicated resolving system which does not provide any clear insight is equivalent to much simpler crucial requirements (121) or (125) on the actual structure of function ; see the central points (ν ι)–(ν ιι ι). Then the subsequent points are devoted to the explicit solution of these equations (125). This is a mere technical task of traditional mathematical analysis and we omit comments at this place.

(ι) The diffiety. Let us introduce space equipped with coordinates . Then

are merely composed functions. The forms

provide a basis of the diffiety ; however, all forms are also lying in as follows from the obvious rule:

and the inclusion .

(ι ι) Standard Filtration. There exists the “natural” filtration of diffiety with respect to the order: submodule involves the forms with . Alternatively saying, is a basis of and

Clearly if as follows from (84). However,

(Figure 4(a)) therefore

Then may be taken for a basis of module (Figure 4(b)).

Moreover

hence constitute a basis of module (Figure 4(c)) and finally

Therefore assuming

from now on, the form may be taken for a basis of module . We have obtained the standard filtration

where forms

provide a basis of module .

Abbreviating from now on, explicit formulae

where and

can be easily found. They will be sufficient in calculations to follow. Recall that we suppose that the inequality (90) hold true, hence .

(ι ιι) Variations. We deal with vector fields

(the notation (75) with indices is retained) on the space . Recall that is a variation if . In terms of coordinates, the conditions are

where the first and third equations are merely recurrences while the middle equation causes serious difficulties (a classical result. Hint: use ). By using the alternative formula

the conditions slightly simplify

(Hint: apply the rule to the forms However, by virtue of Lemma 23 and standard filtration, we have explicit formula

for the variations where and are arbitrary functions. One can then easily obtain explicit formulae for all coefficients in (97) and in (95) by using the left-hand identities (93). They need not be stated here.

(ι ν) Infinitesimal Transformations. We refer to Remark 27: variation   is infinitesimal symmetry if and only if

for appropriate multiplier . In explicit terms, we recall formula

where

and therefore clearly

So denoting

requirement (100) reads

where and

() should be moreover inserted. It follows that requirement (100) is equivalent to the so-called resolving system

Moreover and therefore is of the order 2 at most.

(ν) On the Resolving System. Equations (107) uniquely determine the multiplier and the “horizontal” coefficient in terms of the “vertical” coefficients , , , and . For instance the formula

easily follows. So we may focus on (108).

Equations (108) deserve more effort. They depend only on “vertical” components and can be expressed in terms of functions , , , and if the obvious identities

following from (93) together with the prolongation formula

are applied. By using the lucky identity (direct verification), one can obtain the alternative resolving system

only in terms of the unknown function . Recall that the resolving system is satisfied if and only if the vector field (99) is infinitesimal symmetry.

Our aim is to determine the function satisfying (112)–(114). Alas, the resolving system does not provide any insight into the true structure of function . It will be therefore replaced by other conditions of classical nature, the crucial requirements and the simplified requirements as follows.

(ν ι) Crucial Requirements. We start with simple formulae

Using moreover (94), one can see that there is a unique summand in (113) which involves the factor , namely the summand

It follows that identically and we (temporarily) may denote

The simplest equation (114) of the resolving system then reads

Clearly

where the reduced operator

appears and we obtain three so-called crucial requirements

for the functions , by inspection of the variable . Altogether taken, the last resolving equation (114) is equivalent to three requirements (121). We will see with great pleasure in (ν ιι ι) below that requirements (121) ensure even the remaining equations (112) and (113) of the resolving system.

(ν ιι) The Crucial Requirements Simplified. The right-hand equation (121) reads

and the middle equation (121) reads

whence altogether

The left-hand equation (121) does not change much; it may be expressed by .

Let us summarize our achievements. In order to determine function    given by (124), we have three simplified requirements

for the coefficients   and .

(ν ιι ι) Resolving System is Deleted. Let us recall the primary transcription (108) of the resolving system. We have already seen that (125) implies (114) and hence the equivalent and simplest right-hand equation (108).

Let us turn to the middle equation (108) equivalent to (113). One can directly find formulae

by using (124) and (125). Moreover

by using (124) and right-hand formulae (110). Substitution into middle equation (108) with

gives the identity.

As the right-hand equation (108) equivalent to (112) is concerned, we may use

where

Moreover

and (108) again becomes the identity.

(ι κ) Back to the Crucial Requirements. Passing to the final part of this example, let us eventually solve (125) with the unknown functions and given function . This is already a task of classical mathematical analysis. We abbreviate from now on since this variable frequently occurs in our formulae.

Let us begin with middle equation (125) which reads

whence

since is independent of variable due to the right-hand equation (125). We may insert

and the remaining left-hand equation (125) is expressed by the identity

Functions , , are independent of and thereby subjected to very strong conditions by the inspection of the coefficients of functions

in identity (135). The final result depends on the properties of function and we mention only a few instructive subcases here.

(κ) The Generic Subcase. Functions (136) are in general linearly independent over and identity (135) implies

The unknown functions and can be easily found as follows. We may suppose that

by using (140). Then ; hence due to (139). Moreover which implies ; hence and altogether

Then

follows from (138). Hence, due to (139) and altogether

Recalling moreover (133), we have explicit formulae for the solutions  ,   of crucial requirements (125) and the symmetry problem is resolved. While and are mere polynomials, the total coefficient given by (133) depends on the quadrature and this may be globally rather complicated function. It follows that, in our approach, the elementary and the “transcendental” parts of the solution are in a certain sense separated.

(κ ι) A Special Case of Function . Let us choose . Then series (136) becomes quite explicit; namely,

and these functions are linearly dependent. Identity (135) implies smaller number of requirements; the first term in (137) is combined with (139) into the single equation

without any other change. We can state the final solution

with only one additional parameter if compared to the previous formulae (144).

(κ ιι) Another Special Case. Let us eventually mention the very prominent function ; see [1, 7, 16]. Then the series

stands for (136) and the relevant identity (135) implies the system of equations

We are passing to the solution of the system of (149)–(153) with unknown functions and . Due to (153), we may put

and then (152) is expressed by , whence easily

Moreover (151) reads , whence

Remaining equations (149) and (150) do not admit such simple discussion. Using (154) and (155), identity (149) is equivalent to the system

of three equations and identity (150) is equivalent to the system

if (157) is moreover employed. At the same time, (155) can be improved as

With this improvement, (160) reads and it follows that

Analogously (158) reads

which is equivalent to the system

if (163) is inserted. Altogether, it follows that (158) is equivalent to

whence

At the same time, we have improvements

of the above formulae. Let us eventually turn to the remaining equations (159) and (161). We begin with (161) which can be simplified to

whence

Then the last requirement (159) is easily simplified as

and it follows that

whence easily

The solution is eventually done. It depends on the parameters

in the total number of 15. This is seemingly in contradiction with [1, 7, 16] where 14-dimensional symmetry group (namely, the exceptional simple Lie group ) was declared. However, our final symmetry in fact depends on the sum as follows from (166), (167), and (174) and therefore no contradiction appears. We will not explicitly state the resulting symmetries for obvious reason here. Recall that they are given by (99) where are clarified in (109) and (124). Coefficients appearing in (124) are clarified in (133), (156), and (170) and in (154), (157), (162), (168), (173), and (174).

It should be moreover noted that our approach is of the universal nature while the method of explicit calculations which provides the infinitesimal transformations in [7] rests on a lucky accident; see [7, Theorem 3.2, and the subsequent discussion].

Remark 29. Variations were easily found in . Due to Theorem 26 and Remark 27, infinitesimal symmetries satisfy moreover or, alternatively saying, they preserve the Pfaffian equation  , and this property was just employed. We will now prove the converse without use of Theorem 24. The reasoning is as follows. Let a variation    preserve Pfaffian equation  . Then preserves the space of adjoint variables  , , , , of this Pfaffian equation. In this finite-dimensional space, the variation generates a group which can be prolonged to the higher-order jet variables. It follows that is indeed an infinitesimal transformation.

Remark 30. Let us briefly mention the case as yet excluded by condition (90). In this linear case, clearly

and we may introduce standard filtration

where is a basis of and the forms

provide a basis of module . The symmetries can be easily found. They are the prolonged contact transformations defined by depending moreover on the parameter . Roughly saying, the geometry of the linear second-order equation is identical with the contact geometry of curves in . Quite analogous result can be obtained also for the Monge equation and, in much greater generality, for the system of two Pfaffian equations in four-dimensional space [17].

Remark 31. Let us once more return to the crucial requirement (125) where operators and are applied to unknown functions and . We have employed the simplicity of the second operator in the above solution; see formula (133). However, analogous “complementary” method can be applied to the first operator as follows. Let us introduce new variables

with the obvious inverse transformation (not stated here). Then

in terms of new variables. We again abbreviate . Passing to new coordinates, the left-hand requirement (125) is simplified as . The middle requirement (125) reads

and determines the function in terms of new variables as

where is constant of integration. This is a polynomial in variable   and it follows easily that the remaining right-hand requirement (130) applied to function is equivalent to the system

We will not discuss this alternative approach here in more detail.

Remark 32. Though the symmetries of (77) can be completely determined by applying the common methods, several formally quite different ways of the calculation are possible. It would certainly be of practical interest which of them is the “most economical” one. Let us mention such an alternative way for better clarity. We start with the “opposite” transcription

of (77). The primary concepts are retained, the same underlying space , diffiety , and contact forms , . However, we choose for new coordinates on from now on and the forms

for new basis of . We have moreover

in terms of new coordinates. The standard filtration is formally simplified. The forms

may be taken for new standard basis if the inequality is supposed. This follows from the obvious formulae

simplifying the analogous left-hand side (93). Then, analogously to (95) and (99), we introduce the variations

of diffiety where and may be arbitrary functions. Recall that we have even infinitesimal symmetry of if and only if the requirement (100) is satisfied. However clearly

and one can obtain the resolving equations as follows. First of all, we obtain equations

which determine coefficients and analogously to (107). Moreover

are conditions for the unknown function analogous to (108). The “vertical” coefficients can be expressed in terms of functions and , by using the equation

and the recurrence . As yet the calculations are much easier then for the above case of formulae (110); however, the resulting resolving system of three equations analogous to (112)–(114) is again complicated and will not be explicitly stated here. Remarkable task appears when we investigate the corresponding crucial requirements and try to determine the structure of function in terms of new coordinates. For instance, the “very prominent” and seemingly rather artificial case (κ ιι) turns into the “simplest possible” and quite natural equation in new coordinates.

7. Brief Digression to the Calculus of Variations

The classical Lagrange problem of the calculus of variations deals with an underdetermined system of differential equations (better with a diffiety) together with a variational integral. We are interested in internal symmetries of this variational problem.

Let us start with a diffiety . We choose a standard filtration and the corresponding standard basis . For better clarity, we suppose the controllable case . Let be an independent variable. Let us consider -parametrized solutions of diffiety in the sense

Here is a closed interval with a little confusion: letter denotes both a function on and the common coordinate (that is, a point) in .

Definition 33. A vector field is called a variation of solution of diffiety if . This is a mere slight adaptation of the familiar classical concept.

Lemma 34. A vector field is a variation of if and only if

Proof. A variation satisfies , where

by virtue of (55).

Remark 35. It follows easily that a vector field is a variation of diffiety in the sense of Definition 8 if and only if is a variation of every solution of ; see Lemma 23. Conversely, if is a variation of a solution then there exist many variations of such that at every point of , and they are characterized by the identities along the curve ; see formula (72). We conclude that the concepts “variation of ” and “variations of ” are closely related. Roughly saying, variations of are “restrictions” of variations of to the curve .

Definition 36. A couple where is a diffiety and is a differential form will be identified with a variational problem in the (common) sense that diffiety represents the differential constraints to the variational integral  . A solution of is called an extremal of this variational problem, if

for every variation of which is vanishing at the endpoints . This definition provides the common classical extremals; see Remark 43.

Remark 37. The phrase “variation of ” can be replaced with “variation of ”. The form can be replaced with arbitrary form . The extremals do not change.

Theorem 38. To every standard basis of and given there exists unique form such that

In accordance with (198) we assume that

Then a solution of is extremal if and only if and therefore if and only if

for all vector fields .

Proof (see [9]). For a given , let us look at a top-order summand

If , the summand can be deleted if the primary differential form is replaced with the new form . The extremals do not change. The procedure is unique and terminates in form satisfying (198). Then (200) follows from the identity

where the functions may be quite arbitrary if is a variation, see Lemma 34.

Definition 39. The differential form can be regarded for the internal Poincaré-Cartan form of our variational problem and equations for the Euler-Lagrange system.

We turn to the symmetries.

Definition 40. A symmetry of diffiety is called a symmetry of variational problem  , if .  A variation (infinitesimal symmetry) of is called a variation (infinitesimal symmetry, resp.) of variational problem  , if . Let be a variation of a solution   of diffiety . Then is called a Jacobi vector field of , if moreover . Roughly saying, variations of variational problem are “universal” Jacobi vector fields for all solutions of . In classical theory, Jacobi vector fields are introduced only for the particular case when is an extremal.

We will see in the following example that Poincaré-Cartan forms simplify the calculation of symmetries and variations. On this occasion, we also recall the following admirable result.

Theorem 41 (E. Noether). If is a variation of variational problem and is a Poincaré-Cartan form then . for every extremal .

Proof. We have , and therefore

by virtue of (200).

Remark 42. Many concepts of the classical calculus of variations lose the geometrical meaning if the higher-order symmetries are accepted; for example, this concerns the common concept of a nondegenerate variational problem and even the order of a variational integral. On the other hand, the most important concepts can be appropriately modified; for example, the Hilbert-Weierstrass extremality theory together with the Hamilton-Jacobi equations [18–21] since the Poincaré-Cartan forms make “absolute sense” along the extremals.

Remark 43. In the common classical calculus of variations, extremals are defined by the property , where variations satisfy certain weak boundary conditions at the endpoints (“fixed ends” or transversality) in order to delete some “boundary effects” of the variational integral. Much stronger conditions appear in Definition 36. Therefore

However, can be replaced by the form . Then

For the above special variations , the boundary term vanishes. If is extremal in the sense of Definition 36, then (200) and Remark 15 may be applied and it follows that

In topical Griffiths’ theory [22], extremals are defined by the property

which is clearly equivalent to the condition

where are vector fields vanishing at the endpoints. This condition trivially implies

with variations   vanishing at the endpoints; see Remark 35. Therefore

The converse inclusion

is, however, trivial since the universal form   satisfies even for every extremal in the sense of Definition 36. We conclude that all the mentioned concepts of extremals are identical. (We apologize for this hasty exposition. Roughly saying, the Griffiths’ theory and our approach are almost identical. The Griffiths’ correction    depending on   is made universal here. The classical approach rests on a special choice of boundary conditions for the variations . However, such a special choice is misleading since it does not affect the resulting family of extremals and we prefer a universal choice here as well.)

8. Particular Example of a Variational Integral

A simple illustrative example is necessary at this place. Let us again deal with diffiety of Section 6. So we recall coordinates of the underlying space , the contact forms , generating , the vector field

and the standard basis of . We moreover introduce variational integrals

Assuming and therefore

we introduce the functions

Then

is the Poincaré-Cartan form since the identity

can be directly verified. In accordance with formula (199) where is abbreviated, we have . Let us denote

for better clarity. The following simple result will be needed.

Lemma 44. Identity is equivalent to the equation with appropriate .

Proof. By virtue of (200), the identity is equivalent to the congruence . However, if the rule is applied to the congruence, it follows easily that identically. Therefore ) by using the Poincaré lemma.

Let us mention symmetries   and variations of our variational problem in more detail. In the favourable case , the task is not difficult.

The symmetry   of our variational problem clearly preserves the unique Poincaré-Cartan form and therefore also the vector field determined by the condition . We suppose here. It follows that all differential forms

are preserved, too. Let us moreover suppose . Clearly

Therefore is invariant basis of module in the sense

It follows that the symmetries    of our variational problem   can be comfortably determined. Quite analogous conclusion can be made for the infinitesimal symmetries, of course.

Passing to the variations   of the variational problem, we have explicit formula (99) for the variations of and moreover condition equivalent to . However,

and therefore

Assume . We obtain condition for the unknown function . In more precise notation and in full detail

This is formally a very simple condition concerning the unknown function ; alas, it is not easy to be resolved. Paradoxically, variations cause serious difficulties.

For better clarity, we continue this example with particular choice of the variational integral. Let us consider variational integral . Equation (214) reads

and it follows that

by using (93). We have and therefore

by virtue of (215)–(218). Both the Poincaré-Cartan form and the Euler-Lagrange equation can be expressed in terms of common coordinates, if derivatives (226) are inserted. We omit the final formulae here. Passing to the symmetries , we may simulate the moving frames method and express the differential

in terms of the invariant basis (221). Then all coefficients are invariants of symmetry  ; that is,

In fact we have obtained all invariants. (Hint: for instance, differential

does not provide any novelty.) It follows that the symmetry problem is resolved. Compatibility of the system of (229) ensures the existence of symmetries    of the variational problem    since the Frobenius theorem can be applied to the Pfaffian system (221). In the most favourable case, are even constants. Explicit calculation of invariants is a lengthy but routine procedure. First of all

by using the primary formula (216). Then

may be substituted where the coefficient can be determined analogously as in (226). As the differential

is concerned, we refer to formula in Section 6. The contact forms must be replaced with the standard basis by using the right-hand formulae (93). Then we may use the lucky identity

in order to determine the last summand in (231). In the end, the standard basis in (231) can be easily replaced by the invariant forms and we are done.

9. The Order-Increasing Case

Let us eventually return to the main topic, the differential equations. We will finish this paper with decisive examples of higher-order symmetries, namely, with symmetries of the Monge equation

involving three unknown functions , and . Let us directly turn to the internal theory carried out by using the underlying space with coordinates

diffiety with the basis

and the total derivative

We also introduce functions and differential forms

for the formal reasons. The natural filtration in accordance with the order is such that the forms are taken for the basis of submodule . Let us determine the corresponding standard filtration . Clearly

and therefore

Denoting , we obtain

We will not deal with the case when identically. Let us instead suppose that from now on. Then and we may introduce standard filtration of diffiety where the form

generates and in general the forms

generate module . Notation (53) with indices is retained here. With this preparation, we are passing to the symmetries of diffiety . Theorem 26 and Remark 27 fail since in our case. There exist many standard filtrations of and we may also expect the existence of the order-destroying symmetries.

The preparation is done; however, before passing to quite explicit examples, certain general aspects are worth mentioning. We recall Figure 3 which can be transparently illustrated just at this place for the first time.

First of all, every order-preserving symmetry   on scheme (a) of Figure 3 obviously satisfies certain formulae

where the coefficients cannot be in fact arbitrary since they are subjected to identity (240). In more detail, we have

in accordance with (63). Alternatively (240) implies

where the forms can be expressed in terms of forms . The comparison provides many unpleasant interrelations among coefficients .

However, by using the standard basis, the same symmetry satisfies shorter formulae

with coefficients subjected only to the inequalities and at this place. We employ the fact that both triples and are bases of module . Moreover preserves the natural filtration and therefore also the corresponding standard filtration . Especially, the initial term is preserved and is a mere multiple of .

The order-preserving infinitesimal symmetry   corresponding to scheme (a) satisfies either the system

with coefficients subjected to many identities analogous as above or, alternatively, the equivalent and shorter system

with arbitrary coefficients in terms of the standard basis. For the middle equation use the identity

This follows from the Lie bracket formula which is true if and only if is a variation of diffiety . The Cartan’s general equivalence method [23] can be applied to this order-preserving symmetry problem; however, we will mention the Lie approach later on.

With this result, the simplest possible order-increasing symmetry   on scheme (c) of Figure 3 can be introduced by the equations

Let us prove the invertibility of . Clearly

and it follows that

Inclusions therefore hold true and Lemma 21 can be applied.

In order to state another example to scheme (c), let us consider the equations

Invertibility of such morphism is ensured if for appropriate factors and if moreover

For the proof of invertibility, apply to the inclusion

and verify that .

In both examples, the common general equivalence method [23] fails. The corresponding variations   can be introduced and are rather interesting though they do not generate any symmetry groups. See Remark 46 below.

It is also easy to illustrate scheme (b) of Figure 3 by using the symmetries and infinitesimal symmetries such that

(Hint: Theorem 24 can be trivially applied and the natural filtration is not preserved, if and .) Another example is provided by the equations

“symmetrical” to the order-preserving case. The classical Lie’s infinitesimal symmetries and the Cartan’s equivalence method can be both applied without any change.

We have briefly indicated only the simplest devices here and refer to [2, Section 4] for the universal construction. A complete overview of all possible higher-order symmetries of (235) is lying beyond any actual imagination. For instance, the composition of symmetries and the conjugate groups to a given group provide much more complicated examples than the original components and . The definition equations for such composition of symmetries can be directly found and they look rather depressively for the time being.

10. Concluding Examples on Infinitesimal Symmetries

We deal only with a simplified equation (235), namely, with the equation

for good reasons to be clarified in the Appendix. Let us abbreviate

from now on. The crucial identity (240) then reads

and we recall the standard basis , where

in terms of the simplified notation. The formulae

easily follow. On this occasion, we also recall more general adjustments

of Lemmas 17 and 23. The factor appearing here can be defined by the congruence ) as well.

Several symmetry problems for (260) will be mentioned. We start with examples on infinitesimal symmetries   and demonstrate our approach both using the traditional order-preserving case and then employing two technically quite analogous order-increasing symmetry problems. The calculations are elementary but not of a mere mechanical nature and the concise form of the final results is worth attention. That is, by using the series (268) with the standard basis, the unknown functions , and satisfy quite reasonable and explicitly solvable conditions. Denoting

we simulate the procedure of Section 6 and our method again rests on the explicit formula

for all variations  . We recall that infinitesimal symmetries   moreover satisfy certain additional requirements in order to ensure the conditions of Theorem 24. The choice of such requirements which is arbitrary to a large extent (dotted lines in Figure 3(b)) strongly affects the final result, the resulting symmetries . Altogether taken, reasonings of this Section 10 belong to the Lie’s theory appropriately adapted to the infinite-dimensional spaces. On the contrary, we will conclude this paper with only few remarks on the true (not group-like) higher-order symmetries   in subsequent Section 11. The reasonings can be related to the E. Cartan’s general equivalence method [16, 23] and they would deserve more space than it is possible here.

Let us turn to proper examples.

(ι) The Order-Preserving Symmetry Problem. We again intentionally start with a mere “traditional” case. Let us deal with infinitesimal symmetries satisfying

We use the “hybrid” equations involving both the standard basis and the contact forms. Let us recall the explicit formula (268) for all variations. We have moreover the above equations (269) in order to obtain the true infinitesimal symmetries. In more detail

should be satisfied. Analogously as in Section 6, this is expressed by the resolving system

by using (264) and . It follows that only (271) with inserted and coefficients given by

are the most important.

Let us denote and assume    from now on. Equations (271) are equivalent to

We have unknown functions

and let us pass to the solution of (273), (274), and (275).

The first equation (273) multiplied by function reads

and therefore implies only the identity

if both equations (275) are accepted (direct verification). Alternatively saying, second equation (275) can be regarded for a definition of function   if (278) is taken into account. Let us denote for a moment. Then

and the second equation (274) reads

with the same determinant. Lower-order terms clearly provide the equation

and coefficients of give

for appropriate function ; however trivially . With this result, we obtain

by inspection of coefficients of . It follows that , whence

Analogously the lower-order terms of the first equation (274) give

while the second-order terms do not provide any new requirements.

Let us finally recall (275) with and given by (284) inserted. These equations turn into the compatible system

for the function with the solution

Then (278) is expressed by the crucial requirement

for the functions , , and . One can moreover verify with the help of

that the remaining equations (281) and (285) become identities.

Let us summarize our achievements. Assuming  ,  all infinitesimal symmetries (268) are determined by formula (287), the second equation (284) and   with functions    of variables    satisfying (288).

Traditional methods are sufficient to analyze thoroughly (288). Passing to more details, we have

Analogously as in Section 6, the large series of coefficients

appears. If these functions are -linearly independent, only the solution such that

is possible. It follows that

where are arbitrary constants. This result provides the obvious symmetries which are self-evident at a first glance, the coordinate shifts and the similarity.

For a special choice of function , the symmetry group may be very large and less trivial. We can mention the case . Then the arising system of five equations

for the unknown functions

can be resolved by

where are arbitrary constants.

We omit more examples, in particular the interesting cases (with -linear dependence of functions ) where the infinitesimal symmetries depend on arbitrary functions and the “degenerate” cases when either or identically.

(ι ι) The Order-Increasing Infinitesimal Symmetry. Let us mention variations (268) satisfying moreover the equations

which provide the order-increasing case, if . One can then obtain the resolving system

for the unknown functions

and moreover formula

for the coefficient . We mention only the particular case . Then the resolving system reads and admits the solution

where the functions and may be arbitrarily chosen. Since the above coefficient

does not in general vanish, we have a large family of order-increasing infinitesimal symmetries.

(ι ιι) Another Order-Increasing Case. Let us mention variations (268) satisfying the equations

which provide an order-increasing case if . The resolving system

for the unknown functions

looks more complicated. One can also obtain the formula

for the important coefficient . Let us again mention only the particular case . Then the resolving system is simplified as

It follows immediately that and the resolving system is reduced to the equations

for the unknown functions and of variables . This implies that , where and we obtain two equations

with the solution , where may be arbitrary function while is a fixed particular solution of differential equation

satisfying moreover the identity . We may choose the particular solution . Then the identity turns into the requirement which is satisfied if

Altogether taken, we have obtained the final solution

where

and , , are quite arbitrary functions. The above-mentioned coefficient does not in general vanish. (Indeed, look at the top-order summands

where by virtue of (263); hence, may be substituted.) We again have an order-increasing infinitesimal symmetry.

Remark 45. Variations satisfying (269) preserve the Pfaffian system and therefore generate a group for analogous reasons as in Remark 29. Variations satisfying (297) preserve the Pfaffian system and the case of requirements (303) is quite trivial in this respect. It follows that we have indeed obtained the infinitesimal symmetries  .

11. Concluding Example on Order-Increasing Symmetries

Passing from infinitesimal symmetries to the true symmetries , the linear theory is replaced with highly nonlinear area of Pfaffian equations and the prolongation into involutiveness. In accordance with E. Cartan’s notice, nobody should expect such easily available results as in the Lie’s infinitesimal theory. Our modest aim is twofold: to perform an economical reduction of the symmetry problem to finite dimension and to point out a useful interrelation between appropriate variations   and one-parameter families   of higher-order symmetries. We again deal only with (260).

(ι) Setting the Problem. Let us deal with symmetries such that

Invertibility of is obviously ensured if

where

We tacitly suppose and one can observe that the particular case provides the traditional order-preserving symmetries. Equations (315) can be simplified to the equivalent system of equations

where and . The invertibility is ensured by the inequalities

Equations (318) will be represented by a Pfaffian system in a certain finite-dimensional space; however, let us again simplify the notation by bars; for example,

and so like. Then we have the system

which should be completed by the exterior derivatives

We refer to (264) for terms , appearing here. We have obtained the compatibility problem of (322). The familiar prolongation criterion can be shortly expressed as follows. All coefficients and variables with bars are functions of the primary jet variables. So we may suppose, for example,

(with summands of uncertain lengths) and analogously for with a little adjustment for the differential

Such substitutions into (322) should give identities. However, a short inspection of the summand in differential implies that then necessarily either (the group case) or identically.

(ι ι) A Particular Case. It follows that the assumption is necessary; however, let us again suppose from now on. Then (321) may be retained and (322) become more explicit

We turn to the prolongation procedure in more detail.

(ι ιι) On the Equation (326). The prolongation should satisfy the identity

where . All summands with factor mutually cancel. Then we conclude that necessarily and also . It follows that the problem is reduced to finite dimension:   are functions only of coordinates . Even the explicit formulae can be easily obtained as follows

for the prolongation where moreover is supposed.

(ι ν) On the Equation (325). Calculations modulo are also sufficient here. The prolongation should satisfy

where . We have used the identity

where is moreover substituted. In order to prove the existence of prolongation, the right-hand side terms should be made more explicit. We recall the identity (263) which gives

and uniquely determines the forms and in terms of forms and therefore in terms of contact forms , and , if the rule (265) is applied to the primary equations (321). The result is that

On the other hand, inequalities (266) imply that the factors

on the left-hand side of (329) are linearly independent and we conclude that the prolongation can be realized. Moreover becomes a function of second-order coordinates while and are functions of first-order coordinates as before. The problem is again reduced to finite-dimension; however, we do not state explicit formula for the prolongation here.

Remark 46. In accordance with Lie’s classical theory, the existence of infinitesimal symmetries   (Figure 5(a)) is equivalent to the existence of a one-parameter group   of symmetries (Figures 3(a) and 3(b)) due to the solvability of the Lie system ensured by Theorem 24. Alas, the “genuine” higher-order symmetries (Figure 3(c)) cannot be obtained in this way and they rest on the toilsome mechanisms of Pfaffian systems. We nevertheless propose a hopeful conjecture as follows. Every one-parameter family   of symmetries ensures the existence of many variations   depending on parameter (Figure 5(b)). We believe that the converse can be proved as well: one-parameter families of symmetries can be reconstructed from a “sufficiently large” supply of variations. Indeed, if is regarded as additional variable of the underlying space, then the family turns into a single vector field.

In any case, the existence of many variations is a necessary condition for the existence of “genuine” higher-order symmetries and the following point will be instructive in this respect.

(ν) On the Variations. If a one-parameter family (abbreviation) satisfies (318) then the corresponding family (abbreviation) of variations clearly satisfies the system

where may be regarded as new parameters. Assuming formula (268), one can obtain the resolving system

It follows from right-hand equations (337) that , where do not depend on . Recalling the identity

then the middle equations (337) yield the conditions

and . With this result, (335) turns into identity and (336) reduces to the equation

for the parameter . Equations (339) are trivially satisfied if

There exist many variations corresponding to (318). The necessary condition for the existence of higher-order symmetries is satisfied.

Remark 47. Let us briefly sketch the connection to the general equivalence method [23] by using slightly adapted Cartan’s notation. We consider space (and its counterpart ) with coordinates , resp.) and linearly independent 1-forms . In the classical equivalence problem, a mapping should be determined such that

where is a matrix of a linear group with parameters . In Cartan’s approach, this requirement is made symmetrical:

This provides the invariant differential forms by appropriate simultaneous adjustments of both sides (343). Such procedure fails, if is not a matrix of a linear group which happens just in the case of higher-order symmetries on Figure 3(c). Then the corresponding total system (342) with is invertible only in the infinite-dimensional underlying space and need not be even a square matrix in any finite portion of the system (342). On the other hand, such a finite portion is quite sufficient since Lemma 17 ensures the extension on the total space . The “symmetrization” procedure cannot be applied, invariant differential forms need not exist, and only the common prolongation procedure is available, if the problem is reduced to a finite-dimensional subspace of .

12. Concluding Survey

Our approach to differential equations and our methods differ from the common traditional use. For better clarity, let us briefly report the main novelties as follows: clear interrelation between the external and internal concepts in Remarks 1 and 2; introduction and frequent use of “nonholonomic” series (18); the “absolute” and coordinate-free Definition 4 of ordinary differential equations; the distinction between variations and infinitesimal symmetries in Definition 8; the main tool, the standard bases generalizing the common contact forms in jet spaces; the invariance of constants and , the controllability concept related to the Mayer problem; the distinction between order-preserving, group-like, and true higher-order symmetries in Figure 1; technical Lemmas 17, 19, and 23 and Theorem 24 which provide new universal method of solution of the higher-order symmetry problem; new explicit formula as (136) for the famous and “well-known” symmetry problem of a Monge equation with two unknown functions; the Lagrange variational problem without Lagrange multipliers and with easy proofs; see Theorem 41; particular results of new kind for the Monge equation with three unknown functions; a note on the insufficience of -structures in Remark 47.

Figure 1

(a) Non-contrillable case

(b) Mayer extermal

Figure 2

Figure 3

Figure 4

(a) Vector field

(b) Variations

Figure 5

All these achievements can be carried over the partial differential equations.

On this occasion, the actual extensive theory of the control systems

is worth mentioning. It may be regarded as a mere formally adapted individual subcase of the theory of underdetermined systems of ordinary differential equations. However, the exceptional role of the independent variable   (the change of notation), the state variables  , and the control   is emphasized in applications; see [24–26] and references therein. In particular, only the -preserving and moreover -independent symmetries of the system (344) are accepted. So in our notation (1), such restriction means that we suppose and functions are independent of . This is a fatal restriction of the impact of the theory of control systems. It follows that the results of this theory do not imply the classical results by Lie and Cartan; they are of rather special nature. The lack of new effective methods adapted to the control systems theory should be moreover noted. The absence of explicit solutions of particular examples is also symptomatic. Last but not least, unlike our diffieties, the control systems cannot be reasonably generalized for the partial differential equations.

We believe that the internal and higher-order approach to some nonholonomic theories are possible, for instance, in the case of the higher-order subriemannian geometry [12]. It seems that the advanced results [27] in the theory of geodesics can be appropriately adapted and rephrased in terms of invariants (as in [28]) instead of adjoint tensor fields.

Appendix

A nontrivial automorphism   of the jet space related to the theory of differential equation (260) is worth mentioning [9, pp. 44–46] without additional comments. In terms of usual jet coordinates on the space , we put

and moreover

The morphism is rigorously defined since the transforms

are well-known due to the prolongation (6). The point of construction is as follows. We have

So, assuming the invertibility of ,  differential equations (260) are identified with subspaces   given by equations  . Every such a subspace with given . is clearly isomorphic to the jet space . We conclude that the diffiety   corresponding to given equation (260) is isomorphic to the diffiety   of all curves in three-dimensional space   and therefore admits huge supply of higher-order symmetries; see [4, Section 7] for quite simple examples.

Let us turn to the invertibility problem. We introduce a morphism   which will be identified with the sought inverse . The definition is as follows. Let us introduce functions determined by three linear equations

It follows that functions moreover satisfy

whence the equations

uniquely define function . We finally put

and then

It follows that functions and hence and even the coordinate can be expressed in terms of certain pull-backs . Therefore is invertible and moreover . Indeed, the last three equations read

in full accordance with the initial equations (A.1) and formula (A.2) follows from (A.8).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper was elaborated with the financial support of the European Union’s “Operational Programme Research and Development for Innovations,” no. CZ.1.05/2.1.00/03.0097, as an activity of the regional centre AdMaS “Advanced Materials, Structures and Technologies.”

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