The Lie Group in Infinite Dimension

and Applied Analysis 3


Preface
In the symmetry theory of differential equations, the generalized or: higher-order, Lie this matter of fact, they cause an unpleasant feeling.Indeed, such vector fields as a rule do not generate any one-parameter group of transformations x i G i λ; . . ., x i , w need not have any reasonable locally unique solution.Then Z is a mere formal concept 1-7 not related to any true transformations and the term "infinitesimal symmetry Z" is misleading, no Z-symmetries of differential equations in reality appear.
In order to clarify the situation, we consider one-parameter groups of local transformations in Ê ∞ .We will see that they admit "finite-dimensional approximations" and as a byproduct, the relevant infinitesimal transformations may be exactly characterized by certain "finiteness requirements" of purely algebraical nature.With a little effort, the multidimensional groups can be easily involved, too.This result was briefly discussed in 8, page 243 and systematically mentioned at several places in monograph 9 , but our aim is to make some details more explicit in order to prepare the necessary tools for systematic investigation of groups of generalized symmetries.We intend to continue our previous articles 10- 13 where the algorithm for determination of all individual generalized symmetries was already proposed.
For the convenience of reader, let us transparently describe the crucial approximation result.We consider transformations 2.1 of a local one-parameter group in the space Ê ∞ with coordinates h 1 , h 2 , . ... Equations 2.1 of transformations m λ can be schematically represented by Figure 1 a .
We prove that in appropriate new coordinate system F 1 , F 2 , . . . on Ê ∞ , the same transformations m λ become block triangular as in Figure 1 b .It follows that a certain hierarchy of finite-dimensional subspaces of Ê ∞ is preserved which provides the "approximation" of m λ .The infinitesimal transformation Z dm λ /dλ| λ 0 clearly preserves the same hierarchy which provides certain algebraical "finiteness" of Z.
If the primary space Ê ∞ is moreover equipped with an appropriate structure, for example, the contact forms, it turns into the jet space and the results concerning the transformation groups on Ê ∞ become the theory of higher-order symmetries of differential equations.Unlike the common point symmetries which occupy a number of voluminous monographs see, e.g., 14, 15 and extensive references therein this higher-order theory was not systematically investigated yet.We can mention only the isolated article 16 which involves a direct proof of the "finiteness requirements" for one-parameter groups namely, the result ι of Lemma 5.4 below with two particular examples and monograph 7 involving a theory of generalized infinitesimal symmetries in the formal sense.
Let us finally mention the intentions of this paper.In the classical theory of point or Lie's contact-symmetries of differential equations, the order of derivatives is preserved Figure 2 a .Then the common Lie's and Cartan's methods acting in finite dimensional spaces given ahead of calculations can be applied.On the other extremity, the generalized symmetries need not preserve the order Figure 2 c and even any finite-dimensional space and then the common classical methods fail.For the favourable intermediate case of groups of generalized symmetries, the invariant finite-dimensional subspaces exist, however, they are not known in advance Figure 2 b .We believe that the classical methods can be appropriately adapted for the latter case, and this paper should be regarded as a modest preparation for this task.

Fundamental Approximation Results
Our reasonings will be carried out in the space Ê ∞ with coordinates h 1 , h 2 , . . .9 and we introduce the structural family F of all real-valued, locally defined and C ∞ -smooth functions f f h 1 , . . ., h m f depending on a finite number of coordinates.In future, such functions will contain certain C ∞ -smooth real parameters, too.
We are interested in local groups of transformations m λ in Ê ∞ defined by formulae where of functions H i λ j ; h 1 , . . ., h m i locally attains the maximum for appropriate choice of parameters .This rank and therefore the subset F I, ε ⊂ F does not depend on ε as soon as ε ε I is close enough to zero.This is supposed from now on and we may abbreviate F I F I, ε .We deal with highly nonlinear topics.Then the definition domains cannot be kept fixed in advance.Our results will be true locally, near generic points, on certain open everywhere dense subsets of the underlying space Ê ∞ .With a little effort, the subsets can be exactly characterized, for example, by locally constant rank of matrices, functional independence, existence of implicit function, and so like.We follow the common practice and as a rule omit such routine details from now on.

Lemma 2.2 approximation lemma . The following inclusion is true:
Proof.Clearly Denoting by K I the rank of matrix 2.4 , there exist basical functions such that rank ∂F k /∂h j K I .Then a function f ∈ F lies in F I if and only if f f F 1 , . . ., F K I is a composed function.In more detail is such a composed function if we choose f F given by 2. and analogously for the higher derivatives.

2.12
The basical functions can be taken from the family of functions H i λ; . . .i 1, . . ., I for appropriate choice of various values of λ.Functions 2.12 are enough as well even for a fixed value λ, for example, for λ 0, see Theorem 3.2 below.

Lemma 2.3. For any basical function, one has
Proof.F k ∈ F I implies m λ * F k ∈ F I and 2.9 may be applied with the choice F m λ * F k and λ 1 • • • λ J λ.

Summary 1.
Coordinates h i H i 0; . . .i 1, . . ., I were included into the subfamily F I ⊂ F which is transformed into itself by virtue of 2.13 .So we have a one-parameter group acting on F I .One can even choose F 1 h 1 , . . ., F I h I here and then, if I is large enough, formulae 2.13 provide a "finite-dimensional approximation" of the primary mapping m λ .The block-triangular structure of the infinite matrix of transformations m λ mentioned in Section 1 appears if I → ∞ and the system of functions F 1 , F 2 , . . . is succesively completed.

The Infinitesimal Approach
We introduce the vector field

3.2
In more explicit and classical transcription One can also check the general identity Let us finally reformulate the last sentence in terms of basical functions.Theorem 3.5 approximation theorem .Let Z ∈ be a vector field locally defined on Ê ∞ and F 1 , . . ., F K I ∈ F be a maximal functionally independent subset of the family of all functions Z r h i i 1, . . ., I; r 0, 1, . . . .

3.13
Denoting ZF k F k F 1 , . . ., F K I , then the system

On the Multiparameter Case
The following result does not bring much novelty and we omit the proof.As yet we have closely simulated the primary one-parameter approach, however, the results are a little misleading: the uniformity requirement in Summary 3 may be completely omitted.This follows from the following result 9, page 30 needless here and therefore stated without proof.Theorem 4.2.Let K be a finite-dimensional submodule of the module of vector fields on Ê ∞ such that K, K ⊂ K. Then K ⊂ if and only if there exist generators (over F) of submodule K that are lying in .

Symmetries of the Infinite-Order Jet Space
The previous results can be applied to the groups of generalized symmetries of partial differential equations.Alas, some additional technical tools cannot be easily explained at this place, see the concluding Section 11 below.So we restrict ourselves to the trivial differential equations, that is, to the groups of generalized symmetries in the total infinite-order jet space which do not require any additional preparations.

5.1
Functions f f . . ., x i , w So we intentionally distinguish between true infinitesimal transformations generating a group and the formal concepts; this point of view and the terminology are not commonly used in the current literature.
Remark 5.2.A few notes concerning this unorthodox terminology are useful here.In actual literature, the vector fields 5.6 are as a rule decomposed into the "trivial summand D" and the so-called "evolutionary form V " of the vector field Z, explicitly The summand D is usually neglected in a certain sense 3-7 and the "essential" summand V is identified with the evolutional system of partial differential equations the finite subsystem with I φ empty is enough here since the remaining part is a mere prolongation .This evolutional system is regarded as a "virtual flow" on the "space of solutions" w j w j x 1 , . . ., x n , see 7, especially page 11 .In more generality, some differential constraints may be adjoint.However, in accordance with the ancient classical tradition, functions δw j ∂w j /∂λ are just the variations.There is only one novelty: in classical theory, δw j are introduced only along a given solution while the vector fields Z are "universally" defined on the space.In this "evolutionary approach", the properties of the primary vector field Z are utterly destroyed.It seems that the true sense of this approach lies in the applications to the topical soliton theory.However, then the evolutional system is always completed with boundary conditions and embedded into some normed functional spaces in order to ensure the existence of global "true flows".This is already quite a different story and we return to our topic.
In more explicit terms, morphisms 5.5 are characterized by the implicit recurrence where det D i G i / 0 is supposed and vector field 5.6 is a variation if and only if Recurrence 5.9 easily follows from the inclusion m λ * ω j I ∈ Ω m, n and we omit the proof.Recurrence 5.10 follows from the identity appearing on this occasion also is of a certain sense, see Theorem 5.5 and Section 10 below.It follows that the initial functions G i , G j , z i , z j empty I φ may be in principle arbitrarily prescribed in advance.This is the familiar prolongation procedure in the jet theory.Proof.Inclusion Z ∈ is defined by using the families 5.13 and this trivially implies ι where only the empty multi-indice I φ is involved.Then ι implies ιι by using the rule L Z df dZf.Assuming ιι , we may employ the commutative rule in order to verify identities of the kind and in full generality identities of the kind with unimportant coefficients, therefore ιιι follows.Finally ιιι obviously implies the primary requirement on the families 5.13 .This is not a whole story.The requirements can be expressed only in terms of the structural contact forms.With this final result, the algorithms 10-13 for determination of all individual morphisms can be closely simulated in order to obtain the algorithm for the determination of all groups m λ of morphisms, see Section 10 below.involves only a finite number of linearly independent terms.Some nontrivial preparation is needful for the proof.Let Θ be a finite-dimensional module of 1-forms on the space M m, n but the underlying space is irrelevant here .Let us consider vector fields X such that L fX Θ ⊂ Θ for all functions f.Let moreover Adj Θ be the module of all forms ϕ satisfying ϕ X 0 for all such X.Then Adj Θ has a basis consisting of total differentials of certain functions f 1 , . . ., f K the Frobenius theorem , and there is a basis of module Θ which can be expressed in terms of functions f 1 , . . ., f K .Alternatively saying, an appropriate basis of the Pfaffian system ϑ 0 ϑ ∈ Θ can be expressed only in terms of functions f 1 , . . ., f K .This result frequently appears in Cartan's work, but we may refer only to 9, 18, 19 and to the appendix below for the proof.
Module Adj Θ is intrinsically related to Θ: if a mapping m preserves Θ then m preserves Adj Θ.In particular, assuming is true for a group m λ .In terms of IT of the group m λ , we have equivalent assertion and therefore L r Z Adj Θ ⊂ Adj Θ for all r.The preparation is done.
In order to finish the proof, let us on the contrary assume that Adj Θ does not contain all differentials dx 1 , . . ., dx n .Alternatively saying, the Pfaffian system ϑ 0 ϑ ∈ Θ can be expressed in terms of certain functions f 1 , . . ., f K such that df 1  • • • df K 0 does not imply dx 1 • • • dx n 0. On the other hand, it follows clearly that maximal solutions of the Pfaffian system can be expressed only in terms of functions f 1 , . . ., f K and therefore we do not need all independent variables x 1 , . . ., x n .This is however a contradiction: the Pfaffian system consists of contact forms and involves the equations ω 1 • • • ω n 0. All independent variables are needful if we deal with the common classical solutions w j w j x 1 , . . ., x n .
The result can be rephrased as follows.
Theorem 5.6.Let Ω 0 ⊂ Ω m, n be the submodule of all zeroth-order contact forms ω a j ω j and Z be a variation of the jet structure.Then Z ∈ if and only if dim ⊕L r Z Ω 0 < ∞.

On the Multiparameter Case
Let us temporarily denote by Î the family of all infinitesimal variations 5.6 of the jet structure.

The Order-Preserving Groups in Jet Space
Passing to particular examples from now on, we will briefly comment some well-known classical results for the sake of completeness.
Let Ω l ⊂ Ω m, n be the submodule of all contact forms ω a j I ω j I sum with |I| ≤ l of the order l at most.A morphism 5.5 and the infinitesimal variation 5.6 are called order preserving if respectively, for a certain l 0, 1, . . .equivalently: for all l ∈ AE, see Lemmas 9.1 and 9.2 below .Due to the fundamental Lie-Bäcklund theorem 1, 3, 6, 10-13 , this is possible only in the pointwise case or in the Lie's contact transformation case.In quite explicit terms: assuming 7.1 then either functions G i , G j , z i , z j empty I φ in formulae 5.5 and 5.6 are functions only of the zeroth-order jet variables x i , w j or, in the second case, we have m 1 and all functions G i , G 1 , G 1 i , z i , z 1 , z 1 i contain only the zeroth-and first-order variables x i , w 1 , w 1 i .A somewhat paradoxically, short proofs of this fundamental result are not easily available in current literature.We recall a tricky approach here already applied in 10-13 , to the case of the order-preserving morphisms.The approach is a little formally improved and appropriately adapted to the infinitesimal case.Proof.We suppose Proof.Let us introduce module Θ of m 2n -forms generated by all forms of the kind where are true by virtue of 7.2 and imply L Z Θ ⊂ Θ.
Module Θ vanishes when restricted to certain hyperplanes, namely, just to the hyperplanes of the kind ϑ a i dx i a j dw j 0 7.6 use m ≥ 2 here .This is expressed by Θ ∧ ϑ 0 and it follows that Therefore L Z ϑ again is such a hyperplane: L Z ϑ ∼ 0 mod all dx i and dw j .On the other hand, L Z ϑ ∼ a i dz i a j dz j mod all dx i and dw j 7.8 and it follows that dz i , dz j ∼ 0.
There is a vast literature devoted to the pointwise transformations and symmetries so that any additional comments are needless.On the other hand, the contact transformations are more involved and less popular.They explicitly appear on rather peculiar and dissimilar occasions in actual literature 20, 21 .However, in reality the groups of Lie contact transformations are latently involved in the classical calculus of variations and provide the core of the Hilbert-Weierstrass extremality theory of variational integrals.

Digression to the Calculus of Variations
We establish the following principle.Theorem 8.1 metatheorem .The geometries of nondegenerate local one-parameter groups of Lie contact transformations CT and of nondegenerate first-order one-dimensional variational integrals VI are identical.In particular, the orbits of a given CT group are extremals of appropriate VI and conversely.
Proof.The CT groups act in the jet space M 1, n equipped with the contact module Ω 1, n .Then the abbreviations are possible.Let us recall the classical approach 22, 23 .The Lie contact transformations defined by certain formulae to appear later, involves variables from quite other jet space M n, 1 with coordinates denoted t the independent variable , y 1 , . . ., y n the dependent variables and higher-order jet variables like y i , y i and so on.We are passing to the topic proper.Let us start in the space M 1, n with CT groups.One can check that vector field 5.6 is infinitesimal CT if and only if where the function Q Q x 1 , . . ., x n , w, w 1 , . . ., w n may be arbitrarily chosen.
"Hint: we have, by definition where Alas, the corresponding Lie system not written here is not much inspirational.Let us however consider a function w w x 1 , . . ., x n implicitly defined by an equation V x 1 , . . ., x n , w 0. We may suppose that the transformed function m λ * w satisfies the equation without any loss of generality.In infinitesimal terms However w i ∂w/∂x i −V x i /V w may be inserted here, and we have the crucial Jacobi equation not involving V which can be uniquely rewritten as the Hamilton-Jacobi HJ equation use the identifications 8.13 of coordinates.This is the classical definition of the action V in a Mayer field.We have moreover clarified the additive nature of the level sets V λ: roughly saying, the composition with V μ provides V λ μ see Figure 3 c and this is caused by the additivity of the integral Ł dt calculated along the orbits.
On this occasion, the wave enveloping approach to CT groups is also worth mentioning.In more generality, if function W in Lemma 8.3 moreover depends on a parameter λ, we obtain a mapping m λ which is a certain CT involving a parameter λ and the inverse m λ −1 .In favourable case see below this m λ may be even a CT group.The geometrical sense is as follows.Equation W 0 with x i , w kept fixed represents a wave in the space x i , w Figure 3 a .

Lemma
The total system W D 1 W • • • D n W 0 provides the intersection envelope of infinitely close waves Figure 3 b with the resulting transform, the focus point m or m λ if the parameter λ is present .The reverse waves with the role of variables interchanged gives the inversion.Then the group property holds true if the waves can be composed Figure 3 c within the parameters λ, μ, but this need not be in general the case.
Let us eventually deal with the condition ensuring the group composition property.Without loss of generality, we may consider the λ-depending wave

On the Order-Destroying Groups in Jet Space
We recall that in the order-preserving case, the filtration 3 by using the commutative rule 5.17 .This may be expressed in terms of matrix equations

Lemma 9.2 going-down lemma . Let the group of morphisms m λ preserve a submodule
or, in either of more geometrical transcriptions where A is regarded as a matrix of an operator acting in m-dimensional linear space and depending on parameters λ 1 , . . ., λ n .We do not know explicit solutions A in full generality, however, solutions A such that Ker A does not depend on the parameters λ 1 , . . ., λ n can be easily found and need not be stated here .The same approach can be applied to the more general sufficient requirement Z r w j 0 j 1, . . ., m; fixed r ensuring Z ∈ .If r ≥ n, the requirement is equivalent to the inclusion Z ∈ .
Example 9.4.Let us consider vector field 5.6 where z 1 • • • z m 0. In more detail, we take Then Z r w j 0 and we have to deal with functions Z r x i in order to ensure the inclusion Z ∈ .This is a difficult task.Let us therefore suppose

9.11
Then Zx k 0 k 2, . . ., n and 12 where The second-order summand It follows that all functions Z r x i , Z r w j can be expressed in terms of the finite family of functions x i i 1, . . ., n , w j j 1, . . ., m , u l l 2, . . ., m and therefore Z ∈ .
Remark 9.5.On this occasion, let us briefly mention the groups generated by vector fields Z of the above examples.The Lie system of the vector field 9.4 and 9. 16 .This provides a classical self-contained system of ordinary differential equations where the common existence theorems can be applied.
The above Lie systems admit many nontrivial first integrals F ∈ F, that is, functions F that are constant on the orbits of the group.Conditions F 0 may be interpreted as differential equations in the total jet space, and the above transformation groups turn into the external generalized symmetries of such differential equations, see Section 11 below.

Towards the Main Algorithm
We briefly recall the algorithm 10-13 for determination of all individual automorphisms m of the jet space M m, n in order to compare it with the subsequent calculation of vector field Z ∈ .
Morphisms m of the jet structure were defined by the property m * Ω m, n ⊂ Ω m, n .The inverse m −1 exists if and only if for appropriate term Ω l m of filtration 9.1 .However and it follows that criterion 10.1 can be verified by repeated use of operators L D i .In more detail, we start with equations with uncertain coefficients.Formulae 10.3 determine the module m * Ω 0 .Then we search for lower-order contact forms, especially forms from Ω 0 , lying in m * Ω l with the use of 10.2 .Such forms are ensured if certain linear relations among coefficients exist.The calculation is finished on a certain level l l m and this is the algebraic part of the algorithm.With this favourable choice of coefficients a jj I , functions m * x i , m * w j and therefore the invertible morphism m can be determined by inspection of the bracket in 10.3 .This is the analytic part of algorithm.
Let us turn to the infinitesimal theory.Then the main technical tool is the rule 5.17 in the following transcription: or, when applied to basical forms We are interested in vector fields Z ∈ .They satisfy the recurrence 5.10 together with requirements for appropriate l Z ∈ AE.Due to the recurrence 10.5 these requirements can be effectively investigated.In more detail, we start with equations Formulae 10.7 determine module L Z Ω 0 .Then, choosing l Z ∈ AE, operator L Z is to be repeatedly applied and requirements 10.6 provide certain polynomial relations for the coefficients by using 10.5 .This is the algebraical part of the algorithm.With such coefficients a jj I available, functions z i L Z x i , z j L Z w j and therefore the vector field Z ∈ can be determined by inspection of the bracket in 10.7 or, alternatively, with the use of formulae 5.12 for the particular case I φ empty This is the analytic part of the algorithm.
Altogether taken, the algorithm is not easy and the conviction 7, page 121 that the "exhaustive description of integrable C-fields fields Z ∈ in our notation is given in 16 " is disputable.We can state only one optimistic result at this place.
Theorem 10.1.The jet spaces M 1, n do not admit any true generalized infinitesimal symmetries Z ∈ .
Proof.We suppose m 1 and then 10.7 reads where we state a summand of maximal order.Assuming I φ, the Lie-Bäcklund theorem can be applied and we do not have the true generalized symmetry Z. Assuming I / φ, then  In particular if r 1 we have 10.7 written here in the simplified notation The next requirement r 2 implies the only seemingly stronger inclusion which already ensures 10.12 for all r and therefore Z ∈ easy .We suppose 10.14 from now on.
"Hint for proof of 10.14 : assuming 10.12 and moreover the equality and Lie-Bäcklund theorem can be applied whence L Z Ω 0 ⊂ Ω 0 , l Z 0 which we exclude.It follows that necessarily 10.17 On the other hand dim L Z Ω 0 Ω 0 ≥ 3 and the inclusion 10.14 follows."After this preparation, we are passing to the proper algebra.Clearly We deal only with the more interesting identities 10.20 here.Then It may be seen by direct calculation of L 2 Z ω 2 that the "stronger" inclusion 10.14 is equivalent to the identity ca 1 a 2 , that is, abbreviation a a 1 .Alternatively, 10.24 can be proved by using Lemma 9.2."Hint: denoting Θ L Z Ω 0 Ω 0 , 10.14 implies L Z Θ ⊂ Θ. Moreover L D ω 1 cω 2 ∈ Θ by using 10.22 .Lemma 9.2 can be applied: ω 1 cω 2 ∈ Θ and Θ involves just all multiples of form ω 1 cω 2 .Therefore L Z ω 1 cω 2 ∈ Θ is a multiple of ω 1 cω 2 ." The algebraical part is concluded.We have congruences and equality If Z is a variation then these three conditions together ensure the "stronger inclusion" 10.14 hence Z ∈ .We turn to analysis.Abbreviating for the coefficient c.To cope with levels s ≥ 2, we introduce functions

10.39
The coefficient c is determined by 10.33 and 10.36 in terms of functions Q j .This concludes the analytic part of the algorithm since trivially z j w j 1 z Q j and the vector field Z is determined.The system is compatible: particular solutions with functions Q j quadratic in jet variables and c const.can be found as follows.Assume .40 with constant coefficients A j , B j , C j ∈ Ê.We also suppose c ∈ Ê and then 10.37 is trivially satisfied.
On the other hand, 10.33 provide the requirements 10.41 by using 10.36 .If we put then 10.38 is satisfied a clumsy direct verification .The above requirements turn to a system of six homogeneous linear equations not written here for the six constants A j , B j , C j j 1, 2 with determinant Δ c 2 c 2 − 8 if the values z, Q 1 , Q 2 are inserted and the coefficients of w 1 1 and w 2 1 are compared.The roots c 0 and c ±2 √ 2 of the equation Δ 0 provide rather nontrivial infinitesimal transformation Z, however, we can state only the simplest result for the trivial root c 0 for obvious reason.It reads where A 1 , A 2 are arbitrary constants.
Remark 10.3.It follows that investigation of vector fields Z ∈ cannot be regarded for easy task and some new powerful methods are necessary, for example, better use of differential forms involutive systems with pseudogroup symmetries of the problem moving frames .

A Few Notes on the Symmetries of Differential Equations
The external theory deals with systems of differential equations DE that are firmly localized in the jet spaces.This is the common approach and it runs as follows.A given finite system of DE is infinitely prolonged in order to ensure the compatibility.In general, this prolongation is a toilsome and delicate task, in particular the "singular solutions" are tacitly passed over.The prolongation procedure is expressed in terms of jet variables and as a result a fixed subspace of the infinite-order jet space appears which represents the DE under consideration.Then the external symmetries 2, 3, 6, 7 are such symmetries of the ambient jet space which preserve the subspace.In this sense we may speak of classical symmetries point and contact transformations and higher-order symmetries which destroy the order of derivatives .The internal theory of DE is irrelevant to the jet localization, in particular to the choice of the hierarchy of independent and dependent variables.This point of view is due to E. Cartan and actually the congenial term "diffiety" was introduced in 6, 7 .Alas, these diffieties were defined as objects locally identical with appropriate external DE restricted to the corresponding subspace of the ambient total jet space.This can hardly be regarded as a coordinate-free or jet theory-free approach since the model objects external DE and the intertwining mappings higher-order symmetries essentially need the use of the above hard jet theory mechanisms and concepts.
In reality, the final result of prolongation, the infinitely prolonged DE, can be alternatively characterized by three simple axioms as follows 8, 9, 24-27 .
Let M be a space modelled on Ê ∞ local coordinates h 1 , h 2 , . . .as in Sections 1 and 2 above .Denote by F M the structural module of all smooth functions f on M locally depending on a finite number m f of coordinates .Let Φ M , T M be the F M -modules of all differential 1-forms and vector fields on M, respectively.For every submodule Ω ⊂ Φ M , we have the "orthogonal" submodule Ω ⊥ H ⊂ T M of all X ∈ H such that Ω X 0. Then an F M -submodule Ω ⊂ Φ M is called a diffiety if the following three requirements are locally satisfied.
Here n is the number of independent variables.The independent variables provide the complementary module to Ω in Φ M which is not prescribed in advance.
B dΩ ∼ 0 mod Ω , equivalent L H Ω ⊂ Ω, equivalently: H, H ⊂ H.This Frobenius condition ensures the classical passivity requirement: we deal with the compatible infinite prolongation of differential equations.
C There exists filtration Ω * : This condition may be expressed in terms of a H-polynomial algebra on the graded module ⊕ Ω l /Ω l−1 the Noetherian property and ensures the finite number of dependent variables.Filtration Ω * may be capriciously modified.In particular, various localizations of Ω in jet spaces Ω m, n can be easily obtained.This is exactly counterpart to Theorem 5.6: submodule Θ ⊂ Ω stands here for the previous submodule Ω 0 ⊂ Ω m, n .We postpone the proof of Theorem 11.1 together with applications to some convenient occasion.Remark 11.2.There may exist conical symmetries Z of a diffiety Ω, however, they are all lying in H and generate just the Cauchy characteristics of the diffiety 9, page 155 .
We conclude with two examples of internal theory of underdetermined ordinary differential equations.The reasonings to follow can be carried over quite general diffieties without any change.The prolongation can be represented as the Pfaffian system dx − f t, x, y, y dt 0, dy − y dt 0, dy − y dt 0, . . . .

11.4
Within the framework of diffieties, we introduce space M with coordinates t, x 0 , y 0 , y 1 , y 2 , . . .11.17 Passing to the diffiety, we introduce space M with coordinates t, x 0 , x 1 , y 0 , y

; i 1 ,Definition 5 . 1 .
j I , . . . on M m, n are C ∞ -smooth and depend on a finite number of coordinates.The jet coordinates serve as a mere technical tool.The true jet structure is given just by the module Ω m, n of contact forms , by the "orthogonal" module H m, n Ω ⊥ m, n of formal derivatives . . ., n; D ω L D i D i d dD i denotes the Lie derivative.We are interested in local one-parameter groups of transformations m λ given by certain formulae m λ * x i G i λ; . . ., x i , the jet space M m, n ; see also 1.1 and 1.2 .We speak of a group of morphisms 5.5 of the jet structure if the inclusion m λ * Ω m, n ⊂ Ω m, n holds true.We speak of a (universal) variation 5.6 of the jet structure ifL Z Ω m, n ⊂ Ω m, n .If a variation 5.6 moreover generates a group, speaks of a (generalized or higher-order) infinitesimal symmetry of the jet structure.

Theorem 5 . 5
technical theorem .Let 5.6 be a variation of the jet space.Then Z ∈ if and only if every family L r Z ω j r∈AE j 1, . . ., m 5.20

Theorem 7 . 1
infinitesimal Lie-Bäcklund .Let a variation Z preserve a submodule Ω l ⊂ Ω m, n of contact forms of the order l at most for a certain l ∈ AE.Then Z ∈ and either Z is an infinitesimal point transformation or m 1 and Z is the infinitesimal Lie's contact transformation.

Figure 4 a
. It follows that certain invariant submodules Ω l ⊂ Ω m, n are a priori prescribed which essentially restricts the store of the symmetries the Lie-Bäcklund theorem .The order-destroying groups also preserve certain submodules of Ω m, n due to approximation results, however, they are not known in advance Figure4b and appear after certain saturation Figure4c described in technical theorem 5.1.The saturation is in general a toilsome procedure.It may be simplified by applying two simple principles.Lemma 9.1 going-up lemma .Let a group of morphisms m λ preserve a submodule Θ ⊂ Ω m, n .Then also the submodule

Zcω 2 a ω 1 cω 2 .
10.28 We compare coefficients of forms ω j I on the level s |I | s 0: Z 11 cZ 21 a, Z 12 cZ 22 Zc ac, delete the coefficients a, b, c in order to obtain interrelations only for variables Z jj I .Clearly s 0: Z 12 cZ 22 Zc Z 11 cZ 21 c,

Example 11 . 3 .
Let us deal with the Monge equation

11 . 5 and 6 Clearly 1 • 12 whence to the recurrence ϑ r 1 ZDg ϑ 1 Z ∂f ∂x 0 ϑ 0 1 . 11 . 4 .
submodule Ω ⊂ Φ M with generators dx 0 − fdt, ω r dy r − y r 1 dt r 0, 1, . . .; f f t, x 0 , y 0 , y 1 .11.H Ω ⊥ ⊂ T M is one-dimensional subspace including the vector field find that we have a diffiety.A and B are trivially satisfied.The common order preserving filtrations where Ω l involves dx 0 − fdt and ω r with r ≤ l is enough for C.We introduce a new standard 9 filtration Ω * where the submodule Ω l ⊂ Ω is generated by the formsϑ 0 dx 0 − fdt − ∂f ∂y 1 ω 0 , ω r r ≤ l − 1 .11.8This is indeed a filtration sinceL D ϑ 0 df − Dfdt − D ∂f ∂y L D ω r ω r 1 .Assuming A / 0 from now on this is satisfied if f y 1 y 1 / 0 every module Ω l isgenerated by the forms ϑ r L r D ϑ 0 r ≤ l .The forms ϑ r satisfy the recurrence L D ϑ r ϑ r 1 .Then the formula ϑ r 1 L D ϑ r D dϑ r dϑ r D D dϑ r 11.10 implies the congruence dϑ r ∼ dt ∧ ϑ r 1 mod Ω ∧ Ω .Let of Ω in the common sense L Z Ω ⊂ Ω.This inclusion is equivalent to the congruence L Z ϑ r Z dϑ r dϑ r Z ∼ −ϑ r 1 Z dt Dϑ r Z dt 0 mod Ω 11.Dϑ r Z 11.13 quite analogous to the recurrence 5.10 , see Remark 5.3.It follows that the functions z Zt dt Z , g ϑ 0 Z 11.14 can be quite arbitrarily chosen.Then functions ϑ r Z D r g are determined and we obtain quite explicit formulae for the variation Z.In more detail g ϑ 0 Z dx 0 − fdt − ∂f ∂y 1 ω 0 Z z 0 − fz − ∂f ∂y 1 z 0 − y 1 z , Aω 0 Z ∂f ∂x 0 g A z 0 − y 1 z 11.15 and these equations determine coefficients z 0 and z 0 in terms of functions z and g.Coefficients z r r ≥ 1 follow by prolongation not stated here .If moreover dim {L r Z ϑ 0 } r∈AE < ∞ 11.16 we have infinitesimal symmetry Z ∈ , see Theorem 11.Example Let us deal with the Hilbert-Cartan equation 3 the parameter λ is kept fixed.We suppose An open and common definition domain for all functions H i is tacitly supposed.In more generality, a common definition domain for every finite number of functions H i is quite enough and the germ and sheaf terminology would be more adequate for our reasonings, alas, it looks rather clumsy., . . ., m I } and F is arbitrary C ∞ -smooth function of IJ variables .In functions F ∈ F I, ε , variables λ 1 , . . ., λ J are regarded as mere parameters.Functions 2.3 will be considered on open subsets of Ê ∞ where the rank of the Jacobi Definition 2.1.For every I 1, 2, . . .and 0 < ε < min{ε 1 , . . ., ε I }, let F I, ε ⊂ F be 3 .Parameters λ 1 , . . ., λ J occurring in 2.3 are taken into account here.It follows that 1 , . . ., λ J ; F 1 , . . ., F K I ∈ F I j 1, . . ., J 2.10 I hence Z ∈ .The converse is clearly also true: every vector field Z ∈ generates a local Lie group since the Lie system 3.3 admits finite-dimensional approximations in spaces F I .
1, . . .3.4 by a mere routine induction on r.Lemma 3.1 finiteness lemma .For all r ∈ AE, Z r F I ⊂ F I . in terms of a composed function where i 1, . . ., I and F is a ∞ -smooth function of a finite number of variables.
Ê on the space Ê ∞ .Let moreover Z 1 , . . ., Z d ∈ uniformly in the sense that there is a universal space F I with L Z i F I ⊂ F I for all i 1, . . ., d.Then the Lie equations may be applied and we obtain reasonable finite-dimensional approximations.
in terms of a finite number of jet coordinates.We conclude with simple but practicable remark: due to jet structure, the infinite number of conditions 5.13 can be replaced by a finite number of requirements if Z is a variation.
and it follows that Î is an infinitedimensional Lie algebra coefficients in Ê .On the other hand, if Z ∈ Î and fZ ∈ Î for certain f ∈ F then f ∈ Ê is a constant.Briefly saying: the conical variations of the total jet space do not exist.We omit easy direct proof.It follows that only the common Lie algebras over Ê are engaged if we deal with morphisms of the jet spaces M m, n .Let G ⊂ Î be a finite-dimensional Lie subalgebra.Then G ⊂ if and only if there exists a basis of G that is lying in .The proof is elementary and may be omitted.Briefly saying, Theorem 4.2 coefficients in F turns into quite other and much easier Theorem 6.1 coefficients in Ê .
Assuming m 1, then 7.2 turns into the classical definition of Lie's infinitesimal contact transformation.Assume m ≥ 2. In order to finish the proof we refer to the following result which implies that Z is indeed an infinitesimal point transformation.Let Z be a vector field on the jet space M m, n satisfying 7.2 and m ≥ 2. Then Zx i z i . . ., x i , w j , . . ., Zw j z j . . ., x i , w j , . . .
−∂Q/∂w i , z 1 Q w i z i Q − w i • ∂Q/∂w i ,∂Q/∂w I 0 if |I| ≥ 1 and formula 8.4 follows." Remark 8.2.Let us recall the Mayer fields of extremals for the VI since they provide the true sense of the above construction.The familiar Poincaré-Cartan form x n , w, p 1 , . . .The curves may be interpreted as the orbits of the group m λ .Hint: look at the wellknown classical construction of the solution V of the Cauchy problem 22, 23 in terms of the characteristics.The initial Cauchy data are transferred just along the characteristics, i.e., along the group orbits.Assume moreover the additional condition det ∂ 2 H/∂p i ∂p j / 0. We may introduce variational integral 8.3 with the Lagrange function Ł given by the familiar identities i , y i of the space M n, 1 and variables x i , w, w i of the space M 1, n .Since 8.11 may be regarded as a Hamiltonian system for the extremals of VI, the metatheorem is clarified.is restricted to appropriate subspace y i g i t, y 1 , . . ., y n i 1, . . ., n; the slope field in order to become a total differential φ y i g i dV t, y 1 , . . .
8.3 see 10-13 .Let W x 1 , . . ., x n , w, x 1 , . . ., x n , w be a function of 2n 2 variables.Assume that the system W D 1 W • • • D n W 0 admits a unique solution x i F i . . ., x i , w, w i , . . ., w F 1 . . ., x i , w, w i , . . .* x i F i , m * w F 1 provides a Lie CT and m −1 * x i F i , m −1 * w F 1 is the inverse.
8.19If x i , w are kept fixed, the previous results may be applied.We obtain a group if and only if the HJ equation 8.10 holds true, therefore W w H x 1 , . . ., x n , w, W x 1 , . . .and the wave W − λ 0 corresponds to the Mayer central field of extremals.

4
Summary 4. Conditions 8.21 ensure the existence ofHJ equation 8.20 for the λ-wave 8.19 and therefore the group composition property of waves 8.19 in the nondegenerate case det ∂ 2 H/∂p i ∂p j / 0. Remark 8.4.A reasonable theory of Mayer fields of extremals and Hamilton-Jacobi equations can be developed also for the constrained variational integrals the Lagrange problem within the framework of jet spaces, that is, without the additional Lagrange multipliers 9, Chapter 3 .It follows that there do exist certain groups of generalized Lie's contact transformations with differential constraints.
and Θ is preserved.