# The Lie Group in Infinite Dimension

Authors: V. Tryhuk; V. Chrastinová; O. Dlouhý
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

A Lie group acting on finite-dimensional space is generated by its infinitesimal
transformations and conversely, any Lie algebra of vector fields in finite dimension
generates a Lie group (the first fundamental theorem). This classical result is adjusted
for the infinite-dimensional case. We prove that the (local,C∞ smooth) action
of a Lie group on infinite-dimensional space (a manifold modelled on ℝ∞) may be
regarded as a limit of finite-dimensional approximations and the corresponding Lie
algebra of vector fields may be characterized by certain finiteness requirements. The
result is applied to the theory of generalized (or higher-order) infinitesimal symmetries
of differential equations.


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## Body

## 1. Preface

In the symmetry theory of differential equations, thegeneralized (or:higher-order, Lie-Bäcklund) infinitesimal symmetries(1.1)Z=∑zi∂∂xi+∑zIj∂∂wIj(i=1,…,n;j=1,…,m;I=i1⋯in;i1,…,in=1,…,n),
where the coefficients(1.2)zi=zi(…,xi′,wI′j′,…),zIj=zIj(…,xi′,wI′j′,…)
are functions of independent variables xi, dependent variables wj and a finite number of jet variables wIj=∂nwj/∂xi1⋯∂xin belong to well-established concepts. However, in spite of 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(1.3)x¯i=Gi(λ;…,xi′,wI′j′,…),w¯Ij=GIj(λ;…,xi′,wI′j′,…)
in the underlying infinite-order jet space since the relevant Lie system(1.4)∂Gi∂λ=zi(…,Gi′,GI′j′,…),∂GIj∂λ=zIj(…,Gi′,GI′j′,…)(Gi|λ=0=xi,GIj|λ=0=wIj)
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 inℝ∞. 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 ℝ∞ with coordinates h1,h2,…. Equations (2.1) of transformations m(λ) can be schematically represented by Figure 1(a).(a)(b)We prove that in appropriate new coordinate systemF1,F2,…on ℝ∞, the same transformations m(λ) become block triangular as in Figure 1(b). It follows that a certain hierarchy of finite-dimensional subspaces of ℝ∞ 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 spaceℝ∞ 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 ℝ∞ 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 theclassical 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.(a)(b)(c)


## 2. Fundamental Approximation Results

Our reasonings will be carried out in the spaceℝ∞ with coordinates h1,h2,…[9] and we introduce the structural family ℱ of all real-valued, locally defined and C∞-smooth functions f=f(h1,…,hm(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 transformationsm(λ) in ℝ∞ defined by formulae(2.1)m(λ)*hi=Hi(λ;h1,…,hm(i)),-ɛi<λ<ɛi,ɛi>0(i=1,2,…),
where Hi∈ℱ if the parameter λ is kept fixed. We suppose(2.2)m(0)=id.,m(λ+μ)=m(λ)m(μ)
whenever it makes a sense. An open and common definition domain for all functions Hi is tacitly supposed. (In more generality, a common definition domain for every finite number of functions Hi is quite enough and the germ and sheaf terminology would be more adequate for our reasonings, alas, it looks rather clumsy.)Definition 2.1.
For everyI=1,2,… and 0<ɛ<min{ɛ1,…,ɛI}, let ℱ(I,ɛ)⊂ℱ be the subset of all composed functions
(2.3)F=F(…,m(λj)*hi,…)=F(…,Hi(λj;h1,…,hm(i)),…),
where i=1,…,I;  -ɛ<λj<ɛ;  j=1,…,J=J(I)=max{m(1),…,m(I)} and F is arbitrary C∞-smooth function (of IJ variables). In functions F∈ℱ(I,ɛ), variables λ1,…,λJ are regarded as mere parameters.Functions (2.3) will be considered on open subsets of ℝ∞ where the rank of the Jacobi (IJ×J)-matrix(2.4)(∂∂hj′Hi(λj;h1,…,hm(i)))(i=1,…,I;j,j′=1,…,J)
of functions Hi(λj;h1,…,hm(i)) locally attains the maximum (for appropriate choice of parameters). This rank and therefore the subset ℱ(I,ɛ)⊂ℱ does not depend on ɛ as soon as ɛ=ɛ(I) is close enough to zero. This is supposed from now on and we may abbreviateℱ(I)=ℱ(I,ɛ).We deal with highly nonlinear topics. Then the definition domains cannot be kept fixed in advance. Our results will be truelocally, near generic points, on certain open everywhere dense subsets of the underlying space ℝ∞. 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:(2.5)m(λ)*F(I)⊂F(I).Proof.
Clearly(2.6)m(λ)*Hi(λj;…)=m(λ)*m(λj)*hi=m(λ+λj)*hi=Hi(λ+λj;…)
and therefore
(2.7)m(λ)*F=F(…,Hi(λ+λj;h1,…,hm(i)),…)∈F(I).Denoting byK(I) the rank of matrix (2.4), there exist basical functions(2.8)Fk=Fk(…,Hi(λj;h1,…,hm(i)),…)∈F(I)(k=1,…,K(I))
such that rank(∂Fk/∂hj′)=K(I). Then a function f∈ℱ lies in ℱ(I) if and only if f=f¯(F1,…,FK(I)) is a composed function. In more detail(2.9)F=F¯(λ1,…,λJ;F1,…,FK(I))∈F(I)
is such a composed function if we choose f=F given by (2.3). Parameters λ1,…,λJ occurring in (2.3) are taken into account here. It follows that(2.10)∂F∂λj=∂F¯∂λj(λ1,…,λJ;F1,…,FK(I))∈F(I)(j=1,…,J)
and analogously for the higher derivatives.In particular, we also have(2.11)Hi(λ;h1,…,hm(i))=H¯i(λ;F1,…,FK(I))∈F(I)(i=1,…,I)
for the choice F=Hi(λ;…) in (2.9) whence(2.12)∂rHi∂λr=∂rH¯i∂λr(λ;F1,…,FK(I))∈F(I)(i=1,…,I;r=0,1,…).
The basical functions can be taken from the family of functions Hi(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(2.13)m(λ)*Fk=F¯k(λ;F1,…,FK(I))(k=1,…,K(I)).Proof.
Fk∈ℱ(I) implies m(λ)*Fk∈ℱ(I) and (2.9) may be applied with the choice F=m(λ)*Fk and λ1=⋯=λJ=λ.Summary 1.
Coordinateshi=Hi(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 F1=h1,…,FI=hI 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 F1,F2,… is succesively completed.


## 3. The Infinitesimal Approach

We introduce the vector field(3.1)Z=∑zi∂∂hi=dm(λ)dλ|λ=0(zi=∂Hi∂λ(0;h1,…,hm(i));i=1,2,…),
the infinitesimal transformation(ℐT) of group m(λ). Let us recall the celebrated Lie system (3.2)∂∂λm(λ)*hi=∂Hi∂λ(λ;…)=∂Hi∂μ(λ+μ;…)|μ=0=∂∂μm(λ+μ)*hi|μ=0=m(λ)*∂∂μm(μ)*hi|μ=0=m(λ)*Zhi=m(λ)*zi.
In more explicit (and classical) transcription(3.3)∂Hi∂λ(λ;h1,…,hm(i))=zi(H1(λ;h1,…,hm(1)),…,Hm(i)(λ;h1,…,hm(m(i)))).
One can also check the general identity(3.4)∂r∂λrm(λ)*f=m(λ)*Zrf(f∈F;r=0,1,…)
by a mere routine induction on r.Lemma 3.1 (finiteness lemma).
For allr∈ℕ, Zrℱ(I)⊂ℱ(I).Proof.
Clearly(3.5)ZF=m(λ)*ZF|λ=0=∂∂λm(λ)*F|λ=0∈F(I)
for any function (2.3) by virtue of (2.10): induction on r.Theorem 3.2 (finiteness theorem).
Every functionF∈ℱ(I) admits (locally, near generic points) the representation
(3.6)F=F̃(…,∂rHi∂λr(0;h1,…,hm(i)),…)
in terms of a composed function where i=1,…,I and F̃ is a ℂ∞-smooth function of a finite number of variables.Proof.
Let us temporarily denote(3.7)Hri=∂rHi∂λr(λ;…)=∂r∂λrm(λ)*hi,hri=Hri(0;…)=Zrhi,
where the second equality follows from (3.4) with f=hi,  λ=0. Then
(3.8)Hri=m(λ)*hri=m(λ)*Zrhi
by virtue of (3.4) with general λ.
Ifj=j(i) is large enough, there does exist an identity hj+1i=Gi(h0i,…,hji). Therefore
(3.9)∂j+1Hi∂λj+1=Hj+1i=Gi(H0i,…,Hji)=Gi(Hi,…,∂jHi∂λj)
by applying m(λ)*. This may be regarded as ordinary differential equation with initial values
(3.10)Hi|λ=0=h0i,…,∂jHi∂λj|λ=0=hji.
The solution Hi=H̃i(λ;h0i,…,hji) expressed in terms of initial values reads
(3.11)Hi(λ;h1,…,hm(i))=H̃i(λ;Hi(0;h1,…,hm(i)),…,∂jHi∂λj(0;h1,…,hm(i)))
in full detail. If λ is kept fixed, this is exactly the identity (3.6) for the particular case F=Hi(λ;h1,…,hm(i)). The general case follows by a routine.Definition 3.3.
Let𝔾 be the set of (local) vector fields
(3.12)Z=∑zi∂∂hi(zi∈F,infinitesum)
such that every family of functions {Zrhi}r∈ℕ   (i fixed but arbitrary) can be expressed in terms of a finite number of coordinates.Remark 3.4.
Neither𝔾+𝔾⊂𝔾 nor [𝔾,𝔾]⊂𝔾 as follows from simple examples. However, 𝔾 is a conical set (over ℱ): if Z∈𝔾 then fZ∈𝔾 for any f∈ℱ. Easy direct proof may be omitted here.Summary 2.
IfZ is  ℐ𝒯 of a group then all functions Zrhi  (i=1,…,I;  r=0,1,…) are included into family ℱ(I) hence Z∈𝔾. The converse is clearly also true: every vector field Z∈𝔾 generates a local Lie group since the Lie system (3.3) admits finite-dimensional approximations in spaces ℱ(I).Let us finally reformulate the last sentence in terms of basical functions.Theorem 3.5 (approximation theorem).
LetZ∈𝔾 be a vector field locally defined on ℝ∞ and F1,…,FK(I)∈ℱ be a maximal functionally independent subset of the family of all functions
(3.13)Zrhi(i=1,…,I;r=0,1,…).
Denoting ZFk=F¯k(F1,…,FK(I)), then the system
(3.14)∂∂λm(λ)*Fk=m(λ)*ZFk=F¯k(m(λ)*F1,…,m(λ)*FK(I))(k=1,…,K(I))
may be regarded as a “finite-dimensional approximation" to the Lie system (3.3) of the one-parameter local group m(λ) generated by Z.In particular, assumingF1=h1,…,FI=hI, then the the initial portion (3.15)ddλm(λ)*Fi=ddλm(λ)*hi=ddλHi=zi(H1,…,Hm(i))(i=1,…,I)
of the above system transparently demonstrates the approximation property.


## 4. On the Multiparameter Case

The following result does not bring much novelty and we omit the proof.Theorem 4.1.
LetZ1,…,Zd be commuting local vector fields in the space ℝ∞. Then Z1,…,Zd∈𝔾 if and only if the vector fields Z=a1Z1+⋯+adZd  (a1,…,ad∈ℝ) locally generate an abelian Lie group.In full non-Abelian generality, let us consider a (local) multiparameter group formally given by the same equations (2.1) as above where λ=(λ1,…,λd)∈ℝd are parameters close to the zero point 0=(0,…,0)∈ℝd. The rule (2.2) is generalized as (4.1)m(0)=id.,m(φ(λ,μ))=m(λ)m(μ),
where λ=(λ1,…,λd),  μ=(μ1,…,μd) and φ=(φ1,…,φd) determine the composition of parameters. Appropriately adapting the space ℱ(I) and the concept of basical functions F1,…,FK(I), Lemma 2.2 holds true without any change.Passing to the infinitesimal approach, we introduce vector fieldsZ1,…,Zd which are ℐ𝒯 of the group. We recall (without proof) the Lie equations [17] (4.2)∂∂λjm(λ)*f=∑aij(λ)m(λ)*Zjf(f∈F;j=1,…,d)
with the initial condition m(0)=id. Assuming Z1,…,Zd linearly independent over ℝ, coefficients aij(λ) may be arbitrarily chosen and the solution m(λ) always is a group transformation (the first fundamental theorem). If basical functions F1,…,FK(I) are inserted for f, we have a finite-dimensional approximation which is self-contained in the sense that (4.3)ZjFk=F̃jk(F1,…,FK(I))(j=1,…,d;k=1,…,K(I))
are composed functions in accordance with the definition of the basical functions.Let us conversely consider a Lie algebra of local vector fieldsZ=a1Z1+⋯+adZd  (ai∈ℝ) on the space ℝ∞. Let moreover Z1,…,Zd∈𝔾  uniformly in the sense that there is a universal spaceℱ(I) with ℒZiℱ(I)⊂ℱ(I) for all i=1,…,d. Then the Lie equations may be applied and we obtain reasonable finite-dimensional approximations.Summary 3.
Theorem4.1 holds true even in the non-Abelian and multidimensional case if the inclusions Z1,…,Zd∈𝔾 are uniformly satisfied.As yet we have closely simulated the primary one-parameter approach, however, the results are a little misleading: the uniformity requirement in Summary3 may be completely omitted. This follows from the following result [9, page 30] needless here and therefore stated without proof.Theorem 4.2.
Let𝒦 be a finite-dimensional submodule of the module of vector fields on ℝ∞ such that [𝒦,𝒦]⊂𝒦. Then 𝒦⊂𝔾 if and only if there exist generators (over ℱ) of submodule 𝒦 that are lying in 𝔾.


## 5. Symmetries of the Infinite-Order Jet Space

The previous results can be applied to the groups of generalized symmetries ofpartial 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.LetM(m,n) be the jet space of n-dimensional submanifolds in ℝm+n [9–13]. We recall the familiar (local) jet coordinates (5.1)xi,wIj(I=i1…ir;i,i1,…,ir=1,…,n;r=0,1,…;j=1,…,m).
Functions f=f(…,xi,wIj,…) 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 moduleΩ(m,n)of contact forms(5.2)ω=∑aIjωIj(finitesum,ωIj=dwIj-∑wIijdxi)
or, equivalently, by the “orthogonal" moduleℋ(m,n)=Ω⊥(m,n)of formal derivatives(5.3)D=∑aiDi(Di=∂∂xi+∑wIij∂∂wIj;i=1,…,n;D⌋ωIj=ωIj(D)=0).
Let us state useful formulae (5.4)df=∑Difdxi+∑∂f∂wIjωIj,Di⌋dωIj=ωIij,LDiωIj=ωIij,
where ℒDi=Di⌋d+dDi⌋ denotes the Lie derivative.We are interested in (local) one-parameter groups of transformationsm(λ) given by certain formulae(5.5)m(λ)*xi=Gi(λ;…,xi′,wI′j′,…),m(λ)*wIj=GIj(λ;…,xi′,wI′j′,…)
and in vector fields(5.6)Z=∑zi(…,xi′,wI′j′,…)∂∂xi+∑zIj(…,xi′,wI′j′,…)∂∂wIj
locally defined on the jet space M(m,n); see also (1.1) and (1.2).Definition 5.1.
We speak of agroup 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 ifℒZΩ(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.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
(5.7)Z=D+V(D=∑ziDi∈H(m,n),V=∑QIj∂∂wIj,QIj=zIj-∑wIijzi).
The summand D is usually neglected in a certain sense [3–7] and the “essential" summand V is identified with the evolutional system
(5.8)∂wIj∂λ=QIj(…,xi′,wI′j′,…)(wIj=∂nwj∂xi1⋯∂xin(λ,x1,…,xn))
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" wj=wj(x1,…,xn), see [7, especially page 11]. In more generality, some differential constraints may be adjoint. However, in accordance with the ancient classical tradition, functions δwj=∂wj/∂λ are just the variations. (There is only one novelty: in classical theory, δwj 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(5.9)∑GIijDi′Gi=Di′GIj(i′=1,…,n),
where det(Di′Gi)≠0 is supposed and vector field (5.6) is a variation if and only if(5.10)zIij=DizIj-∑wIi′jDizi′.
Recurrence (5.9) easily follows from the inclusion m(λ)*ωIj∈Ω(m,n) and we omit the proof. Recurrence (5.10) follows from the identity (5.11)LZωIj=LZ(dwIj-∑wIijdxi)=dzIj-∑zIijdxi-∑wIijdzi≅(∑Di′zIj-∑zIi′j-∑wIijDi′zi)dxi′(mod⁡Ω(m,n))
and the inclusion ℒZωIj∈Ω(m,n). The obvious formula(5.12)LZωIj=∑(∂zIj∂wI′j′-∑wIij∂zi∂wI′j′)ωI′j′
appearing on this occasion also is of a certain sense, see Theorem 5.5 and Section 10 below. It follows that the initial functions Gi,  Gj,  zi,  zj (empty I=ϕ) may be in principle arbitrarily prescribed in advance. This is the familiar prolongation procedure in the jet theory.Remark 5.3.
Recurrence (5.10) for the variation Z can be succintly expressed by ωIij(Z)=DiωIj(Z). This remarkable formula admits far going generalizations, see concluding Examples 11.3 and 11.4 below.Let us recall that a vector field (5.6) generates a group (5.5) if and only if Z∈𝔾 hence if and only if every family(5.13){Zrxi}r∈N,{ZrwIj}r∈N
can be expressed 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.Lemma 5.4.
Let (5.6) be a variation of the jet structure. Then the inclusion Z∈𝔾 is equivalent to any of the requirements(ι)
every family of functions(5.14){Zrxi}r∈N,{Zrwj}r∈N(i=1,…,n;j=1,…,m)
can be expressed in terms of a finite number of jet coordinates,(ιι)
every family of differential forms(5.15){LZrdxi}r∈N,{LZrdwj}r∈N(i=1,…,n;j=1,…,m)
involves only a finite number of linearly independent terms,(ιιι)
every family of differential forms(5.16){LZrdxi}r∈N,{LZrdwIj}r∈N(i=1,…,n;j=1,…,m;arbitraryI)
involves only a finite number of linearly independent terms.Proof.
InclusionZ∈𝔾 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 ℒZdf=dZf. Assuming (ιι), we may employ the commutative rule
(5.17)[Di,Z]=DiZ-ZDi=∑aii′Di′(aii′=Dizi′)
in order to verify identities of the kind
(5.18)LZdwij=LZdDiwj=LZLDidwi=LDiLZdwi-∑aii′LDi′wj
and in full generality identities of the kind
(5.19)LZkdwIj=∑aI,kI′LDI′LZk′dwj(sumwithk′≤k,|I′|≤|I|)
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 groupsm(λ)of morphisms, see Section 10 below.Theorem 5.5 (technical theorem).
Let (5.6) be a variation of the jet space. Then Z∈𝔾 if and only if every family
(5.20){LZrωj}r∈N(j=1,…,m)
involves only a finite number of linearly independent terms.Some nontrivial preparation is needful for the proof. LetΘ 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 ℒfXΘ⊂Θ for all functions f. Let moreover AdjΘ be the module of all forms φ satisfying φ(X)=0 for all such X. Then AdjΘ has a basis consisting of total differentials of certain functions f1,…,fK (the Frobenius theorem), and there is a basis of module Θ which can be expressed in terms of functions f1,…,fK. Alternatively saying, (an appropriate basis of) the Pfaffian system ϑ=0 (ϑ∈Θ) can be expressed only in terms of functions f1,…,fK. 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.ModuleAdjΘ is intrinsically related to Θ: if a mapping m preserves Θ then m preserves AdjΘ. In particular, assuming (5.21)m(λ)*Θ⊂Θ,thenm(λ)*AdjΘ⊂AdjΘ
is true for a group m(λ). In terms of ℐT of the group m(λ), we have equivalent assertion (5.22)LZΘ⊂ΘimpliesLZAdjΘ⊂AdjΘ
and therefore ℒZrAdjΘ⊂AdjΘ for all r. The preparation is done.Proof.
LetΘ be the module generated by all differential forms ℒZrωj (j=1,…,m;  r=0,1,…). Assuming finite dimension of module Θ, we have module AdjΘ and clearly ℒZΘ⊂Θ whence ℒZrAdjΘ⊂AdjΘ (r=0,1,…). However AdjΘ involves both the differentials dx1,…,dxn (see below) and the forms ω1,…,ωm. Point (ιι) of previous Lemma 5.4 implies Z∈𝔾. The converse is trivial.
In order to finish the proof, let us on the contrary assume thatAdjΘ  does not contain all differentials dx1,…,dxn. Alternatively saying, the Pfaffian system ϑ=0 (ϑ∈Θ) can be expressed in terms of certain functions f1,…,fK such that df1=⋯=dfK=0 does not imply dx1=⋯=dxn=0. On the other hand, it follows clearly that maximal solutions of the Pfaffian system can be expressed only in terms of functions f1,…,fK and therefore we do not need all independent variables x1,…,xn. This is however a contradiction: the Pfaffian system consists of contact forms and involves the equations ω1=⋯=ωn=0. All independent variables are needful if we deal with the common classical solutions wj=wj(x1,…,xn).The result can be rephrased as follows.Theorem 5.6.
LetΩ0⊂Ω(m,n) be the submodule of all zeroth-order contact forms ω=∑ajωj and Z be a variation of the jet structure. Then Z∈𝔾 if and only if dim⊕ℒZrΩ0<∞.


## 6. On the Multiparameter Case

Let us temporarily denote by𝕍 the family of all infinitesimal variations (5.6) of the jet structure. Then 𝕍+𝕍⊂𝕍, c𝕍⊂𝕍 (c∈ℝ), [𝕍,𝕍]⊂𝕍, and it follows that 𝕍 is an infinite-dimensional Lie algebra (coefficients in ℝ). On the other hand, if Z∈𝕍 and fZ∈𝕍 for certain f∈ℱ then f∈ℝ 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 ℝ are engaged if we deal with morphisms of the jet spaces M(m,n).Theorem 6.1.
Let𝒢⊂𝕍 be a finite-dimensional Lie subalgebra. Then 𝒢⊂𝔾 if and only if there exists a basis of 𝒢 that is lying in 𝔾.The proof is elementary and may be omitted. Briefly saying, Theorem4.2 (coefficients in ℱ) turns into quite other and much easier Theorem 6.1 (coefficients in ℝ).


## 7. 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.LetΩl⊂Ω(m,n) be the submodule of all contact forms ω=∑aIjωIj (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(7.1)m(λ)*Ωl⊂Ωl,LZΩl⊂Ωl,
respectively, for a certain l=0,1,…(equivalently: for all l∈ℕ, 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 Gi,  Gj,   zi,   zj (empty I=ϕ) in formulae (5.5) and (5.6) are functions only of the zeroth-order jet variables xi′,  wj′ or, in the second case, we have m=1 and all functions Gi,  G1,   Gi1,  zi,  z1,  zi1 contain only the zeroth- and first-order variables xi′,  w1,   wi′1.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.Theorem 7.1 (infinitesimal Lie-Bäcklund).
Let a variationZ preserve a submodule Ωl⊂Ω(m,n) of contact forms of the order l at most for a certain l∈ℕ. Then Z∈𝔾 and either Z is an infinitesimal point transformation or m=1 and Z is the infinitesimal Lie's contact transformation.Proof.
We supposeℒZΩl⊂Ωl. Then ℒZrΩ0⊂ℒZrΩl⊂Ωl therefore Z∈𝔾 by virtue of Theorem 5.5. Moreover ℒZΩl-1⊂Ωl-1,…,ℒZΩ0⊂Ω0 by virtue of Lemma 9.2 below. So we have
(7.2)LZωj=∑ajj′ωj′(j,j′=1,…,m).
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.Lemma 7.2.
LetZ be a vector field on the jet space M(m,n) satisfying (7.2) and m≥2. Then
(7.3)Zxi=zi(…,xi′,wj′,…),Zwj=zj(…,xi′,wj′,…)(i=1,…,n;j=1,…,m)
are functions only of the point variables.Proof.
Let us introduce moduleΘ of (m+2n)-forms generated by all forms of the kind
(7.4)ω1∧⋯∧ωm∧(dωj1)n1∧(dωjk)nk=dw1∧⋯dwm∧dx1∧⋯dxn∧∑±dwi1j1′∧⋯∧dwinjn′,
where ∑nk=n. Clearly Θ=(Ω0)m∧(dΩ0)n. The inclusions
(7.5)LZΩ0⊂Ω0,LZdΩ0=dLZΩ0+Ω0⊂dΩ0+Ω0
are true by virtue of (7.2) and imply ℒZΘ⊂Θ.
ModuleΘ vanishes when restricted to certain hyperplanes, namely, just to the hyperplanes of the kind
(7.6)ϑ=∑aidxi+∑ajdwj=0
(use m≥2 here). This is expressed by Θ∧ϑ=0 and it follows that
(7.7)0=LZ(Θ∧ϑ)=LZΘ∧ϑ+Θ∧LZϑ=Θ∧LZϑ.
Therefore ℒZϑ again is such a hyperplane: ℒZϑ≅0 (mod all dxi and dwj). On the other hand,
(7.8)LZϑ≅∑aidzi+∑ajdzj(mod⁡alldxianddwj)
and it follows that dzi,  dzj≅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.


## 8. 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.
The𝒞𝒯 groups act in the jet space M(1,n) equipped with the contact module Ω(1,n). Then the abbreviations
(8.1)wI=wI1,ωI=ωI1=dwI-∑wIidxiZ=∑zi∂∂xi+∑zI1∂∂wI
are possible. Let us recall the classical approach [22, 23]. The Lie contact transformations defined by certain formulae
(8.2)m*xi=Gi(⋅),m*w=G1(⋅),m*wi=Gi1(⋅)((⋅)=(x1,…,xn,w,w1,…,wn))
preserve the Pfaffian equation ω=dw-∑widxi=0 or (equivalently) the submodule Ω0⊂Ω(1,n) of zeroth-order contact forms. Explicit formulae are available in literature. We are interested in one-parameter local 𝒞𝒯 groups of transformations m(λ)(-ɛ<λ<ɛ) which are “nondegenerate" in a sense stated below and then the explicit formulae are not available yet. On the other hand, our 𝒱ℐ with smooth Lagrangian Ł
(8.3)∫Ł(t,y1,…,yn,y1′,…,yn′)dt(yi=yi(t),′=ddt,det⁡(∂2Ł∂yi′∂yj′)≠0)
to appear later, involves variables from quite other jet space M(n,1) with coordinates denoted t (the independent variable), y1,…,yn (the dependent variables) and higher-order jet variables like yi′,  yi′′ and so on.
We are passing to the topic proper. Let us start in the spaceM(1,n) with 𝒞𝒯 groups. One can check that vector field (5.6) is infinitesimal 𝒞𝒯 if and only if
(8.4)Z=-∑Qwi∂∂xi+(Q-∑wiQwi)∂∂w+∑(Qxi+wiQw)∂∂wi+⋯,
where the function Q=Q(x1,…,xn,w,w1,…,wn) may be arbitrarily chosen.
“Hint: we have, by definition(8.5)LZω=Z⌋dω+dω(Z)=∑(ziωi-ωi(Z)dxi)+dQ∈Ω0,
where Q=Q(x1,…,xn,w,w1,…,wn,…)=ω(Z)=z1-∑wizi,
(8.6)dQ=∑DiQdxi+∂Q∂wω+∑∂Q∂wiωi
whence immediately zi=-∂Q/∂wi,  z1=Q+∑wizi=Q-∑wi·∂Q/∂wi,   ∂Q/∂wI=0 if |I|≥1 and formula (8.4) follows.”
Alas, the corresponding Lie system (not written here) is not much inspirational. Let us however consider a functionw=w(x1,…,xn) implicitly defined by an equation V(x1,…,xn,w)=0. We may suppose that the transformed function m(λ)*w satisfies the equation
(8.7)V(x1,…,xn,m(λ)*w)=λ
without any loss of generality. In infinitesimal terms
(8.8)1=∂(V-λ)∂λ=Z(V-λ)=-∑QwiVxi+(Q-∑wiQwi)Vw.
However wi=∂w/∂xi=-Vxi/Vw may be inserted here, and we have the crucial Jacobi equation(8.9)1=Q(x1,…,xn,w,-Vx1Vw,…,-VxnVw)Vw
(not involving V) which can be uniquely rewritten as the Hamilton-Jacobi(ℋ𝒥)equation(8.10)Vw+H(x1,…,xn,w,p1,…,pn)(pi=Vxi)
in the “nondegenerate" case ∑QwiVxi≠1. Let us recall the characteristic curves [22, 23] of the ℋ𝒥 equation given by the system
(8.11)dw1=dxiHpi=-dpiHxi=dV-H+∑piHpi.
The curves may be interpreted as the orbits of the group m(λ). (Hint: look at the well-known 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(∂2ℋ/∂pi∂pj)≠0. We may introduce variational integral (8.3) with the Lagrange function Ł given by the familiar identities
(8.12)Ł+H=∑piyi′
with interrelations
(8.13)t=w,yi=xi,yi′=Hpi,pi=Łyi′(i=1,…,n)
between variables t, yi, yi′ of the space M(n,1) and variables xi, w, wi 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.Remark 8.2.
Let us recall the Mayer fields of extremals for the𝒱ℐ since they provide the true sense of the above construction. The familiar Poincaré-Cartan form
(8.14)φ̆=Łdt+∑Łyi′(dyi-yi′dt)=-Hdt+∑pidyi
is restricted to appropriate subspace yi′=gi(t,y1,…,yn) (i=1,…,n;the slope field) in order to become a total differential
(8.15)φ̆|yi′=gi=dV(t,y1,…,yn)=Vtdt+∑Vyidyi
of the actionV. We obtain the requirements Vt=-ℋ,  Vyi=pi identical with (8.10). In geometrical terms: transformations of a hypersurfaceV=0by means of𝒞𝒯group may be identified with the level setsV=λ (λ∈ℝ)of the action of a Mayer fields of extremals.
The last statement is in accordance with (8.11) where
(8.16)dV=(-H+∑piHpi)dw=(-H+∑piyi′)dt=Łdt,
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 ∫Łdt calculated along the orbits.(a)(b)(c)On this occasion, the wave enveloping approach to𝒞𝒯 groups is also worth mentioning.Lemma 8.3 (see [10–13]).
LetW(x¯1,…,x¯n,w¯,x1,…,xn,w) be a function of 2n+2 variables. Assume that the system W=D1W=⋯=DnW=0 admits a unique solution
(8.17)x¯i=Fi(…,xi′,w,wi′,…),w¯=F1(…,xi′,w,wi′,…)
by applying the implicit function theorem and analogously the system W=D¯1W=⋯=D¯nW=0   (where D¯i=∂/∂x¯i+∑w¯i∂/∂w¯) admits a certain solution
(8.18)xi=F¯i(…,x¯i′,w¯,w¯i′,…),w=F¯1(…,x¯i′,w¯,w¯i′,…).
Then m*xi=Fi,   m*w=F1 provides a Lie 𝒞𝒯 and (m-1)*x¯i=F¯i,  (m-1)*w¯=F¯1 is the inverse.In more generality, if functionW 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) thism(λ)may be even a𝒞𝒯group. The geometrical sense is as follows. Equation W=0 with x¯i,  w¯ kept fixed represents a wave in the space xi,  w (Figure 3(a)).The total systemW=D1W=⋯=DnW=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 theλ-depending wave(8.19)W(x¯1,…,x¯n,w¯,x1,…,xn,w)-λ=0.
If 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(8.20)Ww+H(x1,…,xn,w,Wx1,…,Wxn)=0.
The existence of such function ℋ means that functions Ww,Wx1,…,Wxn  of dashed variables are functionally dependent whence(8.21)det⁡(Www¯Wwx¯i′Wxiw¯Wxix¯i′)=0,det⁡(Wxix¯i′)≠0.
The symmetry x¯i,w¯↔xi,w is not surprising here since the change λ↔-λ provides the inverse mapping: equations (8.22)W(…,x¯i,w¯,…,xi,w)=λ,W(…,xi,w,…,x¯i,w¯)=-λ
are equivalent. In particular, it follows that (8.23)W(…,x¯i,w¯,…,xi,w)=-W(…,xi,w,…,x¯i,w¯),W(…,xi,w,…,xi,w)=0
and the wave W-λ=0 corresponds to the Mayer central field of extremals.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(∂2ℋ/∂pi∂pj)≠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.


## 9. On the Order-Destroying Groups in Jet Space

We recall that in the order-preserving case, the filtration(9.1)Ω(m,n)*:Ω0⊂Ω1⊂⋯⊂Ω(m,n)=∪Ωl
of module Ω(m,n) is preserved (Figure 4(a)). It follows that certain invariant submodules Ωl⊂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 (Figure 4(b)) and appear after certain saturation (Figure 4(c)) described in technical theorem 5.1.(a)(b)(c)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 morphismsm(λ) preserve a submodule Θ⊂Ω(m,n). Then also the submodule
(9.2)Θ+∑LDiΘ⊂Ω(m,n)
is preserved.Proof.
We supposeℒZΘ⊂Θ. Then
(9.3)LZ(Θ+∑LDiΘ)=LZΘ+(LDiLZΘ-∑Dizi′LDi′Θ)⊂Θ+∑LDiΘ
by using the commutative rule (5.17).Lemma 9.2 (going-down lemma).
Let the group of morphismsm(λ) preserve a submodule Θ⊂Ω(m,n). Let Θ′⊂Θ be the submodule of all ω∈Θ satisfying ℒDiω∈Θ (i=1,…,n). Then Θ′ is preserved, too.Proof.
Assumeω∈Θ′ hence ℒDiω∈Θ. Then ℒDiℒZω=ℒZℒDiω+ℒ∑Dizi′·Di′ω∈Θ hence ℒZω∈Θ′ and Θ′ is preserved.We are passing to illustrative examples.Example 9.3.
Let us consider the vector field (the variation of jet structure)(9.4)Z=∑zIj∂∂wIj(zIj=DIzj,DI=Di1⋯Din),
see (5.6) and (5.10) for the particular case zi=0. Then Zrxi=0  (i=1,…,n) and the sufficient requirement Z2wj=0 (j=1,…,m) ensures Z∈𝔾, see (ι) of Lemma 5.4. We will deal with the linear case where
(9.5)zj=∑ai′jj′wi′j′(ai′jj′∈R)
is supposed. Then
(9.6)Z2wj=Zzj=∑ai′jj′zi′j′=∑ai′jj′ai′′j′j′′wi′i′′j′′=0
identically if and only if
(9.7)∑j′(ai′jj′ai′′j′j′′+ai′′jj′ai′j′j′′)=0(i′,i′′=1,…,n;j,j′,j′′=1,…,m).
This may be expressed in terms of matrix equations
(9.8)AiAi′=0(i,i′=1,…,n;Ai=(aijj′))
or, in either of more geometrical transcriptions
(9.9)A2=0,Im⁡A⊂Ker⁡A(A=∑λiAi,λi∈R),
where A is regarded as (a matrix of an) operator acting in m-dimensional linear space and depending on parameters λ1,…,λn. We do not know explicit solutions A in full generality, however, solutions A such that KerA does not depend on the parameters λ1,…,λn  can be easily found (and need not be stated here). The same approach can be applied to the more general sufficient requirement Zrwj=0  (j=1,…,m;  fixed  r) ensuring Z∈𝔾. If r≥n, the requirement is equivalent to the inclusion Z∈𝔾.Example 9.4.
Let us consider vector field (5.6) where z1=⋯=zm=0. In more detail, we take
(9.10)Z=∑zi∂∂xi+∑zij∂∂wij+⋯(zij=-∑wi′jDizi′).
Then Zrwj=0 and we have to deal with functions Zrxi in order to ensure the inclusion Z∈𝔾. This is a difficult task. Let us therefore suppose
(9.11)z1=z(…,xi′,wj′,w1j′,…),zk=ck∈R(k=2,…,n).
Then Zxk=0  (k=2,…,n) and
(9.12)Z2x1=Zz=∑∂z∂xizi+∑∂z∂w1jz1j,
where
(9.13)z1j=-w1jD1z=-w1j(∂z∂x1+∑∂z∂wj′w1j′+∑∂z∂w1j′w11j′).
The second-order summand
(9.14)Z2x1=⋯+∑∂z∂w1jz1j=⋯-∑∂z∂w1jw1j∂z∂w1j′w11j′
identically vanishes for the choice
(9.15)z=f(…,xi′,wj′,ul,…)(ul=w1lw11;l=2,…,m)
as follows by direct verification. Quite analogously
(9.16)Zul=Zw1lw11=z1l1w11-z11w1l(w11)2=(-w1l1w11+w11w1l(w11)2)D1z=0.
It follows that all functions Zrxi,   Zrwj can be expressed in terms of the finite family of functions xi  (i=1,…,n),  wj (j=1,…,m),  ul (l=2,…,m) and therefore Z∈𝔾.Remark 9.5.
On this occasion, let us briefly mention the groups generated by vector fieldsZ of the above examples. The Lie system of the vector field (9.4) and (9.5) reads
(9.17)dGidλ=0,dGjdλ=∑ai′jj′Gi′j′(i=1,…,n;j=1,…,m),
where we omit the prolongations. It is resolved by
(9.18)Gi=xi,Gj=wj+λ∑ai′jj′wi′j′(i=1,…,n;j=1,…,m)
as follows either by direct verification or, alternatively, from the property Z2xi=Zzi=0  (i=1,…,n) which implies
(9.19)d∑ai′jj′Gi′j′dλ=0,∑ai′jj′Gi′j′=∑ai′jj′Gi′j′|λ=0=∑ai′jj′wi′j′.
Quite analogously, the Lie system of the vector field (9.10), (9.11), (9.15) reads
(9.20)dG1dλ=f(…,Gi′,Gj′,G1l′G11,…),dGkdλ=ck,dGjdλ=0(k=2,…,n;j=1,…,m)
and may be completed with the equations
(9.21)d(G1l/G11)dλ=0(l=2,…,m)
following from (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 nontrivialfirst integralsF∈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.


## 10. 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∈𝔾.Morphismsm 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(10.1)Ω0⊂m*Ω(m,n),equivalentlyΩ0⊂m*Ωl(l=l(m))
for appropriate term Ωl(m) of filtration (9.1). However(10.2)m*Ωl+1=m*Ωl+∑LDim*Ωl
and it follows that criterion (10.1) can be verified by repeated use of operators ℒDi. In more detail, we start with equations(10.3)m*ωj=∑aI′jj′ωI′j′(=dm*wj-∑m*wijdm*xi)
with uncertain coefficients. Formulae (10.3) determine the module m*Ω0. Then we search for lower-order contact forms, especially forms from Ω0, lying in m*Ωl 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 aI′jj′, functions m*xi,  m*wj (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:(10.4)LZLDi=LDiLZ-∑Dizi′LDi′
or, when applied to basical forms(10.5)LZωIij=LDiLZωIj-∑Dizi′ωIi′j.
We are interested in vector fields Z∈𝔾. They satisfy the recurrence (5.10) together with requirements(10.6)dim⁡⊕LZrΩ0<∞,equivalentlyLZrΩ0⊂Ωl(Z)(r=0,1,…)
for appropriate l(Z)∈ℕ. Due to the recurrence (10.5) these requirements can be effectively investigated. In more detail, we start with equations(10.7)LZωj=∑aI′jj′ωI′j′(=dzj-∑zijdxi-∑wijdzi).
Formulae (10.7) determine module ℒZΩ0. Then, choosing l(Z)∈ℕ, operator ℒZ 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 aI′jj′ available, functions zi=ℒZxi,  zj=ℒZwj (and therefore the vector field Z∈𝔾) 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(10.8)LZωj=∑(∂zj∂wI′j′-∑wij∂zi∂wI′j′)ωI′j′.
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∈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 spacesM(1,n) do not admit any true generalized infinitesimal symmetries Z∈𝔾.Proof.
We supposem=1 and then (10.7) reads
(10.9)LZω1=∑aI′11ωI′1=⋯+aI′′11ωI′′1(aI′′11≠0),
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
(10.10)LZrω1=⋯+aI′′11ωI′′⋯I′′1(rtermsI′′)
by using rule (10.5) where the last summand may be omitted. It follows that (10.6) is not satisfied hence Z∉𝔾.Example 10.2.
We discuss the simplest possible but still a nontrivial particular example. Assumem=2, n=1 and l(Z)=1. Let us abbreviate
(10.11)x=x1,D=D1,Z=z∂∂x+∑zIj∂∂wIj(j=1,2;I=1⋯1).
Then, due to l(Z)=1, requirement (10.6) reads
(10.12)LZrΩ0⊂Ω1(r=0,1,…).
In particular (if r=1) we have (10.7) written here in the simplified notation
(10.13)LZωj=aj1ω1+aj2ω2+bj1ω11+bj2ω12(j=1,2).
The next requirement (r=2) implies the (only seemingly) stronger inclusion(10.14)LZ2Ω0⊂LZΩ0+Ω0
which already ensures (10.12) for all r and therefore Z∈𝔾 (easy). We suppose (10.14) from now on.
“Hint for proof of (10.14): assuming (10.12) and moreover the equality
(10.15)LZ2Ω0+LZΩ0+Ω0=Ω1,
it follows that
(10.16)LZΩ1⊂LZ3Ω0+LZ2Ω0+LZΩ0⊂Ω1
and Lie-Bäcklund theorem can be applied whence ℒZΩ0⊂Ω0,  l(Z)=0 which we exclude. It follows that necessarily
(10.17)dim⁡(LZ2Ω0+LZΩ0+Ω0)<dim⁡Ω1=4.
On the other hand dim(ℒZΩ0+Ω0)≥3 and the inclusion (10.14) follows.”
After this preparation, we are passing to the proper algebra. Clearly(10.18)LZ2ωj=⋯+bj1LZω11+bj2LZω12=⋯+bj1(b11ω111+b12ω112)+bj2(b21ω111+b22ω112)
by using the commutative rule (10.5). Due to “weaker" inclusion (10.12) with r=2, we obtain identities
(10.19)bj1b11+bj2b21=0,bj1b12+bj2b22=0(j=1,2).
Omitting the trivial solution, they are satisfied if either
(10.20)b11+b22=0,b12=cb11,b11+cb21=0
for appropriate factor c (where b11≠0 and either b12≠0 or b21≠0 is supposed) or
(10.21)b11=b22=0,eitherb12=0orb21=0.
We deal only with the (more interesting) identities (10.20) here. Then
(10.22)LZω1=a11ω1+a12ω2-cb(ω11+cω12),LZω2=a21ω1+a22ω2+b(ω11+cω12)
(abbreviation b=b21) by inserting (10.20) into (10.13). It follows that
(10.23)LZ(ω1+cω2)=a1ω1+a2ω2(a1=a11+ca21,a2=a12+ca22+Zc).
It may be seen by direct calculation of ℒZ2ω2 that the “stronger" inclusion (10.14) is equivalent to the identity ca1=a2, that is,
(10.24)LZ(ω1+cω2)=a(ω1+cω2)
(abbreviation a=a1). Alternatively, (10.24) can be proved by using Lemma 9.2.
“Hint: denotingΘ=ℒZΩ0+Ω0, (10.14) implies ℒZΘ⊂Θ. Moreover ℒD(ω1+cω2)∈Θ by using (10.22). Lemma 9.2 can be applied:  ω1+cω2∈Θ′ and Θ′ involves just all multiples of form ω1+cω2. Therefore ℒZ(ω1+cω2)∈Θ′ is a multiple of ω1+cω2.”
The algebraical part is concluded. We have congruences
(10.25)LZω1≅-cb(ω11+cω12),LZω2≅b(ω11+cω12)(mod⁡Ω0)
and equality
(10.26)LZω1+cLZω2+Zcω2=a(ω1+cω2).
If Z is a variation then these three conditions together ensure the “stronger inclusion" (10.14) hence Z∈𝔾.
We turn to analysis. Abbreviating(10.27)ZI′jj′=∂zj∂wI′j′-w1j∂z∂wI′j′(j,j′=1,2;I′=1⋯1)
and employing (10.8), the above conditions (10.25) and (10.26) read
(10.28)∑ZI′1j′ωI′j′=-cb(ω11+cω12),∑ZI′2j′ωI′j′=b(ω11+cω12)(|I′|≥1),∑(ZI′1j′+cZI′2j′)ωI′j′+Zcω2=a(ω1+cω2).
We compare coefficients of forms ωIj on the level s=|I′|(10.29)s=0:Z11+cZ21=a,Z12+cZ22+Zc=ac,(10.30)s=1:Z111=-cb,Z112=-(c)2b,Z121=b,Z122=bc,Z11j+cZ12j=0,(10.31)s≥2:ZI′jj′=0,ZI′1j′+cZI′2j′=0.
We will successively delete the coefficients a,  b,  c in order to obtain interrelations only for variables ZI′jj′. Clearly
(10.32)s=0:Z12+cZ22+Zc=(Z11+cZ21)c,s=1:Z111+Z122=0,Z111Z122=Z112Z121,
and we moreover have three compatible equations
(10.33)c=-Z111Z121=-Z112Z122,(c)2=-Z112Z121
for the coefficient c. To cope with levels s≥2, we introduce functions
(10.34)Qj=ωj(Z)=zj-w1jz(j=1,2).
Then substitution into (10.27) with the help of (10.31) gives
(10.35)∂Qj∂wI′j′=0(j,j′=1,2;|I′|≥2).
It follows moreover easily that
(10.36)Z1jj′=∂Qj∂w1j′(j≠j′),Z1jj=z+∂Qj∂w1j,Zjj′=∂Qj∂wj′
and we have the final differential equations(10.37)s=0:∂Q1∂w2+c∂Q2∂w2+Zc=(∂Q1∂w1+c∂Q2∂w1)c,(10.38)s=1:2z+∂Q1∂w11+∂Q2∂w12=0,(z+∂Q1∂w11)(z+∂Q2∂w12)=∂Q1∂w12∂Q2∂w11for the unknown functions(10.39)z=z(x,w1,w2,w11,w12),Qj=Qj(x,w1,w2,w11,w12).The coefficientcis determined by(10.33)and(10.36)in terms of functionsQj.This concludes the analytic part of the algorithm since triviallyzj=w1jz+Qjand the vector fieldZis determined.
The system is compatible: particular solutions with functionsQj quadratic in jet variables and c=const. can be found as follows. Assume
(10.40)Qj=Aj(w11)2+2Bjw11w12+Cj(w12)2(j=1,2)
with constant coefficients Aj,Bj,Cj∈ℝ. We also suppose c∈ℝ and then (10.37) is trivially satisfied.
On the other hand, (10.33) provide the requirements
(10.41)z+∂Q1∂w11+c∂Q2∂w11=∂Q1∂w12+c(z+∂Q2∂w12)=∂Q1∂w12+(c)2∂Q2∂w11=0
by using (10.36). If we put
(10.42)z=-∂Q1∂w11-∂Q2∂w12=-(A1+B1)w11-(B1+C2)w12,
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 constantsAj,  Bj,  Cj (j=1,2) with determinant Δ=c2(c2-8) if the values z,  Q1,  Q2 are inserted and the coefficients of w11 and w12 are compared. The roots c=0 and c=±22 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
(10.43)Q1=A1(w11)2,Q2=A2(w11)2,z=-A1w11,z1=0,z2=w11(A2w11+A1w12),
where A1,  A2 are arbitrary constants.Remark 10.3.
It follows that investigation of vector fieldsZ∈𝔾 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).


## 11. A Few Notes on the Symmetries of Differential Equations

Theexternal 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).Theinternal 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 external𝒟ℰ 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 (external𝒟ℰ) 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 prolonged𝒟ℰ, can be alternatively characterized by three simple axioms as follows [8, 9, 24–27].LetM be a space modelled onℝ∞ (local coordinates h1,h2,… 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 anℱ(M)-submodule Ω⊂Φ(M) is called a diffiety if the following three requirements are locally satisfied. (𝒜)
Ω is of codimension n<∞, equivalent ℋ is of dimension n<∞.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 ℒℋΩ⊂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 filtrationΩ*:Ω0⊂Ω1⊂⋯⊂Ω=∪Ωl by finite-dimensional submodules Ωl⊂Ω such that
(11.1)LHΩl⊂Ωl+1(alll),Ωl+1=Ωl+LHΩl(llargeenough).
This condition may be expressed in terms of a ⊙ℋ-polynomial algebra on the graded module ⊕Ωl/Ωl-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.Theinternal symmetries naturally appear. For instance, a vector field Z∈𝒯(M) is called a (universal) variation of diffiety Ω if ℒZΩ⊂Ω and infinitesimal symmetry if moreover Z generates a local group, that is, if and only if Z∈𝔾.Theorem 11.1 (technical theorem).
LetZ be a variation of diffiety Ω. Then Z∈𝔾 if and only if there is a finite-dimensional ℱ(M)-submodule Θ⊂Ω such that
(11.2)⊕LHrΘ=Ω,dim⁡⊕LZrΘ<∞.This is exactly counterpart to Theorem5.6: submodule Θ⊂Ω stands here for the previous submodule Ω0⊂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 symmetriesZ 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.Example 11.3.
Let us deal with theMonge equation(11.3)dxdt=f(t,x,y,dydt).
The prolongation can be represented as the Pfaffian system
(11.4)dx-f(t,x,y,y′)dt=0,dy-y′dt=0,dy′-y′′dt=0,….
Within the framework of diffieties, we introduce space M with coordinates
(11.5)t,x0,y0,y1,y2,…
and submodule Ω⊂Φ(M) with generators
(11.6)dx0-fdt,(ωr=)dyr-yr+1dt(r=0,1,…;f=f(t,x0,y0,y1)).
Clearly ℋ=Ω⊥⊂𝒯(M) is one-dimensional subspace including the vector field
(11.7)D=∂∂t+f∂∂x0+∑yr+1∂∂yr.
One can easily find that we have a diffiety. (𝒜 and ℬ are trivially satisfied. The common order preserving filtrations where Ωl involves dx0-fdt and ωr with r≤l is enough for 𝒞.)
We introduce a new (standard [9])filtrationΩ¯* where the submodule Ω¯l⊂Ω is generated by the forms
(11.8)ϑ0=dx0-fdt-∂f∂y1ω0,ωr(r≤l-1).
This is indeed a filtration since
(11.9)LDϑ0=df-Dfdt-D∂f∂y1⋅ω0-∂f∂y1ω1=∂f∂x0(dx0-fdt)+(∂f∂y0-D∂f∂y1)ω0=∂f∂x0ϑ0+Aω0(A=∂f∂y0+∂f∂x0∂f∂y1-D∂f∂y1)
and (trivially) ℒDωr=ωr+1. Assuming A≠0 from now on (this is satisfied if fy1y1≠0) every moduleΩ¯lis generated by the formsϑr=ℒDrϑ0 (r≤l).
The formsϑr satisfy the recurrence ℒDϑr=ϑr+1. Then the formula
(11.10)ϑr+1=LDϑr=D⌋dϑr+dϑr(D)=D⌋dϑr
implies the congruence dϑr≅dt∧ϑr+1 (mod Ω∧Ω). Let
(11.11)Z=z∂∂t+z0∂∂x0+∑zr∂∂yr
be a variation of Ω in the common sense ℒZΩ⊂Ω. This inclusion is equivalent to the congruence
(11.12)LZϑr=Z⌋dϑr+dϑr(Z)≅-ϑr+1(Z)dt+Dϑr(Z)dt=0(mod⁡Ω)
whence to the recurrence
(11.13)ϑr+1(Z)=Dϑr(Z)
quite analogous to the recurrence (5.10), see Remark 5.3. It follows that the functions
(11.14)z=Zt=dt(Z),g=ϑ0(Z)
can be quite arbitrarily chosen. Then functions ϑr(Z)=Drg are determined and we obtain quite explicit formulae for the variationZ. In more detail
(11.15)g=ϑ0(Z)=(dx0-fdt-∂f∂y1ω0)(Z)=z0-fz-∂f∂y1(z0-y1z),Dg=ϑ1(Z)(∂f∂x0ϑ0+Aω0)(Z)=∂f∂x0g+A(z0-y1z)
and these equations determine coefficients z0 and z0 in terms of functions z and g. Coefficients zr (r≥1) follow by prolongation (not stated here). If moreover
(11.16)dim⁡{LZrϑ0}r∈N<∞
we have infinitesimal symmetry Z∈𝔾, see Theorem 11.1.Example 11.4.
Let us deal with theHilbert-Cartan equation [3]
(11.17)dydt=(d2xdt2)2.
Passing to the diffiety, we introduce space M with coordinates
(11.18)t,x0,x1,y0,y1,y2,…
and submodule Ω⊂Φ(M) generated by forms
(11.19)dx0-x1dt,dx1-y1dt,(ωr=)dyr-yr+1dt(r=0,1,…).
The submodule ℋ=Ω⊥⊂𝒯(M) is generated by the vector field
(11.20)D=∂∂t+x1∂∂x0+y1∂∂x1+∑yr+1∂∂yr.
We introduce the form
(11.21)ϑ0=dx0-x1dt+B{dx1-y1dt-12y1ω0}(B=1/y1D(1/y1))
and moreover the forms
(11.22)ϑ1=LDϑ0=(1+DB){⋯},ϑ2=LDϑ1=D2B{⋯}-Cω0(C=(1+DB)D12y1),ϑ3=⋯+Cω1,ϑ4=⋯+Cω2,⋮
Assuming C≠0, we have a standard filtration Ω¯* where the submodules Ω¯l⊂Ω are generated by forms ϑr (r≤l). Explicit formulae for variations
(11.23)Z=z∂∂t+z0∂∂x0+z1∂∂x1+∑zr∂∂yr
can be obtained analogously as in Example 11.3 (and are omitted here). Functions z and g=ϑ0(Z) can be arbitrarily chosen. Condition (11.16) ensures Z∈𝔾.



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