The Lie Group in Infinite Dimension

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.

1. Preface

In the symmetry theory of differential equations, the generalized (or: higher-order, Lie-Bäcklund) infinitesimal symmetries 𝑍=𝑧_𝑖𝜕𝜕𝑥_𝑖+𝑧^𝑗𝐼𝜕𝜕𝑤^𝑗𝐼𝑖=1,…,𝑛;𝑗=1,…,𝑚;𝐼=𝑖₁⋯𝑖_𝑛;𝑖₁,…,𝑖_𝑛=1,…,𝑛,(1.1) where the coefficients𝑧_𝑖=𝑧_𝑖…,𝑥^𝑖′,𝑤^𝑗′_𝐼′,…,𝑧^𝑗𝐼=𝑧^𝑗𝐼…,𝑥^𝑖′,𝑤^𝑗′_𝐼′,…(1.2) are functions of independent variables 𝑥_𝑖, dependent variables 𝑤^𝑗 and a finite number of jet variables 𝑤^𝑗𝐼=𝜕^𝑛𝑤^𝑗/𝜕𝑥_𝑖1⋯𝜕𝑥_𝑖𝑛 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 transformations𝑥_𝑖=𝐺_𝑖𝜆;…,𝑥^𝑖′,𝑤^𝑗′_𝐼′,…,𝑤^𝑗𝐼=𝐺^𝑗𝐼𝜆;…,𝑥^𝑖′,𝑤^𝑗′_𝐼′,…(1.3) in the underlying infinite-order jet space since the relevant Lie system𝜕𝐺_𝑖𝜕𝜆=𝑧^𝑖…,𝐺^𝑖′,𝐺^𝑗′_𝐼′,…,𝜕𝐺^𝑗𝐼𝜕𝜆=𝑧^𝑗𝐼…,𝐺^𝑖′,𝐺^𝑗′_𝐼′,…𝐺_𝑖||_𝜆=0=𝑥_𝑖,𝐺^𝑗𝐼||_𝜆=0=𝑤^𝑗𝐼(1.4) 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 ℝ^∞. 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)

Figure 1 

We prove that in appropriate new coordinate system 𝐹¹,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 space ℝ^∞ 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 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.

(a)

(b)

(c)

Figure 2 

2. Fundamental Approximation Results

Our reasonings will be carried out in the space ℝ^∞ 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,…,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 ℝ^∞ defined by formulae𝐦(𝜆)^∗ℎ^𝑖=𝐻^𝑖𝜆;ℎ¹,…,ℎ^𝑚(𝑖),−𝜀^𝑖<𝜆<𝜀^𝑖,𝜀^𝑖>0(𝑖=1,2,…),(2.1) where 𝐻^𝑖∈ℱ if the parameter 𝜆 is kept fixed. We suppose𝐦(0)=id.,𝐦(𝜆+𝜇)=𝐦(𝜆)𝐦(𝜇)(2.2) whenever it makes a sense. 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.)

Definition 2.1. For every 𝐼=1,2,… and 0<𝜀<min{𝜀¹,…,𝜀^𝐼}, let ℱ(𝐼,𝜀)⊂ℱ be the subset of all composed functions 𝐹=𝐹…,𝐦𝜆_𝑗^∗ℎ^𝑖,…=𝐹…,𝐻^𝑖𝜆_𝑗;ℎ¹,…,ℎ^𝑚(𝑖),…,(2.3) where 𝑖=1,…,𝐼;  −𝜀<𝜆_𝑗<𝜀;  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,𝜀), variables 𝜆₁,…,𝜆_𝐽 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𝜕ℎ𝜕^𝑗′𝐻^𝑖𝜆_𝑗;ℎ¹,…,ℎ^𝑚(𝑖)𝑖=1,…,𝐼;𝑗,𝑗^=1,…,𝐽(2.4) of functions 𝐻^𝑖(𝜆_𝑗;ℎ¹,…,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 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: 𝐦(𝜆)^∗ℱ(𝐼)⊂ℱ(𝐼).(2.5)

Proof. Clearly 𝐦(𝜆)^∗𝐻^𝑖𝜆_𝑗;…=𝐦(𝜆)^∗𝐦𝜆_𝑗^∗ℎ^𝑖=𝐦𝜆+𝜆_𝑗^∗ℎ^𝑖=𝐻^𝑖𝜆+𝜆_𝑗;…(2.6) and therefore 𝐦(𝜆)^∗𝐹=𝐹…,𝐻^𝑖𝜆+𝜆_𝑗;ℎ¹,…,ℎ^𝑚(𝑖),…∈ℱ(𝐼).(2.7)

Denoting by 𝐾(𝐼) the rank of matrix (2.4), there exist basical functions 𝐹^𝑘=𝐹^𝑘…,𝐻^𝑖𝜆_𝑗;ℎ¹,…,ℎ^𝑚(𝑖),…∈ℱ(𝐼)(𝑘=1,…,𝐾(𝐼))(2.8) such that rank(𝜕𝐹^𝑘/𝜕ℎ^𝑗′)=𝐾(𝐼). Then a function 𝑓∈ℱ lies in ℱ(𝐼) if and only if 𝑓=𝑓(𝐹¹,…,𝐹^𝐾(𝐼)) is a composed function. In more detail𝐹=𝐹𝜆₁,…,𝜆_𝐽;𝐹¹,…,𝐹^𝐾(𝐼)∈ℱ(𝐼)(2.9) is such a composed function if we choose 𝑓=𝐹 given by (2.3). Parameters 𝜆₁,…,𝜆_𝐽 occurring in (2.3) are taken into account here. It follows that𝜕𝐹𝜕𝜆_𝑗=𝜕𝐹𝜕𝜆_𝑗𝜆₁,…,𝜆_𝐽;𝐹¹,…,𝐹^𝐾(𝐼)∈ℱ(𝐼)(𝑗=1,…,𝐽)(2.10) and analogously for the higher derivatives.

In particular, we also have 𝐻^𝑖𝜆;ℎ¹,…,ℎ^𝑚(𝑖)=𝐻^𝑖𝜆;𝐹¹,…,𝐹^𝐾(𝐼)∈ℱ(𝐼)(𝑖=1,…,𝐼)(2.11) for the choice 𝐹=𝐻^𝑖(𝜆;…) in (2.9) whence𝜕^𝑟𝐻^𝑖𝜕𝜆^𝑟=𝜕^𝑟𝐻^𝑖𝜕𝜆^𝑟𝜆;𝐹¹,…,𝐹^𝐾(𝐼)∈ℱ(𝐼)(𝑖=1,…,𝐼;𝑟=0,1,…).(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 𝐦(𝜆)^∗𝐹^𝑘=𝐹^𝑘𝜆;𝐹¹,…,𝐹^𝐾(𝐼)(𝑘=1,…,𝐾(𝐼)).(2.13)

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 𝜆₁=⋯=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 𝐹¹=ℎ¹,…,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 𝐹¹,𝐹²,… is succesively completed.

3. The Infinitesimal Approach

We introduce the vector field 𝑍=𝑧^𝑖𝜕𝜕ℎ^𝑖=d𝐦(𝜆)d𝜆|||_𝜆=0𝑧^𝑖=𝜕𝐻^𝑖𝜕𝜆0;ℎ¹,…,ℎ^𝑚(𝑖);𝑖=1,2,…,(3.1) the infinitesimal transformation (ℐ𝑇) of group 𝐦(𝜆). Let us recall the celebrated Lie system 𝜕𝜕𝜆𝐦(𝜆)^∗ℎ^𝑖=𝜕𝐻^𝑖𝜕𝜆(𝜆;…)=𝜕𝜕𝐻𝜇^𝑖(𝜆+𝜇;…)||||_𝜇=0=𝜕𝜕𝜇𝐦(𝜆+𝜇)^∗ℎ^𝑖||||_𝜇=0=𝐦(𝜆)^∗𝜕𝜕𝜇𝐦(𝜇)^∗ℎ^𝑖||||_𝜇=0=𝐦(𝜆)^∗𝑍ℎ^𝑖=𝐦(𝜆)^∗𝑧^𝑖.(3.2) In more explicit (and classical) transcription𝜕𝐻^𝑖𝜕𝜆𝜆;ℎ¹,…,ℎ^𝑚(𝑖)=𝑧^𝑖𝐻¹𝜆;ℎ¹,…,ℎ^𝑚(1),…,𝐻^𝑚(𝑖)𝜆;ℎ¹,…,ℎ^{𝑚(𝑚(𝑖))}.(3.3) One can also check the general identity𝜕^𝑟𝜕𝜆^𝑟𝐦(𝜆)^∗𝑓=𝐦(𝜆)^∗𝑍^𝑟𝑓(𝑓∈ℱ;𝑟=0,1,…)(3.4) by a mere routine induction on 𝑟.

Lemma 3.1 (finiteness lemma). For all 𝑟∈ℕ, 𝑍^𝑟ℱ(𝐼)⊂ℱ(𝐼).

Proof. Clearly 𝑍𝐹=𝐦(𝜆)^∗𝑍𝐹||_𝜆=0=𝜕𝜕𝜆𝐦(𝜆)^∗𝐹|||_𝜆=0∈ℱ(𝐼)(3.5) for any function (2.3) by virtue of (2.10): induction on 𝑟.

Theorem 3.2 (finiteness theorem). Every function 𝐹∈ℱ(𝐼) admits (locally, near generic points) the representation 𝐹=𝐹…,𝜕^𝑟𝐻^𝑖𝜕𝜆^𝑟0;ℎ¹,…,ℎ^𝑚(𝑖),…(3.6) in terms of a composed function where 𝑖=1,…,𝐼 and 𝐹 is a ℂ^∞-smooth function of a finite number of variables.

Proof. Let us temporarily denote 𝐻^𝑖𝑟=𝜕^𝑟𝐻^𝑖𝜕𝜆^𝑟(𝜆;…)=𝜕^𝑟𝜕𝜆^𝑟𝐦(𝜆)^∗ℎ^𝑖,ℎ^𝑖𝑟=𝐻^𝑖𝑟(0;…)=𝑍^𝑟ℎ^𝑖,(3.7) where the second equality follows from (3.4) with 𝑓=ℎ^𝑖,  𝜆=0. Then 𝐻^𝑖𝑟=𝐦(𝜆)^∗ℎ^𝑖𝑟=𝐦(𝜆)^∗𝑍^𝑟ℎ^𝑖(3.8) by virtue of (3.4) with general 𝜆.
If 𝑗=𝑗(𝑖) is large enough, there does exist an identity ℎ^𝑖𝑗+1=𝐺^𝑖(ℎ^𝑖0,…,ℎ^𝑖𝑗). Therefore 𝜕^𝑗+1𝐻^𝑖𝜕𝜆^𝑗+1=𝐻^𝑖𝑗+1=𝐺^𝑖𝐻^𝑖0,…,𝐻^𝑖𝑗=𝐺^𝑖𝐻^𝑖,…,𝜕^𝑗𝐻^𝑖𝜕𝜆^𝑗(3.9) by applying 𝐦(𝜆)^∗. This may be regarded as ordinary differential equation with initial values 𝐻^𝑖||_𝜆=0=ℎ^𝑖0,…,𝜕^𝑗𝐻^𝑖𝜕𝜆^𝑗||||_𝜆=0=ℎ^𝑖𝑗.(3.10) The solution 𝐻^𝑖=𝐻^𝑖(𝜆;ℎ^𝑖0,…,ℎ^𝑖𝑗) expressed in terms of initial values reads 𝐻^𝑖𝜆;ℎ¹,…,ℎ^𝑚(𝑖)=𝐻^𝑖𝜆;𝐻^𝑖0;ℎ¹,…,ℎ^𝑚(𝑖),…,𝜕^𝑗𝐻^𝑖𝜕𝜆^𝑗0;ℎ¹,…,ℎ^𝑚(𝑖)(3.11) in full detail. If 𝜆 is kept fixed, this is exactly the identity (3.6) for the particular case 𝐹=𝐻^𝑖(𝜆;ℎ¹,…,ℎ^𝑚(𝑖)). The general case follows by a routine.

Definition 3.3. Let 𝔾 be the set of (local) vector fields 𝑍=𝑧^𝑖𝜕𝜕ℎ^𝑖𝑧^𝑖∈ℱ,infinitesum(3.12) such that every family of functions {𝑍^𝑟ℎ^𝑖}_𝑟∈ℕ   (𝑖 fixed but arbitrary) can be expressed in terms of a finite number of coordinates.

Remark 3.4. Neither 𝔾+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. If 𝑍 is  ℐ𝒯 of a group then all functions 𝑍^𝑟ℎ^𝑖  (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). Let 𝑍∈𝔾 be a vector field locally defined on ℝ^∞ and 𝐹¹,…,𝐹^𝐾(𝐼)∈ℱ be a maximal functionally independent subset of the family of all functions 𝑍^𝑟ℎ^𝑖(𝑖=1,…,𝐼;𝑟=0,1,…).(3.13) Denoting 𝑍𝐹^𝑘=𝐹^𝑘(𝐹¹,…,𝐹^𝐾(𝐼)), then the system 𝜕𝜕𝜆𝐦(𝜆)^∗𝐹^𝑘=𝐦(𝜆)^∗𝑍𝐹^𝑘=𝐹^𝑘𝐦(𝜆)^∗𝐹¹,…,𝐦(𝜆)^∗𝐹^𝐾(𝐼)(𝑘=1,…,𝐾(𝐼))(3.14) 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, assuming 𝐹¹=ℎ¹,…,𝐹^𝐼=ℎ^𝐼, then the the initial portion dd𝜆𝐦(𝜆)^∗𝐹^𝑖=dd𝜆𝐦(𝜆)^∗ℎ^𝑖=dd𝜆𝐻^𝑖=𝑧^𝑖𝐻¹,…,𝐻^𝑚(𝑖)(𝑖=1,…,𝐼)(3.15) 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. Let 𝑍₁,…,𝑍_𝑑 be commuting local vector fields in the space ℝ^∞. Then 𝑍₁,…,𝑍_𝑑∈𝔾 if and only if the vector fields 𝑍=𝑎₁𝑍₁+⋯+𝑎_𝑑𝑍_𝑑  (𝑎₁,…,𝑎_𝑑∈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,…,𝜆_𝑑)∈ℝ^𝑑 are parameters close to the zero point 0=(0,…,0)∈ℝ^𝑑. The rule (2.2) is generalized as 𝐦(0)=id.,𝐦(𝜑(𝜆,𝜇))=𝐦(𝜆)𝐦(𝜇),(4.1) where 𝜆=(𝜆₁,…,𝜆_𝑑),  𝜇=(𝜇₁,…,𝜇_𝑑) and 𝜑=(𝜑₁,…,𝜑_𝑑) determine the composition of parameters. Appropriately adapting the space ℱ(𝐼) and the concept of basical functions 𝐹¹,…,𝐹^𝐾(𝐼), Lemma 2.2 holds true without any change.

Passing to the infinitesimal approach, we introduce vector fields 𝑍₁,…,𝑍_𝑑 which are ℐ𝒯 of the group. We recall (without proof) the Lie equations [17] 𝜕𝜕𝜆_𝑗𝐦(𝜆)^∗𝑓=𝑎^𝑗𝑖(𝜆)𝐦(𝜆)^∗𝑍_𝑗𝑓(𝑓∈ℱ;𝑗=1,…,𝑑)(4.2) with the initial condition 𝐦(0)=id. Assuming 𝑍₁,…,𝑍_𝑑 linearly independent over ℝ, coefficients 𝑎^𝑗𝑖(𝜆) may be arbitrarily chosen and the solution 𝐦(𝜆) always is a group transformation (the first fundamental theorem). If basical functions 𝐹¹,…,𝐹^𝐾(𝐼) are inserted for 𝑓, we have a finite-dimensional approximation which is self-contained in the sense that 𝑍_𝑗𝐹^𝑘=𝐹^𝑘𝑗𝐹¹,…,𝐹^𝐾(𝐼)(𝑗=1,…,𝑑;𝑘=1,…,𝐾(𝐼))(4.3) are composed functions in accordance with the definition of the basical functions.

Let us conversely consider a Lie algebra of local vector fields 𝑍=𝑎₁𝑍₁+⋯+𝑎_𝑑𝑍_𝑑  (𝑎_𝑖∈ℝ) on the space ℝ^∞. Let moreover 𝑍₁,…,𝑍_𝑑∈𝔾  uniformly in the sense that there is a universal space ℱ(𝐼) with ℒ_𝑍𝑖ℱ(𝐼)⊂ℱ(𝐼) for all 𝑖=1,…,𝑑. Then the Lie equations may be applied and we obtain reasonable finite-dimensional approximations.

Summary 3. Theorem 4.1 holds true even in the non-Abelian and multidimensional case if the inclusions 𝑍₁,…,𝑍_𝑑∈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 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 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 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.

Let 𝐌(𝑚,𝑛) be the jet space of 𝑛-dimensional submanifolds in ℝ^𝑚+𝑛 [9–13]. We recall the familiar (local) jet coordinates 𝑥_𝑖,𝑤^𝑗𝐼𝐼=𝑖₁…𝑖_𝑟;𝑖,𝑖₁,…,𝑖_𝑟=1,…,𝑛;𝑟=0,1,…;𝑗=1,…,𝑚.(5.1) Functions 𝑓=𝑓(…,𝑥_𝑖,𝑤^𝑗𝐼,…) 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 𝜔=𝑎^𝑗𝐼𝜔^𝑗𝐼finitesum,𝜔^𝑗𝐼=d𝑤^𝑗𝐼−𝑤^𝑗𝐼𝑖d𝑥_𝑖(5.2) or, equivalently, by the “orthogonal" module ℋ(𝑚,𝑛)=Ω^⟂(𝑚,𝑛) of formal derivatives 𝐷=𝑎_𝑖𝐷_𝑖𝐷^𝑖=𝜕𝜕𝑥_𝑖+𝑤^𝑗𝐼𝑖𝜕𝜕𝑤^𝑗𝐼;𝑖=1,…,𝑛;𝐷⌋𝜔^𝑗𝐼=𝜔^𝑗𝐼(𝐷)=0.(5.3) Let us state useful formulae d𝑓=𝐷_𝑖𝑓d𝑥_𝑖+𝜕𝑓𝜕𝑤^𝑗𝐼𝜔^𝑗𝐼,𝐷_𝑖⌋d𝜔^𝑗𝐼=𝜔^𝑗𝐼𝑖,ℒ_𝐷𝑖𝜔^𝑗𝐼=𝜔^𝑗𝐼𝑖,(5.4) where ℒ_𝐷𝑖=𝐷_𝑖⌋d+d𝐷_𝑖⌋ denotes the Lie derivative.

We are interested in (local) one-parameter groups of transformations 𝐦(𝜆) given by certain formulae𝐦(𝜆)^∗𝑥_𝑖=𝐺_𝑖𝜆;…,𝑥^𝑖′,𝑤^𝑗′_𝐼′,…,𝐦(𝜆)^∗𝑤^𝑗𝐼=𝐺^𝑗𝐼𝜆;…,𝑥^𝑖′,𝑤^𝑗′_𝐼′,…(5.5) and in vector fields𝑍=𝑧_𝑖…,𝑥^𝑖′,𝑤^𝑗′_𝐼′,…𝜕𝜕𝑥_𝑖+𝑧^𝑗𝐼…,𝑥^𝑖′,𝑤^𝑗′_𝐼′,…𝜕𝜕𝑤^𝑗𝐼(5.6) locally defined on the jet space 𝐌(𝑚,𝑛); see also (1.1) and (1.2).

Definition 5.1. 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_𝑍Ω(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 𝑍=𝐷+V𝐷=𝑧_𝑖𝐷_𝑖∈ℋ(𝑚,𝑛),𝑉=𝑄^𝑗𝐼𝜕𝜕𝑤^𝑗𝐼,𝑄^𝑗𝐼=𝑧^𝑗𝐼−𝑤^𝑗𝐼𝑖𝑧_𝑖.(5.7) The summand 𝐷 is usually neglected in a certain sense [3–7] and the “essential" summand 𝑉 is identified with the evolutional system 𝜕𝑤^𝑗𝐼𝜕𝜆=𝑄^𝑗𝐼…,𝑥^𝑖′,𝑤^𝑗′_𝐼′,…𝑤^𝑗𝐼=𝜕^𝑛𝑤^𝑗𝜕𝑥_𝑖1⋯𝜕𝑥_𝑖𝑛𝜆,𝑥₁,…,𝑥_𝑛(5.8) 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(𝑥₁,…,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𝐺^𝑗𝐼𝑖𝐷_𝑖′𝐺_𝑖=𝐷_𝑖′𝐺^𝑗𝐼𝑖^=1,…,𝑛,(5.9) where det(𝐷_𝑖𝐺_𝑖)≠0 is supposed and vector field (5.6) is a variation if and only if𝑧^𝑗𝐼𝑖=𝐷_𝑖𝑧^𝑗𝐼−𝑤^𝑗𝐼𝑖_′𝐷_𝑖𝑧_𝑖′.(5.10) Recurrence (5.9) easily follows from the inclusion 𝐦(𝜆)^∗𝜔^𝑗𝐼∈Ω(𝑚,𝑛) and we omit the proof. Recurrence (5.10) follows from the identity ℒ_𝑍𝜔^𝑗𝐼=ℒ_𝑍d𝑤^𝑗𝐼−𝑤^𝑗𝐼𝑖d𝑥_𝑖=d𝑧^𝑗𝐼−𝑧^𝑗𝐼𝑖d𝑥_𝑖−𝑤^𝑗𝐼𝑖d𝑧_𝑖≅𝐷_𝑖′z^𝑗𝐼−𝑧^𝑗𝐼𝑖_′−𝑤^𝑗𝐼𝑖𝐷_𝑖′𝑧_𝑖d𝑥^𝑖′(modΩ(𝑚,𝑛))(5.11) and the inclusion ℒ_𝑍𝜔^𝑗𝐼∈Ω(𝑚,𝑛). The obvious formulaℒ_𝑍𝜔^𝑗𝐼=⎛⎜⎜⎝𝜕𝑧^𝑗𝐼𝜕𝑤^𝑗′_𝐼′−𝑤^𝑗𝐼𝑖𝜕𝑧_𝑖𝜕𝑤^𝑗′_𝐼′⎞⎟⎟⎠𝜔_{𝑗′𝐼′}(5.12) 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.

Remark 5.3. Recurrence (5.10) for the variation 𝑍 can be succintly expressed by 𝜔^𝑗𝐼𝑖(𝑍)=𝐷_𝑖𝜔^𝑗𝐼(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𝑍^𝑟𝑥_𝑖_𝑟∈ℕ,𝑍^𝑟𝑤^𝑗𝐼_𝑟∈ℕ(5.13) 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 𝑍^𝑟𝑥_𝑖_𝑟∈ℕ,𝑍^𝑟𝑤^𝑗_𝑟∈ℕ(𝑖=1,…,𝑛;𝑗=1,…,𝑚)(5.14) can be expressed in terms of a finite number of jet coordinates,(𝜄𝜄)every family of differential forms ℒ^𝑟𝑍d𝑥_𝑖_𝑟∈ℕ,ℒ^𝑟𝑍d𝑤^𝑗_𝑟∈ℕ(𝑖=1,…,𝑛;𝑗=1,…,𝑚)(5.15) involves only a finite number of linearly independent terms,(𝜄𝜄𝜄)every family of differential forms ℒ^𝑟𝑍d𝑥_𝑖_𝑟∈ℕ,ℒ^𝑟𝑍d𝑤^𝑗𝐼_𝑟∈ℕ(𝑖=1,…,𝑛;𝑗=1,…,𝑚;arbitrary𝐼)(5.16) involves only a finite number of linearly independent terms.

Proof. Inclusion 𝑍∈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 ℒ_𝑍d𝑓=d𝑍𝑓. Assuming (𝜄𝜄), we may employ the commutative rule 𝐷_𝑖,𝑍=𝐷_𝑖𝑍−𝑍𝐷_𝑖=𝑎^𝑖′_𝑖𝐷_𝑖′𝑎^𝑖′𝑖=𝐷_𝑖𝑧_𝑖′(5.17) in order to verify identities of the kind ℒ_𝑍d𝑤^𝑗𝑖=ℒ_Zd𝐷_𝑖𝑤^𝑗=ℒ_𝑍ℒ_𝐷𝑖d𝑤^𝑖=ℒ_𝐷𝑖ℒ_𝑍d𝑤^𝑖−𝑎^𝑖′_𝑖ℒ_𝐷𝑖′𝑤^𝑗(5.18) and in full generality identities of the kind ℒ^𝑘𝑍d𝑤^𝑗𝐼=𝑎^𝐼′_𝐼,𝑘ℒ_𝐷𝐼′ℒ^𝑘′_𝑍d𝑤^𝑗sumwith𝑘^≤𝑘,||𝐼^||≤||𝐼||(5.19) 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.

Theorem 5.5 (technical theorem). Let (5.6) be a variation of the jet space. Then 𝑍∈𝔾 if and only if every family ℒ^𝑟𝑍𝜔^𝑗_𝑟∈ℕ(𝑗=1,…,𝑚)(5.20) 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_𝑓𝑋Θ⊂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,…,𝑓^𝐾 (the Frobenius theorem), and there is a basis of module Θ which can be expressed in terms of functions 𝑓¹,…,𝑓^𝐾. Alternatively saying, (an appropriate basis of) the Pfaffian system 𝜗=0 (𝜗∈Θ) can be expressed only in terms of functions 𝑓¹,…,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 AdjΘ is intrinsically related to Θ: if a mapping 𝐦 preserves Θ then 𝐦 preserves AdjΘ. In particular, assuming 𝐦(𝜆)^∗Θ⊂Θ,then𝐦(𝜆)^∗AdjΘ⊂AdjΘ(5.21) is true for a group 𝐦(𝜆). In terms of ℐ𝑇 of the group 𝐦(𝜆), we have equivalent assertion ℒ_𝑍Θ⊂Θimpliesℒ_𝑍AdjΘ⊂AdjΘ(5.22) and therefore ℒ^𝑟𝑍AdjΘ⊂AdjΘ for all 𝑟. The preparation is done.

Proof. Let Θ be the module generated by all differential forms ℒ^𝑟𝑍𝜔^𝑗 (𝑗=1,…,𝑚; 𝑟=0,1,…). Assuming finite dimension of module Θ, we have module AdjΘ and clearly ℒ_𝑍Θ⊂Θ whence ℒ^𝑟𝑍AdjΘ⊂AdjΘ (𝑟=0,1,…). However AdjΘ involves both the differentials d𝑥₁,…,d𝑥_𝑛 (see below) and the forms 𝜔¹,…,𝜔^𝑚. 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 that AdjΘ  does not contain all differentials d𝑥₁,…,d𝑥_𝑛. Alternatively saying, the Pfaffian system 𝜗=0 (𝜗∈Θ) can be expressed in terms of certain functions 𝑓¹,…,𝑓^𝐾 such that d𝑓¹=⋯=d𝑓^𝐾=0 does not imply d𝑥₁=⋯=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 𝑓¹,…,𝑓^𝐾 and therefore we do not need all independent variables 𝑥₁,…,𝑥_𝑛. This is however a contradiction: the Pfaffian system consists of contact forms and involves the equations 𝜔¹=⋯=𝜔^𝑛=0. All independent variables are needful if we deal with the common classical solutions 𝑤^𝑗=𝑤^𝑗(𝑥₁,…,𝑥_𝑛).

The result can be rephrased as follows.

Theorem 5.6. Let Ω₀⊂Ω(𝑚,𝑛) be the submodule of all zeroth-order contact forms 𝜔=∑𝑎^𝑗𝜔^𝑗 and 𝑍 be a variation of the jet structure. Then 𝑍∈𝔾 if and only if dim⊕ℒ^𝑟𝑍Ω₀<∞.

6. On the Multiparameter Case

Let us temporarily denote by 𝕍 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. Let 𝒢⊂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, Theorem 4.2 (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.

Let Ω_𝑙⊂Ω(𝑚,𝑛) be the submodule of all contact forms 𝜔=∑𝑎^𝑗𝐼𝜔^𝑗𝐼 (sum with |𝐼|≤𝑙) of the order 𝑙 at most. A morphism (5.5) and the infinitesimal variation (5.6) are called order preserving if𝐦(𝜆)^∗Ω_𝑙⊂Ω_𝑙,ℒ_𝑍Ω_𝑙⊂Ω_𝑙,(7.1) 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 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, 𝐺¹, 𝐺^1𝑖, 𝑧_𝑖, 𝑧¹, 𝑧^1𝑖 contain only the zeroth- and first-order variables 𝑥_𝑖, 𝑤¹, 𝑤^1𝑖.

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 variation 𝑍 preserve a submodule Ω_𝑙⊂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 suppose ℒ_𝑍Ω_𝑙⊂Ω_𝑙. Then ℒ^𝑟𝑍Ω₀⊂ℒ^𝑟𝑍Ω_𝑙⊂Ω_𝑙 therefore 𝑍∈𝔾 by virtue of Theorem 5.5. Moreover ℒ_𝑍Ω_𝑙−1⊂Ω_𝑙−1,…,ℒ_𝑍Ω₀⊂Ω₀ by virtue of Lemma 9.2 below. So we have ℒ_𝑍𝜔^𝑗=𝑎^𝑗𝑗′𝜔^𝑗′𝑗,𝑗^=1,…,𝑚.(7.2) 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. 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′,…(𝑖=1,…,𝑛;𝑗=1,…,𝑚)(7.3) are functions only of the point variables.

Proof. Let us introduce module Θ of (𝑚+2𝑛)-forms generated by all forms of the kind 𝜔¹∧⋯∧𝜔^𝑚∧d𝜔^𝑗1^𝑛1∧d𝜔^𝑗𝑘^𝑛𝑘=d𝑤¹∧⋯d𝑤^𝑚∧d𝑥₁∧⋯d𝑥_𝑛∧±d𝑤^𝑗′1_𝑖1∧⋯∧d𝑤^𝑗′𝑛_𝑖𝑛,(7.4) where ∑𝑛_𝑘=𝑛. Clearly Θ=(Ω₀)^𝑚∧(dΩ₀)^𝑛. The inclusions ℒ_𝑍Ω₀⊂Ω₀,ℒ_𝑍dΩ₀=dℒ_𝑍Ω₀+Ω₀⊂dΩ₀+Ω₀(7.5) are true by virtue of (7.2) and imply ℒ_𝑍Θ⊂Θ.
Module Θ vanishes when restricted to certain hyperplanes, namely, just to the hyperplanes of the kind 𝜗=𝑎_𝑖d𝑥_𝑖+𝑎^𝑗d𝑤^𝑗=0(7.6) (use 𝑚≥2 here). This is expressed by Θ∧𝜗=0 and it follows that 0=ℒ_𝑍(Θ∧𝜗)=ℒ_𝑍Θ∧𝜗+Θ∧ℒ_𝑍𝜗=Θ∧ℒ_𝑍𝜗.(7.7) Therefore ℒ_𝑍𝜗 again is such a hyperplane: ℒ_𝑍𝜗≅0 (mod all d𝑥_𝑖 and d𝑤^𝑗). On the other hand, ℒ_𝑍𝜗≅𝑎_𝑖d𝑧_𝑖+𝑎^𝑗d𝑧^𝑗modalld𝑥_𝑖andd𝑤^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.

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. The 𝒞𝒯 groups act in the jet space 𝐌(1,𝑛) equipped with the contact module Ω(1,𝑛). Then the abbreviations 𝑤_𝐼=𝑤^1𝐼,𝜔_𝐼=𝜔^1𝐼=d𝑤_𝐼−𝑤_𝐼𝑖d𝑥_𝑖𝑍=𝑧_𝑖𝜕𝜕𝑥_𝑖+𝑧^1𝐼𝜕𝜕𝑤_𝐼(8.1) are possible. Let us recall the classical approach [22, 23]. The Lie contact transformations defined by certain formulae 𝐦^∗𝑥_𝑖=𝐺_𝑖(⋅),𝐦^∗𝑤=𝐺¹(⋅),𝐦^∗𝑤_𝑖=𝐺^1𝑖(⋅)(⋅)=𝑥₁,…,𝑥_𝑛,𝑤,𝑤₁,…,𝑤_𝑛(8.2) preserve the Pfaffian equation 𝜔=d𝑤−∑𝑤_𝑖d𝑥_𝑖=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)(−𝜀<𝜆<𝜀) 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 Ł𝑡,𝑦¹,…,𝑦_𝑛,𝑦^1,…,𝑦^𝑛𝑑𝑡𝑦_𝑖=𝑦_𝑖(𝑡),^=𝑑𝑑𝑡,det𝜕²Ł𝜕𝑦^𝑖𝜕𝑦^𝑗≠0(8.3) to appear later, involves variables from quite other jet space 𝐌(𝑛,1) with coordinates denoted 𝑡 (the independent variable), 𝑦₁,…,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 𝑍=−𝑄_𝑤𝑖𝜕𝜕𝑥_𝑖+𝑄−𝑤_𝑖𝑄_𝑤𝑖𝜕𝜕𝑤+𝑄^𝑥𝑖+𝑤_𝑖𝑄_𝑤𝜕𝜕𝑤_𝑖+⋯,(8.4) where the function 𝑄=𝑄(𝑥₁,…,𝑥_𝑛,𝑤,𝑤₁,…,𝑤_𝑛) may be arbitrarily chosen.
“Hint: we have, by definition ℒ_𝑍𝜔=𝑍⌋d𝜔+d𝜔(𝑍)=𝑧_𝑖𝜔_𝑖−𝜔_𝑖(𝑍)d𝑥_𝑖+d𝑄∈Ω₀,(8.5) where 𝑄=𝑄(𝑥₁,…,𝑥_𝑛,𝑤,𝑤₁,…,𝑤_𝑛,…)=𝜔(𝑍)=𝑧¹−∑𝑤_𝑖𝑧_𝑖, d𝑄=𝐷_𝑖𝑄d𝑥_𝑖+𝜕𝑄𝜕𝑤𝜔+𝜕𝜕𝑤𝑄_𝑖𝜔_𝑖(8.6) whence immediately 𝑧_𝑖=−𝜕𝑄/𝜕𝑤_𝑖, 𝑧¹=𝑄+∑𝑤_𝑖𝑧_𝑖=𝑄−∑𝑤_𝑖⋅𝜕𝑄/𝜕𝑤_𝑖,   𝜕𝑄/𝜕𝑤_𝐼=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 function 𝑤=𝑤(𝑥₁,…,𝑥_𝑛) implicitly defined by an equation 𝑉(𝑥₁,…,𝑥_𝑛,𝑤)=0. We may suppose that the transformed function 𝐦(𝜆)^∗𝑤 satisfies the equation 𝑉𝑥₁,…,𝑥_𝑛,𝐦(𝜆)^∗𝑤=𝜆(8.7) without any loss of generality. In infinitesimal terms 1=𝜕(𝑉−𝜆)𝜕𝜆=𝑍(𝑉−𝜆)=−𝑄^𝑤𝑖𝑉_𝑥𝑖+𝑄−𝑤_𝑖𝑄_𝑤𝑖𝑉^𝑤.(8.8) However 𝑤_𝑖=𝜕𝑤/𝜕𝑥_𝑖=−𝑉_𝑥𝑖/𝑉_𝑤 may be inserted here, and we have the crucial Jacobi equation 1=𝑄𝑥₁,…,𝑥_𝑛,𝑤,−𝑉^𝑥1𝑉_𝑤,…,−𝑉^𝑥𝑛𝑉_𝑤𝑉^𝑤(8.9) (not involving 𝑉) which can be uniquely rewritten as the Hamilton-Jacobi (ℋ𝒥) equation 𝑉_𝑤+ℋ𝑥₁,…,𝑥_𝑛,𝑤,𝑝₁,…,𝑝_𝑛𝑝_𝑖=𝑉_𝑥𝑖(8.10) in the “nondegenerate" case ∑𝑄_𝑤𝑖𝑉_𝑥𝑖≠1. Let us recall the characteristic curves [22, 23] of the ℋ𝒥 equation given by the system d𝑤1=ℋd𝑥^𝑖_𝑝𝑖=−d𝑝^𝑖ℋ_𝑥𝑖=d𝑉−ℋ+∑𝑝_𝑖ℋ_𝑝𝑖.(8.11) 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/𝜕𝑝_𝑖𝜕𝑝_𝑗)≠0. We may introduce variational integral (8.3) with the Lagrange function Ł given by the familiar identities Ł+ℋ=𝑝_𝑖𝑦^𝑖(8.12) with interrelations 𝑡=𝑤,𝑦_𝑖=𝑥_𝑖,𝑦^𝑖=ℋ_𝑝𝑖,𝑝_𝑖=Ł_𝑦^′𝑖(𝑖=1,…,𝑛)(8.13) between variables 𝑡, 𝑦_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.

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 𝜑̆=Łd𝑡+Ł_𝑦^′𝑖d𝑦_𝑖−𝑦^𝑖d𝑡=−ℋd𝑡+𝑝_𝑖d𝑦_𝑖(8.14) is restricted to appropriate subspace 𝑦^𝑖=𝑔_𝑖(𝑡,𝑦₁,…,𝑦_𝑛) (𝑖=1,…,𝑛;the slope field) in order to become a total differential 𝜑̆||_{𝑦′𝑖=𝑔𝑖}=d𝑉𝑡,𝑦₁,…,𝑦_𝑛=𝑉_𝑡d𝑡+𝑉_𝑦𝑖d𝑦_𝑖(8.15) of the action 𝑉. We obtain the requirements 𝑉_𝑡=−ℋ, 𝑉_𝑦𝑖=𝑝_𝑖 identical with (8.10). In geometrical terms: transformations of a hypersurface 𝑉=0 by means of 𝒞𝒯 group may be identified with the level sets 𝑉=𝜆 (𝜆∈ℝ) of the action of a Mayer fields of extremals.
The last statement is in accordance with (8.11) where d𝑉=−ℋ+𝑝_𝑖ℋ_𝑝𝑖d𝑤=−ℋ+𝑝^𝑖𝑦^𝑖d𝑡=Łd𝑡,(8.16) 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)

Figure 3 

On this occasion, the wave enveloping approach to 𝒞𝒯 groups is also worth mentioning.

Lemma 8.3 (see [10–13]). Let 𝑊(𝑥₁,…,𝑥_𝑛,𝑤,𝑥₁,…,𝑥_𝑛,𝑤) be a function of 2𝑛+2 variables. Assume that the system 𝑊=𝐷₁𝑊=⋯=𝐷_𝑛𝑊=0 admits a unique solution 𝑥_𝑖=𝐹_𝑖…,𝑥_𝑖,𝑤,𝑤_𝑖,…,𝑤=𝐹¹…,𝑥_𝑖,𝑤,𝑤_𝑖,…(8.17) by applying the implicit function theorem and analogously the system 𝑊=𝐷₁𝑊=⋯=𝐷_𝑛𝑊=0   (where 𝐷_𝑖=𝜕/𝜕𝑥_𝑖+∑𝑤_𝑖𝜕/𝜕𝑤) admits a certain solution 𝑥_𝑖=𝐹_𝑖…,𝑥_𝑖′,𝑤,𝑤_𝑖′,…,𝑤=𝐹¹…,𝑥_𝑖′,𝑤,𝑤_𝑖′,….(8.18) Then 𝐦^∗𝑥_𝑖=𝐹_𝑖,   𝐦^∗𝑤=𝐹¹ provides a Lie 𝒞𝒯 and (𝐦⁻¹)^∗𝑥_𝑖=𝐹_𝑖, (𝐦⁻¹)^∗𝑤=𝐹¹ is the inverse.

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)).

The total system 𝑊=𝐷₁𝑊=⋯=𝐷_𝑛𝑊=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 wave𝑊𝑥₁,…,𝑥_𝑛,𝑤,𝑥₁,…,𝑥_𝑛,𝑤−𝜆=0.(8.19) 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, therefore𝑊_𝑤+ℋ𝑥₁,…,𝑥_𝑛,𝑤,𝑊_𝑥1,…,𝑊_𝑥𝑛=0.(8.20) The existence of such function ℋ means that functions 𝑊_𝑤,𝑊_𝑥1,…,𝑊_𝑥𝑛  of dashed variables are functionally dependent whencedet𝑊^𝑤𝑤𝑊_{𝑤𝑥𝑖′}𝑊_𝑥𝑖𝑤𝑊_{𝑥𝑖𝑥𝑖′}=0,det𝑊^{𝑥𝑖𝑥𝑖′}≠0.(8.21) The symmetry 𝑥_𝑖,𝑤↔𝑥_𝑖,𝑤 is not surprising here since the change 𝜆↔−𝜆 provides the inverse mapping: equations 𝑊…,𝑥_𝑖,𝑤,…,𝑥_𝑖,𝑤=𝜆,𝑊…,𝑥_𝑖,𝑤,…,𝑥_𝑖,𝑤=−𝜆(8.22) are equivalent. In particular, it follows that 𝑊…,𝑥_𝑖,𝑤,…,𝑥_𝑖,𝑤=−𝑊…,𝑥_𝑖,𝑤,…,𝑥_𝑖,𝑤,𝑊…,𝑥_𝑖,𝑤,…,𝑥_𝑖,𝑤=0(8.23) 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/𝜕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.

9. On the Order-Destroying Groups in Jet Space

We recall that in the order-preserving case, the filtrationΩ(𝑚,𝑛)_∗∶Ω₀⊂Ω₁⊂⋯⊂Ω(𝑚,𝑛)=∪Ω_𝑙(9.1) of module Ω(𝑚,𝑛) is preserved (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 (Figure 4(b)) and appear after certain saturation (Figure 4(c)) described in technical theorem 5.1.

(a)

(b)

(c)

Figure 4 

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 Θ+ℒ_𝐷𝑖Θ⊂Ω(𝑚,𝑛)(9.2) is preserved.

Proof. We suppose ℒ_𝑍Θ⊂Θ. Then ℒ_𝑍Θ+ℒ^𝐷_𝑖Θ=ℒ_𝑍Θ+ℒ_𝐷𝑖ℒ_𝑍Θ−𝐷_𝑖𝑧_𝑖ℒ_𝐷𝑖′Θ⊂Θ+ℒ_𝐷𝑖Θ(9.3) by using the commutative rule (5.17).

Lemma 9.2 (going-down lemma). Let the group of morphisms 𝐦(𝜆) preserve a submodule Θ⊂Ω(𝑚,𝑛). Let Θ′⊂Θ be the submodule of all 𝜔∈Θ satisfying ℒ_𝐷𝑖𝜔∈Θ (𝑖=1,…,𝑛). Then Θ′ is preserved, too.

Proof. Assume 𝜔∈Θ^ hence ℒ_𝐷𝑖𝜔∈Θ. Then ℒ_𝐷𝑖ℒ_𝑍𝜔=ℒ_𝑍ℒ_𝐷𝑖𝜔+ℒ_{∑𝐷𝑖𝑧𝑖′⋅𝐷𝑖′}𝜔∈Θ hence ℒ_𝑍𝜔∈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) 𝑍=𝑧^𝑗𝐼𝜕𝜕𝑤^𝑗𝐼𝑧^𝑗𝐼=𝐷_𝐼𝑧^𝑗,𝐷_𝐼=𝐷_𝑖1⋯𝐷_𝑖𝑛,(9.4) see (5.6) and (5.10) for the particular case 𝑧_𝑖=0. Then 𝑍^𝑟𝑥_𝑖=0  (𝑖=1,…,𝑛) and the sufficient requirement 𝑍²𝑤^𝑗=0 (𝑗=1,…,𝑚) ensures 𝑍∈𝔾, see (𝜄) of Lemma 5.4. We will deal with the linear case where 𝑧^𝑗=𝑎^𝑗𝑗′_𝑖′𝑤^𝑗′_𝑖′𝑎^𝑗𝑗′_𝑖^′∈ℝ(9.5) is supposed. Then 𝑍²𝑤^𝑗=𝑍𝑧^𝑗=𝑎^𝑗𝑗′_𝑖′𝑧^𝑗′_𝑖′=𝑎^𝑗𝑗′_𝑖′𝑎^{𝑗′𝑗′′}_𝑖′′𝑤^𝑗′′_{𝑖′𝑖′′}=0(9.6) identically if and only if _𝑗′𝑎^𝑗𝑗′_𝑖^′𝑎^{𝑗′𝑗′′}_𝑖′′+𝑎^𝑗𝑗′_𝑖′′𝑎^{𝑗′𝑗′′}_𝑖′=0𝑖^,𝑖^=1,…,𝑛;𝑗,𝑗^,𝑗^=1,…,𝑚.(9.7) This may be expressed in terms of matrix equations 𝐴_𝑖𝐴_𝑖′=0𝑖,𝑖^=1,…,𝑛;𝐴_𝑖=𝑎^𝑗𝑗′_𝑖(9.8) or, in either of more geometrical transcriptions 𝐴²=0,Im𝐴⊂Ker𝐴𝐴=𝜆_𝑖𝐴_𝑖,𝜆_𝑖∈ℝ,(9.9) where 𝐴 is regarded as (a matrix of an) operator acting in 𝑚-dimensional linear space and depending on parameters 𝜆₁,…,𝜆_𝑛. We do not know explicit solutions 𝐴 in full generality, however, solutions 𝐴 such that Ker𝐴 does not depend on the parameters 𝜆₁,…,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^𝑟𝑤^𝑗=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=⋯=𝑧^𝑚=0. In more detail, we take 𝑍=𝑧_𝑖𝜕𝜕𝑥_𝑖+𝑧^𝑗𝑖𝜕𝜕𝑤^𝑗𝑖+⋯𝑧^𝑗𝑖=−𝑤^𝑗𝑖_′𝐷_𝑖𝑧_𝑖′.(9.10) Then 𝑍^𝑟𝑤^𝑗=0 and we have to deal with functions 𝑍^𝑟𝑥_𝑖 in order to ensure the inclusion 𝑍∈𝔾. This is a difficult task. Let us therefore suppose 𝑧₁=𝑧…,𝑥_𝑖′,𝑤^𝑗′,𝑤^𝑗′₁,…,𝑧_𝑘=𝑐_𝑘∈ℝ(𝑘=2,…,𝑛).(9.11) Then 𝑍𝑥_𝑘=0  (𝑘=2,…,𝑛) and 𝑍²𝑥₁=𝑍𝑧=𝜕𝑧𝜕𝑥_𝑖𝑧_𝑖+𝜕𝑧𝜕𝑤^𝑗1𝑧^𝑗1,(9.12) where 𝑧^𝑗1=−𝑤^𝑗1𝐷₁𝑧=−𝑤^𝑗1𝜕𝜕𝑥𝑧₁+𝜕𝑧𝜕𝑤^𝑗′𝑤^𝑗′₁+𝜕𝑧𝜕𝑤^𝑗′₁𝑤^𝑗′₁₁.(9.13) The second-order summand 𝑍²𝑥₁=⋯+𝜕𝑧𝜕𝑤^𝑗1𝑧^𝑗1=⋯−𝜕𝑧𝜕𝑤^𝑗1𝑤^𝑗1𝜕𝑧𝜕𝑤^𝑗′₁𝑤^𝑗′₁₁(9.14) identically vanishes for the choice 𝑧=𝑓…,𝑥_𝑖′,𝑤^𝑗′,𝑢^𝑙,…𝑢^𝑙=𝑤^𝑙1𝑤¹¹;𝑙=2,…,𝑚(9.15) as follows by direct verification. Quite analogously 𝑍𝑢^𝑙=𝑍𝑤^𝑙1𝑤¹¹=𝑧^𝑙11𝑤¹¹−𝑧¹¹𝑤^𝑙1𝑤¹¹²=−𝑤^𝑙11𝑤¹¹+𝑤¹¹𝑤^𝑙1𝑤¹¹²𝐷¹𝑧=0.(9.16) It follows that all functions 𝑍^𝑟𝑥_𝑖, 𝑍^𝑟𝑤^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∈𝔾.

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 (9.5) reads d𝐺_𝑖d𝜆=0,dd𝐺𝜆^𝑗=𝑎^{𝑗𝑗′𝑖′}𝐺^𝑗′_𝑖′(𝑖=1,…,𝑛;𝑗=1,…,𝑚),(9.17) where we omit the prolongations. It is resolved by 𝐺_𝑖=𝑥_𝑖,𝐺^𝑗=𝑤^𝑗+𝜆𝑎^𝑗𝑗′_𝑖′𝑤^𝑗′_𝑖′(𝑖=1,…,𝑛;𝑗=1,…,𝑚)(9.18) as follows either by direct verification or, alternatively, from the property 𝑍²𝑥_𝑖=𝑍𝑧_𝑖=0  (𝑖=1,…,𝑛) which implies d∑𝑎^𝑗𝑗′_𝑖′𝐺^𝑗′_𝑖′d𝜆=0,𝑎^{𝑗𝑗′𝑖′}𝐺^𝑗′_𝑖′=𝑎^𝑗𝑗′_𝑖′𝐺^𝑗′_𝑖′|||_𝜆=0=𝑎^𝑗𝑗′_𝑖′𝑤^𝑗′_𝑖′.(9.19) Quite analogously, the Lie system of the vector field (9.10), (9.11), (9.15) reads d𝐺₁d𝜆=𝑓…,𝐺^𝑖′,𝐺^𝑗′,𝐺^𝑙′1𝐺¹¹,…,d𝐺^𝑘d𝜆=𝑐^𝑘,d𝐺^𝑗d𝜆=0(𝑘=2,…,𝑛;𝑗=1,…,𝑚)(9.20) and may be completed with the equations d𝐺^𝑙1/𝐺¹¹d𝜆=0(𝑙=2,…,𝑚)(9.21) 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 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.

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.

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 ifΩ₀⊂𝐦^∗Ω(𝑚,𝑛),equivalentlyΩ₀⊂𝐦^∗Ω_𝑙(𝑙=𝑙(𝐦))(10.1) for appropriate term Ω_𝑙(𝐦) of filtration (9.1). However𝐦^∗Ω_𝑙+1=𝐦^∗Ω_𝑙+ℒ_𝐷𝑖𝐦^∗Ω_𝑙(10.2) and it follows that criterion (10.1) can be verified by repeated use of operators ℒ_𝐷𝑖. In more detail, we start with equations𝐦^∗𝜔^𝑗=𝑎^𝑗𝑗′_𝐼′𝜔^𝑗′_𝐼′=d𝐦^∗𝑤^𝑗−𝐦^∗𝑤^𝑗𝑖d𝐦^∗𝑥_𝑖(10.3) 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^∗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:ℒ_𝑍ℒ_𝐷𝑖=ℒ_𝐷𝑖ℒ_𝑍−𝐷_𝑖𝑧_𝑖ℒ_𝐷𝑖′(10.4) or, when applied to basical formsℒ_𝑍𝜔^𝑗𝐼𝑖=ℒ_𝐷𝑖ℒ_𝑍𝜔^𝑗𝐼−𝐷_𝑖𝑧_𝑖𝜔^𝑗𝐼𝑖_′.(10.5) We are interested in vector fields 𝑍∈𝔾. They satisfy the recurrence (5.10) together with requirementsdim⊕ℒ^𝑟𝑍Ω₀<∞,equivalentlyℒ^𝑟𝑍Ω₀⊂Ω_𝑙(𝑍)(𝑟=0,1,…)(10.6) for appropriate 𝑙(𝑍)∈ℕ. Due to the recurrence (10.5) these requirements can be effectively investigated. In more detail, we start with equationsℒ_𝑍𝜔^𝑗=𝑎^𝑗𝑗′_𝐼′𝜔^𝑗′_𝐼′=d𝑧^𝑗−𝑧^𝑗𝑖d𝑥_𝑖−𝑤^𝑗𝑖d𝑧_𝑖.(10.7) Formulae (10.7) determine module ℒ_𝑍Ω₀. Then, choosing 𝑙(𝑍)∈ℕ, 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_𝑍𝑥_𝑖,  𝑧^𝑗=ℒ_𝑍𝑤^𝑗 (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 emptyℒ_𝑍𝜔^𝑗=⎛⎜⎜⎝𝜕𝑧^𝑗𝜕𝑤^𝑗′_𝐼′−𝑤^𝑗𝑖𝜕𝑧_𝑖𝜕𝑤^𝑗′_𝐼′⎞⎟⎟⎠𝜔_{𝑗′𝐼′}.(10.8) 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 spaces 𝐌(1,𝑛) do not admit any true generalized infinitesimal symmetries 𝑍∈𝔾.

Proof. We suppose 𝑚=1 and then (10.7) reads ℒ_𝑍𝜔¹=𝑎^𝐼11_′𝜔^1𝐼_′=⋯+𝑎^𝐼11_′′𝜔^1𝐼_′′𝑎^𝐼11′′≠0,(10.9) 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 ℒ^𝑟𝑍𝜔¹=⋯+𝑎^𝐼11_′′𝜔^1𝐼_{′′⋯𝐼′′}𝑟terms𝐼^(10.10) 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. Assume 𝑚=2, 𝑛=1 and 𝑙(𝑍)=1. Let us abbreviate 𝑥=𝑥₁,𝐷=𝐷₁,𝑍=𝑧𝜕𝜕𝑥+𝑧^𝑗𝐼𝜕𝜕𝑤^𝑗𝐼(𝑗=1,2;𝐼=1⋯1).(10.11) Then, due to 𝑙(𝑍)=1, requirement (10.6) reads ℒ^𝑟𝑍Ω₀⊂Ω₁(𝑟=0,1,…).(10.12) In particular (if 𝑟=1) we have (10.7) written here in the simplified notation ℒ_𝑍𝜔^𝑗=𝑎^𝑗1𝜔¹+𝑎^𝑗2𝜔²+𝑏^𝑗1𝜔¹¹+𝑏^𝑗2𝜔²¹(𝑗=1,2).(10.13) The next requirement (𝑟=2) implies the (only seemingly) stronger inclusion ℒ^2𝑍Ω₀⊂ℒ_𝑍Ω₀+Ω₀(10.14) 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 ℒ^2𝑍Ω₀+ℒ_𝑍Ω₀+Ω₀=Ω₁,(10.15) it follows that ℒ_𝑍Ω₁⊂ℒ^3𝑍Ω₀+ℒ^2𝑍Ω₀+ℒ_𝑍Ω₀⊂Ω₁(10.16) and Lie-Bäcklund theorem can be applied whence ℒ_𝑍Ω₀⊂Ω₀,  𝑙(𝑍)=0 which we exclude. It follows that necessarily dimℒ^2𝑍Ω₀+ℒ_𝑍Ω₀+Ω₀<dimΩ₁=4.(10.17) On the other hand dim(ℒ_𝑍Ω₀+Ω₀)≥3 and the inclusion (10.14) follows.”
After this preparation, we are passing to the proper algebra. Clearly ℒ^2𝑍𝜔^𝑗=⋯+𝑏^𝑗1ℒ_𝑍𝜔¹¹+𝑏^𝑗2ℒ_𝑍𝜔²¹=⋯+𝑏^𝑗1𝑏¹¹𝜔¹¹¹+𝑏¹²𝜔²¹¹+𝑏^𝑗2𝑏²¹𝜔¹¹¹+𝑏²²𝜔²¹¹(10.18) by using the commutative rule (10.5). Due to “weaker" inclusion (10.12) with 𝑟=2, we obtain identities 𝑏^𝑗1𝑏¹¹+𝑏^𝑗2𝑏²¹=0,𝑏^𝑗1𝑏¹²+𝑏^𝑗2𝑏²²=0(𝑗=1,2).(10.19) Omitting the trivial solution, they are satisfied if either 𝑏¹¹+𝑏²²=0,𝑏¹²=𝑐𝑏¹¹,𝑏¹¹+𝑐𝑏²¹=0(10.20) for appropriate factor 𝑐 (where 𝑏¹¹≠0 and either 𝑏¹²≠0 or 𝑏²¹≠0 is supposed) or 𝑏¹¹=𝑏²²=0,either𝑏¹²=0or𝑏²¹=0.(10.21) We deal only with the (more interesting) identities (10.20) here. Then ℒℒ_𝑍𝑍𝜔¹=𝑎¹¹𝜔¹+𝑎¹²𝜔²−𝑐𝑏𝜔¹¹+𝑐𝜔²¹,𝜔²=𝑎²¹𝜔¹+𝑎²²𝜔²+𝑏𝜔¹¹+𝑐𝜔²¹(10.22) (abbreviation 𝑏=𝑏²¹) by inserting (10.20) into (10.13). It follows that ℒ_𝑍𝜔¹+𝑐𝜔²=𝑎¹𝜔¹+𝑎²𝜔²𝑎¹=𝑎¹¹+𝑐𝑎²¹,𝑎²=𝑎¹²+𝑐𝑎²²+𝑍𝑐.(10.23) It may be seen by direct calculation of ℒ^2𝑍𝜔² that the “stronger" inclusion (10.14) is equivalent to the identity 𝑐𝑎¹=𝑎², that is, ℒ_𝑍𝜔¹+𝑐𝜔²=𝑎𝜔¹+𝑐𝜔²(10.24) (abbreviation 𝑎=𝑎¹). Alternatively, (10.24) can be proved by using Lemma 9.2.
“Hint: denoting Θ=ℒ_𝑍Ω₀+Ω₀, (10.14) implies ℒ_𝑍Θ⊂Θ. Moreover ℒ_𝐷(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 ℒ_𝑍(𝜔¹+𝑐𝜔²)∈Θ^ is a multiple of 𝜔¹+𝑐𝜔².”
The algebraical part is concluded. We have congruences ℒ_𝑍𝜔¹≅−𝑐𝑏𝜔¹¹+𝑐𝜔²¹,ℒ_𝑍𝜔²≅𝑏𝜔¹¹+𝑐𝜔²¹modΩ₀(10.25) and equality ℒ_𝑍𝜔¹+𝑐ℒ_𝑍𝜔²+𝑍𝑐𝜔²=𝑎𝜔¹+𝑐𝜔².(10.26) If 𝑍 is a variation then these three conditions together ensure the “stronger inclusion" (10.14) hence 𝑍∈𝔾.
We turn to analysis. Abbreviating 𝑍^𝑗𝑗′_𝐼′=𝜕𝑧^𝑗𝜕𝑤^𝑗′_𝐼′−𝑤^𝑗1𝜕𝑧𝜕𝑤^𝑗′_𝐼′𝑗,𝑗^=1,2;𝐼^=1⋯1(10.27) and employing (10.8), the above conditions (10.25) and (10.26) read 𝑍^{1𝑗′𝐼′}𝜔^𝑗′_𝐼′=−𝑐𝑏𝑍𝜔¹¹+𝑐𝜔²¹,𝑍^2𝑗′_𝐼′𝜔^𝑗′_𝐼′=𝑏𝜔¹¹+𝑐𝜔²¹||𝐼^||≥1_{1𝑗′𝐼′},+𝑐𝑍^2𝑗′_𝐼′𝜔^𝑗′_𝐼^′+𝑍𝑐𝜔²=𝑎𝜔¹+𝑐𝜔².(10.28) We compare coefficients of forms 𝜔^𝑗𝐼 on the level 𝑠=|𝐼^|𝑠𝑠==01∶∶𝑍𝑍¹¹₁₁₁+𝑐𝑍²¹=𝑎,𝑍¹²+𝑐𝑍²²+𝑍𝑐=𝑎𝑐,(10.29)𝑠≥2∶𝑍_{𝑗𝑗′𝐼′}=−𝑐𝑏,𝑍¹¹²=−(𝑐)²𝑏,𝑍¹²¹=𝑏,𝑍¹²²=𝑏𝑐,𝑍^11𝑗+𝑐𝑍^12𝑗=0,(10.30)=0,𝑍^1𝑗′_𝐼′+𝑐𝑍^2𝑗′_𝐼′=0.(10.31) We will successively delete the coefficients 𝑎, 𝑏, 𝑐 in order to obtain interrelations only for variables 𝑍^{𝑗𝑗𝐼}. Clearly 𝑠𝑠==01∶∶𝑍𝑍¹²₁₁₁+𝑐𝑍²²+𝑍𝑐=𝑍¹¹+𝑐𝑍²¹𝑐,+𝑍¹²²=0,𝑍¹¹¹𝑍¹²²=𝑍¹¹²𝑍¹²¹,(10.32) and we moreover have three compatible equations 𝑐=−𝑍¹¹¹𝑍¹²¹=−𝑍¹¹²𝑍¹²²,(𝑐)²=−𝑍¹¹²𝑍¹²¹(10.33) for the coefficient 𝑐. To cope with levels 𝑠≥2, we introduce functions 𝑄^𝑗=𝜔^𝑗(𝑍)=𝑧^𝑗−𝑤^𝑗1𝑧(𝑗=1,2).(10.34) Then substitution into (10.27) with the help of (10.31) gives 𝜕𝑄^𝑗𝜕𝑤^𝑗′_𝐼′=0𝑗,𝑗^=1,2;||𝐼^||≥2.(10.35) It follows moreover easily that 𝑍^𝑗𝑗′₁=𝜕𝑄^𝑗𝜕𝑤^𝑗′₁𝑗≠𝑗^,𝑍^1𝑗𝑗=𝑧+𝜕𝑄^𝑗𝜕𝑤^𝑗1,𝑍^𝑗𝑗′=𝜕𝑄^𝑗𝜕𝑤^𝑗′(10.36) and we have the final differential equations 𝑠=0∶𝜕𝑄¹𝜕𝑤²+𝑐𝜕𝑄²𝜕𝑤²+𝑍𝑐=𝜕𝑄¹𝜕𝑤¹+𝑐𝜕𝑄²𝑠=1∶2𝑧+𝜕𝑄₁𝜕𝑤¹𝑐,(10.37)𝜕𝑤¹¹+𝜕𝑄²𝜕𝑤²¹=0,𝑧+𝜕𝑄¹𝜕𝑤¹¹𝑧+𝜕𝜕𝑄𝑤²²¹=𝜕𝜕𝑄𝑤¹²¹𝜕𝑄²𝜕𝑤¹¹(10.38)for the unknown functions 𝑧=𝑧𝑥,𝑤¹,𝑤²,𝑤¹¹,𝑤²¹,𝑄^𝑗=𝑄^𝑗𝑥,𝑤¹,𝑤²,𝑤¹¹,𝑤²¹.(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^𝑗1𝑧+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 𝑄^𝑗=𝐴^𝑗𝑤¹¹²+2𝐵^𝑗𝑤¹¹𝑤²¹+𝐶^𝑗𝑤²¹²(𝑗=1,2)(10.40) with constant coefficients 𝐴^𝑗,𝐵^𝑗,𝐶^𝑗∈ℝ. We also suppose 𝑐∈ℝ and then (10.37) is trivially satisfied.
On the other hand, (10.33) provide the requirements 𝑧+𝜕𝑄¹𝜕𝑤¹¹+𝑐𝜕𝑄²𝜕𝑤¹¹=𝜕𝑄¹𝜕𝑤²¹+𝑐𝑧+𝜕𝑄²𝜕𝑤²¹=𝜕𝜕𝑄𝑤¹²¹+(𝑐)²𝜕𝑄²𝜕𝑤¹¹=0(10.41) by using (10.36). If we put 𝑧=−𝜕𝑄¹𝜕𝑤¹¹−𝜕𝑄²𝜕𝑤²¹=−𝐴¹+𝐵¹𝑤¹¹−𝐵¹+𝐶²𝑤²¹,(10.42) 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=𝑐²(𝑐²−8) if the values 𝑧, 𝑄¹, 𝑄² are inserted and the coefficients of 𝑤¹¹ and 𝑤²¹ 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 𝑄¹=𝐴¹𝑤¹¹²,𝑄²=𝐴²𝑤¹¹²,𝑧=−𝐴¹𝑤¹¹,𝑧¹=0,𝑧²=𝑤¹¹𝐴²𝑤¹¹+𝐴¹𝑤²¹,(10.43) where 𝐴¹, 𝐴² are arbitrary constants.

Remark 10.3. It follows that investigation of vector fields 𝑍∈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

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 ℝ^∞ (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 an ℱ(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 ℒ_ℋΩ⊂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_∗∶Ω₀⊂Ω₁⊂⋯⊂Ω=∪Ω_𝑙 by finite-dimensional submodules Ω_𝑙⊂Ω such that ℒ_ℋΩ_𝑙⊂Ω_𝑙+1(all𝑙),Ω_𝑙+1=Ω_𝑙+ℒ_ℋΩ_𝑙(𝑙largeenough).(11.1) 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.

The internal symmetries naturally appear. For instance, a vector field 𝑍∈𝒯(𝐌) is called a (universal) variation of diffiety Ω if ℒ_𝑍Ω⊂Ω and infinitesimal symmetry if moreover 𝑍 generates a local group, that is, if and only if 𝑍∈𝔾.

Theorem 11.1 (technical theorem). Let 𝑍 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 ⊕ℒ^𝑟ℋΘ=Ω,dim⊕ℒ^𝑟𝑍Θ<∞.(11.2)

This is exactly counterpart to Theorem 5.6: 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 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.

Example 11.3. Let us deal with the Monge equation 𝑑𝑥𝑑𝑡=𝑓𝑡,𝑥,𝑦,𝑑𝑑𝑦𝑡.(11.3) 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,𝑥₀,𝑦₀,𝑦₁,𝑦₂,…(11.5) and submodule Ω⊂Φ(𝐌) with generators d𝑥₀−𝑓d𝑡,𝜔_𝑟=d𝑦_𝑟−𝑦_𝑟+1d𝑡𝑟=0,1,…;𝑓=𝑓𝑡,𝑥₀,𝑦₀,𝑦₁.(11.6) Clearly ℋ=Ω^⟂⊂𝒯(𝐌) is one-dimensional subspace including the vector field 𝐷=𝜕𝜕𝑡+𝑓𝜕𝜕𝑥₀+𝑦_𝑟+1𝜕𝜕𝑦_𝑟.(11.7) One can easily find that we have a diffiety. (𝒜 and ℬ are trivially satisfied. The common order preserving filtrations where Ω_𝑙 involves d𝑥₀−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 forms 𝜗₀=d𝑥₀−𝑓d𝑡−𝜕𝑓𝜕𝑦₁𝜔₀,𝜔_𝑟(𝑟≤𝑙−1).(11.8) This is indeed a filtration since ℒ_𝐷𝜗₀=d𝑓−𝐷𝑓d𝑡−𝐷𝜕𝑓𝜕𝑦₁⋅𝜔₀−𝜕𝑓𝜕𝑦₁𝜔₁=𝜕𝑓𝜕𝑥₀d𝑥₀−𝑓d𝑡+𝜕𝑓𝜕𝑦₀−𝐷𝜕𝑓𝜕𝑦₁𝜔⁰=𝜕𝑓𝜕𝑥₀𝜗₀+𝐴𝜔₀𝐴=𝜕𝜕𝑦𝑓₀+𝜕𝑓𝜕𝑥₀𝜕𝑓𝜕𝑦₁−𝐷𝜕𝑓𝜕𝑦₁(11.9) and (trivially) ℒ_𝐷𝜔_𝑟=𝜔_𝑟+1. Assuming 𝐴≠0 from now on (this is satisfied if 𝑓_𝑦1𝑦1≠0) every module Ω_𝑙 is generated by the forms 𝜗_𝑟=ℒ^𝑟𝐷𝜗₀ (𝑟≤𝑙).
The forms 𝜗_𝑟 satisfy the recurrence ℒ_𝐷𝜗_𝑟=𝜗_𝑟+1. Then the formula 𝜗_𝑟+1=ℒ_𝐷𝜗_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 𝑍=𝑧𝜕𝜕𝑡+𝑧⁰𝜕𝜕𝑥₀+𝑧_𝑟𝜕𝜕𝑦_𝑟(11.11) be a variation of Ω in the common sense ℒ_𝑍Ω⊂Ω. This inclusion is equivalent to the congruence ℒ_𝑍𝜗_𝑟=𝑍⌋d𝜗_𝑟+d𝜗_𝑟(𝑍)≅−𝜗_𝑟+1(𝑍)d𝑡+𝐷𝜗_𝑟(𝑍)d𝑡=0(modΩ)(11.12) whence to the recurrence 𝜗_𝑟+1(𝑍)=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=𝜗₀(𝑍)(11.14) can be quite arbitrarily chosen. Then functions 𝜗_𝑟(𝑍)=𝐷^𝑟𝑔 are determined and we obtain quite explicit formulae for the variation 𝑍. In more detail 𝑔=𝜗₀(𝑍)=d𝑥₀−𝑓d𝑡−𝜕𝑓𝜕𝑦₁𝜔₀(𝑍)=𝑧⁰−𝑓𝑧−𝜕𝑓𝜕𝑦₁𝑧₀𝐷𝑔=𝜗₁−𝑦₁𝑧,(𝑍)𝜕𝑓𝜕𝑥₀𝜗₀+𝐴𝜔₀(𝑍)=𝜕𝜕𝑥𝑓₀𝑔+𝐴𝑧₀−𝑦₁𝑧(11.15) and these equations determine coefficients 𝑧⁰ and 𝑧₀ in terms of functions 𝑧 and 𝑔. Coefficients 𝑧_𝑟 (𝑟≥1) follow by prolongation (not stated here). If moreover dim{ℒ^𝑟𝑍𝜗₀}_𝑟∈ℕ<∞(11.16) we have infinitesimal symmetry 𝑍∈𝔾, see Theorem 11.1.

Example 11.4. Let us deal with the Hilbert-Cartan equation [3] 𝑑𝑦𝑑𝑡=𝑑²𝑥𝑑𝑡²².(11.17) Passing to the diffiety, we introduce space 𝐌 with coordinates 𝑡,𝑥₀,𝑥₁,𝑦₀,𝑦₁,𝑦₂,…(11.18) and submodule Ω⊂Φ(𝐌) generated by forms d𝑥₀−𝑥₁d𝑡,d𝑥₁−√𝑦₁d𝑡,𝜔_𝑟=d𝑦_𝑟−𝑦_𝑟+1d𝑡(𝑟=0,1,…).(11.19) The submodule ℋ=Ω^⟂⊂𝒯(𝐌) is generated by the vector field 𝐷=𝜕𝜕𝑡+𝑥¹𝜕𝜕𝑥₀+√𝑦₁𝜕𝜕𝑥₁+𝑦_𝑟+1𝜕𝜕𝑦_𝑟.(11.20) We introduce the form 𝜗₀=d𝑥₀−𝑥₁d𝑡+𝐵d𝑥₁−√𝑦₁d𝑡−12√𝑦₁𝜔₀⎛⎜⎜⎝𝐵=1/√𝑦₁𝐷1/√𝑦₁⎞⎟⎟⎠(11.21) and moreover the forms 𝜗𝜗₁=ℒ_𝐷𝜗₀=(1+𝐷𝐵){⋯},₂=ℒ_𝐷𝜗₁=𝐷²𝐵{⋯}−𝐶𝜔₀𝜗₃𝐶=(1+𝐷𝐵)𝐷2√1𝑦₁,𝜗₄=⋯+𝐶𝜔₁,⋮=⋯+𝐶𝜔₂,(11.22) Assuming 𝐶≠0, we have a standard filtration Ω_∗ where the submodules Ω_𝑙⊂Ω are generated by forms 𝜗_𝑟 (𝑟≤𝑙). Explicit formulae for variations 𝑍=𝑧𝜕𝜕𝑡+𝑧⁰𝜕𝜕𝑥₀+𝑧¹𝜕𝜕𝑥₁+𝑧_𝑟𝜕𝜕𝑦_𝑟(11.23) 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.

Appendix

For the convenience of reader, we survey some results [9, 18, 19] on the modules Adj. Our reasonings are carried out in the space ℝ^𝑛 and will be true locally near generic points.

Let Θ be a given module of 1-forms and 𝐴(Θ) the module of all vector fields 𝑋 such that ℒ_𝑓𝑋Θ⊂Θ for all functions 𝑓, see [9]. Clearly ℒ_[𝑋,𝑍]Θ=ℒ_𝑋ℒ_𝑍−ℒ_𝑍ℒ_𝑋Θ⊂Θ(𝑋,𝑍∈𝐴(Θ))(A.1) and it follows that identity 𝑓[𝑋,𝑌]=[𝑋,𝑍]+𝑋𝑓⋅𝑌(𝑋,𝑌∈𝐴(Θ);𝑍=𝑓𝑌)(A.2) implies ℒ_{𝑓[𝑋,𝑌]}Θ⊂Θ whence [𝐴(Θ),𝐴(Θ)]⊂𝐴(Θ).

Let Θ be of a finite dimension 𝐼. The Frobenius theorem can be applied, and it follows that module AdjΘ=𝐴(Θ)^⟂ (of all forms 𝜑 satisfying 𝜑(𝐴(Θ))=0) has a certain basis d𝑓¹,…,d𝑓^𝐾 (𝐾≥𝐼).

On the other hand, identity ℒ_𝑓𝑋𝜗=𝑓𝑋⌋d𝜗+d(𝑓𝜗(𝑋))=𝑓ℒ_𝑋𝜗+𝜗(𝑋)𝜗(A.3) implies that 𝑋∈𝐴(Θ) if and only if 𝜗(𝑋)=0,𝑋⌋d𝜗∈Θ(𝜗∈Θ)(A.4) which is the classical definition, see [2]. In particular Θ⊂AdjΘ so we may suppose the generators 𝜗^𝑖=d𝑓^𝑖+𝑔^𝑖𝐼+1d𝑓^𝐼+1+⋯+𝑔^𝑖𝐾d𝑓^𝐾∈Θ(𝑖=1,…,𝐼)(A.5) of module Θ. Recall that 𝑋𝑓^𝑘=0   (𝑘=1,…,𝐾;𝑋∈𝐴(Θ)) whence ℒ_𝑋𝜗^𝑖=𝑋𝑔^𝑖𝐼+1d𝑓^𝐼+1+⋯+𝑋𝑔^𝑖𝐾d𝑓^𝐾∈Θ(A.6) and this implies 𝑋𝑔^𝑖𝐼+1=⋯=𝑋𝑔^𝑖𝐾=0. It follows that d𝑔^𝑖𝐼+1,…,d𝑔^𝑖𝐾∈AdjΘ(𝑖=1,…,𝐼)(A.7) and therefore all coefficients 𝑔^𝑖𝑘 depend only on variables 𝑓¹,…,𝑓^𝐾.

Acknowledgment

This research has been conducted at the Department of Mathematics as part of the research project CEZ: Progressive reliable and durable structures, MSM 0021630519.