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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 919538, 35 pages
http://dx.doi.org/10.1155/2011/919538
Research Article

The Lie Group in Infinite Dimension

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

Received 6 December 2010; Accepted 12 January 2011

Academic Editor: Miroslava RůžičkovÑ

Copyright Β© 2011 V. Tryhuk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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, 𝐢∞ smooth) action of a Lie group on infinite-dimensional space (a manifold modelled on β„βˆž) 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-BΓ€cklund) infinitesimal symmetries 𝑧𝑍=π‘–πœ•πœ•π‘₯𝑖+ξ“π‘§π‘—πΌπœ•πœ•π‘€π‘—πΌξ€·π‘–=1,…,𝑛;𝑗=1,…,π‘š;𝐼=𝑖1⋯𝑖𝑛;𝑖1,…,𝑖𝑛,=1,…,𝑛(1.1) where the coefficients𝑧𝑖=𝑧𝑖…,π‘₯𝑖′,𝑀𝑗′𝐼′,…,𝑧𝑗𝐼=𝑧𝑗𝐼…,π‘₯𝑖′,𝑀𝑗′𝐼′,…(1.2) are functions of independent variables π‘₯𝑖, dependent variables 𝑀𝑗 and a finite number of jet variables 𝑀𝑗𝐼=πœ•π‘›π‘€π‘—/πœ•π‘₯𝑖1β‹―πœ•π‘₯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 transformationsπ‘₯𝑖=πΊπ‘–ξ‚€πœ†;…,π‘₯𝑖′,𝑀𝑗′𝐼′,,…𝑀𝑗𝐼=πΊπ‘—πΌξ‚€πœ†;…,π‘₯𝑖′,𝑀𝑗′𝐼′,…(1.3) in the underlying infinite-order jet space since the relevant Lie systemπœ•πΊπ‘–πœ•πœ†=𝑧𝑖…,𝐺𝑖′,𝐺𝑗′𝐼′,,β€¦πœ•πΊπ‘—πΌπœ•πœ†=𝑧𝑗𝐼…,𝐺𝑖′,𝐺𝑗′𝐼′𝐺,…𝑖||πœ†=0=π‘₯𝑖,𝐺𝑗𝐼||πœ†=0=𝑀𝑗𝐼(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 β€œinfinitesimal 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 β„βˆž. We will see that they admit β€œfinite-dimensional approximations" and as a byproduct, the relevant infinitesimal transformations may be exactly characterized by certain β€œfiniteness 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 β„βˆž with coordinates β„Ž1,β„Ž2,…. Equations (2.1) of transformations 𝐦(πœ†) can be schematically represented by Figure 1(a).

fig1
Figure 1

We prove that in appropriate new coordinate system 𝐹1,𝐹2,…on β„βˆž, the same transformations 𝐦(πœ†) become block triangular as in Figure 1(b). It follows that a certain hierarchy of finite-dimensional subspaces of β„βˆž is preserved which provides the β€œapproximation" of 𝐦(πœ†). The infinitesimal transformation 𝑍=d𝐦(πœ†)/dπœ†|πœ†=0 clearly preserves the same hierarchy which provides certain algebraical β€œfiniteness" of 𝑍.

If the primary space β„βˆž 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 β„βˆž 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 β€œfiniteness requirements" for one-parameter groups (namely, the result (πœ„) 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.

fig2
Figure 2

2. Fundamental Approximation Results

Our reasonings will be carried out in the space β„βˆž with coordinates β„Ž1,β„Ž2,…[9] and we introduce the structural family β„± of all real-valued, locally defined and 𝐢∞-smooth functions 𝑓=𝑓(β„Ž1,…,β„Žπ‘š(𝑓)) depending on a finite number of coordinates. In future, such functions will contain certain 𝐢∞-smooth real parameters, too.

We are interested in (local) groups of transformations 𝐦(πœ†) in β„βˆž defined by formulae𝐦(πœ†)βˆ—β„Žπ‘–=π»π‘–ξ€·πœ†;β„Ž1,…,β„Žπ‘š(𝑖)ξ€Έ,βˆ’πœ€π‘–<πœ†<πœ€π‘–,πœ€π‘–>0(𝑖=1,2,…),(2.1) where π»π‘–βˆˆβ„± if the parameter πœ† is kept fixed. We suppose𝐦(0)=id.,𝐦(πœ†+πœ‡)=𝐦(πœ†)𝐦(πœ‡)(2.2) 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 every 𝐼=1,2,… and 0<πœ€<min{πœ€1,…,πœ€πΌ}, let β„±(𝐼,πœ€)βŠ‚β„± be the subset of all composed functions ξ€·ξ€·πœ†πΉ=𝐹…,π¦π‘—ξ€Έβˆ—β„Žπ‘–ξ€Έξ€·,…=𝐹…,π»π‘–ξ€·πœ†π‘—;β„Ž1,…,β„Žπ‘š(𝑖)ξ€Έξ€Έ,,…(2.3) where 𝑖=1,…,𝐼; β€‰βˆ’πœ€<πœ†π‘—<πœ€;  𝑗=1,…,𝐽=𝐽(𝐼)=max{π‘š(1),…,π‘š(𝐼)} and 𝐹 is arbitrary 𝐢∞-smooth function (of 𝐼𝐽 variables). In functions πΉβˆˆβ„±(𝐼,πœ€), variables πœ†1,…,πœ†π½ are regarded as mere parameters.

Functions (2.3) will be considered on open subsets of β„βˆž where the rank of the Jacobi (𝐼𝐽×𝐽)-matrixξ‚΅πœ•πœ•β„Žπ‘—β€²π»π‘–ξ€·πœ†π‘—;β„Ž1,…,β„Žπ‘š(𝑖)𝑖=1,…,𝐼;𝑗,π‘—ξ…žξ€Έ=1,…,𝐽(2.4) of functions 𝐻𝑖(πœ†π‘—;β„Ž1,…,β„Žπ‘š(𝑖)) locally attains the maximum (for appropriate choice of parameters). This rank and therefore the subset β„±(𝐼,πœ€)βŠ‚β„± does not depend on πœ€ as soon as πœ€=πœ€(𝐼) is close enough to zero. This is supposed from now on and we may abbreviate β„±(𝐼)=β„±(𝐼,πœ€).

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 β„βˆž. 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(πœ†)βˆ—β„±(𝐼)βŠ‚β„±(𝐼).(2.5)

Proof. Clearly 𝐦(πœ†)βˆ—π»π‘–ξ€·πœ†π‘—ξ€Έ;…=𝐦(πœ†)βˆ—π¦ξ€·πœ†π‘—ξ€Έβˆ—β„Žπ‘–ξ€·=π¦πœ†+πœ†π‘—ξ€Έβˆ—β„Žπ‘–=π»π‘–ξ€·πœ†+πœ†π‘—ξ€Έ;…(2.6) and therefore 𝐦(πœ†)βˆ—ξ€·πΉ=𝐹…,π»π‘–ξ€·πœ†+πœ†π‘—;β„Ž1,…,β„Žπ‘š(𝑖)ξ€Έξ€Έ,β€¦βˆˆβ„±(𝐼).(2.7)

Denoting by 𝐾(𝐼) the rank of matrix (2.4), there exist basical functions πΉπ‘˜=πΉπ‘˜ξ€·β€¦,π»π‘–ξ€·πœ†π‘—;β„Ž1,…,β„Žπ‘š(𝑖)ξ€Έξ€Έ,β€¦βˆˆβ„±(𝐼)(π‘˜=1,…,𝐾(𝐼))(2.8) such that rank(πœ•πΉπ‘˜/πœ•β„Žπ‘—β€²)=𝐾(𝐼). Then a function π‘“βˆˆβ„± lies in β„±(𝐼) if and only if 𝑓=𝑓(𝐹1,…,𝐹𝐾(𝐼)) is a composed function. In more detail𝐹=πΉξ€·πœ†1,…,πœ†π½;𝐹1,…,𝐹𝐾(𝐼)ξ€Έβˆˆβ„±(𝐼)(2.9) is such a composed function if we choose 𝑓=𝐹 given by (2.3). Parameters πœ†1,…,πœ†π½ occurring in (2.3) are taken into account here. It follows thatπœ•πΉπœ•πœ†π‘—=πœ•πΉπœ•πœ†π‘—ξ€·πœ†1,…,πœ†π½;𝐹1,…,𝐹𝐾(𝐼)ξ€Έβˆˆβ„±(𝐼)(𝑗=1,…,𝐽)(2.10) and analogously for the higher derivatives.

In particular, we also have π»π‘–ξ€·πœ†;β„Ž1,…,β„Žπ‘š(𝑖)ξ€Έ=π»π‘–ξ€·πœ†;𝐹1,…,𝐹𝐾(𝐼)ξ€Έβˆˆβ„±(𝐼)(𝑖=1,…,𝐼)(2.11) for the choice 𝐹=𝐻𝑖(πœ†;…) in (2.9) whenceπœ•π‘Ÿπ»π‘–πœ•πœ†π‘Ÿ=πœ•π‘Ÿπ»π‘–πœ•πœ†π‘Ÿξ€·πœ†;𝐹1,…,𝐹𝐾(𝐼)ξ€Έβˆˆβ„±(𝐼)(𝑖=1,…,𝐼;π‘Ÿ=0,1,…).(2.12) The basical functions can be taken from the family of functions 𝐻𝑖(πœ†;…)  (𝑖=1,…,𝐼) for appropriate choice of various values of πœ†. Functions (2.12) are enough as well even for a fixed value πœ†, for example, for πœ†=0, see Theorem 3.2 below.

Lemma 2.3. For any basical function, one has 𝐦(πœ†)βˆ—πΉπ‘˜=πΉπ‘˜ξ€·πœ†;𝐹1,…,𝐹𝐾(𝐼)ξ€Έ(π‘˜=1,…,𝐾(𝐼)).(2.13)

Proof. πΉπ‘˜βˆˆβ„±(𝐼) implies 𝐦(πœ†)βˆ—πΉπ‘˜βˆˆβ„±(𝐼) and (2.9) may be applied with the choice 𝐹=𝐦(πœ†)βˆ—πΉπ‘˜ and πœ†1=β‹―=πœ†π½=πœ†.

Summary 1. Coordinates β„Žπ‘–=𝐻𝑖(0;…)  (𝑖=1,…,𝐼) were included into the subfamily β„±(𝐼)βŠ‚β„± which is transformed into itself by virtue of (2.13). So we have a one-parameter group acting on β„±(𝐼). One can even choose 𝐹1=β„Ž1,…,𝐹𝐼=β„ŽπΌ here and then, if 𝐼 is large enough, formulae (2.13) provide a β€œfinite-dimensional approximation" of the primary mapping 𝐦(πœ†). The block-triangular structure of the infinite matrix of transformations 𝐦(πœ†) mentioned in Section 1 appears if πΌβ†’βˆž and the system of functions 𝐹1,𝐹2,… is succesively completed.

3. The Infinitesimal Approach

We introduce the vector field 𝑧𝑍=π‘–πœ•πœ•β„Žπ‘–=d𝐦(πœ†)|||dπœ†πœ†=0𝑧𝑖=πœ•π»π‘–ξ€·πœ•πœ†0;β„Ž1,…,β„Žπ‘š(𝑖)ξ€Έξ‚Ά;𝑖=1,2,…,(3.1) the infinitesimal transformation (ℐ𝑇) of group 𝐦(πœ†). Let us recall the celebrated Lie system πœ•π¦πœ•πœ†(πœ†)βˆ—β„Žπ‘–=πœ•π»π‘–πœ•πœ†(πœ†;…)=πœ•π»π‘–||||πœ•πœ‡(πœ†+πœ‡;…)πœ‡=0=πœ•πœ•πœ‡π¦(πœ†+πœ‡)βˆ—β„Žπ‘–||||πœ‡=0=𝐦(πœ†)βˆ—πœ•πœ•πœ‡π¦(πœ‡)βˆ—β„Žπ‘–||||πœ‡=0=𝐦(πœ†)βˆ—π‘β„Žπ‘–=𝐦(πœ†)βˆ—π‘§π‘–.(3.2) In more explicit (and classical) transcriptionπœ•π»π‘–ξ€·πœ•πœ†πœ†;β„Ž1,…,β„Žπ‘š(𝑖)ξ€Έ=𝑧𝑖𝐻1ξ€·πœ†;β„Ž1,…,β„Žπ‘š(1)ξ€Έ,…,π»π‘š(𝑖)ξ€·πœ†;β„Ž1,…,β„Žπ‘š(π‘š(𝑖)).ξ€Έξ€Έ(3.3) One can also check the general identityπœ•π‘Ÿπœ•πœ†π‘Ÿπ¦(πœ†)βˆ—π‘“=𝐦(πœ†)βˆ—π‘π‘Ÿπ‘“(π‘“βˆˆβ„±;π‘Ÿ=0,1,…)(3.4) by a mere routine induction on π‘Ÿ.

Lemma 3.1 (finiteness lemma). For all π‘Ÿβˆˆβ„•, π‘π‘Ÿβ„±(𝐼)βŠ‚β„±(𝐼).

Proof. Clearly 𝑍𝐹=𝐦(πœ†)βˆ—||π‘πΉπœ†=0=πœ•πœ•πœ†π¦(πœ†)βˆ—πΉ|||πœ†=0βˆˆβ„±(𝐼)(3.5) for any function (2.3) by virtue of (2.10): induction on π‘Ÿ.

Theorem 3.2 (finiteness theorem). Every function πΉβˆˆβ„±(𝐼) admits (locally, near generic points) the representation ξ‚πΉξ‚΅πœ•πΉ=…,π‘Ÿπ»π‘–πœ•πœ†π‘Ÿξ€·0;β„Ž1,…,β„Žπ‘š(𝑖)ξ€Έξ‚Ά,…(3.6) in terms of a composed function where 𝑖=1,…,𝐼 and 𝐹 is a β„‚βˆž-smooth function of a finite number of variables.

Proof. Let us temporarily denote π»π‘–π‘Ÿ=πœ•π‘Ÿπ»π‘–πœ•πœ†π‘Ÿπœ•(πœ†;…)=π‘Ÿπœ•πœ†π‘Ÿπ¦(πœ†)βˆ—β„Žπ‘–,β„Žπ‘–π‘Ÿ=π»π‘–π‘Ÿ(0;…)=π‘π‘Ÿβ„Žπ‘–,(3.7) where the second equality follows from (3.4) with 𝑓=β„Žπ‘–,β€‰β€‰πœ†=0. Then π»π‘–π‘Ÿ=𝐦(πœ†)βˆ—β„Žπ‘–π‘Ÿ=𝐦(πœ†)βˆ—π‘π‘Ÿβ„Žπ‘–(3.8) by virtue of (3.4) with general πœ†.
If 𝑗=𝑗(𝑖) is large enough, there does exist an identity β„Žπ‘–π‘—+1=𝐺𝑖(β„Žπ‘–0,…,β„Žπ‘–π‘—). Therefore πœ•π‘—+1π»π‘–πœ•πœ†π‘—+1=𝐻𝑖𝑗+1=𝐺𝑖𝐻𝑖0,…,𝐻𝑖𝑗=πΊπ‘–ξ‚΅π»π‘–πœ•,…,π‘—π»π‘–πœ•πœ†π‘—ξ‚Ά(3.9) by applying 𝐦(πœ†)βˆ—. This may be regarded as ordinary differential equation with initial values 𝐻𝑖||πœ†=0=β„Žπ‘–0πœ•,…,π‘—π»π‘–πœ•πœ†π‘—||||πœ†=0=β„Žπ‘–π‘—.(3.10) The solution 𝐻𝑖=𝐻𝑖(πœ†;β„Žπ‘–0,…,β„Žπ‘–π‘—) expressed in terms of initial values reads π»π‘–ξ€·πœ†;β„Ž1,…,β„Žπ‘š(𝑖)ξ€Έ=ξ‚π»π‘–ξ‚΅πœ†;𝐻𝑖0;β„Ž1,…,β„Žπ‘š(𝑖)ξ€Έπœ•,…,π‘—π»π‘–πœ•πœ†π‘—ξ€·0;β„Ž1,…,β„Žπ‘š(𝑖)ξ€Έξ‚Ά(3.11) in full detail. If πœ† is kept fixed, this is exactly the identity (3.6) for the particular case 𝐹=𝐻𝑖(πœ†;β„Ž1,…,β„Žπ‘š(𝑖)). The general case follows by a routine.

Definition 3.3. Let 𝔾 be the set of (local) vector fields 𝑧𝑍=π‘–πœ•πœ•β„Žπ‘–ξ€·π‘§π‘–ξ€Έβˆˆβ„±,infinitesum(3.12) such that every family of functions {π‘π‘Ÿβ„Žπ‘–}π‘Ÿβˆˆβ„•β€‰β€‰ (𝑖 fixed but arbitrary) can be expressed in terms of a finite number of coordinates.

Remark 3.4. Neither 𝔾+π”ΎβŠ‚π”Ύ nor [𝔾,𝔾]βŠ‚π”Ύ as follows from simple examples. However, 𝔾 is a conical set (over β„±): if π‘βˆˆπ”Ύ then π‘“π‘βˆˆπ”Ύ for any π‘“βˆˆβ„±. Easy direct proof may be omitted here.

Summary 2. If 𝑍 is  ℐ𝒯 of a group then all functions π‘π‘Ÿβ„Žπ‘–β€‰β€‰(𝑖=1,…,𝐼; π‘Ÿ=0,1,…) are included into family β„±(𝐼) hence π‘βˆˆπ”Ύ. The converse is clearly also true: every vector field π‘βˆˆπ”Ύ generates a local Lie group since the Lie system (3.3) admits finite-dimensional approximations in spaces β„±(𝐼).

Let us finally reformulate the last sentence in terms of basical functions.

Theorem 3.5 (approximation theorem). Let π‘βˆˆπ”Ύ be a vector field locally defined on β„βˆž and 𝐹1,…,𝐹𝐾(𝐼)βˆˆβ„± be a maximal functionally independent subset of the family of all functions π‘π‘Ÿβ„Žπ‘–(𝑖=1,…,𝐼;π‘Ÿ=0,1,…).(3.13) Denoting π‘πΉπ‘˜=πΉπ‘˜(𝐹1,…,𝐹𝐾(𝐼)), then the system πœ•πœ•πœ†π¦(πœ†)βˆ—πΉπ‘˜=𝐦(πœ†)βˆ—π‘πΉπ‘˜=πΉπ‘˜ξ€·π¦(πœ†)βˆ—πΉ1,…,𝐦(πœ†)βˆ—πΉπΎ(𝐼)ξ€Έ(π‘˜=1,…,𝐾(𝐼))(3.14) may be regarded as a β€œfinite-dimensional approximation" to the Lie system (3.3) of the one-parameter local group 𝐦(πœ†) generated by 𝑍.

In particular, assuming 𝐹1=β„Ž1,…,𝐹𝐼=β„ŽπΌ, then the the initial portion ddπœ†π¦(πœ†)βˆ—πΉπ‘–=ddπœ†π¦(πœ†)βˆ—β„Žπ‘–=d𝐻dπœ†π‘–=𝑧𝑖𝐻1,…,π»π‘š(𝑖)ξ€Έ(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,…,𝑍𝑑 be commuting local vector fields in the space β„βˆž. Then 𝑍1,…,π‘π‘‘βˆˆπ”Ύ if and only if the vector fields 𝑍=π‘Ž1𝑍1+β‹―+π‘Žπ‘‘π‘π‘‘β€‰β€‰(π‘Ž1,…,π‘Žπ‘‘βˆˆβ„) 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 πœ†=(πœ†1,…,πœ†π‘‘)βˆˆβ„π‘‘ are parameters close to the zero point 0=(0,…,0)βˆˆβ„π‘‘. The rule (2.2) is generalized as 𝐦(0)=id.,𝐦(πœ‘(πœ†,πœ‡))=𝐦(πœ†)𝐦(πœ‡),(4.1) where πœ†=(πœ†1,…,πœ†π‘‘),β€‰β€‰πœ‡=(πœ‡1,…,πœ‡π‘‘) and πœ‘=(πœ‘1,…,πœ‘π‘‘) determine the composition of parameters. Appropriately adapting the space β„±(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 fields 𝑍1,…,𝑍𝑑 which are ℐ𝒯 of the group. We recall (without proof) the Lie equations [17] πœ•πœ•πœ†π‘—π¦(πœ†)βˆ—ξ“π‘Žπ‘“=𝑗𝑖(πœ†)𝐦(πœ†)βˆ—π‘π‘—π‘“(π‘“βˆˆβ„±;𝑗=1,…,𝑑)(4.2) with the initial condition 𝐦(0)=id. Assuming 𝑍1,…,𝑍𝑑 linearly independent over ℝ, coefficients π‘Žπ‘—π‘–(πœ†) may be arbitrarily chosen and the solution 𝐦(πœ†) 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 π‘π‘—πΉπ‘˜=ξ‚πΉπ‘˜π‘—ξ€·πΉ1,…,𝐹𝐾(𝐼)ξ€Έ(𝑗=1,…,𝑑;π‘˜=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=π‘Ž1𝑍1+β‹―+π‘Žπ‘‘π‘π‘‘β€‰β€‰(π‘Žπ‘–βˆˆβ„) on the space β„βˆž. Let moreover 𝑍1,…,π‘π‘‘βˆˆπ”Ύβ€‰β€‰uniformly in the sense that there is a universal space β„±(𝐼) with ℒ𝑍𝑖ℱ(𝐼)βŠ‚β„±(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,…,π‘π‘‘βˆˆπ”Ύ 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 β„βˆž such that [𝒦,𝒦]βŠ‚π’¦. Then π’¦βŠ‚π”Ύ if and only if there exist generators (over β„±) 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(π‘š,𝑛) be the jet space of 𝑛-dimensional submanifolds in β„π‘š+𝑛 [9–13]. We recall the familiar (local) jet coordinates π‘₯𝑖,𝑀𝑗𝐼𝐼=𝑖1β€¦π‘–π‘Ÿ;𝑖,𝑖1,…,π‘–π‘Ÿξ€Έ.=1,…,𝑛;π‘Ÿ=0,1,…;𝑗=1,…,π‘š(5.1) Functions 𝑓=𝑓(…,π‘₯𝑖,𝑀𝑗𝐼,…) on 𝐌(π‘š,𝑛) are 𝐢∞-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 Ξ©(π‘š,𝑛) of contact forms ξ“π‘Žπœ”=π‘—πΌπœ”π‘—πΌξ‚€ο¬nitesum,πœ”π‘—πΌ=dπ‘€π‘—πΌβˆ’ξ“π‘€π‘—πΌπ‘–dπ‘₯𝑖(5.2) or, equivalently, by the β€œorthogonal" module β„‹(π‘š,𝑛)=Ξ©βŸ‚(π‘š,𝑛) of formal derivatives ξ“π‘Žπ·=𝑖𝐷𝑖𝐷𝑖=πœ•πœ•π‘₯𝑖+ξ“π‘€π‘—πΌπ‘–πœ•πœ•π‘€π‘—πΌ;𝑖=1,…,𝑛;π·βŒ‹πœ”π‘—πΌ=πœ”π‘—πΌξƒͺ(𝐷)=0.(5.3) Let us state useful formulae 𝐷d𝑓=𝑖𝑓dπ‘₯𝑖+ξ“πœ•π‘“πœ•π‘€π‘—πΌπœ”π‘—πΌ,π·π‘–βŒ‹dπœ”π‘—πΌ=πœ”π‘—πΌπ‘–,β„’π·π‘–πœ”π‘—πΌ=πœ”π‘—πΌπ‘–,(5.4) where ℒ𝐷𝑖=π·π‘–βŒ‹d+dπ·π‘–βŒ‹ denotes the Lie derivative.

We are interested in (local) one-parameter groups of transformations 𝐦(πœ†) given by certain formulae𝐦(πœ†)βˆ—π‘₯𝑖=πΊπ‘–ξ‚€πœ†;…,π‘₯𝑖′,𝑀𝑗′𝐼′,…,𝐦(πœ†)βˆ—π‘€π‘—πΌ=πΊπ‘—πΌξ‚€πœ†;…,π‘₯𝑖′,𝑀𝑗′𝐼′,…(5.5) and in vector fields𝑧𝑍=𝑖…,π‘₯𝑖′,π‘€π‘—β€²πΌβ€²ξ‚πœ•,β€¦πœ•π‘₯𝑖+𝑧𝑗𝐼…,π‘₯𝑖′,π‘€π‘—β€²πΌβ€²ξ‚πœ•,β€¦πœ•π‘€π‘—πΌ(5.6) locally defined on the jet space 𝐌(π‘š,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(πœ†)βˆ—Ξ©(π‘š,𝑛)βŠ‚Ξ©(π‘š,𝑛) holds true. We speak of a (universal) variation (5.6) of the jet structure ifℒ𝑍Ω(π‘š,𝑛)βŠ‚Ξ©(π‘š,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 β€œtrivial summand 𝐷" and the so-called β€œevolutionary form 𝑉" of the vector field 𝑍, explicitly 𝑧𝑍=𝐷+V𝐷=π‘–π·π‘–ξ“π‘„βˆˆβ„‹(π‘š,𝑛),𝑉=π‘—πΌπœ•πœ•π‘€π‘—πΌ,𝑄𝑗𝐼=π‘§π‘—πΌβˆ’ξ“π‘€π‘—πΌπ‘–π‘§π‘–ξƒͺ.(5.7) The summand 𝐷 is usually neglected in a certain sense [3–7] and the β€œessential" summand 𝑉 is identified with the evolutional system πœ•π‘€π‘—πΌπœ•πœ†=𝑄𝑗𝐼…,π‘₯𝑖′,𝑀𝑗′𝐼′𝑀,…𝑗𝐼=πœ•π‘›π‘€π‘—πœ•π‘₯𝑖1β‹―πœ•π‘₯π‘–π‘›ξ€·πœ†,π‘₯1,…,π‘₯𝑛ξƒͺ(5.8) of partial differential equations (the finite subsystem with 𝐼=πœ™ empty is enough here since the remaining part is a mere prolongation). This evolutional system is regarded as a β€œvirtual flow" on the β€œspace of solutions" 𝑀𝑗=𝑀𝑗(π‘₯1,…,π‘₯𝑛), see [7, especially page 11]. In more generality, some differential constraints may be adjoint. However, in accordance with the ancient classical tradition, functions 𝛿𝑀𝑗=πœ•π‘€π‘—/πœ•πœ† are just the variations. (There is only one novelty: in classical theory, 𝛿𝑀𝑗 are introduced only along a given solution while the vector fields 𝑍 are β€œuniversally" defined on the space.) In this β€œevolutionary 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 β€œtrue 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𝐺𝑗𝐼𝑖𝐷𝑖′𝐺𝑖=π·π‘–β€²πΊπ‘—πΌξ€·π‘–ξ…žξ€Έ=1,…,𝑛,(5.9) where det(π·π‘–ξ…žπΊπ‘–)β‰ 0 is supposed and vector field (5.6) is a variation if and only if𝑧𝑗𝐼𝑖=π·π‘–π‘§π‘—πΌβˆ’ξ“π‘€π‘—πΌπ‘–β€²π·π‘–π‘§π‘–β€².(5.10) Recurrence (5.9) easily follows from the inclusion 𝐦(πœ†)βˆ—πœ”π‘—πΌβˆˆΞ©(π‘š,𝑛) and we omit the proof. Recurrence (5.10) follows from the identity β„’π‘πœ”π‘—πΌ=ℒ𝑍dπ‘€π‘—πΌβˆ’ξ“π‘€π‘—πΌπ‘–dπ‘₯𝑖=dπ‘§π‘—πΌβˆ’ξ“π‘§π‘—πΌπ‘–dπ‘₯π‘–βˆ’ξ“π‘€π‘—πΌπ‘–d𝑧𝑖≅𝐷𝑖′zπ‘—πΌβˆ’ξ“π‘§π‘—πΌπ‘–β€²βˆ’ξ“π‘€π‘—πΌπ‘–π·π‘–β€²π‘§π‘–ξ‚dπ‘₯𝑖′(modΞ©(π‘š,𝑛))(5.11) and the inclusion β„’π‘πœ”π‘—πΌβˆˆΞ©(π‘š,𝑛). The obvious formulaβ„’π‘πœ”π‘—πΌ=ξ“βŽ›βŽœβŽœβŽπœ•π‘§π‘—πΌπœ•π‘€π‘—β€²πΌβ€²βˆ’ξ“π‘€π‘—πΌπ‘–πœ•π‘§π‘–πœ•π‘€π‘—β€²πΌβ€²βŽžβŽŸβŽŸβŽ πœ”π‘—β€²πΌβ€²(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 𝐺𝑖, 𝐺𝑗, 𝑧𝑖, 𝑧𝑗 (empty 𝐼=πœ™) 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 πœ”π‘—πΌπ‘–(𝑍)=π·π‘–πœ”π‘—πΌ(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 π‘βˆˆπ”Ύ hence if and only if every familyξ€½π‘π‘Ÿπ‘₯π‘–ξ€Ύπ‘Ÿβˆˆβ„•,ξ€½π‘π‘Ÿπ‘€π‘—πΌξ€Ύπ‘Ÿβˆˆβ„•(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 π‘βˆˆπ”Ύ is equivalent to any of the requirements (πœ„) every family of functions ξ€½π‘π‘Ÿπ‘₯π‘–ξ€Ύπ‘Ÿβˆˆβ„•,ξ€½π‘π‘Ÿπ‘€π‘—ξ€Ύπ‘Ÿβˆˆβ„•(𝑖=1,…,𝑛;𝑗=1,…,π‘š)(5.14) can be expressed in terms of a finite number of jet coordinates,(πœ„πœ„)every family of differential forms ξ€½β„’π‘Ÿπ‘dπ‘₯π‘–ξ€Ύπ‘Ÿβˆˆβ„•,ξ€½β„’π‘Ÿπ‘dπ‘€π‘—ξ€Ύπ‘Ÿβˆˆβ„•(𝑖=1,…,𝑛;𝑗=1,…,π‘š)(5.15) involves only a finite number of linearly independent terms,(πœ„πœ„πœ„)every family of differential forms ξ€½β„’π‘Ÿπ‘dπ‘₯π‘–ξ€Ύπ‘Ÿβˆˆβ„•,ξ€½β„’π‘Ÿπ‘dπ‘€π‘—πΌξ€Ύπ‘Ÿβˆˆβ„•(𝑖=1,…,𝑛;𝑗=1,…,π‘š;arbitrary𝐼)(5.16) involves only a finite number of linearly independent terms.

Proof. Inclusion π‘βˆˆπ”Ύ is defined by using the families (5.13) and this trivially implies (πœ„) where only the empty multi-indice 𝐼=πœ™ is involved. Then (πœ„) implies (πœ„πœ„) by using the rule ℒ𝑍d𝑓=d𝑍𝑓. Assuming (πœ„πœ„), we may employ the commutative rule 𝐷𝑖,𝑍=π·π‘–π‘βˆ’π‘π·π‘–=ξ“π‘Žπ‘–β€²π‘–π·π‘–β€²ξ‚€π‘Žπ‘–β€²π‘–=𝐷𝑖𝑧𝑖′(5.17) in order to verify identities of the kind ℒ𝑍d𝑀𝑗𝑖=β„’Zd𝐷𝑖𝑀𝑗=ℒ𝑍ℒ𝐷𝑖d𝑀𝑖=ℒ𝐷𝑖ℒ𝑍dπ‘€π‘–βˆ’ξ“π‘Žπ‘–β€²π‘–β„’π·π‘–β€²π‘€π‘—(5.18) and in full generality identities of the kind β„’π‘˜π‘d𝑀𝑗𝐼=ξ“π‘ŽπΌβ€²πΌ,π‘˜β„’π·πΌβ€²β„’π‘˜β€²π‘d𝑀𝑗sumwithπ‘˜ξ…ž||πΌβ‰€π‘˜,ξ…ž||≀||𝐼||ξ€Έ(5.19) with unimportant coefficients, therefore (πœ„πœ„πœ„) follows. Finally (πœ„πœ„πœ„) 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(πœ†) of morphisms, see Section 10 below.

Theorem 5.5 (technical theorem). Let (5.6) be a variation of the jet space. Then π‘βˆˆπ”Ύ if and only if every family ξ€½β„’π‘Ÿπ‘πœ”π‘—ξ€Ύπ‘Ÿβˆˆβ„•(𝑗=1,…,π‘š)(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(π‘š,𝑛) but the underlying space is irrelevant here). Let us consider vector fields 𝑋 such that β„’π‘“π‘‹Ξ˜βŠ‚Ξ˜ for all functions 𝑓. Let moreover AdjΘ be the module of all forms πœ‘ satisfying πœ‘(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 πœ—=0 (πœ—βˆˆΞ˜) 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.

Module AdjΘ is intrinsically related to Θ: if a mapping 𝐦 preserves Θ then 𝐦 preserves AdjΘ. In particular, assuming 𝐦(πœ†)βˆ—Ξ˜βŠ‚Ξ˜,then𝐦(πœ†)βˆ—AdjΞ˜βŠ‚AdjΘ(5.21) is true for a group 𝐦(πœ†). In terms of ℐ𝑇 of the group 𝐦(πœ†), we have equivalent assertion β„’π‘Ξ˜βŠ‚Ξ˜impliesℒ𝑍AdjΞ˜βŠ‚AdjΘ(5.22) and therefore β„’π‘Ÿπ‘AdjΞ˜βŠ‚AdjΘ for all π‘Ÿ. The preparation is done.

Proof. Let Θ be the module generated by all differential forms β„’π‘Ÿπ‘πœ”π‘— (𝑗=1,…,π‘š; π‘Ÿ=0,1,…). Assuming finite dimension of module Θ, we have module AdjΘ and clearly β„’π‘Ξ˜βŠ‚Ξ˜ whence β„’π‘Ÿπ‘AdjΞ˜βŠ‚AdjΘ (π‘Ÿ=0,1,…). However AdjΘ involves both the differentials dπ‘₯1,…,dπ‘₯𝑛 (see below) and the forms πœ”1,…,πœ”π‘š. Point (πœ„πœ„) of previous Lemma 5.4 implies π‘βˆˆπ”Ύ. The converse is trivial.
In order to finish the proof, let us on the contrary assume that AdjΞ˜β€‰β€‰does not contain all differentials dπ‘₯1,…,dπ‘₯𝑛. Alternatively saying, the Pfaffian system πœ—=0 (πœ—βˆˆΞ˜) can be expressed in terms of certain functions 𝑓1,…,𝑓𝐾 such that d𝑓1=β‹―=d𝑓𝐾=0 does not imply dπ‘₯1=β‹―=dπ‘₯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,…,𝑓𝐾 and therefore we do not need all independent variables π‘₯1,…,π‘₯𝑛. This is however a contradiction: the Pfaffian system consists of contact forms and involves the equations πœ”1=β‹―=πœ”π‘›=0. All independent variables are needful if we deal with the common classical solutions 𝑀𝑗=𝑀𝑗(π‘₯1,…,π‘₯𝑛).

The result can be rephrased as follows.

Theorem 5.6. Let Ξ©0βŠ‚Ξ©(π‘š,𝑛) be the submodule of all zeroth-order contact forms βˆ‘π‘Žπœ”=π‘—πœ”π‘— and 𝑍 be a variation of the jet structure. Then π‘βˆˆπ”Ύ if and only if dimβŠ•β„’π‘Ÿπ‘Ξ©0<∞.

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+π•βŠ‚π•, π‘π•βŠ‚π• (π‘βˆˆβ„), [𝕍,𝕍]βŠ‚π•, and it follows that 𝕍 is an infinite-dimensional Lie algebra (coefficients in ℝ). On the other hand, if π‘βˆˆπ• and π‘“π‘βˆˆπ• for certain π‘“βˆˆβ„± then π‘“βˆˆβ„ 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(π‘š,𝑛).

Theorem 6.1. Let π’’βŠ‚π• be a finite-dimensional Lie subalgebra. Then π’’βŠ‚π”Ύ if and only if there exists a basis of 𝒒 that is lying in 𝔾.

The proof is elementary and may be omitted. Briefly saying, Theorem 4.2 (coefficients in β„±) 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 Ξ©π‘™βŠ‚Ξ©(π‘š,𝑛) be the submodule of all contact forms βˆ‘π‘Žπœ”=π‘—πΌπœ”π‘—πΌ (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(πœ†)βˆ—Ξ©π‘™βŠ‚Ξ©π‘™,β„’π‘Ξ©π‘™βŠ‚Ξ©π‘™,(7.1) respectively, for a certain 𝑙=0,1,…(equivalently: for all π‘™βˆˆβ„•, see Lemmas 9.1 and 9.2 below). Due to the fundamental Lie-BΓ€cklund 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=πœ™) in formulae (5.5) and (5.6) are functions only of the zeroth-order jet variables π‘₯π‘–ξ…ž, π‘€π‘—ξ…ž or, in the second case, we have π‘š=1 and all functions 𝐺𝑖, 𝐺1, 𝐺1𝑖, 𝑧𝑖, 𝑧1, 𝑧1𝑖 contain only the zeroth- and first-order variables π‘₯π‘–ξ…ž, 𝑀1, 𝑀1π‘–ξ…ž.

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-BΓ€cklund). Let a variation 𝑍 preserve a submodule Ξ©π‘™βŠ‚Ξ©(π‘š,𝑛) of contact forms of the order 𝑙 at most for a certain π‘™βˆˆβ„•. Then π‘βˆˆπ”Ύ and either 𝑍 is an infinitesimal point transformation or π‘š=1 and 𝑍 is the infinitesimal Lie's contact transformation.

Proof. We suppose β„’π‘Ξ©π‘™βŠ‚Ξ©π‘™. Then β„’π‘Ÿπ‘Ξ©0βŠ‚β„’π‘Ÿπ‘Ξ©π‘™βŠ‚Ξ©π‘™ therefore π‘βˆˆπ”Ύ by virtue of Theorem 5.5. Moreover β„’π‘Ξ©π‘™βˆ’1βŠ‚Ξ©π‘™βˆ’1,…,ℒ𝑍Ω0βŠ‚Ξ©0 by virtue of Lemma 9.2 below. So we have β„’π‘πœ”π‘—=ξ“π‘Žπ‘—π‘—β€²πœ”π‘—β€²ξ€·π‘—,π‘—ξ…žξ€Έ.=1,…,π‘š(7.2) Assuming π‘š=1, then (7.2) turns into the classical definition of Lie's infinitesimal contact transformation. Assume π‘šβ‰₯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(π‘š,𝑛) satisfying (7.2) and π‘šβ‰₯2. Then 𝑍π‘₯𝑖=𝑧𝑖…,π‘₯𝑖′,𝑀𝑗′,…,𝑍𝑀𝑗=𝑧𝑗…,π‘₯𝑖′,𝑀𝑗′,…(𝑖=1,…,𝑛;𝑗=1,…,π‘š)(7.3) are functions only of the point variables.

Proof. Let us introduce module Θ of (π‘š+2𝑛)-forms generated by all forms of the kind πœ”1βˆ§β‹―βˆ§πœ”π‘šβˆ§ξ€·dπœ”π‘—1𝑛1βˆ§ξ€·dπœ”π‘—π‘˜ξ€Έπ‘›π‘˜=d𝑀1βˆ§β‹―dπ‘€π‘šβˆ§dπ‘₯1βˆ§β‹―dπ‘₯π‘›βˆ§ξ“Β±d𝑀𝑗′1𝑖1βˆ§β‹―βˆ§d𝑀𝑗′𝑛𝑖𝑛,(7.4) where βˆ‘π‘›π‘˜=𝑛. Clearly Θ=(Ξ©0)π‘šβˆ§(dΞ©0)𝑛. The inclusions ℒ𝑍Ω0βŠ‚Ξ©0,ℒ𝑍dΞ©0=dℒ𝑍Ω0+Ξ©0βŠ‚dΞ©0+Ξ©0(7.5) are true by virtue of (7.2) and imply β„’π‘Ξ˜βŠ‚Ξ˜.
Module Θ vanishes when restricted to certain hyperplanes, namely, just to the hyperplanes of the kind ξ“π‘Žπœ—=𝑖dπ‘₯𝑖+ξ“π‘Žπ‘—d𝑀𝑗=0(7.6) (use π‘šβ‰₯2 here). This is expressed by Ξ˜βˆ§πœ—=0 and it follows that 0=ℒ𝑍(Ξ˜βˆ§πœ—)=β„’π‘Ξ˜βˆ§πœ—+Ξ˜βˆ§β„’π‘πœ—=Ξ˜βˆ§β„’π‘πœ—.(7.7) Therefore β„’π‘πœ— again is such a hyperplane: β„’π‘πœ—β‰…0 (mod all dπ‘₯𝑖 and d𝑀𝑗). On the other hand, β„’π‘ξ“π‘Žπœ—β‰…π‘–d𝑧𝑖+ξ“π‘Žπ‘—d𝑧𝑗modalldπ‘₯𝑖andd𝑀𝑗(7.8) 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 (π’žπ’―) and of nondegenerate first-order one-dimensional variational integrals (𝒱ℐ) are identical. In particular, the orbits of a given π’žπ’― group are extremals of appropriate 𝒱ℐ and conversely.

Proof. The π’žπ’― groups act in the jet space 𝐌(1,𝑛) equipped with the contact module Ξ©(1,𝑛). Then the abbreviations 𝑀𝐼=𝑀1𝐼,πœ”πΌ=πœ”1𝐼=dπ‘€πΌβˆ’ξ“π‘€πΌπ‘–dπ‘₯𝑖𝑧𝑍=π‘–πœ•πœ•π‘₯𝑖+𝑧1πΌπœ•πœ•π‘€πΌ(8.1) are possible. Let us recall the classical approach [22, 23]. The Lie contact transformations defined by certain formulae π¦βˆ—π‘₯𝑖=𝐺𝑖(β‹…),π¦βˆ—π‘€=𝐺1(β‹…),π¦βˆ—π‘€π‘–=𝐺1𝑖π‘₯(β‹…)(β‹…)=1,…,π‘₯𝑛,𝑀,𝑀1,…,𝑀𝑛(8.2) preserve the Pfaffian equation βˆ‘π‘€πœ”=dπ‘€βˆ’π‘–dπ‘₯𝑖=0 or (equivalently) the submodule Ξ©0βŠ‚Ξ©(1,𝑛) of zeroth-order contact forms. Explicit formulae are available in literature. We are interested in one-parameter local π’žπ’― groups of transformations 𝐦(πœ†)(βˆ’πœ€<πœ†<πœ€) which are β€œnondegenerate" in a sense stated below and then the explicit formulae are not available yet. On the other hand, our 𝒱ℐ with smooth Lagrangian Ł ξ€œΕξ€·π‘‘,𝑦1,…,𝑦𝑛,π‘¦ξ…ž1,…,π‘¦ξ…žπ‘›ξ€Έξƒ©π‘¦π‘‘π‘‘π‘–=𝑦𝑖(𝑑),ξ…ž=π‘‘ξƒ©πœ•π‘‘π‘‘,det2Επœ•π‘¦ξ…žπ‘–πœ•π‘¦ξ…žπ‘—ξƒͺξƒͺβ‰ 0(8.3) to appear later, involves variables from quite other jet space 𝐌(𝑛,1) with coordinates denoted 𝑑 (the independent variable), 𝑦1,…,𝑦𝑛 (the dependent variables) and higher-order jet variables like π‘¦ξ…žπ‘–, π‘¦π‘–ξ…žξ…ž and so on.
We are passing to the topic proper. Let us start in the space 𝐌(1,𝑛) with π’žπ’― groups. One can check that vector field (5.6) is infinitesimal π’žπ’― if and only if 𝑄𝑍=βˆ’π‘€π‘–πœ•πœ•π‘₯𝑖+ξ‚€ξ“π‘€π‘„βˆ’π‘–π‘„π‘€π‘–ξ‚πœ•+ξ“ξ€·π‘„πœ•π‘€π‘₯𝑖+π‘€π‘–π‘„π‘€ξ€Έπœ•πœ•π‘€π‘–+β‹―,(8.4) where the function 𝑄=𝑄(π‘₯1,…,π‘₯𝑛,𝑀,𝑀1,…,𝑀𝑛) may be arbitrarily chosen.
β€œHint: we have, by definition β„’π‘ξ“ξ€·π‘§πœ”=π‘βŒ‹dπœ”+dπœ”(𝑍)=π‘–πœ”π‘–βˆ’πœ”π‘–(𝑍)dπ‘₯𝑖+dπ‘„βˆˆΞ©0,(8.5) where 𝑄=𝑄(π‘₯1,…,π‘₯𝑛,𝑀,𝑀1,…,𝑀𝑛,…)=πœ”(𝑍)=𝑧1βˆ’βˆ‘π‘€π‘–π‘§π‘–, 𝐷d𝑄=𝑖𝑄dπ‘₯𝑖+πœ•π‘„ξ“πœ•π‘€πœ”+πœ•π‘„πœ•π‘€π‘–πœ”π‘–(8.6) whence immediately 𝑧𝑖=βˆ’πœ•π‘„/πœ•π‘€π‘–, 𝑧1βˆ‘π‘€=𝑄+π‘–π‘§π‘–βˆ‘π‘€=π‘„βˆ’π‘–β‹…πœ•π‘„/πœ•π‘€π‘–, β€‰β€‰πœ•π‘„/πœ•π‘€πΌ=0 if |𝐼|β‰₯1 and formula (8.4) follows.”
Alas, the corresponding Lie system (not written here) is not much inspirational. Let us however consider a function 𝑀=𝑀(π‘₯1,…,π‘₯𝑛) implicitly defined by an equation 𝑉(π‘₯1,…,π‘₯𝑛,𝑀)=0. We may suppose that the transformed function 𝐦(πœ†)βˆ—π‘€ satisfies the equation 𝑉π‘₯1,…,π‘₯𝑛,𝐦(πœ†)βˆ—π‘€ξ€Έ=πœ†(8.7) without any loss of generality. In infinitesimal terms 1=πœ•(π‘‰βˆ’πœ†)ξ“π‘„πœ•πœ†=𝑍(π‘‰βˆ’πœ†)=βˆ’π‘€π‘–π‘‰π‘₯𝑖+ξ‚€ξ“π‘€π‘„βˆ’π‘–π‘„π‘€π‘–ξ‚π‘‰π‘€.(8.8) However 𝑀𝑖=πœ•π‘€/πœ•π‘₯𝑖=βˆ’π‘‰π‘₯𝑖/𝑉𝑀 may be inserted here, and we have the crucial Jacobi equation ξ‚΅π‘₯1=𝑄1,…,π‘₯𝑛𝑉,𝑀,βˆ’π‘₯1𝑉𝑀𝑉,…,βˆ’π‘₯𝑛𝑉𝑀𝑉𝑀(8.9) (not involving 𝑉) which can be uniquely rewritten as the Hamilton-Jacobi (β„‹π’₯) equation 𝑉𝑀π‘₯+β„‹1,…,π‘₯𝑛,𝑀,𝑝1,…,𝑝𝑛𝑝𝑖=𝑉π‘₯𝑖(8.10) in the β€œnondegenerate" case βˆ‘π‘„π‘€π‘–π‘‰π‘₯𝑖≠1. Let us recall the characteristic curves [22, 23] of the β„‹π’₯ equation given by the system d𝑀1=dπ‘₯𝑖ℋ𝑝𝑖=βˆ’d𝑝𝑖ℋπ‘₯𝑖=dπ‘‰βˆ‘π‘βˆ’β„‹+𝑖ℋ𝑝𝑖.(8.11) The curves may be interpreted as the orbits of the group 𝐦(πœ†). (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(πœ•2β„‹/πœ•π‘π‘–πœ•π‘π‘—)β‰ 0. We may introduce variational integral (8.3) with the Lagrange function Ł given by the familiar identities 𝑝Ł+β„‹=π‘–π‘¦ξ…žπ‘–(8.12) with interrelations 𝑑=𝑀,𝑦𝑖=π‘₯𝑖,π‘¦ξ…žπ‘–=ℋ𝑝𝑖,𝑝𝑖=Ł𝑦′𝑖(𝑖=1,…,𝑛)(8.13) between variables 𝑑, 𝑦𝑖, π‘¦ξ…žπ‘– of the space 𝐌(𝑛,1) and variables π‘₯𝑖, 𝑀, 𝑀𝑖 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 PoincarΓ©-Cartan form ξ“ΕΜ†πœ‘=Łd𝑑+𝑦′𝑖dπ‘¦π‘–βˆ’π‘¦ξ…žπ‘–ξ€Έξ“π‘d𝑑=βˆ’β„‹d𝑑+𝑖d𝑦𝑖(8.14) is restricted to appropriate subspace π‘¦ξ…žπ‘–=𝑔𝑖(𝑑,𝑦1,…,𝑦𝑛) (𝑖=1,…,𝑛;the slope field) in order to become a total differential ||Μ†πœ‘π‘¦β€²π‘–=𝑔𝑖=d𝑉𝑑,𝑦1,…,𝑦𝑛=𝑉𝑑𝑉d𝑑+𝑦𝑖d𝑦𝑖(8.15) of the action 𝑉. We obtain the requirements 𝑉𝑑=βˆ’β„‹, 𝑉𝑦𝑖=𝑝𝑖 identical with (8.10). In geometrical terms: transformations of a hypersurface 𝑉=0 by means of π’žπ’― group may be identified with the level sets 𝑉=πœ† (πœ†βˆˆβ„) of the action of a Mayer fields of extremals.
The last statement is in accordance with (8.11) where 𝑝d𝑉=βˆ’β„‹+𝑖ℋ𝑝𝑖𝑝d𝑀=βˆ’β„‹+π‘–π‘¦ξ…žπ‘–ξ‚d𝑑=Łd𝑑,(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=πœ†: roughly saying, the composition with 𝑉=πœ‡ provides 𝑉=πœ†+πœ‡ (see Figure 3(c)) and this is caused by the additivity of the integral βˆ«Επ‘‘π‘‘ calculated along the orbits.

fig3
Figure 3

On this occasion, the wave enveloping approach to π’žπ’― groups is also worth mentioning.

Lemma 8.3 (see [10–13]). Let π‘Š(π‘₯1,…,π‘₯𝑛,𝑀,π‘₯1,…,π‘₯𝑛,𝑀) be a function of 2𝑛+2 variables. Assume that the system π‘Š=𝐷1π‘Š=β‹―=π·π‘›π‘Š=0 admits a unique solution π‘₯𝑖=𝐹𝑖…,π‘₯π‘–ξ…ž,𝑀,π‘€π‘–ξ…žξ€Έ,,…𝑀=𝐹1…,π‘₯π‘–ξ…ž,𝑀,π‘€π‘–ξ…žξ€Έ,…(8.17) by applying the implicit function theorem and analogously the system π‘Š=𝐷1π‘Š=β‹―=π·π‘›π‘Š=0   (where 𝐷𝑖=πœ•/πœ•π‘₯𝑖+βˆ‘π‘€π‘–πœ•/πœ•π‘€) admits a certain solution π‘₯𝑖=𝐹𝑖…,π‘₯𝑖′,𝑀,𝑀𝑖′,…,𝑀=𝐹1…,π‘₯𝑖′,𝑀,𝑀𝑖′.,…(8.18) Then π¦βˆ—π‘₯𝑖=𝐹𝑖, β€‰β€‰π¦βˆ—π‘€=𝐹1 provides a Lie π’žπ’― and (π¦βˆ’1)βˆ—π‘₯𝑖=𝐹𝑖, (π¦βˆ’1)βˆ—π‘€=𝐹1 is the inverse.

In more generality, if function π‘Š in Lemma 8.3 moreover depends on a parameter πœ†, we obtain a mapping 𝐦(πœ†) which is a certain π’žπ’― involving a parameter πœ† and the inverse 𝐦(πœ†)βˆ’1. In favourable case (see below) this 𝐦(πœ†) may be even a π’žπ’― group. The geometrical sense is as follows. Equation π‘Š=0 with π‘₯𝑖, 𝑀 kept fixed represents a wave in the space π‘₯𝑖, 𝑀 (Figure 3(a)).

The total system π‘Š=𝐷1π‘Š=β‹―=π·π‘›π‘Š=0 provides the intersection (envelope) of infinitely close waves (Figure 3(b)) with the resulting transform, the focus point 𝐦 (or 𝐦(πœ†) if the parameter πœ† 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 πœ†, πœ‡, 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 πœ†-depending waveπ‘Šξ€·π‘₯1,…,π‘₯𝑛,𝑀,π‘₯1,…,π‘₯𝑛,π‘€βˆ’πœ†=0.(8.19) If π‘₯𝑖, 𝑀 are kept fixed, the previous results may be applied. We obtain a group if and only if the β„‹π’₯ equation (8.10) holds true, thereforeπ‘Šπ‘€ξ€·π‘₯+β„‹1,…,π‘₯𝑛,𝑀,π‘Šπ‘₯1,…,π‘Šπ‘₯𝑛=0.(8.20) The existence of such function β„‹ means that functions π‘Šπ‘€,π‘Šπ‘₯1,…,π‘Šπ‘₯𝑛  of dashed variables are functionally dependent whenceξ‚΅π‘Šdetπ‘€π‘€π‘Šπ‘€π‘₯π‘–β€²π‘Šπ‘₯π‘–π‘€π‘Šπ‘₯𝑖π‘₯π‘–β€²ξ‚Άξ€·π‘Š=0,detπ‘₯𝑖π‘₯𝑖′≠0.(8.21) The symmetry π‘₯𝑖,𝑀↔π‘₯𝑖,𝑀 is not surprising here since the change πœ†β†”βˆ’πœ† provides the inverse mapping: equations π‘Šξ€·β€¦,π‘₯𝑖,𝑀,…,π‘₯𝑖,𝑀=πœ†,π‘Šβ€¦,π‘₯𝑖,𝑀,…,π‘₯𝑖,𝑀=βˆ’πœ†(8.22) are equivalent. In particular, it follows that π‘Šξ€·β€¦,π‘₯𝑖,𝑀,…,π‘₯𝑖,𝑀=βˆ’π‘Šβ€¦,π‘₯𝑖,𝑀,…,π‘₯𝑖,𝑀,π‘Šβ€¦,π‘₯𝑖,𝑀,…,π‘₯𝑖,𝑀=0(8.23) and the wave π‘Šβˆ’πœ†=0 corresponds to the Mayer central field of extremals.

Summary 4. Conditions (8.21) ensure the existence ofβ„‹π’₯ equation (8.20) for the πœ†-wave (8.19) and therefore the group composition property of waves (8.19) in the nondegenerate case det(πœ•2β„‹/πœ•π‘π‘–πœ•π‘π‘—)β‰ 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 filtrationΞ©(π‘š,𝑛)βˆ—βˆΆΞ©0βŠ‚Ξ©1βŠ‚β‹―βŠ‚Ξ©(π‘š,𝑛)=βˆͺΩ𝑙(9.1) of module Ξ©(π‘š,𝑛) is preserved (Figure 4(a)). It follows that certain invariant submodules Ξ©π‘™βŠ‚Ξ©(π‘š,𝑛) are a priori prescribed which essentially restricts the store of the symmetries (the Lie-BΓ€cklund theorem). The order-destroying groups also preserve certain submodules of Ξ©(π‘š,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.

fig4
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(πœ†) preserve a submodule Ξ˜βŠ‚Ξ©(π‘š,𝑛). Then also the submodule ξ“β„’Ξ˜+π·π‘–Ξ˜βŠ‚Ξ©(π‘š,𝑛)(9.2) is preserved.

Proof. We suppose β„’π‘Ξ˜βŠ‚Ξ˜. Then β„’π‘ξ‚€ξ“β„’Ξ˜+π·π‘–Ξ˜ξ‚=β„’π‘ξ‚€β„’Ξ˜+π·π‘–β„’π‘ξ“π·Ξ˜βˆ’π‘–π‘§π‘–ξ…žβ„’π·π‘–β€²Ξ˜ξ‚ξ“β„’βŠ‚Ξ˜+π·π‘–Ξ˜(9.3) by using the commutative rule (5.17).

Lemma 9.2 (going-down lemma). Let the group of morphisms 𝐦(πœ†) preserve a submodule Ξ˜βŠ‚Ξ©(π‘š,𝑛). Let Ξ˜β€²βŠ‚Ξ˜ be the submodule of all πœ”βˆˆΞ˜ satisfying β„’π·π‘–πœ”βˆˆΞ˜ (𝑖=1,…,𝑛). Then Ξ˜β€² is preserved, too.

Proof. Assume πœ”βˆˆΞ˜ξ…ž hence β„’π·π‘–πœ”βˆˆΞ˜. Then β„’π·π‘–β„’π‘πœ”=β„’π‘β„’π·π‘–πœ”+β„’βˆ‘π·π‘–π‘§π‘–β€²β‹…π·π‘–β€²πœ”βˆˆΞ˜ hence β„’π‘πœ”βˆˆΞ˜ξ…ž and Ξ˜ξ…ž is preserved.

We are passing to illustrative examples.

Example 9.3. Let us consider the vector field (the variation of jet structure) 𝑧𝑍=π‘—πΌπœ•πœ•π‘€π‘—πΌξ€·π‘§π‘—πΌ=𝐷𝐼𝑧𝑗,𝐷𝐼=𝐷𝑖1⋯𝐷𝑖𝑛,(9.4) see (5.6) and (5.10) for the particular case 𝑧𝑖=0. Then π‘π‘Ÿπ‘₯𝑖=0  (𝑖=1,…,𝑛) and the sufficient requirement 𝑍2𝑀𝑗=0 (𝑗=1,…,π‘š) ensures π‘βˆˆπ”Ύ, see (πœ„) of Lemma 5.4. We will deal with the linear case where 𝑧𝑗=ξ“π‘Žπ‘—π‘—β€²π‘–β€²π‘€π‘—β€²π‘–β€²ξ‚€π‘Žπ‘—π‘—β€²π‘–β€²ξ‚βˆˆβ„(9.5) is supposed. Then 𝑍2𝑀𝑗=𝑍𝑧𝑗=ξ“π‘Žπ‘—π‘—β€²π‘–β€²π‘§π‘—β€²π‘–β€²=ξ“π‘Žπ‘—π‘—β€²π‘–β€²π‘Žπ‘—β€²π‘—β€²β€²π‘–β€²β€²π‘€π‘—β€²β€²π‘–β€²π‘–β€²β€²=0(9.6) identically if and only if ξ“π‘—β€²ξ‚€π‘Žπ‘—π‘—β€²π‘–β€²π‘Žπ‘—β€²π‘—β€²β€²π‘–β€²β€²+π‘Žπ‘—π‘—β€²π‘–β€²β€²π‘Žπ‘—β€²π‘—β€²β€²π‘–β€²ξ‚ξ€·π‘–=0ξ…ž,π‘–ξ…žξ…ž=1,…,𝑛;𝑗,π‘—ξ…ž,π‘—ξ…žξ…žξ€Έ.=1,…,π‘š(9.7) This may be expressed in terms of matrix equations 𝐴𝑖𝐴𝑖′=0𝑖,π‘–ξ…ž=1,…,𝑛;𝐴𝑖=ξ‚€π‘Žπ‘—π‘—β€²π‘–ξ‚ξ‚(9.8) or, in either of more geometrical transcriptions 𝐴2ξ‚€ξ“πœ†=0,Imπ΄βŠ‚Ker𝐴𝐴=𝑖𝐴𝑖,πœ†π‘–ξ‚βˆˆβ„,(9.9) where 𝐴 is regarded as (a matrix of an) operator acting in π‘š-dimensional linear space and depending on parameters πœ†1,…,πœ†π‘›. We do not know explicit solutions 𝐴 in full generality, however, solutions 𝐴 such that Ker𝐴 does not depend on the parameters πœ†1,…,πœ†π‘› can be easily found (and need not be stated here). The same approach can be applied to the more general sufficient requirement π‘π‘Ÿπ‘€π‘—=0  (𝑗=1,…,π‘š; fixed π‘Ÿ) ensuring π‘βˆˆπ”Ύ. If π‘Ÿβ‰₯𝑛, the requirement is equivalent to the inclusion π‘βˆˆπ”Ύ.

Example 9.4. Let us consider vector field (5.6) where 𝑧1=β‹―=π‘§π‘š=0. In more detail, we take 𝑧𝑍=π‘–πœ•πœ•π‘₯𝑖+ξ“π‘§π‘—π‘–πœ•πœ•π‘€π‘—π‘–ξ‚€π‘§+⋯𝑗𝑖𝑀=βˆ’π‘—π‘–β€²π·π‘–π‘§π‘–β€²ξ‚.(9.10) Then π‘π‘Ÿπ‘€π‘—=0 and we have to deal with functions π‘π‘Ÿπ‘₯𝑖 in order to ensure the inclusion π‘βˆˆπ”Ύ. This is a difficult task. Let us therefore suppose 𝑧1ξ‚€=𝑧…,π‘₯𝑖′,𝑀𝑗′,𝑀𝑗′1,…,π‘§π‘˜=π‘π‘˜βˆˆβ„(π‘˜=2,…,𝑛).(9.11) Then 𝑍π‘₯π‘˜=0  (π‘˜=2,…,𝑛) and 𝑍2π‘₯1=𝑍𝑧=πœ•π‘§πœ•π‘₯𝑖𝑧𝑖+ξ“πœ•π‘§πœ•π‘€π‘—1𝑧𝑗1,(9.12) where 𝑧𝑗1=βˆ’π‘€π‘—1𝐷1𝑧=βˆ’π‘€π‘—1ξƒ©πœ•π‘§πœ•π‘₯1+ξ“πœ•π‘§πœ•π‘€π‘—β€²π‘€π‘—β€²1+ξ“πœ•π‘§πœ•π‘€π‘—β€²1𝑀𝑗′11ξƒͺ.(9.13) The second-order summand 𝑍2π‘₯1=β‹―+πœ•π‘§πœ•π‘€π‘—1𝑧𝑗1=β‹―βˆ’πœ•π‘§πœ•π‘€π‘—1𝑀𝑗1πœ•π‘§πœ•π‘€π‘—β€²1𝑀𝑗′11(9.14) identically vanishes for the choice 𝑧=𝑓…,π‘₯𝑖′,𝑀𝑗′,𝑒𝑙𝑒,…𝑙=𝑀𝑙1𝑀11ξƒͺ;𝑙=2,…,π‘š(9.15) as follows by direct verification. Quite analogously 𝑍𝑒𝑙𝑀=𝑍𝑙1𝑀11=𝑧𝑙11𝑀11βˆ’π‘§11𝑀𝑙1𝑀11ξ€Έ2=ξƒ©βˆ’π‘€π‘™11𝑀11+𝑀11𝑀𝑙1𝑀11ξ€Έ2ξƒͺ𝐷1𝑧=0.(9.16) It follows that all functions π‘π‘Ÿπ‘₯𝑖, π‘π‘Ÿπ‘€π‘— can be expressed in terms of the finite family of functions π‘₯𝑖 (𝑖=1,…,𝑛), 𝑀𝑗 (𝑗=1,…,π‘š), 𝑒𝑙 (𝑙=2,…,π‘š) and therefore π‘βˆˆπ”Ύ.

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 d𝐺𝑖dπœ†=0,d𝐺𝑗=ξ“π‘Ždπœ†π‘—π‘—β€²π‘–β€²πΊπ‘—β€²π‘–β€²(𝑖=1,…,𝑛;𝑗=1,…,π‘š),(9.17) where we omit the prolongations. It is resolved by 𝐺𝑖=π‘₯𝑖,𝐺𝑗=π‘€π‘—ξ“π‘Ž+πœ†π‘—π‘—β€²π‘–β€²π‘€π‘—β€²π‘–β€²(𝑖=1,…,𝑛;𝑗=1,…,π‘š)(9.18) as follows either by direct verification or, alternatively, from the property 𝑍2π‘₯𝑖=𝑍𝑧𝑖=0  (𝑖=1,…,𝑛) which implies dβˆ‘π‘Žπ‘—π‘—β€²π‘–β€²πΊπ‘—β€²π‘–β€²ξ“π‘Ždπœ†=0,𝑗𝑗′𝑖′𝐺𝑗′𝑖′=ξ“π‘Žπ‘—π‘—β€²π‘–β€²πΊπ‘—β€²π‘–β€²|||πœ†=0=ξ“π‘Žπ‘—π‘—β€²π‘–β€²π‘€π‘—β€²π‘–β€².(9.19) Quite analogously, the Lie system of the vector field (9.10), (9.11), (9.15) reads d𝐺1dπœ†=𝑓…,𝐺𝑖′,𝐺𝑗′,𝐺𝑙′1𝐺11ξƒͺ,,…dπΊπ‘˜dπœ†=π‘π‘˜,d𝐺𝑗dπœ†=0(π‘˜=2,…,𝑛;𝑗=1,…,π‘š)(9.20) and may be completed with the equations d𝐺𝑙1/𝐺11ξ€Έdπœ†=0(𝑙=2,…,π‘š)(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 πΉβˆˆβ„±, 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(π‘š,𝑛) in order to compare it with the subsequent calculation of vector field π‘βˆˆπ”Ύ.

Morphisms 𝐦 of the jet structure were defined by the property π¦βˆ—Ξ©(π‘š,𝑛)βŠ‚Ξ©(π‘š,𝑛). The inverse π¦βˆ’1 exists if and only ifΞ©0βŠ‚π¦βˆ—Ξ©(π‘š,𝑛),equivalentlyΞ©0βŠ‚π¦βˆ—Ξ©π‘™(𝑙=𝑙(𝐦))(10.1) for appropriate term Ω𝑙(𝐦) of filtration (9.1). Howeverπ¦βˆ—Ξ©π‘™+1=π¦βˆ—Ξ©π‘™+ξ“β„’π·π‘–π¦βˆ—Ξ©π‘™(10.2) and it follows that criterion (10.1) can be verified by repeated use of operators ℒ𝐷𝑖. In more detail, we start with equationsπ¦βˆ—πœ”π‘—=ξ“π‘Žπ‘—π‘—β€²πΌβ€²πœ”π‘—β€²πΌβ€²ξ‚€=dπ¦βˆ—π‘€π‘—βˆ’ξ“π¦βˆ—π‘€π‘—π‘–dπ¦βˆ—π‘₯𝑖(10.3) with uncertain coefficients. Formulae (10.3) determine the module π¦βˆ—Ξ©0. Then we search for lower-order contact forms, especially forms from Ξ©0, lying in π¦βˆ—Ξ©π‘™ 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 π‘Žπ‘—π‘—β€²πΌβ€², functions π¦βˆ—π‘₯𝑖,β€‰β€‰π¦βˆ—π‘€π‘— (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:ℒ𝑍ℒ𝐷𝑖=β„’π·π‘–β„’π‘βˆ’ξ“π·π‘–π‘§π‘–ξ…žβ„’π·π‘–β€²(10.4) or, when applied to basical formsβ„’π‘πœ”π‘—πΌπ‘–=β„’π·π‘–β„’π‘πœ”π‘—πΌβˆ’ξ“π·π‘–π‘§π‘–ξ…žπœ”π‘—πΌπ‘–β€².(10.5) We are interested in vector fields π‘βˆˆπ”Ύ. They satisfy the recurrence (5.10) together with requirementsdimβŠ•β„’π‘Ÿπ‘Ξ©0<∞,equivalentlyβ„’π‘Ÿπ‘Ξ©0βŠ‚Ξ©π‘™(𝑍)(π‘Ÿ=0,1,…)(10.6) for appropriate 𝑙(𝑍)βˆˆβ„•. Due to the recurrence (10.5) these requirements can be effectively investigated. In more detail, we start with equationsβ„’π‘πœ”π‘—=ξ“π‘Žπ‘—π‘—β€²πΌβ€²πœ”π‘—β€²πΌβ€²ξ‚€=dπ‘§π‘—βˆ’ξ“π‘§π‘—π‘–dπ‘₯π‘–βˆ’ξ“π‘€π‘—π‘–d𝑧𝑖.(10.7) Formulae (10.7) determine module ℒ𝑍Ω0. Then, choosing 𝑙(𝑍)βˆˆβ„•, 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 π‘Žπ‘—π‘—β€²πΌβ€² available, functions 𝑧𝑖=ℒ𝑍π‘₯𝑖,  𝑧𝑗=ℒ𝑍𝑀𝑗 (and therefore the vector field π‘βˆˆπ”Ύ) 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=πœ™ emptyβ„’π‘πœ”π‘—=ξ“βŽ›βŽœβŽœβŽπœ•π‘§π‘—πœ•π‘€π‘—β€²πΌβ€²βˆ’ξ“π‘€π‘—π‘–πœ•π‘§π‘–πœ•π‘€π‘—β€²πΌβ€²βŽžβŽŸβŽŸβŽ πœ”π‘—β€²πΌβ€².(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 β€œexhaustive description of integrable 𝐢-fields (fields π‘βˆˆβ„€ 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 π‘βˆˆπ”Ύ.

Proof. We suppose π‘š=1 and then (10.7) reads β„’π‘πœ”1=ξ“π‘ŽπΌ11β€²πœ”1𝐼′=β‹―+π‘ŽπΌ11β€²β€²πœ”1πΌβ€²β€²ξ‚€π‘ŽπΌ11′′≠0,(10.9) where we state a summand of maximal order. Assuming πΌξ…žξ…ž=πœ™, the Lie-BΓ€cklund theorem can be applied and we do not have the true generalized symmetry 𝑍. Assuming πΌξ…žξ…žβ‰ πœ™, then β„’π‘Ÿπ‘πœ”1=β‹―+π‘ŽπΌ11β€²β€²πœ”1πΌβ€²β€²β‹―πΌβ€²β€²ξ€·π‘ŸtermsπΌξ…žξ…žξ€Έ(10.10) by using rule (10.5) where the last summand may be omitted. It follows that (10.6) is not satisfied hence π‘βˆ‰π”Ύ.

Example 10.2. We discuss the simplest possible but still a nontrivial particular example. Assume π‘š=2, 𝑛=1 and 𝑙(𝑍)=1. Let us abbreviate π‘₯=π‘₯1,𝐷=𝐷1πœ•,𝑍=𝑧+ξ“π‘§πœ•π‘₯π‘—πΌπœ•πœ•π‘€π‘—πΌ(𝑗=1,2;𝐼=1β‹―1).(10.11) Then, due to 𝑙(𝑍)=1, requirement (10.6) reads β„’π‘Ÿπ‘Ξ©0βŠ‚Ξ©1(π‘Ÿ=0,1,…).(10.12) In particular (if π‘Ÿ=1) we have (10.7) written here in the simplified notation β„’π‘πœ”π‘—=π‘Žπ‘—1πœ”1+π‘Žπ‘—2πœ”2+𝑏𝑗1πœ”11+𝑏𝑗2πœ”21(𝑗=1,2).(10.13) The next requirement (π‘Ÿ=2) implies the (only seemingly) stronger inclusion β„’2𝑍Ω0βŠ‚β„’π‘Ξ©0+Ξ©0(10.14) which already ensures (10.12) for all π‘Ÿ and therefore π‘βˆˆπ”Ύ (easy). We suppose (10.14) from now on.
β€œHint for proof of (10.14): assuming (10.12) and moreover the equality β„’2𝑍Ω0+ℒ𝑍Ω0+Ξ©0=Ξ©1,(10.15) it follows that ℒ𝑍Ω1βŠ‚β„’3𝑍Ω0+β„’2𝑍Ω0+ℒ𝑍Ω0βŠ‚Ξ©1(10.16) and Lie-BΓ€cklund theorem can be applied whence ℒ𝑍Ω0βŠ‚Ξ©0,  𝑙(𝑍)=0 which we exclude. It follows that necessarily ξ€·β„’dim2𝑍Ω0+ℒ𝑍Ω0+Ξ©0ξ€Έ<dimΞ©1=4.(10.17) On the other hand dim(ℒ𝑍Ω0+Ξ©0)β‰₯3 and the inclusion (10.14) follows.”
After this preparation, we are passing to the proper algebra. Clearly β„’2π‘πœ”π‘—=β‹―+𝑏𝑗1β„’π‘πœ”11+𝑏𝑗2β„’π‘πœ”21=β‹―+𝑏𝑗1𝑏11πœ”111+𝑏12πœ”211ξ€Έ+𝑏𝑗2𝑏21πœ”111+𝑏22πœ”211ξ€Έ(10.18) by using the commutative rule (10.5). Due to β€œweaker" inclusion (10.12) with π‘Ÿ=2, we obtain identities 𝑏𝑗1𝑏11+𝑏𝑗2𝑏21=0,𝑏𝑗1𝑏12+𝑏𝑗2𝑏22=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 𝑏11β‰ 0 and either 𝑏12β‰ 0 or 𝑏21β‰ 0 is supposed) or 𝑏11=𝑏22=0,either𝑏12=0or𝑏21=0.(10.21) We deal only with the (more interesting) identities (10.20) here. Then β„’π‘πœ”1=π