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

The Lie Group in Infinite Dimension

V. Tryhuk,Β V. Chrastinová,Β and O. Dlouhý

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

Received 6 December 2010; Accepted 12 January 2011

Academic Editor: MiroslavaΒ Růžičková

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 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 πœ• πœ• π‘₯ 𝑖 +  𝑧 𝑗 𝐼 πœ• πœ• 𝑀 𝑗 𝐼 ξ€· 𝑖 = 1 , … , 𝑛 ; 𝑗 = 1 , … , π‘š ; 𝐼 = 𝑖 1 β‹― 𝑖 𝑛 ; 𝑖 1 , … , 𝑖 𝑛 ξ€Έ , = 1 , … , 𝑛 ( 1 . 1 ) where the coefficients 𝑧 𝑖 = 𝑧 𝑖 ξ‚€ … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€² 𝐼 β€²  , … , 𝑧 𝑗 𝐼 = 𝑧 𝑗 𝐼 ξ‚€ … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€² 𝐼 β€²  , … ( 1 . 2 ) are functions of independent variables π‘₯ 𝑖 , dependent variables 𝑀 𝑗 and a finite number of jet variables 𝑀 𝑗 𝐼 = πœ• 𝑛 𝑀 𝑗 / πœ• π‘₯ 𝑖 1 β‹― πœ• π‘₯ 𝑖 𝑛 belong to well-established concepts. However, in spite of this matter of fact, they cause an unpleasant feeling. Indeed, such vector fields as a rule do not generate any one-parameter group of transformations π‘₯ 𝑖 = 𝐺 𝑖 ξ‚€ πœ† ; … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€² 𝐼 β€²  , , … 𝑀 𝑗 𝐼 = 𝐺 𝑗 𝐼 ξ‚€ πœ† ; … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€² 𝐼 β€²  , … ( 1 . 3 ) in the underlying infinite-order jet space since the relevant Lie system πœ• 𝐺 𝑖 πœ• πœ† = 𝑧 𝑖 ξ‚€ … , 𝐺 𝑖 β€² , 𝐺 𝑗 β€² 𝐼 β€²  , , … πœ• 𝐺 𝑗 𝐼 πœ• πœ† = 𝑧 𝑗 𝐼 ξ‚€ … , 𝐺 𝑖 β€² , 𝐺 𝑗 β€² 𝐼 β€²  ξ€· 𝐺 , … 𝑖 | | πœ† = 0 = π‘₯ 𝑖 , 𝐺 𝑗 𝐼 | | πœ† = 0 = 𝑀 𝑗 𝐼 ξ€Έ ( 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 β„Ž 1 , β„Ž 2 , … . 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 𝐹 1 , 𝐹 2 , … on ℝ ∞ , 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 ℝ ∞ is preserved which provides the “approximation" of 𝐦 ( πœ† ) . The infinitesimal transformation 𝑍 = d 𝐦 ( πœ† ) / d πœ† | πœ† = 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 ( 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 ℝ ∞ with coordinates β„Ž 1 , β„Ž 2 , … [9] and we introduce the structural family β„± of all real-valued, locally defined and 𝐢 ∞ -smooth functions 𝑓 = 𝑓 ( β„Ž 1 , … , β„Ž π‘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 ℝ ∞ defined by formulae 𝐦 ( πœ† ) βˆ— β„Ž 𝑖 = 𝐻 𝑖 ξ€· πœ† ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ξ€Έ , βˆ’ πœ€ 𝑖 < πœ† < πœ€ 𝑖 , πœ€ 𝑖 > 0 ( 𝑖 = 1 , 2 , … ) , ( 2 . 1 ) where 𝐻 𝑖 ∈ β„± if the parameter πœ† is kept fixed. We suppose 𝐦 ( 0 ) = i d . , 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 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 < πœ€ < m i n { πœ€ 1 , … , πœ€ 𝐼 } , let β„± ( 𝐼 , πœ€ ) βŠ‚ β„± be the subset of all composed functions ξ€· ξ€· πœ† 𝐹 = 𝐹 … , 𝐦 𝑗 ξ€Έ βˆ— β„Ž 𝑖 ξ€Έ ξ€· , … = 𝐹 … , 𝐻 𝑖 ξ€· πœ† 𝑗 ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ξ€Έ ξ€Έ , , … ( 2 . 3 ) where 𝑖 = 1 , … , 𝐼 ;   βˆ’ πœ€ < πœ† 𝑗 < πœ€ ;   𝑗 = 1 , … , 𝐽 = 𝐽 ( 𝐼 ) = m a x { π‘š ( 1 ) , … , π‘š ( 𝐼 ) } and 𝐹 is arbitrary 𝐢 ∞ -smooth function (of 𝐼 𝐽 variables). In functions 𝐹 ∈ β„± ( 𝐼 , πœ€ ) , variables πœ† 1 , … , πœ† 𝐽 are regarded as mere parameters.

Functions (2.3) will be considered on open subsets of ℝ ∞ where the rank of the Jacobi ( 𝐼 𝐽 Γ— 𝐽 ) -matrix ξ‚΅ πœ• πœ• β„Ž 𝑗 β€² 𝐻 𝑖 ξ€· πœ† 𝑗 ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ξ€Έ ξ‚Ά ξ€· 𝑖 = 1 , … , 𝐼 ; 𝑗 , 𝑗 ξ…ž ξ€Έ = 1 , … , 𝐽 ( 2 . 4 ) of functions 𝐻 𝑖 ( πœ† 𝑗 ; β„Ž 1 , … , β„Ž π‘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 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 ( πœ† ) βˆ— β„± ( 𝐼 ) βŠ‚ β„± ( 𝐼 ) . ( 2 . 5 )

Proof. Clearly 𝐦 ( πœ† ) βˆ— 𝐻 𝑖 ξ€· πœ† 𝑗 ξ€Έ ; … = 𝐦 ( πœ† ) βˆ— 𝐦 ξ€· πœ† 𝑗 ξ€Έ βˆ— β„Ž 𝑖 ξ€· = 𝐦 πœ† + πœ† 𝑗 ξ€Έ βˆ— β„Ž 𝑖 = 𝐻 𝑖 ξ€· πœ† + πœ† 𝑗 ξ€Έ ; … ( 2 . 6 ) and therefore 𝐦 ( πœ† ) βˆ— ξ€· 𝐹 = 𝐹 … , 𝐻 𝑖 ξ€· πœ† + πœ† 𝑗 ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ξ€Έ ξ€Έ , … ∈ β„± ( 𝐼 ) . ( 2 . 7 )

Denoting by 𝐾 ( 𝐼 ) the rank of matrix (2.4), there exist basical functions 𝐹 π‘˜ = 𝐹 π‘˜ ξ€· … , 𝐻 𝑖 ξ€· πœ† 𝑗 ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ξ€Έ ξ€Έ , … ∈ β„± ( 𝐼 ) ( π‘˜ = 1 , … , 𝐾 ( 𝐼 ) ) ( 2 . 8 ) such that r a n k ( πœ• 𝐹 π‘˜ / πœ• β„Ž 𝑗 β€² ) = 𝐾 ( 𝐼 ) . Then a function 𝑓 ∈ β„± lies in β„± ( 𝐼 ) if and only if 𝑓 = 𝑓 ( 𝐹 1 , … , 𝐹 𝐾 ( 𝐼 ) ) is a composed function. In more detail 𝐹 = 𝐹 ξ€· πœ† 1 , … , πœ† 𝐽 ; 𝐹 1 , … , 𝐹 𝐾 ( 𝐼 ) ξ€Έ ∈ β„± ( 𝐼 ) ( 2 . 9 ) is such a composed function if we choose 𝑓 = 𝐹 given by (2.3). Parameters πœ† 1 , … , πœ† 𝐽 occurring in (2.3) are taken into account here. It follows that πœ• 𝐹 πœ• πœ† 𝑗 = πœ• 𝐹 πœ• πœ† 𝑗 ξ€· πœ† 1 , … , πœ† 𝐽 ; 𝐹 1 , … , 𝐹 𝐾 ( 𝐼 ) ξ€Έ ∈ β„± ( 𝐼 ) ( 𝑗 = 1 , … , 𝐽 ) ( 2 . 1 0 ) and analogously for the higher derivatives.

In particular, we also have 𝐻 𝑖 ξ€· πœ† ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ξ€Έ = 𝐻 𝑖 ξ€· πœ† ; 𝐹 1 , … , 𝐹 𝐾 ( 𝐼 ) ξ€Έ ∈ β„± ( 𝐼 ) ( 𝑖 = 1 , … , 𝐼 ) ( 2 . 1 1 ) for the choice 𝐹 = 𝐻 𝑖 ( πœ† ; … ) in (2.9) whence πœ• π‘Ÿ 𝐻 𝑖 πœ• πœ† π‘Ÿ = πœ• π‘Ÿ 𝐻 𝑖 πœ• πœ† π‘Ÿ ξ€· πœ† ; 𝐹 1 , … , 𝐹 𝐾 ( 𝐼 ) ξ€Έ ∈ β„± ( 𝐼 ) ( 𝑖 = 1 , … , 𝐼 ; π‘Ÿ = 0 , 1 , … ) . ( 2 . 1 2 ) The basical functions can be taken from the family of functions 𝐻 𝑖 ( 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— 𝐹 π‘˜ = 𝐹 π‘˜ ξ€· πœ† ; 𝐹 1 , … , 𝐹 𝐾 ( 𝐼 ) ξ€Έ ( π‘˜ = 1 , … , 𝐾 ( 𝐼 ) ) . ( 2 . 1 3 )

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

Summary 1. Coordinates β„Ž 𝑖 = 𝐻 𝑖 ( 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 𝐹 1 = β„Ž 1 , … , 𝐹 𝐼 = β„Ž 𝐼 here and then, if 𝐼 is large enough, formulae (2.13) provide a “finite-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 𝐼 β†’ ∞ and the system of functions 𝐹 1 , 𝐹 2 , … is succesively completed.

3. The Infinitesimal Approach

We introduce the vector field  𝑧 𝑍 = 𝑖 πœ• πœ• β„Ž 𝑖 = d 𝐦 ( πœ† ) | | | d πœ† πœ† = 0 ξ‚΅ 𝑧 𝑖 = πœ• 𝐻 𝑖 ξ€· πœ• πœ† 0 ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ξ€Έ ξ‚Ά ; 𝑖 = 1 , 2 , … , ( 3 . 1 ) the infinitesimal transformation ( ℐ 𝑇 ) of group 𝐦 ( πœ† ) . Let us recall the celebrated Lie system πœ• 𝐦 πœ• πœ† ( πœ† ) βˆ— β„Ž 𝑖 = πœ• 𝐻 𝑖 πœ• πœ† ( πœ† ; … ) = πœ• 𝐻 𝑖 | | | | πœ• πœ‡ ( πœ† + πœ‡ ; … ) πœ‡ = 0 = πœ• πœ• πœ‡ 𝐦 ( πœ† + πœ‡ ) βˆ— β„Ž 𝑖 | | | | πœ‡ = 0 = 𝐦 ( πœ† ) βˆ— πœ• πœ• πœ‡ 𝐦 ( πœ‡ ) βˆ— β„Ž 𝑖 | | | | πœ‡ = 0 = 𝐦 ( πœ† ) βˆ— 𝑍 β„Ž 𝑖 = 𝐦 ( πœ† ) βˆ— 𝑧 𝑖 . ( 3 . 2 ) In more explicit (and classical) transcription πœ• 𝐻 𝑖 ξ€· πœ• πœ† πœ† ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ξ€Έ = 𝑧 𝑖 ξ€· 𝐻 1 ξ€· πœ† ; β„Ž 1 , … , β„Ž π‘š ( 1 ) ξ€Έ , … , 𝐻 π‘š ( 𝑖 ) ξ€· πœ† ; β„Ž 1 , … , β„Ž π‘š ( π‘š ( 𝑖 ) ) . ξ€Έ ξ€Έ ( 3 . 3 ) One can also check the general identity πœ• π‘Ÿ πœ• πœ† π‘Ÿ 𝐦 ( πœ† ) βˆ— 𝑓 = 𝐦 ( πœ† ) βˆ— 𝑍 π‘Ÿ 𝑓 ( 𝑓 ∈ β„± ; π‘Ÿ = 0 , 1 , … ) ( 3 . 4 ) by a mere routine induction on π‘Ÿ .

Lemma 3.1 (finiteness lemma). For all π‘Ÿ ∈ β„• , 𝑍 π‘Ÿ β„± ( 𝐼 ) βŠ‚ β„± ( 𝐼 ) .

Proof. Clearly 𝑍 𝐹 = 𝐦 ( πœ† ) βˆ— | | 𝑍 𝐹 πœ† = 0 = πœ• πœ• πœ† 𝐦 ( πœ† ) βˆ— 𝐹 | | | πœ† = 0 ∈ β„± ( 𝐼 ) ( 3 . 5 ) for any function (2.3) by virtue of (2.10): induction on π‘Ÿ .

Theorem 3.2 (finiteness theorem). Every function 𝐹 ∈ β„± ( 𝐼 ) admits (locally, near generic points) the representation  𝐹 ξ‚΅ πœ• 𝐹 = … , π‘Ÿ 𝐻 𝑖 πœ• πœ† π‘Ÿ ξ€· 0 ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ξ€Έ ξ‚Ά , … ( 3 . 6 ) in terms of a composed function where 𝑖 = 1 , … , 𝐼 and  𝐹 is a β„‚ ∞ -smooth function of a finite number of variables.

Proof. Let us temporarily denote 𝐻 𝑖 π‘Ÿ = πœ• π‘Ÿ 𝐻 𝑖 πœ• πœ† π‘Ÿ πœ• ( πœ† ; … ) = π‘Ÿ πœ• πœ† π‘Ÿ 𝐦 ( πœ† ) βˆ— β„Ž 𝑖 , β„Ž 𝑖 π‘Ÿ = 𝐻 𝑖 π‘Ÿ ( 0 ; … ) = 𝑍 π‘Ÿ β„Ž 𝑖 , ( 3 . 7 ) where the second equality follows from (3.4) with 𝑓 = β„Ž 𝑖 ,   πœ† = 0 . Then 𝐻 𝑖 π‘Ÿ = 𝐦 ( πœ† ) βˆ— β„Ž 𝑖 π‘Ÿ = 𝐦 ( πœ† ) βˆ— 𝑍 π‘Ÿ β„Ž 𝑖 ( 3 . 8 ) by virtue of (3.4) with general πœ† .
If 𝑗 = 𝑗 ( 𝑖 ) is large enough, there does exist an identity β„Ž 𝑖 𝑗 + 1 = 𝐺 𝑖 ( β„Ž 𝑖 0 , … , β„Ž 𝑖 𝑗 ) . Therefore πœ• 𝑗 + 1 𝐻 𝑖 πœ• πœ† 𝑗 + 1 = 𝐻 𝑖 𝑗 + 1 = 𝐺 𝑖 ξ‚€ 𝐻 𝑖 0 , … , 𝐻 𝑖 𝑗  = 𝐺 𝑖 ξ‚΅ 𝐻 𝑖 πœ• , … , 𝑗 𝐻 𝑖 πœ• πœ† 𝑗 ξ‚Ά ( 3 . 9 ) by applying 𝐦 ( πœ† ) βˆ— . This may be regarded as ordinary differential equation with initial values 𝐻 𝑖 | | πœ† = 0 = β„Ž 𝑖 0 πœ• , … , 𝑗 𝐻 𝑖 πœ• πœ† 𝑗 | | | | πœ† = 0 = β„Ž 𝑖 𝑗 . ( 3 . 1 0 ) The solution 𝐻 𝑖 =  𝐻 𝑖 ( πœ† ; β„Ž 𝑖 0 , … , β„Ž 𝑖 𝑗 ) expressed in terms of initial values reads 𝐻 𝑖 ξ€· πœ† ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ξ€Έ =  𝐻 𝑖 ξ‚΅ πœ† ; 𝐻 𝑖 ξ€· 0 ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ξ€Έ πœ• , … , 𝑗 𝐻 𝑖 πœ• πœ† 𝑗 ξ€· 0 ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ξ€Έ ξ‚Ά ( 3 . 1 1 ) in full detail. If πœ† is kept fixed, this is exactly the identity (3.6) for the particular case 𝐹 = 𝐻 𝑖 ( πœ† ; β„Ž 1 , … , β„Ž π‘š ( 𝑖 ) ) . The general case follows by a routine.

Definition 3.3. Let 𝔾 be the set of (local) vector fields  𝑧 𝑍 = 𝑖 πœ• πœ• β„Ž 𝑖 ξ€· 𝑧 𝑖 ξ€Έ ∈ β„± , i n fi n i t e s u m ( 3 . 1 2 ) such that every family of functions { 𝑍 π‘Ÿ β„Ž 𝑖 } π‘Ÿ ∈ β„•    ( 𝑖 fixed but arbitrary) can be expressed in terms of a finite number of coordinates.

Remark 3.4. Neither 𝔾 + 𝔾 βŠ‚ 𝔾 nor [ 𝔾 , 𝔾 ] βŠ‚ 𝔾 as follows from simple examples. However, 𝔾 is a conical set (over β„± ): if 𝑍 ∈ 𝔾 then 𝑓 𝑍 ∈ 𝔾 for any 𝑓 ∈ β„± . Easy direct proof may be omitted here.

Summary 2. If 𝑍 is   ℐ 𝒯 of a group then all functions 𝑍 π‘Ÿ β„Ž 𝑖   ( 𝑖 = 1 , … , 𝐼 ; π‘Ÿ = 0 , 1 , … ) are included into family β„± ( 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 β„± ( 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 𝐹 1 , … , 𝐹 𝐾 ( 𝐼 ) ∈ β„± be a maximal functionally independent subset of the family of all functions 𝑍 π‘Ÿ β„Ž 𝑖 ( 𝑖 = 1 , … , 𝐼 ; π‘Ÿ = 0 , 1 , … ) . ( 3 . 1 3 ) Denoting 𝑍 𝐹 π‘˜ = 𝐹 π‘˜ ( 𝐹 1 , … , 𝐹 𝐾 ( 𝐼 ) ) , then the system πœ• πœ• πœ† 𝐦 ( πœ† ) βˆ— 𝐹 π‘˜ = 𝐦 ( πœ† ) βˆ— 𝑍 𝐹 π‘˜ = 𝐹 π‘˜ ξ€· 𝐦 ( πœ† ) βˆ— 𝐹 1 , … , 𝐦 ( πœ† ) βˆ— 𝐹 𝐾 ( 𝐼 ) ξ€Έ ( π‘˜ = 1 , … , 𝐾 ( 𝐼 ) ) ( 3 . 1 4 ) may be regarded as a “finite-dimensional approximation" to the Lie system (3.3) of the one-parameter local group 𝐦 ( πœ† ) generated by 𝑍 .

In particular, assuming 𝐹 1 = β„Ž 1 , … , 𝐹 𝐼 = β„Ž 𝐼 , then the the initial portion d d πœ† 𝐦 ( πœ† ) βˆ— 𝐹 𝑖 = d d πœ† 𝐦 ( πœ† ) βˆ— β„Ž 𝑖 = d 𝐻 d πœ† 𝑖 = 𝑧 𝑖 ξ€· 𝐻 1 , … , 𝐻 π‘š ( 𝑖 ) ξ€Έ ( 𝑖 = 1 , … , 𝐼 ) ( 3 . 1 5 ) 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 𝑍 1 , … , 𝑍 𝑑 be commuting local vector fields in the space ℝ ∞ . Then 𝑍 1 , … , 𝑍 𝑑 ∈ 𝔾 if and only if the vector fields 𝑍 = π‘Ž 1 𝑍 1 + β‹― + π‘Ž 𝑑 𝑍 𝑑   ( π‘Ž 1 , … , π‘Ž 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 πœ† = ( πœ† 1 , … , πœ† 𝑑 ) ∈ ℝ 𝑑 are parameters close to the zero point 0 = ( 0 , … , 0 ) ∈ ℝ 𝑑 . The rule (2.2) is generalized as 𝐦 ( 0 ) = i d . , 𝐦 ( πœ‘ ( πœ† , πœ‡ ) ) = 𝐦 ( πœ† ) 𝐦 ( πœ‡ ) , ( 4 . 1 ) where πœ† = ( πœ† 1 , … , πœ† 𝑑 ) ,   πœ‡ = ( πœ‡ 1 , … , πœ‡ 𝑑 ) and πœ‘ = ( πœ‘ 1 , … , πœ‘ 𝑑 ) determine the composition of parameters. Appropriately adapting the space β„± ( 𝐼 ) and the concept of basical functions 𝐹 1 , … , 𝐹 𝐾 ( 𝐼 ) , Lemma 2.2 holds true without any change.

Passing to the infinitesimal approach, we introduce vector fields 𝑍 1 , … , 𝑍 𝑑 which are ℐ 𝒯 of the group. We recall (without proof) the Lie equations [17] πœ• πœ• πœ† 𝑗 𝐦 ( πœ† ) βˆ—  π‘Ž 𝑓 = 𝑗 𝑖 ( πœ† ) 𝐦 ( πœ† ) βˆ— 𝑍 𝑗 𝑓 ( 𝑓 ∈ β„± ; 𝑗 = 1 , … , 𝑑 ) ( 4 . 2 ) with the initial condition 𝐦 ( 0 ) = i d . Assuming 𝑍 1 , … , 𝑍 𝑑 linearly independent over ℝ , coefficients π‘Ž 𝑗 𝑖 ( πœ† ) may be arbitrarily chosen and the solution 𝐦 ( πœ† ) always is a group transformation (the first fundamental theorem). If basical functions 𝐹 1 , … , 𝐹 𝐾 ( 𝐼 ) are inserted for 𝑓 , we have a finite-dimensional approximation which is self-contained in the sense that 𝑍 𝑗 𝐹 π‘˜ =  𝐹 π‘˜ 𝑗 ξ€· 𝐹 1 , … , 𝐹 𝐾 ( 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‘Ž 1 𝑍 1 + β‹― + π‘Ž 𝑑 𝑍 𝑑    ( π‘Ž 𝑖 ∈ ℝ ) on the space ℝ ∞ . Let moreover 𝑍 1 , … , 𝑍 𝑑 ∈ 𝔾   uniformly in the sense that there is a universal space β„± ( 𝐼 ) with β„’ 𝑍 𝑖 β„± ( 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 𝑍 1 , … , 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 ] βŠ‚ 𝒦 . Then 𝒦 βŠ‚ 𝔾 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 ℝ π‘š + 𝑛 [9–13]. We recall the familiar (local) jet coordinates π‘₯ 𝑖 , 𝑀 𝑗 𝐼 ξ€· 𝐼 = 𝑖 1 … 𝑖 π‘Ÿ ; 𝑖 , 𝑖 1 , … , 𝑖 π‘Ÿ ξ€Έ . = 1 , … , 𝑛 ; π‘Ÿ = 0 , 1 , … ; 𝑗 = 1 , … , π‘š ( 5 . 1 ) Functions 𝑓 = 𝑓 ( … , π‘₯ 𝑖 , 𝑀 𝑗 𝐼 , … ) 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  π‘Ž πœ” = 𝑗 𝐼 πœ” 𝑗 𝐼 ξ‚€ fi n i t e s u m , πœ” 𝑗 𝐼 = d 𝑀 𝑗 𝐼 βˆ’  𝑀 𝑗 𝐼 𝑖 d π‘₯ 𝑖  ( 5 . 2 ) or, equivalently, by the “orthogonal" module β„‹ ( π‘š , 𝑛 ) = Ξ© βŸ‚ ( π‘š , 𝑛 ) of formal derivatives  π‘Ž 𝐷 = 𝑖 𝐷 𝑖  𝐷 𝑖 = πœ• πœ• π‘₯ 𝑖 +  𝑀 𝑗 𝐼 𝑖 πœ• πœ• 𝑀 𝑗 𝐼 ; 𝑖 = 1 , … , 𝑛 ; 𝐷 βŒ‹ πœ” 𝑗 𝐼 = πœ” 𝑗 𝐼 ξƒͺ ( 𝐷 ) = 0 . ( 5 . 3 ) Let us state useful formulae  𝐷 d 𝑓 = 𝑖 𝑓 d π‘₯ 𝑖 +  πœ• 𝑓 πœ• 𝑀 𝑗 𝐼 πœ” 𝑗 𝐼 , 𝐷 𝑖 βŒ‹ d πœ” 𝑗 𝐼 = πœ” 𝑗 𝐼 𝑖 , β„’ 𝐷 𝑖 πœ” 𝑗 𝐼 = πœ” 𝑗 𝐼 𝑖 , ( 5 . 4 ) where β„’ 𝐷 𝑖 = 𝐷 𝑖 βŒ‹ d + d 𝐷 𝑖 βŒ‹ denotes the Lie derivative.

We are interested in (local) one-parameter groups of transformations 𝐦 ( πœ† ) given by certain formulae 𝐦 ( πœ† ) βˆ— π‘₯ 𝑖 = 𝐺 𝑖 ξ‚€ πœ† ; … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€² 𝐼 β€²  , … , 𝐦 ( πœ† ) βˆ— 𝑀 𝑗 𝐼 = 𝐺 𝑗 𝐼 ξ‚€ πœ† ; … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€² 𝐼 β€²  , … ( 5 . 5 ) and in vector fields  𝑧 𝑍 = 𝑖 ξ‚€ … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€² 𝐼 β€²  πœ• , … πœ• π‘₯ 𝑖 +  𝑧 𝑗 𝐼 ξ‚€ … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€² 𝐼 β€²  πœ• , … πœ• 𝑀 𝑗 𝐼 ( 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 if β„’ 𝑍 Ξ© ( 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 “trivial summand 𝐷 " and the so-called “evolutionary form 𝑉 " of the vector field 𝑍 , explicitly   𝑧 𝑍 = 𝐷 + V 𝐷 = 𝑖 𝐷 𝑖  𝑄 ∈ β„‹ ( π‘š , 𝑛 ) , 𝑉 = 𝑗 𝐼 πœ• πœ• 𝑀 𝑗 𝐼 , 𝑄 𝑗 𝐼 = 𝑧 𝑗 𝐼 βˆ’  𝑀 𝑗 𝐼 𝑖 𝑧 𝑖 ξƒͺ . ( 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 πœ• 𝑀 𝑗 𝐼 πœ• πœ† = 𝑄 𝑗 𝐼 ξ‚€ … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€² 𝐼 β€²   𝑀 , … 𝑗 𝐼 = πœ• 𝑛 𝑀 𝑗 πœ• π‘₯ 𝑖 1 β‹― πœ• π‘₯ 𝑖 𝑛 ξ€· πœ† , π‘₯ 1 , … , π‘₯ 𝑛 ξ€Έ 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 “virtual flow" on the “space of solutions" 𝑀 𝑗 = 𝑀 𝑗 ( 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 𝛿 𝑀 𝑗 = πœ• 𝑀 𝑗 / πœ• πœ† are just the variations. (There is only one novelty: in classical theory, 𝛿 𝑀 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 𝑗 𝐼 𝑖 𝐷 𝑖 β€² 𝐺 𝑖 = 𝐷 𝑖 β€² 𝐺 𝑗 𝐼 ξ€· 𝑖 ξ…ž ξ€Έ = 1 , … , 𝑛 , ( 5 . 9 ) where d e t ( 𝐷 𝑖 ξ…ž 𝐺 𝑖 ) β‰  0 is supposed and vector field (5.6) is a variation if and only if 𝑧 𝑗 𝐼 𝑖 = 𝐷 𝑖 𝑧 𝑗 𝐼 βˆ’  𝑀 𝑗 𝐼 𝑖 β€² 𝐷 𝑖 𝑧 𝑖 β€² . ( 5 . 1 0 ) Recurrence (5.9) easily follows from the inclusion 𝐦 ( πœ† ) βˆ— πœ” 𝑗 𝐼 ∈ Ξ© ( π‘š , 𝑛 ) and we omit the proof. Recurrence (5.10) follows from the identity β„’ 𝑍 πœ” 𝑗 𝐼 = β„’ 𝑍 ξ‚€ d 𝑀 𝑗 𝐼 βˆ’  𝑀 𝑗 𝐼 𝑖 d π‘₯ 𝑖  = d 𝑧 𝑗 𝐼 βˆ’  𝑧 𝑗 𝐼 𝑖 d π‘₯ 𝑖 βˆ’  𝑀 𝑗 𝐼 𝑖 d 𝑧 𝑖 β‰… ξ‚€  𝐷 𝑖 β€² z 𝑗 𝐼 βˆ’  𝑧 𝑗 𝐼 𝑖 β€² βˆ’  𝑀 𝑗 𝐼 𝑖 𝐷 𝑖 β€² 𝑧 𝑖  d π‘₯ 𝑖 β€² ( m o d Ξ© ( π‘š , 𝑛 ) ) ( 5 . 1 1 ) and the inclusion β„’ 𝑍 πœ” 𝑗 𝐼 ∈ Ξ© ( π‘š , 𝑛 ) . The obvious formula β„’ 𝑍 πœ” 𝑗 𝐼 =  βŽ› ⎜ ⎜ ⎝ πœ• 𝑧 𝑗 𝐼 πœ• 𝑀 𝑗 β€² 𝐼 β€² βˆ’  𝑀 𝑗 𝐼 𝑖 πœ• 𝑧 𝑖 πœ• 𝑀 𝑗 β€² 𝐼 β€² ⎞ ⎟ ⎟ ⎠ πœ” 𝑗 β€² 𝐼 β€² ( 5 . 1 2 ) 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 = 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” 𝑗 𝐼 𝑖 ( 𝑍 ) = 𝐷 𝑖 πœ” 𝑗 𝐼 ( 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 ξ€½ 𝑍 π‘Ÿ π‘₯ 𝑖 ξ€Ύ π‘Ÿ ∈ β„• , ξ€½ 𝑍 π‘Ÿ 𝑀 𝑗 𝐼 ξ€Ύ π‘Ÿ ∈ β„• ( 5 . 1 3 ) 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 ( πœ„ ) every family of functions ξ€½ 𝑍 π‘Ÿ π‘₯ 𝑖 ξ€Ύ π‘Ÿ ∈ β„• , ξ€½ 𝑍 π‘Ÿ 𝑀 𝑗 ξ€Ύ π‘Ÿ ∈ β„• ( 𝑖 = 1 , … , 𝑛 ; 𝑗 = 1 , … , π‘š ) ( 5 . 1 4 ) can be expressed in terms of a finite number of jet coordinates,( πœ„ πœ„ )every family of differential forms ξ€½ β„’ π‘Ÿ 𝑍 d π‘₯ 𝑖 ξ€Ύ π‘Ÿ ∈ β„• , ξ€½ β„’ π‘Ÿ 𝑍 d 𝑀 𝑗 ξ€Ύ π‘Ÿ ∈ β„• ( 𝑖 = 1 , … , 𝑛 ; 𝑗 = 1 , … , π‘š ) ( 5 . 1 5 ) involves only a finite number of linearly independent terms,( πœ„ πœ„ πœ„ )every family of differential forms ξ€½ β„’ π‘Ÿ 𝑍 d π‘₯ 𝑖 ξ€Ύ π‘Ÿ ∈ β„• , ξ€½ β„’ π‘Ÿ 𝑍 d 𝑀 𝑗 𝐼 ξ€Ύ π‘Ÿ ∈ β„• ( 𝑖 = 1 , … , 𝑛 ; 𝑗 = 1 , … , π‘š ; a r b i t r a r y 𝐼 ) ( 5 . 1 6 ) 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 ( 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 β„’ 𝑍 d 𝑓 = d 𝑍 𝑓 . Assuming ( πœ„ πœ„ ) , we may employ the commutative rule ξ€Ί 𝐷 𝑖 ξ€» , 𝑍 = 𝐷 𝑖 𝑍 βˆ’ 𝑍 𝐷 𝑖 =  π‘Ž 𝑖 β€² 𝑖 𝐷 𝑖 β€² ξ‚€ π‘Ž 𝑖 β€² 𝑖 = 𝐷 𝑖 𝑧 𝑖 β€²  ( 5 . 1 7 ) in order to verify identities of the kind β„’ 𝑍 d 𝑀 𝑗 𝑖 = β„’ Z d 𝐷 𝑖 𝑀 𝑗 = β„’ 𝑍 β„’ 𝐷 𝑖 d 𝑀 𝑖 = β„’ 𝐷 𝑖 β„’ 𝑍 d 𝑀 𝑖 βˆ’  π‘Ž 𝑖 β€² 𝑖 β„’ 𝐷 𝑖 β€² 𝑀 𝑗 ( 5 . 1 8 ) and in full generality identities of the kind β„’ π‘˜ 𝑍 d 𝑀 𝑗 𝐼 =  π‘Ž 𝐼 β€² 𝐼 , π‘˜ β„’ 𝐷 𝐼 β€² β„’ π‘˜ β€² 𝑍 d 𝑀 𝑗 ξ€· s u m w i t h π‘˜ ξ…ž | | 𝐼 ≀ π‘˜ , ξ…ž | | ≀ | | 𝐼 | | ξ€Έ ( 5 . 1 9 ) 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 𝑍 ∈ 𝔾 if and only if every family ξ€½ β„’ π‘Ÿ 𝑍 πœ” 𝑗 ξ€Ύ π‘Ÿ ∈ β„• ( 𝑗 = 1 , … , π‘š ) ( 5 . 2 0 ) 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„’ 𝑓 𝑋 Θ βŠ‚ Θ for all functions 𝑓 . Let moreover A d j Θ be the module of all forms πœ‘ satisfying πœ‘ ( 𝑋 ) = 0 for all such 𝑋 . Then A d j Θ has a basis consisting of total differentials of certain functions 𝑓 1 , … , 𝑓 𝐾 (the Frobenius theorem), and there is a basis of module Θ which can be expressed in terms of functions 𝑓 1 , … , 𝑓 𝐾 . Alternatively saying, (an appropriate basis of) the Pfaffian system πœ— = 0 ( πœ— ∈ Θ ) can be expressed only in terms of functions 𝑓 1 , … , 𝑓 𝐾 . This result frequently appears in Cartan's work, but we may refer only to [9, 18, 19] and to the appendix below for the proof.

Module A d j Θ is intrinsically related to Θ : if a mapping 𝐦 preserves Θ then 𝐦 preserves A d j Θ . In particular, assuming 𝐦 ( πœ† ) βˆ— Θ βŠ‚ Θ , t h e n 𝐦 ( πœ† ) βˆ— A d j Θ βŠ‚ A d j Θ ( 5 . 2 1 ) is true for a group 𝐦 ( πœ† ) . In terms of ℐ 𝑇 of the group 𝐦 ( πœ† ) , we have equivalent assertion β„’ 𝑍 Θ βŠ‚ Θ i m p l i e s β„’ 𝑍 A d j Θ βŠ‚ A d j Θ ( 5 . 2 2 ) and therefore β„’ π‘Ÿ 𝑍 A d j Θ βŠ‚ A d j Θ for all π‘Ÿ . The preparation is done.

Proof. Let Θ be the module generated by all differential forms β„’ π‘Ÿ 𝑍 πœ” 𝑗 ( 𝑗 = 1 , … , π‘š ; π‘Ÿ = 0 , 1 , … ). Assuming finite dimension of module Θ , we have module A d j Θ and clearly β„’ 𝑍 Θ βŠ‚ Θ whence β„’ π‘Ÿ 𝑍 A d j Θ βŠ‚ A d j Θ ( π‘Ÿ = 0 , 1 , … ). However A d j Θ involves both the differentials d π‘₯ 1 , … , d π‘₯ 𝑛 (see below) and the forms πœ” 1 , … , πœ” π‘š . Point ( πœ„ πœ„ ) 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 A d j Θ   does not contain all differentials d π‘₯ 1 , … , d π‘₯ 𝑛 . Alternatively saying, the Pfaffian system πœ— = 0 ( πœ— ∈ Θ ) can be expressed in terms of certain functions 𝑓 1 , … , 𝑓 𝐾 such that d 𝑓 1 = β‹― = d 𝑓 𝐾 = 0 does not imply d π‘₯ 1 = β‹― = d π‘₯ 𝑛 = 0 . On the other hand, it follows clearly that maximal solutions of the Pfaffian system can be expressed only in terms of functions 𝑓 1 , … , 𝑓 𝐾 and therefore we do not need all independent variables 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” 𝑛 = 0 . All independent variables are needful if we deal with the common classical solutions 𝑀 𝑗 = 𝑀 𝑗 ( π‘₯ 1 , … , π‘₯ 𝑛 ) .

The result can be rephrased as follows.

Theorem 5.6. Let Ξ© 0 βŠ‚ Ξ© ( π‘š , 𝑛 ) be the submodule of all zeroth-order contact forms βˆ‘ π‘Ž πœ” = 𝑗 πœ” 𝑗 and 𝑍 be a variation of the jet structure. Then 𝑍 ∈ 𝔾 if and only if d i m βŠ• β„’ π‘Ÿ 𝑍 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 + 𝕍 βŠ‚ 𝕍 , 𝑐 𝕍 βŠ‚ 𝕍 ( 𝑐 ∈ ℝ ), [ 𝕍 , 𝕍 ] βŠ‚ 𝕍 , and it follows that 𝕍 is an infinite-dimensional Lie algebra (coefficients in ℝ ). On the other hand, if 𝑍 ∈ 𝕍 and 𝑓 𝑍 ∈ 𝕍 for certain 𝑓 ∈ β„± 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 ( π‘š , 𝑛 ) .

Theorem 6.1. Let 𝒒 βŠ‚ 𝕍 be a finite-dimensional Lie subalgebra. Then 𝒒 βŠ‚ 𝔾 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© 𝑙 βŠ‚ Ξ© ( π‘š , 𝑛 ) be the submodule of all contact forms βˆ‘ π‘Ž πœ” = 𝑗 𝐼 πœ” 𝑗 𝐼 (sum with | 𝐼 | ≀ 𝑙 ) of the order 𝑙 at most. A morphism (5.5) and the infinitesimal variation (5.6) are called order preserving if 𝐦 ( πœ† ) βˆ— Ξ© 𝑙 βŠ‚ Ξ© 𝑙 , β„’ 𝑍 Ξ© 𝑙 βŠ‚ Ξ© 𝑙 , ( 7 . 1 ) respectively, for a certain 𝑙 = 0 , 1 , … (equivalently: for all 𝑙 ∈ β„• , 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 = πœ™ ) in formulae (5.5) and (5.6) are functions only of the zeroth-order jet variables π‘₯ 𝑖 ξ…ž , 𝑀 𝑗 ξ…ž or, in the second case, we have π‘š = 1 and all functions 𝐺 𝑖 , 𝐺 1 , 𝐺 1 𝑖 , 𝑧 𝑖 , 𝑧 1 , 𝑧 1 𝑖 contain only the zeroth- and first-order variables π‘₯ 𝑖 ξ…ž , 𝑀 1 , 𝑀 1 𝑖 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-Bäcklund). Let a variation 𝑍 preserve a submodule Ξ© 𝑙 βŠ‚ Ξ© ( π‘š , 𝑛 ) of contact forms of the order 𝑙 at most for a certain 𝑙 ∈ β„• . Then 𝑍 ∈ 𝔾 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„’ 𝑍 Ξ© 𝑙 βŠ‚ Ξ© 𝑙 . Then β„’ π‘Ÿ 𝑍 Ξ© 0 βŠ‚ β„’ π‘Ÿ 𝑍 Ξ© 𝑙 βŠ‚ Ξ© 𝑙 therefore 𝑍 ∈ 𝔾 by virtue of Theorem 5.5. Moreover β„’ 𝑍 Ξ© 𝑙 βˆ’ 1 βŠ‚ Ξ© 𝑙 βˆ’ 1 , … , β„’ 𝑍 Ξ© 0 βŠ‚ Ξ© 0 by virtue of Lemma 9.2 below. So we have β„’ 𝑍 πœ” 𝑗 =  π‘Ž 𝑗 𝑗 β€² πœ” 𝑗 β€² ξ€· 𝑗 , 𝑗 ξ…ž ξ€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‘š β‰₯ 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‘š β‰₯ 2 . Then 𝑍 π‘₯ 𝑖 = 𝑧 𝑖 ξ‚€ … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€²  , … , 𝑍 𝑀 𝑗 = 𝑧 𝑗 ξ‚€ … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€²  , … ( 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 + 2 𝑛 ) -forms generated by all forms of the kind πœ” 1 ∧ β‹― ∧ πœ” π‘š ∧ ξ€· d πœ” 𝑗 1 ξ€Έ 𝑛 1 ∧ ξ€· d πœ” 𝑗 π‘˜ ξ€Έ 𝑛 π‘˜ = d 𝑀 1 ∧ β‹― d 𝑀 π‘š ∧ d π‘₯ 1 ∧ β‹― d π‘₯ 𝑛 ∧  Β± d 𝑀 𝑗 β€² 1 𝑖 1 ∧ β‹― ∧ d 𝑀 𝑗 β€² 𝑛 𝑖 𝑛 , ( 7 . 4 ) where βˆ‘ 𝑛 π‘˜ = 𝑛 . Clearly Θ = ( Ξ© 0 ) π‘š ∧ ( d Ξ© 0 ) 𝑛 . The inclusions β„’ 𝑍 Ξ© 0 βŠ‚ Ξ© 0 , β„’ 𝑍 d Ξ© 0 = d β„’ 𝑍 Ξ© 0 + Ξ© 0 βŠ‚ d Ξ© 0 + Ξ© 0 ( 7 . 5 ) are true by virtue of (7.2) and imply β„’ 𝑍 Θ βŠ‚ Θ .
Module Θ vanishes when restricted to certain hyperplanes, namely, just to the hyperplanes of the kind  π‘Ž πœ— = 𝑖 d π‘₯ 𝑖 +  π‘Ž 𝑗 d 𝑀 𝑗 = 0 ( 7 . 6 ) (use π‘š β‰₯ 2 here). This is expressed by Θ ∧ πœ— = 0 and it follows that 0 = β„’ 𝑍 ( Θ ∧ πœ— ) = β„’ 𝑍 Θ ∧ πœ— + Θ ∧ β„’ 𝑍 πœ— = Θ ∧ β„’ 𝑍 πœ— . ( 7 . 7 ) Therefore β„’ 𝑍 πœ— again is such a hyperplane: β„’ 𝑍 πœ— β‰… 0 (mod all d π‘₯ 𝑖 and d 𝑀 𝑗 ). On the other hand, β„’ 𝑍  π‘Ž πœ— β‰… 𝑖 d 𝑧 𝑖 +  π‘Ž 𝑗 d 𝑧 𝑗 ξ€· m o d a l l d π‘₯ 𝑖 a n d d 𝑀 𝑗 ξ€Έ ( 7 . 8 ) and it follows that d 𝑧 𝑖 , d 𝑧 𝑗 β‰… 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’ž 𝒯 ) and of nondegenerate first-order one-dimensional variational integrals ( 𝒱 ℐ ) are identical. In particular, the orbits of a given π’ž 𝒯 group are extremals of appropriate 𝒱 ℐ and conversely.

Proof. The π’ž 𝒯 groups act in the jet space 𝐌 ( 1 , 𝑛 ) equipped with the contact module Ξ© ( 1 , 𝑛 ) . Then the abbreviations 𝑀 𝐼 = 𝑀 1 𝐼 , πœ” 𝐼 = πœ” 1 𝐼 = d 𝑀 𝐼 βˆ’  𝑀 𝐼 𝑖 d π‘₯ 𝑖  𝑧 𝑍 = 𝑖 πœ• πœ• π‘₯ 𝑖 +  𝑧 1 𝐼 πœ• πœ• 𝑀 𝐼 ( 8 . 1 ) are possible. Let us recall the classical approach [22, 23]. The Lie contact transformations defined by certain formulae 𝐦 βˆ— π‘₯ 𝑖 = 𝐺 𝑖 ( β‹… ) , 𝐦 βˆ— 𝑀 = 𝐺 1 ( β‹… ) , 𝐦 βˆ— 𝑀 𝑖 = 𝐺 1 𝑖 ξ€· ξ€· π‘₯ ( β‹… ) ( β‹… ) = 1 , … , π‘₯ 𝑛 , 𝑀 , 𝑀 1 , … , 𝑀 𝑛 ξ€Έ ξ€Έ ( 8 . 2 ) preserve the Pfaffian equation βˆ‘ 𝑀 πœ” = d 𝑀 βˆ’ 𝑖 d π‘₯ 𝑖 = 0 or (equivalently) the submodule Ξ© 0 βŠ‚ Ξ© ( 1 , 𝑛 ) of zeroth-order contact forms. Explicit formulae are available in literature. We are interested in one-parameter local π’ž 𝒯 groups of transformations 𝐦 ( πœ† ) ( βˆ’ πœ€ < πœ† < πœ€ ) which are “nondegenerate" in a sense stated below and then the explicit formulae are not available yet. On the other hand, our 𝒱 ℐ with smooth Lagrangian Ł ξ€œ Ł ξ€· 𝑑 , 𝑦 1 , … , 𝑦 𝑛 , 𝑦 ξ…ž 1 , … , 𝑦 ξ…ž 𝑛 ξ€Έ  𝑦 𝑑 𝑑 𝑖 = 𝑦 𝑖 ( 𝑑 ) , ξ…ž = 𝑑  πœ• 𝑑 𝑑 , d e t 2 Ł πœ• 𝑦 ξ…ž 𝑖 πœ• 𝑦 ξ…ž 𝑗 ξƒͺ ξƒͺ β‰  0 ( 8 . 3 ) to appear later, involves variables from quite other jet space 𝐌 ( 𝑛 , 1 ) with coordinates denoted 𝑑 (the independent variable), 𝑦 1 , … , 𝑦 𝑛 (the dependent variables) and higher-order jet variables like 𝑦 ξ…ž 𝑖 , 𝑦 𝑖 ξ…ž ξ…ž and so on.
We are passing to the topic proper. Let us start in the space 𝐌 ( 1 , 𝑛 ) with π’ž 𝒯 groups. One can check that vector field (5.6) is infinitesimal π’ž 𝒯 if and only if  𝑄 𝑍 = βˆ’ 𝑀 𝑖 πœ• πœ• π‘₯ 𝑖 + ξ‚€  𝑀 𝑄 βˆ’ 𝑖 𝑄 𝑀 𝑖  πœ• +  ξ€· 𝑄 πœ• 𝑀 π‘₯ 𝑖 + 𝑀 𝑖 𝑄 𝑀 ξ€Έ πœ• πœ• 𝑀 𝑖 + β‹― , ( 8 . 4 ) where the function 𝑄 = 𝑄 ( π‘₯ 1 , … , π‘₯ 𝑛 , 𝑀 , 𝑀 1 , … , 𝑀 𝑛 ) may be arbitrarily chosen.
“Hint: we have, by definition β„’ 𝑍  ξ€· 𝑧 πœ” = 𝑍 βŒ‹ d πœ” + d πœ” ( 𝑍 ) = 𝑖 πœ” 𝑖 βˆ’ πœ” 𝑖 ( 𝑍 ) d π‘₯ 𝑖 ξ€Έ + d 𝑄 ∈ Ξ© 0 , ( 8 . 5 ) where 𝑄 = 𝑄 ( π‘₯ 1 , … , π‘₯ 𝑛 , 𝑀 , 𝑀 1 , … , 𝑀 𝑛 , … ) = πœ” ( 𝑍 ) = 𝑧 1 βˆ’ βˆ‘ 𝑀 𝑖 𝑧 𝑖 ,  𝐷 d 𝑄 = 𝑖 𝑄 d π‘₯ 𝑖 + πœ• 𝑄  πœ• 𝑀 πœ” + πœ• 𝑄 πœ• 𝑀 𝑖 πœ” 𝑖 ( 8 . 6 ) whence immediately 𝑧 𝑖 = βˆ’ πœ• 𝑄 / πœ• 𝑀 𝑖 , 𝑧 1 βˆ‘ 𝑀 = 𝑄 + 𝑖 𝑧 𝑖 βˆ‘ 𝑀 = 𝑄 βˆ’ 𝑖 β‹… πœ• 𝑄 / πœ• 𝑀 𝑖 ,    πœ• 𝑄 / πœ• 𝑀 𝐼 = 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— 𝑀 satisfies the equation 𝑉 ξ€· π‘₯ 1 , … , π‘₯ 𝑛 , 𝐦 ( πœ† ) βˆ— 𝑀 ξ€Έ = πœ† ( 8 . 7 ) without any loss of generality. In infinitesimal terms 1 = πœ• ( 𝑉 βˆ’ πœ† )  𝑄 πœ• πœ† = 𝑍 ( 𝑉 βˆ’ πœ† ) = βˆ’ 𝑀 𝑖 𝑉 π‘₯ 𝑖 + ξ‚€  𝑀 𝑄 βˆ’ 𝑖 𝑄 𝑀 𝑖  𝑉 𝑀 . ( 8 . 8 ) However 𝑀 𝑖 = πœ• 𝑀 / πœ• π‘₯ 𝑖 = βˆ’ 𝑉 π‘₯ 𝑖 / 𝑉 𝑀 may be inserted here, and we have the crucial Jacobi equation ξ‚΅ π‘₯ 1 = 𝑄 1 , … , π‘₯ 𝑛 𝑉 , 𝑀 , βˆ’ π‘₯ 1 𝑉 𝑀 𝑉 , … , βˆ’ π‘₯ 𝑛 𝑉 𝑀 ξ‚Ά 𝑉 𝑀 ( 8 . 9 ) (not involving 𝑉 ) which can be uniquely rewritten as the Hamilton-Jacobi ( β„‹ π’₯ ) equation 𝑉 𝑀 ξ€· π‘₯ + β„‹ 1 , … , π‘₯ 𝑛 , 𝑀 , 𝑝 1 , … , 𝑝 𝑛 𝑝 ξ€Έ ξ€· 𝑖 = 𝑉 π‘₯ 𝑖 ξ€Έ ( 8 . 1 0 ) in the “nondegenerate" case βˆ‘ 𝑄 𝑀 𝑖 𝑉 π‘₯ 𝑖 β‰  1 . Let us recall the characteristic curves [22, 23] of the β„‹ π’₯ equation given by the system d 𝑀 1 = d π‘₯ 𝑖 β„‹ 𝑝 𝑖 = βˆ’ d 𝑝 𝑖 β„‹ π‘₯ 𝑖 = d 𝑉 βˆ‘ 𝑝 βˆ’ β„‹ + 𝑖 β„‹ 𝑝 𝑖 . ( 8 . 1 1 ) 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 d e t ( πœ• 2 β„‹ / πœ• 𝑝 𝑖 πœ• 𝑝 𝑗 ) β‰  0 . We may introduce variational integral (8.3) with the Lagrange function Ł given by the familiar identities  𝑝 Ł + β„‹ = 𝑖 𝑦 ξ…ž 𝑖 ( 8 . 1 2 ) with interrelations 𝑑 = 𝑀 , 𝑦 𝑖 = π‘₯ 𝑖 , 𝑦 ξ…ž 𝑖 = β„‹ 𝑝 𝑖 , 𝑝 𝑖 = Ł 𝑦 β€² 𝑖 ( 𝑖 = 1 , … , 𝑛 ) ( 8 . 1 3 ) between variables 𝑑 , 𝑦 𝑖 , 𝑦 ξ…ž 𝑖 of the space 𝐌 ( 𝑛 , 1 ) and variables π‘₯ 𝑖 , 𝑀 , 𝑀 𝑖 of the space 𝐌 ( 1 , 𝑛 ) . Since (8.11) may be regarded as a Hamiltonian system for the extremals of 𝒱 ℐ , the metatheorem is clarified.

Remark 8.2. Let us recall the Mayer fields of extremals for the 𝒱 ℐ since they provide the true sense of the above construction. The familiar Poincaré-Cartan form  Ł Μ† πœ‘ = Ł d 𝑑 + 𝑦 β€² 𝑖 ξ€· d 𝑦 𝑖 βˆ’ 𝑦 ξ…ž 𝑖 ξ€Έ  𝑝 d 𝑑 = βˆ’ β„‹ d 𝑑 + 𝑖 d 𝑦 𝑖 ( 8 . 1 4 ) is restricted to appropriate subspace 𝑦 ξ…ž 𝑖 = 𝑔 𝑖 ( 𝑑 , 𝑦 1 , … , 𝑦 𝑛 ) ( 𝑖 = 1 , … , 𝑛 ; the slope field) in order to become a total differential | | Μ† πœ‘ 𝑦 β€² 𝑖 = 𝑔 𝑖 ξ€· = d 𝑉 𝑑 , 𝑦 1 , … , 𝑦 𝑛 ξ€Έ = 𝑉 𝑑  𝑉 d 𝑑 + 𝑦 𝑖 d 𝑦 𝑖 ( 8 . 1 5 ) of the action 𝑉 . We obtain the requirements 𝑉 𝑑 = βˆ’ β„‹ , 𝑉 𝑦 𝑖 = 𝑝 𝑖 identical with (8.10). In geometrical terms: transformations of a hypersurface 𝑉 = 0 by means of π’ž 𝒯 group may be identified with the level sets 𝑉 = πœ† ( πœ† ∈ ℝ ) of the action of a Mayer fields of extremals.
The last statement is in accordance with (8.11) where ξ‚€  𝑝 d 𝑉 = βˆ’ β„‹ + 𝑖 β„‹ 𝑝 𝑖  ξ‚€  𝑝 d 𝑀 = βˆ’ β„‹ + 𝑖 𝑦 ξ…ž 𝑖  d 𝑑 = Ł d 𝑑 , ( 8 . 1 6 ) 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 ∫ Ł 𝑑 𝑑 calculated along the orbits.

fig3
Figure 3

On this occasion, the wave enveloping approach to π’ž 𝒯 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 2 𝑛 + 2 variables. Assume that the system π‘Š = 𝐷 1 π‘Š = β‹― = 𝐷 𝑛 π‘Š = 0 admits a unique solution π‘₯ 𝑖 = 𝐹 𝑖 ξ€· … , π‘₯ 𝑖 ξ…ž , 𝑀 , 𝑀 𝑖 ξ…ž ξ€Έ , , … 𝑀 = 𝐹 1 ξ€· … , π‘₯ 𝑖 ξ…ž , 𝑀 , 𝑀 𝑖 ξ…ž ξ€Έ , … ( 8 . 1 7 ) by applying the implicit function theorem and analogously the system π‘Š = 𝐷 1 π‘Š = β‹― = 𝐷 𝑛 π‘Š = 0    (where 𝐷 𝑖 = πœ• / πœ• π‘₯ 𝑖 + βˆ‘ 𝑀 𝑖 πœ• / πœ• 𝑀 ) admits a certain solution π‘₯ 𝑖 = 𝐹 𝑖 ξ€· … , π‘₯ 𝑖 β€² , 𝑀 , 𝑀 𝑖 β€² ξ€Έ , … , 𝑀 = 𝐹 1 ξ€· … , π‘₯ 𝑖 β€² , 𝑀 , 𝑀 𝑖 β€² ξ€Έ . , … ( 8 . 1 8 ) Then 𝐦 βˆ— π‘₯ 𝑖 = 𝐹 𝑖 ,    𝐦 βˆ— 𝑀 = 𝐹 1 provides a Lie π’ž 𝒯 and ( 𝐦 βˆ’ 1 ) βˆ— π‘₯ 𝑖 = 𝐹 𝑖 , ( 𝐦 βˆ’ 1 ) βˆ— 𝑀 = 𝐹 1 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’ž 𝒯 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’ž 𝒯 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 = 𝐷 1 π‘Š = β‹― = 𝐷 𝑛 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 wave π‘Š ξ€· π‘₯ 1 , … , π‘₯ 𝑛 , 𝑀 , π‘₯ 1 , … , π‘₯ 𝑛 ξ€Έ , 𝑀 βˆ’ πœ† = 0 . ( 8 . 1 9 ) 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, therefore π‘Š 𝑀 ξ€· π‘₯ + β„‹ 1 , … , π‘₯ 𝑛 , 𝑀 , π‘Š π‘₯ 1 , … , π‘Š π‘₯ 𝑛 ξ€Έ = 0 . ( 8 . 2 0 ) The existence of such function β„‹ means that functions π‘Š 𝑀 , π‘Š π‘₯ 1 , … , π‘Š π‘₯ 𝑛   of dashed variables are functionally dependent whence ξ‚΅ π‘Š d e t 𝑀 𝑀 π‘Š 𝑀 π‘₯ 𝑖 β€² π‘Š π‘₯ 𝑖 𝑀 π‘Š π‘₯ 𝑖 π‘₯ 𝑖 β€² ξ‚Ά ξ€· π‘Š = 0 , d e t π‘₯ 𝑖 π‘₯ 𝑖 β€² ξ€Έ β‰  0 . ( 8 . 2 1 ) The symmetry π‘₯ 𝑖 , 𝑀 ↔ π‘₯ 𝑖 , 𝑀 is not surprising here since the change πœ† ↔ βˆ’ πœ† provides the inverse mapping: equations π‘Š ξ€· … , π‘₯ 𝑖 , 𝑀 , … , π‘₯ 𝑖 ξ€Έ ξ€· , 𝑀 = πœ† , π‘Š … , π‘₯ 𝑖 , 𝑀 , … , π‘₯ 𝑖 , 𝑀 ξ€Έ = βˆ’ πœ† ( 8 . 2 2 ) are equivalent. In particular, it follows that π‘Š ξ€· … , π‘₯ 𝑖 , 𝑀 , … , π‘₯ 𝑖 ξ€Έ ξ€· , 𝑀 = βˆ’ π‘Š … , π‘₯ 𝑖 , 𝑀 , … , π‘₯ 𝑖 , 𝑀 ξ€Έ ξ€· , π‘Š … , π‘₯ 𝑖 , 𝑀 , … , π‘₯ 𝑖 ξ€Έ , 𝑀 = 0 ( 8 . 2 3 ) and the wave π‘Š βˆ’ πœ† = 0 corresponds to the Mayer central field of extremals.

Summary 4. Conditions (8.21) ensure the existence of β„‹ 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 d e t ( πœ• 2 β„‹ / πœ• 𝑝 𝑖 πœ• 𝑝 𝑗 ) 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 filtration Ξ© ( π‘š , 𝑛 ) βˆ— ∢ Ξ© 0 βŠ‚ Ξ© 1 βŠ‚ β‹― βŠ‚ Ξ© ( π‘š , 𝑛 ) = βˆͺ Ξ© 𝑙 ( 9 . 1 ) of module Ξ© ( π‘š , 𝑛 ) is preserved (Figure 4(a)). It follows that certain invariant submodules Ξ© 𝑙 βŠ‚ Ξ© ( π‘š , 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 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 Θ βŠ‚ Ξ© ( π‘š , 𝑛 ) . Then also the submodule  β„’ Θ + 𝐷 𝑖 Θ βŠ‚ Ξ© ( π‘š , 𝑛 ) ( 9 . 2 ) is preserved.

Proof. We suppose β„’ 𝑍 Θ βŠ‚ Θ . Then β„’ 𝑍 ξ‚€  β„’ Θ + 𝐷 𝑖 Θ  = β„’ 𝑍 ξ‚€ β„’ Θ + 𝐷 𝑖 β„’ 𝑍  𝐷 Θ βˆ’ 𝑖 𝑧 𝑖 ξ…ž β„’ 𝐷 𝑖 β€² Θ   β„’ βŠ‚ Θ + 𝐷 𝑖 Θ ( 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 Θ βŠ‚ Ξ© ( π‘š , 𝑛 ) . Let Θ β€² βŠ‚ Θ be the submodule of all πœ” ∈ Θ satisfying β„’ 𝐷 𝑖 πœ” ∈ Θ ( 𝑖 = 1 , … , 𝑛 ). Then Θ β€² is preserved, too.

Proof. Assume πœ” ∈ Θ ξ…ž hence β„’ 𝐷 𝑖 πœ” ∈ Θ . Then β„’ 𝐷 𝑖 β„’ 𝑍 πœ” = β„’ 𝑍 β„’ 𝐷 𝑖 πœ” + β„’ βˆ‘ 𝐷 𝑖 𝑧 𝑖 β€² β‹… 𝐷 𝑖 β€² πœ” ∈ Θ hence β„’ 𝑍 πœ” ∈ Θ ξ…ž and Θ ξ…ž is preserved.

We are passing to illustrative examples.

Example 9.3. Let us consider the vector field (the variation of jet structure)  𝑧 𝑍 = 𝑗 𝐼 πœ• πœ• 𝑀 𝑗 𝐼 ξ€· 𝑧 𝑗 𝐼 = 𝐷 𝐼 𝑧 𝑗 , 𝐷 𝐼 = 𝐷 𝑖 1 β‹― 𝐷 𝑖 𝑛 ξ€Έ , ( 9 . 4 ) see (5.6) and (5.10) for the particular case 𝑧 𝑖 = 0 . Then 𝑍 π‘Ÿ π‘₯ 𝑖 = 0   ( 𝑖 = 1 , … , 𝑛 ) and the sufficient requirement 𝑍 2 𝑀 𝑗 = 0 ( 𝑗 = 1 , … , π‘š ) ensures 𝑍 ∈ 𝔾 , see ( πœ„ ) of Lemma 5.4. We will deal with the linear case where 𝑧 𝑗 =  π‘Ž 𝑗 𝑗 β€² 𝑖 β€² 𝑀 𝑗 β€² 𝑖 β€² ξ‚€ π‘Ž 𝑗 𝑗 β€² 𝑖 β€²  ∈ ℝ ( 9 . 5 ) is supposed. Then 𝑍 2 𝑀 𝑗 = 𝑍 𝑧 𝑗 =  π‘Ž 𝑗 𝑗 β€² 𝑖 β€² 𝑧 𝑗 β€² 𝑖 β€² =  π‘Ž 𝑗 𝑗 β€² 𝑖 β€² π‘Ž 𝑗 β€² 𝑗 β€² β€² 𝑖 β€² β€² 𝑀 𝑗 β€² β€² 𝑖 β€² 𝑖 β€² β€² = 0 ( 9 . 6 ) identically if and only if  𝑗 β€² ξ‚€ π‘Ž 𝑗 𝑗 β€² 𝑖 β€² π‘Ž 𝑗 β€² 𝑗 β€² β€² 𝑖 β€² β€² + π‘Ž 𝑗 𝑗 β€² 𝑖 β€² β€² π‘Ž 𝑗 β€² 𝑗 β€² β€² 𝑖 β€²  ξ€· 𝑖 = 0 ξ…ž , 𝑖 ξ…ž ξ…ž = 1 , … , 𝑛 ; 𝑗 , 𝑗 ξ…ž , 𝑗 ξ…ž ξ…ž ξ€Έ . = 1 , … , π‘š ( 9 . 7 ) This may be expressed in terms of matrix equations 𝐴 𝑖 𝐴 𝑖 β€² ξ‚€ = 0 𝑖 , 𝑖 ξ…ž = 1 , … , 𝑛 ; 𝐴 𝑖 = ξ‚€ π‘Ž 𝑗 𝑗 β€² 𝑖   ( 9 . 8 ) or, in either of more geometrical transcriptions 𝐴 2 ξ‚€  πœ† = 0 , I m 𝐴 βŠ‚ K e r 𝐴 𝐴 = 𝑖 𝐴 𝑖 , πœ† 𝑖  ∈ ℝ , ( 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† 𝑛 . We do not know explicit solutions 𝐴 in full generality, however, solutions 𝐴 such that K e r 𝐴 does not depend on the parameters πœ† 1 , … , πœ† 𝑛 can be easily found (and need not be stated here). The same approach can be applied to the more general sufficient requirement 𝑍 π‘Ÿ 𝑀 𝑗 = 0   ( 𝑗 = 1 , … , π‘š ; fixed π‘Ÿ ) ensuring 𝑍 ∈ 𝔾 . If π‘Ÿ β‰₯ 𝑛 , the requirement is equivalent to the inclusion 𝑍 ∈ 𝔾 .

Example 9.4. Let us consider vector field (5.6) where 𝑧 1 = β‹― = 𝑧 π‘š = 0 . In more detail, we take  𝑧 𝑍 = 𝑖 πœ• πœ• π‘₯ 𝑖 +  𝑧 𝑗 𝑖 πœ• πœ• 𝑀 𝑗 𝑖 ξ‚€ 𝑧 + β‹― 𝑗 𝑖  𝑀 = βˆ’ 𝑗 𝑖 β€² 𝐷 𝑖 𝑧 𝑖 β€²  . ( 9 . 1 0 ) Then 𝑍 π‘Ÿ 𝑀 𝑗 = 0 and we have to deal with functions 𝑍 π‘Ÿ π‘₯ 𝑖 in order to ensure the inclusion 𝑍 ∈ 𝔾 . This is a difficult task. Let us therefore suppose 𝑧 1 ξ‚€ = 𝑧 … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€² , 𝑀 𝑗 β€² 1  , … , 𝑧 π‘˜ = 𝑐 π‘˜ ∈ ℝ ( π‘˜ = 2 , … , 𝑛 ) . ( 9 . 1 1 ) Then 𝑍 π‘₯ π‘˜ = 0   ( π‘˜ = 2 , … , 𝑛 ) and 𝑍 2 π‘₯ 1  = 𝑍 𝑧 = πœ• 𝑧 πœ• π‘₯ 𝑖 𝑧 𝑖 +  πœ• 𝑧 πœ• 𝑀 𝑗 1 𝑧 𝑗 1 , ( 9 . 1 2 ) where 𝑧 𝑗 1 = βˆ’ 𝑀 𝑗 1 𝐷 1 𝑧 = βˆ’ 𝑀 𝑗 1  πœ• 𝑧 πœ• π‘₯ 1 +  πœ• 𝑧 πœ• 𝑀 𝑗 β€² 𝑀 𝑗 β€² 1 +  πœ• 𝑧 πœ• 𝑀 𝑗 β€² 1 𝑀 𝑗 β€² 1 1 ξƒͺ . ( 9 . 1 3 ) The second-order summand 𝑍 2 π‘₯ 1  = β‹― + πœ• 𝑧 πœ• 𝑀 𝑗 1 𝑧 𝑗 1  = β‹― βˆ’ πœ• 𝑧 πœ• 𝑀 𝑗 1 𝑀 𝑗 1 πœ• 𝑧 πœ• 𝑀 𝑗 β€² 1 𝑀 𝑗 β€² 1 1 ( 9 . 1 4 ) identically vanishes for the choice ξ‚€ 𝑧 = 𝑓 … , π‘₯ 𝑖 β€² , 𝑀 𝑗 β€² , 𝑒 𝑙   𝑒 , … 𝑙 = 𝑀 𝑙 1 𝑀 1 1 ξƒͺ ; 𝑙 = 2 , … , π‘š ( 9 . 1 5 ) as follows by direct verification. Quite analogously 𝑍 𝑒 𝑙 𝑀 = 𝑍 𝑙 1 𝑀 1 1 = 𝑧 𝑙 1 1 𝑀 1 1 βˆ’ 𝑧 1 1 𝑀 𝑙 1 ξ€· 𝑀 1 1 ξ€Έ 2 =  βˆ’ 𝑀 𝑙 1 1 𝑀 1 1 + 𝑀 1 1 𝑀 𝑙 1 ξ€· 𝑀 1 1 ξ€Έ 2 ξƒͺ 𝐷 1 𝑧 = 0 . ( 9 . 1 6 ) It follows that all functions 𝑍 π‘Ÿ π‘₯ 𝑖 , 𝑍 π‘Ÿ 𝑀 𝑗 can be expressed in terms of the finite family of functions π‘₯ 𝑖 ( 𝑖 = 1 , … , 𝑛 ) , 𝑀 𝑗 ( 𝑗 = 1 , … , π‘š ) , 𝑒 𝑙 ( 𝑙 = 2 , … , π‘š ) 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 d 𝐺 𝑖 d πœ† = 0 , d 𝐺 𝑗 =  π‘Ž d πœ† 𝑗 𝑗 β€² 𝑖 β€² 𝐺 𝑗 β€² 𝑖 β€² ( 𝑖 = 1 , … , 𝑛 ; 𝑗 = 1 , … , π‘š ) , ( 9 . 1 7 ) where we omit the prolongations. It is resolved by 𝐺 𝑖 = π‘₯ 𝑖 , 𝐺 𝑗 = 𝑀 𝑗  π‘Ž + πœ† 𝑗 𝑗 β€² 𝑖 β€² 𝑀 𝑗 β€² 𝑖 β€² ( 𝑖 = 1 , … , 𝑛 ; 𝑗 = 1 , … , π‘š ) ( 9 . 1 8 ) as follows either by direct verification or, alternatively, from the property 𝑍 2 π‘₯ 𝑖 = 𝑍 𝑧 𝑖 = 0   ( 𝑖 = 1 , … , 𝑛 ) which implies d βˆ‘ π‘Ž 𝑗 𝑗 β€² 𝑖 β€² 𝐺 𝑗 β€² 𝑖 β€²  π‘Ž d πœ† = 0 , 𝑗 𝑗 β€² 𝑖 β€² 𝐺 𝑗 β€² 𝑖 β€² =  π‘Ž 𝑗 𝑗 β€² 𝑖 β€² 𝐺 𝑗 β€² 𝑖 β€² | | | πœ† = 0 =  π‘Ž 𝑗 𝑗 β€² 𝑖 β€² 𝑀 𝑗 β€² 𝑖 β€² . ( 9 . 1 9 ) Quite analogously, the Lie system of the vector field (9.10), (9.11), (9.15) reads d 𝐺 1  d πœ† = 𝑓 … , 𝐺 𝑖 β€² , 𝐺 𝑗 β€² , 𝐺 𝑙 β€² 1 𝐺 1 1 ξƒͺ , , … d 𝐺 π‘˜ d πœ† = 𝑐 π‘˜ , d 𝐺 𝑗 d πœ† = 0 ( π‘˜ = 2 , … , 𝑛 ; 𝑗 = 1 , … , π‘š ) ( 9 . 2 0 ) and may be completed with the equations d ξ€· 𝐺 𝑙 1 / 𝐺 1 1 ξ€Έ d πœ† = 0 ( 𝑙 = 2 , … , π‘š ) ( 9 . 2 1 ) 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 ∈ β„± , 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 𝑍 ∈ 𝔾 .

Morphisms 𝐦 of the jet structure were defined by the property 𝐦 βˆ— Ξ© ( π‘š , 𝑛 ) βŠ‚ Ξ© ( π‘š , 𝑛 ) . The inverse 𝐦 βˆ’ 1 exists if and only if Ξ© 0 βŠ‚ 𝐦 βˆ— Ξ© ( π‘š , 𝑛 ) , e q u i v a l e n t l y Ξ© 0 βŠ‚ 𝐦 βˆ— Ξ© 𝑙 ( 𝑙 = 𝑙 ( 𝐦 ) ) ( 1 0 . 1 ) for appropriate term Ξ© 𝑙 ( 𝐦 ) of filtration (9.1). However 𝐦 βˆ— Ξ© 𝑙 + 1 = 𝐦 βˆ— Ξ© 𝑙 +  β„’ 𝐷 𝑖 𝐦 βˆ— Ξ© 𝑙 ( 1 0 . 2 ) and it follows that criterion (10.1) can be verified by repeated use of operators β„’ 𝐷 𝑖 . In more detail, we start with equations 𝐦 βˆ— πœ” 𝑗 =  π‘Ž 𝑗 𝑗 β€² 𝐼 β€² πœ” 𝑗 β€² 𝐼 β€² ξ‚€ = d 𝐦 βˆ— 𝑀 𝑗 βˆ’  𝐦 βˆ— 𝑀 𝑗 𝑖 d 𝐦 βˆ— π‘₯ 𝑖  ( 1 0 . 3 ) with uncertain coefficients. Formulae (10.3) determine the module 𝐦 βˆ— Ξ© 0 . Then we search for lower-order contact forms, especially forms from Ξ© 0 , lying in 𝐦