# Automorphisms of Ordinary Differential Equations

Authors: Václav Tryhuk; Veronika Chrastinová
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The paper deals with the local theory of internal symmetries of underdetermined
systems of ordinary differential equations in full generality. The symmetries need not
preserve the choice of the independent variable, the hierarchy of dependent variables,
and the order of derivatives. Internal approach to the symmetries of one-dimensional
constrained variational integrals is moreover proposed without the use of multipliers.


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## Body

## 1. Preface

The theory of symmetries ofdetermined systems (the solution depends on constants) of ordinary differential equations was ultimately established in Lie’s and Cartan’s era in the most possible generality and the technical tools (infinitesimal transformations and moving frames) are well known. Recall that the calculations are performed in finite-dimensional spaces given in advance and the results are expressed in terms of Lie groups or Lie-Cartan pseudogroups.We deal withunderdetermined systems (more unknown functions than the number of equations) of ordinary differential equations here. Then the symmetry problem is rather involved. Even the system of three first-order quasilinear equations with four unknown functions (equivalently, three Pfaffian equations with five variables) treated in the famous Cartan’s article [1] and repeatedly referred to in actual literature was not yet clearly explained in detail. Paradoxically, the common tools (the calculations in given finite-order jet space) are quite sufficient for this particular example. We will later see that they are insufficient to analyze the seemingly easier symmetry problem of one first-order equation with three unknown functions (alternatively, two Pfaffian equations with five variables) in full generality since the order of derivatives need not be preserved in this case and the finite-order jet spaces may be destroyed. Recall that even the higher-order symmetries (automorphisms) of empty systems of differential equations (i.e., of the infinite order jet spaces without any additional differential constraints) are nontrivial [2–4] and cannot be included into the classical Lie-Cartan theory of transformation groups. Such symmetries need not preserve any finite-dimensional space and therefore the invariant differential forms (the Maurer-Cartan forms, the moving coframes) need not exist.Let us outline the very core of the subject for better clarity by using the common jet terminology. We start with thehigher-order transformations of curves wi=wi(x)(i=1,…,m) lying in the space ℝm+1 with coordinates x,w1,…,wm. The transformations are defined by certain formulae
(1)x-=W(x,…,wsj,…),w-i=Wi(x,…,wsj,…)(i=1,…,m),
where the C∞-smooth real-valued functions W,Wi depend on a finite number of the familiar jet variables
(2)wsj=dswjdxs(j=1,…,m;s=0,1,…).
The resulting curve w-i=w-i(x-)(i=1,…,m) again lying in ℝm+1 appears as follows. We put
(3)x-=W(x,…,dswj(x)dxs,…)=w(x)
and assuming
(4)w′(x)=(DW)(x,…,dswj(x)dxs,…)≠0(D=∂∂x+∑‍ws+1j∂∂wsj),
there exists the inverse function x=w-1(x-) which provides the desired result
(5)w-i(x-)=Wi(w-1(x-),…,dswjdxs(w-1(x-)),…).
One can also easily obtain the well-known prolongation formula(6)w-si=Wsi(x,…,wsj,…)(Ws+1i=DWsiDW;i=1,…,m;s=0,1,…;W0i=Wi)
for the derivatives w-ri=drw-i/dx-r by using the Pfaffian equations
(7)dw-ri-w-r+1idx-=0(i=1,…,m;r=0,1,…).
Functions W satisfying (4) and Wi may be arbitrary here.At this place, in order to obtain coherent theory, introduction of the familiarinfinite-order jet space ofx-parametrized curves briefly designated as M(m) with coordinates x,wri(i=1,…,m;r=0,1,…) is necessary. Then formulae ((1), (6)) determine a mapping m:M(m)→M(m), a morphism of the jet space M(m). If the inverse m-1 given by certain formulae
(8)x=W-(x-,…,w-sj,…),wri=W-ri(x-,…,w-sj,…)(i=1,…,m;r=0,1,…)
exists, we speak of an automorphism (in alternative common terms, symmetry) m of the jet space M(m). It should be noted that we tacitly deal with the local theory in the sense that all formulae and identities, all mappings, and transformation groups to follow are in fact considered only on certain open subsets of the relevant underlying spaces which is not formally declared by the notation. Expressively saying, in order to avoid the clumsy purism, we follow the reasonable 19th century practice and do not rigorously indicate the true definition domains.After this preparation, asystem of differential equations is traditionally identified with the subspace M⊂M(m) given by certain equations
(9)DrGk=0(k=1,…K;r=0,1,…;Gk=Gk(x,…,wsj,…)).
(We tacitly suppose that M⊂M(m) is a “reasonable subspace” and omit the technical details.) This is the infinitely prolonged system. The total derivative vector field D defined on M(m) is tangent to the subspace M⊂M(m) and may be regarded as a vector field on M, as well. The morphism m:M(m)→M(m) transforms M⊂M(m) into the subspace mM⊂mM(m)⊂M(m) given by the equations
(10)(DrGk)(W,…,Wsj,…)=0(k=1,…,K;r=0,1,…).
This is again a system of differential equations. In our paper, we are interested only in the particular case when mM=M. Then, if the inverse m-1 locally exists on a neighbourhood of the subspace M⊂M(m) in the total jet space, we speak of the external symmetry  m of the system of differential equations (9). Let us, however, deal with the natural restriction m-:M→M of the mapping m to the subspace M. If there exists the inverse m--1:M→M of the restriction, we speak of the internal symmetry. Internal symmetries do not depend on the localizations of M in M(m). More precisely, differential equations can be introduced without any reference to jet spaces and the internal symmetries can be defined without the use of localizations. On this occasion, we are also interested in groups of internal symmetries. They are generated by special vector fields, the infinitesimal symmetries.In the actual literature, differential equations are as a rule considered in finite-dimensional jet spaces. Then the internal and external symmetries become rather delicate and differ from our concepts since the higher-order symmetries are not taken into account. We will not discuss such conceptual confusion in this paper with the belief that the following two remarks (and Remark5) should be quite sufficient in this respect.Remark 1 (on the symmetries).
The true structure of the jet spaceM(m) is determined by the contact module  Ω(m) which involves all contact forms(11)ω=∑‍ariωri(ωri=dwri-wr+1idx,finitesum,arbitrarycoefficients).
Then the above morphisms  m:M(m)→M(m) given in ((1), (6)) are characterized by the property m*Ω(m)⊂Ω(m). Recall that invertible morphisms are automorphisms. Let us introduce the subspace i:M⊂M(m) of all points (9). This M is equipped with the restriction Ω=i*Ω(m) of the contact module. Recall that we are interested only in the case mM=M (abbreviation of miM=iM). Let m-:M→M be the restriction of m. If m is a morphism then m- is a morphism in the sense that m-*Ω⊂Ω. Recall that we have the internal symmetry, if m- is moreover invertible. If also m is invertible, we have the external symmetry  m-. The internal symmetries can be defined without any reference to m and M(m) as follows. Let m-:M→M be any invertible mapping such that m-*Ω⊂Ω. This m- can be always extended to a morphism  m:M(m)→M(m) of the ambient jet space. (Hint, recurrence (6) holds true both in M(m) and in M.) So we may conclude that such m- is just the internal symmetry. Moreover, if there exists invertible extension m of m-, then m- is even the external symmetry but the latter concept already depends on the localization i of M in M(m).Remark 2 (on infinitesimal symmetries).
Let us consider a vector field(12)Z=z∂∂x+∑‍zri∂∂wri(infinitesum,arbitrarycoefficients)
on the jet space M(m). Let us moreover suppose ℒZΩ(m)⊂Ω(m) from now on (where ℒZ denotes the Lie derivative see also Definition 8). In common terminology, such vector fields Z are called generalized (higher-order, Lie-Bäcklund) infinitesimal symmetries of the jet space M(m). However Z need not in general generate any true group of transformations and we therefore prefer the “unorthodox” term a variationZ here. (See Section 7 and especially Remark 35 where the reasons for this term are clarified.) The common term infinitesimal symmetry is retained only for the favourable case when Z generates a local one-parameter Lie group [5]. Let us consider the above subspace i:M⊂M(m). If Z is tangent to M, then there exists the natural restriction Z- of Z to M. Clearly ℒZ-Ω⊂Ω and we speak of the (internal) variation  Z-. If Z- moreover generates a group in M, we have the (internal) infinitesimal symmetry  Z-. The internal concepts on M can be easily introduced without any reference to the ambient space M(m). This is not the case for the concept of the external infinitesimal symmetry  Z- which supposes that appropriate extension  Z of Z- on the ambient space M(m)  generates a Lie group.We deal only with theinternal symmetries and infinitesimal symmetries in this paper. It is to be noted once more that infinite-dimensional underlying spaces are necessary if we wish to obtain a coherent theory. The common technical tools invented in the finite-dimensional spaces will be only slightly adapted; alas, the ingenious methods proposed, for example, in [6–8] seem to be not suitable for this aim and so we undertake the elementary approach [9] here.


## 2. Technical Tools

We introduce infinite-dimensional manifoldM modelled on the space ℝ∞ with local coordinates h1,h2,… in full accordance with [9]. The manifold M is equipped with the structural algebra  ℱ(M) of C∞-smooth functions expressed as f=f(h1,…,hm(f)) in terms of coordinates. Transformations (mappings) m:M→M are (locally) given by certain formulae
(13)m*hi=Hi(h1,…,hm(i))(Hi∈ℱ(M);i=1,2,…),
and analogous (invertible) formulae describe the change of coordinates at the overlapping coordinate systems.LetΦ(M) be the ℱ(M)-module of differential 1-forms
(14)φ=∑‍fidgi(fi,gi∈ℱ(M);finitesum).
The familiar rules of exterior calculus can be applied without any change, in particular m*φ=∑‍m*fidm*gi for the above transformation m.Let𝒯(M) be the ℱ(M)-module of vector fields Z. In terms of coordinates we have
(15)Z=∑‍zi∂∂hi(zi=zi(h1,…,hm(i))∈ℱ(M),infinitesum),
where the coefficients zi may be quite arbitrary. We identify Z with the linear functional on Φ(M) determined by the familiar duality pairing
(16)dhi(Z)=Z⌋dhi=Zhi=zi(i=1,2,…).
With this principle in mind, if certain forms φ1,φ2,…∈Φ(M) generate the ℱ(M)-module, then the values
(17)φi(Z)=Z⌋φi=z-i∈ℱ(M)(i=1,2,…)
uniquely determine the vector field Z and (17) can be very expressively (and unorthodoxly) recorded by
(18)Z=∑‍z-i∂∂φi(z-i=φi(Z)∈ℱ(M),infinitesum).
This is a mere symbolical record, not the true infinite series. However, if φ1,φ2,… is a basis of the module Φ(M) in the sense that every φ∈Φ(M) admits a unique representation φ=∑‍fiφi(fi∈ℱ(M), finite sum) then the coefficients z-i can be quite arbitrary and (18) may be regarded as a true infinite series. The arising vector fields ∂/∂φ1,∂/∂φ2,… provide a weak basis (infinite expansions, see [9]) of 𝒯(M) dual to the basis φ1,φ1,… of Φ(M). In this transcription, (15) is alternatively expressed as
(19)Z=∑‍zi∂∂dhi(zi∈ℱ(M),infinitesum).
We recall the Lie derivative  ℒZ=Z⌋d+dZ⌋ acting on exterior differential forms. The image m*Z of a vector field defined by the property
(20)m*(m*Z)f=Zm*f(f∈ℱ(M))
need not exist. It is defined if m is invertible.We consider various submodulesΩ⊂Φ(M) of differential forms together with the relevant orthogonal submodules Ω⊥⊂𝒯(M) consisting of all vector fields Z∈𝒯(M) such that ω(Z)=0(ω∈Ω). The existence of (local) ℱ(M)-bases in all submodules of Φ(M) to appear in our reasonings is tacitly postulated. Dimension of an ℱ(M)-module is the number of elements of an ℱ(M)-basis. Omitting some “exceptional points,” it may be confused with the dimension of the corresponding ℝ-module (the localization) at a fixed place P∈M. On this occasion, it should be noted that the image
(21)m*ZP((m*ZP)Qf=ZP(m*f),mP=Q,P∈M)
of a tangent vector ZP at P exists as a vector at the place Q.Let us also remark with regret that any rigorous exposition of classical analysis in the infinite-dimensional spaceℝ∞ is not yet available; however, certain adjustments of finite-dimensional results are not difficult. For instance, the following invertibility theorem will latently occur in the proof of Theorem 20.Theorem 3.
A mappingm:M→M is invertible if and only if any of the following equivalent conditions is satisfied: the pull-back m*:ℱ(M)→ℱ(M) is invertible, the pull-back m*:Φ(M)→Φ(M) is invertible, and if φ1,φ2,… is a (fixed, equivalently: arbitrary) basis of module Φ(M), then m*φ1,m*φ2,… again is a basis.Hint. A nonlinear version of the familiar Gauss elimination procedure for infinite dimension [9] provides a direct proof with difficulties concerning the definition domain of the resulting inverse mapping. Nevertheless if m is moreover a morphism of a diffiety (see Definition 8) then the prolongation procedure ensures the local existence of m-1 in the common sense.


## 3. Fundamental Concepts

We introduce a somewhat unusual intrinsical approach to underdetermined systems of ordinary differential equations in terms of the above underlying spaceM, a submodule Ω⊂Φ(M) of differential 1-forms, and its orthogonal submodule ℋ=Ω⊥⊂𝒯(M) of vector fields.Definition 4.
A codimension one submoduleΩ⊂Φ(M) is called a diffiety if there exists a good filtration(22)Ω*:Ω0⊂Ω1⊂⋯⊂Ω=∪Ωl
by finite-dimensional submodules Ωl⊂Ω(l=0,1,…) such that
(23)ℒℋΩl⊂Ωl+1(alll),ℒℋΩl+Ωl=Ωl+1(llargeenough).
To every subset Θ⊂Φ(M), let ℒℋΘ⊂Φ(M) denote the submodule with generators ℒZϑ(Z∈ℋ,ϑ∈Θ). Since Θ⊂ℒℋΘ (easy), the second requirement (23) can be a little formally simplified as ℒℋΩl=Ωl+1.Remark 5.
This is a global coordinate-free definition; however, we again deal only with the local theory from now on in the sense that the definition domains (of filtrations (22), of independent variable x to follow, and so on) are not specified. It should be noted on this occasion that the common geometrical approach [6–8] to differential equations rests on the use of the rigid structure of finite-order jets. Many classical concepts then become incorrect, if the higher-order mappings are allowed but we cannot adequately discuss this important topic here. Rather subtle difficulties are also passed over already in the common approach to the fundamental jet theory. For instance, smooth curves in the plane ℝ2 with coordinates x,y are parametrized either by x (i.e., y=y(x)) or by coordinate y (i.e., x=x(y)) in the common so-called “geometrical” approach [6–8]. However, then already the Lie’s classical achievements concerning contact transformations [10, 11] with curves parametrized either by p=dy/dx or by q=dx/dy cannot be involved. Quite analogously, the “higher-order” parameterizations and mappings [2–5] are in fact rejected in the common “rigid” jet theory with a mere point symmetries.Definition 6.
Let a differentialdx(x∈ℱ(M)) generate together with Ω the total module Φ(M) of all differential 1-forms. Then x is called the independent variable to diffiety Ω. The vector field D=Dx (abbreviation) such that
(24)D∈ℋ,Dx=dx(D)=1(symbolicallyD=∂∂dx+∑ω∈Ω‍0·∂∂ω)
is called total (or formal) derivative of Ω with respect to the independent variable x. This vector field D is a basis of the one-dimensional module ℋ=Ω⊥ for every fixed particular choice of the independent variable x.Remark 7.
Let us state some simple properties of diffieties. The proofs are quite easy and may be omitted. A formφ∈Φ(M) is lying in Ω if and only if φ(D)=0. In particular ℒDΩ⊂Ω in accordance with the identities
(25)ℒDω=D⌋dω+dω(D)=D⌋dω,(ℒDω)(D)=dω(D,D)=0(ω∈Ω).
(This trivial property clarifies the more restrictive condition (23).) Moreover clearly
(26)Dfdg-Dgdf,df-Dfdx∈Ω(f,g∈ℱ(M))
and in particular
(27)Dhidhj-Dhjdhi,dhi-Dhidx∈Ω(i,j=1,2,…)
for all coordinates. We have very useful ℱ(M)-generators of diffiety Ω. The independent variable and the filtrations (22) can be capriciously modified. In particular the c-lift [9]
(28)Ω*+c=Ω~*:Ω~0⊂Ω~1⊂⋯⊂Ω=∪Ω~l(Ω~l=Ωl+c,c=0,1,…)
with c large enough ensures that Ω~l+1=ℒℋΩ~l+Ω~l for all l≥0. We will be, however, interested just in the reverse concept “Ω*-c” latently involved in the “standard adaptation” of filtrations to appear later on.Definition 8.
A transformationm:M→M is called a morphism of the diffiety Ω if m*Ω⊂Ω. Invertible morphisms are automorphisms (or symmetries) of Ω. A vector field Z∈𝒯(M) satisfying ℒZΩ⊂Ω is called the variation of Ω. If moreover Z (locally) generates a one-parameter group of transformations, we speak of the infinitesimal symmetry  Z of diffiety Ω.Remark 9.
Let us mention the transformation groups in more detail. A local one-parameter group of transformationsm(λ):M→M is given by certain formulae
(29)m(λ)*hi=Hi(h1,…,hm(i);λ)(i=1,2,…;-ε<λ<ε)
in terms of local coordinates, where m(λ+μ)=m(λ)m(μ),m(0)=id is supposed. Then the special vector field (15) defined by
(30)zi=∂∂λm(λ)*hi|λ=0=∂∂λHi(h1,…,hm(i);0)(i=1,2,…)
is called the infinitesimal transformation of the group (29). In the opposite direction, we recall that a general vector field (15) generates the local group (29) if and only if the Lie system(31)∂Hi∂λ=zi(H1,…,Hn(i)),Hi(h1,…,hm(i);0)=hi(i=1,2,…)
is satisfied. Alas, a given vector field (19) need not in general generate any transformation group since the Lie system need not admit any solution (29).With all fundamental concepts available, let us eventually recall the familiar and thoroughly discussed in [9] interrelation between the diffieties and the corresponding classical concept of differential equations for the convenience of reader. In brief terms, the idea is quite simple. A given system of differential equations is represented by a system of Pfaffian equations ω=0 and the module Ω generated by such 1-forms ω is just the diffiety. More precisely, we deal with the infinite prolongations as follows.In one direction, let a system of underdetermined ordinary differential equations be given. We may deal with the first-order system(32)dwjdx=fj(x,w1,…,wm,dwJ+1dx,…,dwmdx)(j=1,…,J)
without any true loss of generality. Then (32) completed with
(33)dwskdx=ws+1k(w0k=wk;k=J+1,…,m;s=0,1,…)
provides the infinite prolongation. The corresponding diffiety Ω is generated by the forms
(34)dwj-fjdx,dwsk-ws+1kdx(j=1,…,J;k=J+1,…,m;s=0,1,…)
in the space M with coordinates
(35)wj(j=1,…,J),wsk(k=J+1,…,m;s=0,1,…;w0k=wk).
Clearly
(36)Dx=∂∂x+∑‍fj∂∂wj+∑‍ws+1k∂∂wsk∈ℋ
is the total derivative and the submodules Ωl⊂Ω of all forms (34) with s≤l determine a quite simple filtration (22) with respect to the order of contact forms. (Hint: use the formulae
(37)ℒD(dwj-fjdx)=∑‍∂fj∂wj(dwj-fjdx)+∑‍∂fj∂w1j(dw1j-w2jdx)
and ℒD(dwsj-ws+1jdx)=dws+1j-ws+2jdx.) However, there exist many other and more useful filtrations; see the examples to follow later on.The particular caseJ=0 of the empty system (32) can be easily related to the case of the jet space M(m) of all x-parametrized curves in ℝm+1 of the Section 1. The relevant diffiety is identified with the module Ω(m) of all contact forms (11), of course.In the reverse direction, let a diffietyΩ be given on the space M. In accordance with (27), the forms dhi-Dhidx(i=1,2,…) generate Ω. So we have the Pfaffian system dhi-Dhidx=0(i=1,2,…) and therefore the system of differential equations
(38)dhidx=gi(x,h1,…,hm(i))(i=1,2,…;gi=Dhi)
of rather unpleasant kind. Then, due to the existence of a filtration (22) and (23), one can obtain also the above classical system of differential equations (32) together with the prolongation (33) by means of appropriate change of coordinates [9]. This is, however, a lengthy procedure and a shorter approach can be described as follows. Let the second requirement (23) be satisfied, if l≥L. Suppose that the forms ωj=∑‍aijdhi(j=1,…,J=dim⁡ΩL) generate module ΩL. Then all forms
(39)ℒDkωj(j=1,…,J;k=0,1,…)
generate the diffiety Ω. The corresponding Pfaffian system ℒDkωj=0 is equivalent to certain infinite prolongation of differential equations, namely,
(40)ωj=∑‍aijdhi=0isequivalentto∑‍aijdhidx=0,ℒDωj=∑‍Daijdhi+∑‍aijdDhi=0isequivalenttoddx∑‍aijdhidx=0
(direct verification), and in general
(41)ℒDkωj=∑‍Dkaijdhi+⋯+∑‍aijdDkhi=0isequivalenttodkdxk∑‍aijdhidx=0.
We have the infinite prolongation of the classical system ∑‍aijdhi/dx=0(j=1,…,J) and this is just the system that corresponds to diffiety Ω.Altogether taken, differential equations uniquely determine the corresponding diffieties; however, a given diffiety leads to many rather dissimilar but equivalent systems of differential equations with regard to the additional choice of dependent and independent variables.Remark 10.
Definitions4–8 make good sense even if M is a finite-dimensional manifold and then provide the well-known intrinsical approach to determined systems of differential equations. They are identified with vector fields (better, fields of directions) in the finite-dimensional space M. Choosing a certain independent variable x, the equations are represented by the vector field Dx or, more visually, by the corresponding Dx-flow. The general theory becomes trivial; we may, for example, choose Ωl=Ω for all l in filtration (22).


## 4. On the Structure of Diffieties

Definition 11.
To every submoduleΘ⊂Ω of a diffiety Ω⊂Φ(M), let Ker⁡Θ⊂Θ be the submodule of all ϑ∈Θ such that ℒℋϑ∈Θ. Filtration (22) and (23) is called a standard one, if
(42)Ker⁡Ωl+1=Ωl(l≥0),Ker⁡2Ω0=Ker⁡Ω0≠Ω0.
For every ω∈Ω, the first condition ensures that the inclusions ω∈Ωl,ℒDω∈Ωl+1 are equivalent and the second condition ensures that ℒDω∈Ω0 implies ℒD2ω∈Ω0.Theorem 12.
Appropriate adaptation of some lower-order terms of a given filtration (22) and (23) provides a standard filtration in a unique manner [9]. Equivalently and in more detail, there exists unique standard filtration Ω-*:Ω-0⊂Ω-1⊂⋯⊂Ω=∪Ω-l such that Ωl=Ω-l+c for appropriate c∈ℕ and all l large enough. Equivalently and briefly, there exists unique standard filtration Ω-* such that Ω*+c′=Ω-*+c′′ for appropriate c′,c′′∈ℕ.Proof.
The mappingℒD:Ωl→Ωl+1 naturally induces certain ℱ(M)-homomorphism
(43)D:Ωl/Ωl-1⟶Ωl+1/Ωl(l≥0,formallyΩ-1=0)
of factor modules denoted by D for better clarity. Homomorphisms D are surjective and therefore even bijective for all l large enough, say for l≥L. However, the injectivity of D implies Ker⁡Ωl=Ωl-1(l≥L). It follows that we have strongly decreasing sequence
(44)⋯⊃ΩL(=Ker⁡ΩL+1)⊃ΩL-1(=Ker⁡ΩL)⊃Ker⁡ΩL-1⊃Ker⁡2ΩL-1⊃⋯,
which necessarily terminates with the stationarity Ker⁡CΩL-1=Ker⁡C+1ΩL-1. Denoting
(45)Ω-0=Ker⁡C-1ΩL-1,…,Ω-C-1=Ker⁡ΩL-1,Ω-C=ΩL-1,Ω-C+1=ΩL,…,
we have the sought strongly increasing standard filtration
(46)Ω-*:Ω-0⊂Ω-1⊂⋯⊂Ω-C(=ΩL-1)⊂Ω-C+1(=ΩL)⊂⋯⊂Ω=∪Ω-l
of diffiety Ω. In particular Ker⁡2Ω-0=Ker⁡C+1ΩL-1=Ker⁡CΩL-1=Ker⁡Ω-0.Proof of Theorem12 was of the algorithmical nature and provides a useful standard basis of diffiety Ω as follows. Assume that the forms
(47)τ1,…,τK∈Ω-0provideabasisofthesubmoduleKer⁡Ω-0⊂Ω-0
(recall that Ker⁡2Ω-0=Ker⁡Ω-0 whence ℒDKer⁡Ω-0⊂Ker⁡Ω-0) and moreover the classes of forms
(48)π1,…,πj0∈Ω-0provideabasisofΩ-0/Ker⁡Ω-0
(recall that D:Ω-0/Ker⁡Ω-0→Ω-1/Ω-0 is injective mapping), the classes of forms
(49)πj0+1,…,πj1∈Ω-1provideabasisofΩ-1/(Ω-0+ℒℋΩ-0)
(recall that D:Ω-1/Ω-0→Ω-2/Ω-1 is injective mapping), and in general the classes of forms
(50)πjl-1+1,…,πjl∈Ω-lprovideabasisofΩ-l/(Ω-l-1+ℒℋΩ-l-1).
Alternatively saying, the following forms constitute a basis:
(51)τ1,…,τKofKer⁡Ω-0,togetherwithπ1,…,πj0ofΩ-0,togetherwithℒDπ1,…,ℒDπj0,πj0+1,…,πj1ofΩ-1,togetherwithℒD2π1,…,ℒD2πj0,ℒDπj0+1,…,ℒDπj1,πj1+1,…,πj2ofΩ-2,
and so on. Let us denote
(52)πrj=ℒDrπj(j=jl-1+1,…,jl).
In terms of this notation
(53)τ1,…,τKisabasisofKer⁡Ω-0andtogetherwiththeformsτ1,…,τK,π01,…,π0j0,π11,…,π1j0,…,πl1,…,πlj0,π0j0+1,…,π0j1,…,πl-1j0+1,…,πl-1j1…π0jl-1+1,…,π0jl,
we have the standard basis of Ω-l.ClearlyjL=jL+1=⋯ and it follows that there is only a finite number μ(Ω)=jL of initial forms(54)π1=π01,…,πj0=π0j0,πj0+1=π0j0+1,…,πjL-1+1=π0jL-1+1,…,πjL=π0jL
with the lower zero indice. The following forms πrj(r>0) satisfy the recurrence and the (equivalent) congruence
(55)ℒDπrj=πr+1j,dπrj≅dx∧πr+1j(mod⁡Ω∧Ω).
In this sense, the linearly independent forms  πrj  are generalizations of the classical contact forms  ωrj=dwrj-wr+1jdx  of the jet theory.Theorem 13.
LetΩ-* be a standard filtration of diffiety Ω. Then the submodule Ker⁡Ω-0⊂Ω is generated by all differentials df∈Ω.Proof.
First assumedf∈Ω. Then Df=df(D)=0 whence ℒDdf=dDf=0. Clearly df∈Ω-l for appropriate l. This implies df∈Ker⁡Ω-l-1, if l≥0 therefore df∈Ker⁡Ω-0. It follows that Ker⁡Ω-0 contains all differentials df∈Ω.
Conversely letτ∈Ker⁡Ω-0. Due to the equality Ker⁡Ω-0=Ker⁡2Ω0, we have ℒDτ∈Ker⁡Ω-0 whence dτ≅dx∧ℒDτ(mod⁡Ω∧Ω), consequently
(56)dτ≅∑‍arsjiπrj∧πsi(mod⁡Ker⁡Ω-0,sumoveri≤j).
It follows that arsji=0 identically by using d(dτ)=0 and (55). (Hint: look at assumed top order product πRj∧πsi where R≥allr. Then d2τ involves only one summand with dx∧πR+1j∧πsi which is impossible since d2=0.) Therefore d(Ker⁡Ω-0)≅0(mod⁡Ker⁡Ω-0) and the Frobenius theorem can be applied. Module Ker⁡Ω-0 has a basis consisting of total differentials.Definition 14.
We may denoteℛ(Ω)=Ker⁡Ω-0 since this module does not depend on the choice of the filtration (22). Together with the original basis τ1,…,τK occurring in (53), there exists alternative basis dt1,…,dtK with differentials. In the particular case ℛ(Ω)=0, hence, K=0, we speak of a controllable diffiety Ω.Remark 15.
The controllability is a familiar concept of the theory of underdetermined ordinary differential equations or Pfaffian systems in finite-dimensional spaces [12]; however, some aspects due to diffieties are worth mentioning here. If ℛ(Ω)≠0 is a nontrivial module, the underlying space M is fibered by the leaves tk=ck∈ℝ(k=1,…,K) depending on K>0 parameters. A curve p:I→M(I⊂ℝ) is called a solution of diffiety Ω, if p*ω=0(ω∈Ω). Since dtk∈ℛ(Ω)⊂Ω, we have
(57)p*dtk=dp*tk=0,p*tk=ck∈ℝ(k=1,…,K),
therefore every solution of diffiety Ω is contained in a certain leaf (the Figure 2(a)).In the controllable case, such foliation of the spaceM does not exist. However, the construction of the standard filtration need not be of the “universal nature.” There may exist some “exceptional points” where the terms πri of the standard basis are not independent. We may even obtain a solution p consisting of such exceptional points and then there appears the “infinitesimal leaf” of the noncontrollability along p which means that p is a Mayer extremal (the Figure 2(b)). We refer to article [13] inspired by the beautiful paper [14]. In the present paper, such exceptional points are tacitly excluded. They produce singularities of the symmetry groups and deserve a special, not yet available approach. It should be noted that the noncontrollable case also causes some technical difficulties. We may however suppose ℛ(Ω)=0 without much loss of generality since the noncontrollable diffiety can be restricted to a leaf and regarded as a diffiety depending on parameters c1,…,cK.Theorem 16.
The total numberμ(Ω) of initial forms does not depend on the choice of the good filtration (22).Proof.
Filtration (22) differs from the standard filtration Ω-* only in lower terms whence
(58)dim⁡Ωl=dim⁡Ω-l+const.=μ(Ω)l+const.(llargeenough).
Let another filtration Ω~*:Ω~0⊂Ω~1⊂⋯⊂Ω=∪Ω~l of diffiety Ω provide (corresponding standard filtration and therefore) certain number μ~(Ω) of (other) initial forms. Then
(59)dim⁡Ωl=dim⁡Ω~l+const.=μ~(Ω)l+const.(llargeenough).
However Ωl⊂Ω~L(l)⊂ΩM(l) for appropriate L(l) and M(l) whence
(60)Ωl+k⊂Ω~L(l)+k⊂ΩM(l)+k(l,klargeenough)
by using (23) and the equality μ(Ω)=μ~(Ω) easily follows.


## 5. On the Morphisms and Variations

A huge literature on thepoint symmetries (scheme (a) of Figure 3, the order of derivatives is preserved) of differential equations is available. On the contrary, we can mention only a few fundamental principles for the generalized (or higher-order) symmetries (scheme (c) of Figure 3) since the general theory deserves quite another paper. Our modest aim is to clarify a little the mechanisms of the particular examples to follow. We will also deal with generalized (or higher-order) groups of symmetries and the relevant generalized infinitesimal symmetries (scheme (b) Figure 3) with ambiguous higher-order invariant subspaces (the dotted lines). Figure 3 should be therefore regarded as a rough description of the topics to follow and we also refer to Section 9 for more transparent details. The main difficulty of the higher-order theory lies in the fact that the dotted domains are not known in advance. Modules Ωl represent the “natural” filtration with respect to the primary order of contact forms in the ambient jet space, see the examples. They depend on the accidental inclusion M⊂M(m) mentioned in Section 1 and do not have any true geometrical sense in the internal approach. It is to be therefore surprisingly observed that the seemingly “exotic” at the first glance concept of higher-order transformations of Section 1 should be regarded for reasonable and the only possible in the coordinate free theory. On the other hand, an important distinction between the group-like morphisms with large number of finite-dimensional invariant subspaces (scheme (a) and (b)) and the genuine order-destroying morphisms without such subspaces (scheme (c)) is of the highest importance.We are passing to rigorous exposition. Let us recall the diffietyΩ⊂Φ(M) on the space M, the independent variable x∈ℱ(M) with the corresponding vector field D=Dx∈Ω⊥=ℋ, the controllability submodule ℛ(Ω)⊂Ω with the basis dt1,…,dtK, and a standard basis πrj(j=1,…,μ(Ω);r=0,1,…) of diffiety Ω.Let us begin with morphisms.Lemma 17.
Ifm:M→M is a morphism of Ω then m*ℛ(Ω)⊂ℛ(Ω) and the recurrence
(61)DWm*πr+1j≅ℒDm*πrj(W=m*x;j=1,…,μ(Ω);r=0,1,…)
modulo ℛ(Ω) holds true.Proof.
Ifm is a morphism then m*Ω⊂Ω therefore m*ℛ(Ω)⊂ℛ(Ω) (use Theorem 13) and m*πrj≅∑‍arsjiπsi(mod⁡ℛ(Ω)). It follows that
(62)m*dπrj≅m*(dx∧πr+1j)≅DWdx∧m*πr+1j,dm*πrj≅d∑‍arsjiπsi≅∑‍Darsjidx∧πsi+∑‍arsjidx∧πs+1i≅dx∧ℒDm*πrj
modulo ℛ(Ω) and Ω∧Ω. This implies (61) by comparing both factors of dx.Remark 18.
On this occasion, the following useful principles of calculation are worth mentioning:(63)ifα,β∈Ωsatisfydα≅dx∧β(mod⁡Ω∧Ω)thenDWm*β=ℒDm*α,(64)ifu,v∈ℱ(M),du-vdx∈ΩthenDWm*v=Dm*u,
and in general
(65)m*Df·Dm*g=m*Dg·Dm*f(f,g∈ℱ(M)).
In terms of notation (21), we conclude that m*DP=DW(P)·DQ and therefore
(66)m*(1DWD)=D(W=m*x),
if the morphism m of diffiety Ω is invertible.Let us turn to invertible morphisms.Lemma 19.
The inverse of a morphismm again is a morphism.Proof.
Assumeω∈Ω,m-1*ω≅fdx(mod⁡Ω). Then
(67)ω=m*m-1*ω≅m*(fdx)=m*f·dW∈Ω,
where dW=dm*x=m*dx≠0. Hence m*f=0,f=0 and therefore m-1*Ω⊂Ω.We havem*Ω⊂Ω if m:M→M is a morphism and moreover m-1*Ω⊂Ω hence Ω*⊂m*Ω in the invertible case. The converse and rather useful assertion is as follows.Theorem 20.
A morphismm of diffiety Ω is invertible if and only if m*Ω=Ω.This may be obtained easily from the following result.Lemma 21.
Letm*ℛ(Ω)=ℛ(Ω) and π0j∈m*Ω(j=1,…,μ(Ω)). Then m is invertible.Proof.
Proof of the Lemma21 is analogous as in [2, Theorem 2] and we briefly recall only the main principles here. It is sufficient to prove the invertibility of m*:Ω→Ω.
Assumingπrj∈m*Ω then πr+1j=ℒDπrj∈m*Ω by virtue of recurrence (61). It follows that Ω⊂m*Ω and m* is surjective. We prove that m*:Ω→Ω  iseven injectivity by using the well-known algebraical interrelation between filtrations and gradations.
Let us introduce filtrationsΩ* (Ω-*, resp.) as follows: the submodule Ωl⊂Ω(Ω-l⊂Ω) is generated by ℛ(Ω) and all forms πrj(m*πrj) where r≤l. We also introduce the gradations
(68)ℳ=⊕ℳl(ℳl=Ωl/Ωl-1),ℳ-=⊕ℳ-l(ℳ-l=Ω-l/Ω-l-1)(l=0,1,…)
(formally Ω-1=Ω--1=0). It follows that the naturally induced mapping m*:ℳ→ℳ is surjective and it is sufficient to prove that this inducedm*is also injective.
We are passing to the most delicate part of the proof. The surjectivity ofm*:Ω→Ω implies that Ω0⊂Ω-L for L large enough. Therefore Ωl⊂Ω-L+l by applying the recursion (61) which implies
(69)dim⁡Ωl=μ(Ω)l+const.≤dim⁡Ω-L+l(μ(Ω)=dim⁡ℳl;l=0,1,…).
On the other hand, assume the noninjectivity therefore the existence of a nontrivial identity
(70)0=∑‍arim*πri=⋯+∑‍aRim*πRi(top-order terms).
Then 0=⋯+(DW)-l∑‍aRim*πR+li(l=0,1,…) by applying operator ℒD and recurrence (61). Due to the existence of such identities, it follows that
(71)dim⁡ℳ-l<dim⁡ℳl=μ(Ω),dim⁡Ω-L+l≤(μ(Ω)-1)l+const.
and this is a contradiction.Remark 22.
Recall that ifm:M→M is a mapping and Ω⊂Φ(M) a submodule, then m*Ω⊂Φ(M) denotes the submodule with generators m*ω(ω∈Ω) in accordance with the common practice in the algebraical module theory. Let in particular Ω be a diffiety and assume ℛ(Ω)=0 for simplicity. Then module m*Ω is generated by all forms m*πrj and therefore by all forms ℒDrm*π0j, see Lemma 17. It follows that the invertibility of the morphism m depends only on the properties of the forms m*π0j, see Lemma 21. In this sense, the invertibility problem is reduced to the finite-dimensional reasonings.We turn to the variations.Lemma 23.
A vector fieldZ∈𝒯(M) is a variation of diffiety Ω if and only if
(72)πr+1j(Z)=Dπrj(Z)(j=1,…,μ(Ω);r=0,1,…)
and all Ztk(k=1,…,K;fixedk) are functions only of variables t1,…,tK.Proof.
We supposeℒZΩ⊂Ω which is equivalent to the congruences
(73)ℒZdtk=dZtk≅DZtkdx=0(mod⁡Ω),ℒZπrj=Z⌋dπrj+dπrj(Z)≅(-πr+1j(Z)+Dπrj(Z))dx=0(mod⁡Ω)
by using ((26) and (55)). So we have obtained (72) and moreover identities DZtk=0(k=1,…,K).
It is sufficient to prove that the latter identities implydZtk=0(mod⁡dt1,…,dtK). However, every differential df(f∈ℱ(M)) can be represented as
(74)df=Dfdx+∑‍fk′dtk′+∑‍frjπrj
in terms of the standard basis. Assuming in particular f=Ztk(fixedk=1,…,K), we have already obtained the equation Df=0 and then identities frj=0 easily follow by applying the common rule d(df)=0 together with (26). This concludes the proof.Theorem 24.
A variationZ of diffiety Ω is infinitesimal symmetry of Ω if and only if all forms ℒZkπ0j(k=0,1,…) are contained in a finite-dimensional module.We omit lengthy proof and refer to more general results [5, Lemma 5.4, Theorem 5.6, and especially Theorem 11.1]. In future examples, we apply other and quite elementary arguments in order to avoid the nontrivial Theorem 24.Remark 25.
It follows from Lemma23 that variations Z of diffiety Ω can be represented by the universal series
(75)Z=∑‍ck∂∂dtk+z∂∂dx+∑‍Drpj∂∂πrj,
where ck=ck(t1,…,tK) are arbitrary composed functions and z=Zx,pj=π0j(Z) are arbitrary functions in ℱ(M). We have explicit formulae for all variations (in common terms, for all Lie-Bäcklund infinitesimal symmetries) of a given system of ordinary differential equations. Recall that these variations Z need not generate any true group, and though the criterion in Theorem 24 is formally simple, it is not easy to be applied. Lemma 17 can be regarded as a counterpart to Lemma 23 since it ensures quite analogous result for the morphism m or, better saying, for the pullback m*:Φ(M)→Φ(M) of a morphism. In more detail, the quite arbitrary choice of the initial terms m*π0j of recurrence (61) is in principle possible but provides a mere formal result (corresponding to the formal nature of variations Z) and does not ensure the existence of true morphism m. We may refer to articles [2, 3] where the formal part (the algebra) is distinguished from the nonformal part (the analysis) in the higher-order algorithms.We conclude this Section with the only gratifying result [9, point (ν) on page 40].Theorem 26.
The standard filtration is unique in the caseμ(Ω)=1.Proof.
Let us take a fixed filtration (22) and the corresponding standard filtration (46). Since μ(Ω)=1, we have only one initial form π01 and therefore τ1,…,τK,π01,…,πl1 is a basis of Ω-l; see (53). Let us take another standard filtration Ω~*. Then the module Ω~0 has certain basis
(76)τ1,…,τK(commonforms),π^01=∑‍arπr1=⋯+aRπR1(aR≠0,top-order  term).
These forms together with all π~s1=ℒDsπ^01=⋯+aRπR+s1(s≥0) generate the module Ω and this is possible only if R=0. We conclude that π~01=aRπR1=a0π01 which implies Ω~0=Ω-0 hence Ω~l=Ω-l for all l.Remark 27.
It follows that in the particular caseμ(Ω)=1, every symmetry and infinitesimal symmetry preserves all terms of the (unique) standard filtration. So we have a large family of finite-dimensional subspaces of the underlying space M which are preserved too. The classical methods acting in finite-dimensional spaces uniquely determined in advance can be applied and are quite sufficient in this case  μ(Ω)=1.Remark 28.
In more generality, one could also consider two diffietiesΩ and Ω~ on the underlying spaces M and M~, respectively. Though we do not deal with the isomorphism problems of two diffieties Ω and Ω~ here, let us mention that such isomorphism is defined as invertible mapping m:M~→M of underlying spaces satisfying m*Ω=Ω~. Quite equivalent “absolute equivalence” problem was introduced in [15] and resolved just for the case μ(Ω)=μ(Ω~)=1 (in our terminology) by using finite-dimensional methods. We have discovered alternative approach here: the isomorphism m identifies the unique standard filtrations of Ω and of Ω~. On this occasion, it is worth mentioning Cartan’s pessimistic notice (rather unusual in his work) to the case μ(Ω)>1: “Je dois ajourter que la géneralization de la théorie de l’equivalence absolu aux systémes differentiels dont la solution générale dépend de deux functions arbitraires d’un argument n’est pas immédiate et souléve d’asses grosses difficultiés.” The same notice can be literally repeated also for the theory of the higher-order symmetries treated in this paper.


## 6. The Order-Preserving Case of Infinitesimal Symmetries

We are passing to the first example which intentionally concerns the well-known “towering” problem in order to examine our method reliably. Let us deal with infinitesimal symmetries of differential equation(77)d2udx2=F(dvdx)
involving two unknown functions u=u(x) and v=v(x). In external theory, (77) is identified with the subspace i:M⊂M(2) defined by the conditions
(78)Dr(u2-F)=ur+2-DrF(v1)=0(r=0,1,…;D=∂∂x+∑‍ur+1∂∂ur+∑‍vr+1∂∂vr)
in the jet space M(2). We use simplified notation of coordinates and contact forms
(79)ur=wr1,vr=wr2,αr=ωr1,βr=ωr2(r=0,1,…)
here. We are, however, interested in internal theory, that is, in the diffiety Ω corresponding to (77). Diffiety Ω appears if the contact forms
(80)αr=dur-ur+1dx,βr=dvr-vr+1dx(r=0,1,…)
are restricted to the subspace i:M⊂M(2). In accordance with the common practice, let us again simplify as
(81)ur=i*ur,vr=i*vr,αr=i*αr,βr=i*βr
the notation of the restrictions to M, and moreover D will be regarded as a vector field on M from now on.Let us outline the lengthy path of future reasonings for the convenience of reader. We begin with preparatory points (ι)–(ιιι). The underlying space M together with the diffiety Ω is introduced and the standard basis π0,π1,…(μ(Ω)=1, abbreviation πr=πr1) of diffiety Ω is determined. The standard basis is related to the “common” basis of Ω by means of formulae (93). We obtain explicit representation (99) for the variations  Z with two arbitrary functions  z=Zx and p=π0(Z) as the final result. Variations Z generating the true group (i.e., the infinitesimal symmetries  Z of Ω) satisfy certain strong conditions discovered in points (ιν) and (ν). The conditions are expressed by the resolving system (107) and (108) or, alternatively, by (112)–(114) only in terms of the functions p, Dp, D2p, and D3p. This rather complicated resolving system which does not provide any clear insight is equivalent to much simpler crucial requirements (121) or (125) on the actual structure of function p; see the central points (νι)–(νιιι). Then the subsequent points are devoted to the explicit solution of these equations (125). This is a mere technical task of traditional mathematical analysis and we omit comments at this place.(ι) The diffiety. Let us introduce space M equipped with coordinates x,u0,u1,vr(r=0,1,…). Then
(82)ur+2=DrF(v1)(r=0,1,…;D=∂∂x+u1∂∂u0+F∂∂u1+∑‍vr+1∂∂vr)
are merely composed functions. The forms
(83)α0=du0-u1dx,α1=du1-Fdx,βr=dvr-vr+1dx(r=0,1,…)
provide a basis of the diffiety Ω; however, all forms αr=dur-ur+1dx(r=2,3,…) are also lying in Ω as follows from the obvious rule:
(84)ℒDαr=αr+1,ℒDβr=βr+1(r=0,1,…)
and the inclusion ℒDΩ⊂Ω.(ιι) Standard Filtration. There exists the “natural” filtration Ω* of diffiety Ω with respect to the order: submodule Ωl⊂Ω involves the forms αr,βr with r≤l. Alternatively saying, α0,β0 is a basis of Ω0 and
(85)α0,β0,α1,β1,β2,…,βlisabasisofΩl(l≥1).
Clearly Ker⁡Ωl+1=Ωl if l≥1 as follows from (84). However,
(86)ℒDα1=ℒD(du1-Fdx)=dF-DFdx=F′(dv1-v2dx)=F′β1∈Ω1.
(Figure 4(a)) therefore
(87)ℒD(α1-F′β0)=F′β1-DF′β0-F′ℒDβ0=-DF′β0∈Ω0.
Then α0,α=α1-F′β1,β0 may be taken for a basis of module Ker⁡Ω1 (Figure 4(b)).Moreover(88)ℒDα0=α1=α+F′β0,ℒDα=-DF′β0∈Ker⁡Ω1,
hence α0,α constitute a basis of module Ker⁡2Ω1 (Figure 4(c)) and finally
(89)ℒD(F′α+DF′α)=DF′α+D2F′α0+F′ℒDα+DF′ℒDα0=2DF′α+D2F′α0∈Ker⁡2Ω1.
Therefore assuming
(90)DF′=F′′v3≠0(henceF′′≠0)
from now on, the form π0=F′α+DF′α0 may be taken for a basis of module Ker⁡3Ω0. We have obtained the standard filtration
(91)Ω-*:Ω-0=Ker⁡3Ω1⊂Ω-1=Ker⁡2Ω1⊂Ω-2=Ker⁡Ω1⊂Ω-3=Ω1⊂⋯(ℛ(Ω)=0),
where forms
(92)πr=ℒDrπ0((F′)2β0r=0,…,l;π0=F′α+DF′α0=DF′α0+F′α1-(F′)2β0)
provide a basis of module Ω-l.Abbreviatingf=F′ from now on, explicit formulae
(93)π0=fα+Dfα0,π1=2Dfα+D2fα0,π2=3D2fα+D3fα0+Cβ0,Cα=D2fπ0-Dfπ1,Cα0=-2Dfπ0+fπ1,C2β0=Aπ0+Bπ1+Cπ2,
where α=α1-fβ0 and
(94)A=2Df·D3f-3(D2f)2,B=3Df·D2f-fD3f,C=fD2f-2(Df)2
can be easily found. They will be sufficient in calculations to follow. Recall that we suppose that the inequality (90) hold true, hence C=⋯+ff′v3=⋯+F′F′′v3≠0.(ιιι) Variations. We deal with vector fields
(95)Z=z∂∂x+z01∂∂u0+z11∂∂u1+∑‍zr2∂∂vr
(the notation (75) with indices is retained) on the space M. Recall that Z is a variation if ℒZΩ⊂Ω. In terms of coordinates, the conditions are
(96)z11=Dz01-u1Dz,fz01=Dz11-FDz,zr+12=Dzr2-vr+1Dz(r=0,1,…),
where the first and third equations are merely recurrences while the middle equation causes serious difficulties (a classical result. Hint: use ℒZα0∈Ω,ℒZα1∈Ω,ℒZβr∈Ω). By using the alternative formula
(97)Z=z∂∂dx+a0∂∂α0+a1∂∂α1+∑‍br∂∂βr,
the conditions slightly simplify
(98)a1=Da0,Da1=fb1,br+1=Dbr(r=0,1,…).
(Hint: apply the rule ℒZφ=Z⌋dφ+dφ(Z) to the forms φ=α0,α1,βr.) However, by virtue of Lemma 23 and standard filtration, we have explicit formula
(99)Z=z∂∂dx+∑‍Drp∂∂πr(z=Zx,p=π0(Z))
for the variations where z and p are arbitrary functions. One can then easily obtain explicit formulae for all coefficients a0,a1,br in (97) and z01,z1,zr2 in (95) by using the left-hand identities (93). They need not be stated here.(ιν) Infinitesimal Transformations. We refer to Remark 27: variation  Z is infinitesimal symmetry if and only if(100)ℒZπ0=Z⌋dπ0+dp=λπ0
for appropriate multiplier λ∈ℱ(M). In explicit terms, we recall formula
(101)π0=fα+Dfα0=fα+f′v2α0,
where
(102)dπ0=dx∧π1(mod⁡Ω∧Ω),dα=d(α1-fβ0)≅-df∧β0=-f′β1∧β0(mod⁡dx),
and therefore clearly
(103)dπ0=dx∧π1+(f′β1∧α-ff′β1∧β0)+(f′′v2β1+f′β2)∧α0=dx∧π1+β1∧(f′α-ff′β0+f′′v2α0)+f′β2∧α0.
So denoting
(104)z=Zx=dx(Z),a0=α0(Z),a=α(Z),br=βr(Z),p=π0(Z)(r=0,1,…),
requirement (100) reads
(105)zπ1+b1(f′α-ff′α0+f′′v2β0)+f′b2α0-(f′a-ff′a0+f′′v2b0)β1-f′a0β2+dp=λ(fα+Dfα0),
where π1=2Dfα+D2fα0 and
(106)dp≅pu0α0+pu1α1+∑‍pvrβr=pu0α0+pu1α+(fpu1+pv0)β0+∑r>0‍pvrβr
(mod⁡dx) should be moreover inserted. It follows that requirement (100) is equivalent to the so-called resolving system(107)2zDf+f′b1+pu1=λf,zD2f+f′′v2b1+pu0=λDf,(108)ff′b1=fpu1+pv0,f′a-ff′b0+f′′v2a0=pv1,f′a0=pv2.
Moreover pvr=0(r≥2) and therefore p=p(u0,u1,v0,v1,v2) is of the order 2 at most.(ν) On the Resolving System. Equations (107) uniquely determine the multiplier λ and the “horizontal” coefficient z=Zx in terms of the “vertical” coefficients a0, a, br, and p. For instance the formula
(109)z=1C((f′b1+pu1)Df-(f′′v2b1+pu0)f)
easily follows. So we may focus on (108).Equations (108) deserve more effort. They depend only on “vertical” components and can be expressed in terms of functions p, Dp, D2p, and D3p if the obvious identities
(110)p=af+a0Df,Dp=2aDf+a0D2f,D2p=3aD2f+a0D3f+b0C,Ca=D2f·p-Df·Dp,Ca0=-2Df·p+fDp,C2b0=Ap+BDp+CD2p,
following from (93) together with the prolongation formula
(111)C2b1+2b0CDC=D(C2b0)=D(Ap+BDp+CD2p)
are applied. By using the lucky identity DC=-B (direct verification), one can obtain the alternative resolving system(112)ff′(C(DA·p+(A+DB)Dp+CD3p)+2B(Ap+BDp+CD2p))=C3(fpu1+pv0),(113)C((f′D2f-2v2f′′Df)p+(v2f′′f-f′Df)Dp)-ff′(Ap+BDp+CD2p)=C2pv1,(114)-2f′Df·p+ff′Dp=Cpv2
only in terms of the unknown function p. Recall that the resolving system is satisfied if and only if the vector field (99) is infinitesimal symmetry.Our aim is to determine the functionp satisfying (112)–(114). Alas, the resolving system does not provide any insight into the true structure of function p. It will be therefore replaced by other conditions of classical nature, the crucial requirements and the simplified requirements as follows.(νι) Crucial Requirements. We start with simple formulae
(115)Df=f′v2,D2f=f′′v22+f′v3,D3f=f′′′v23+3f′′v2v3+f′v4,Dp=⋯+pv2v3,D2p=⋯+pv2v2v32+pv2v4(the top-order terms).
Using moreover (94), one can see that there is a unique summand in (113) which involves the factor v33, namely the summand
(116)-ff′·C·D2p≅-ff′·ff′v3·pv2v2v32.
It follows that pv2v2=0 identically and we (temporarily) may denote
(117)p=M(x,u0,u1,v0,v1)+N(x,u0,u1,v0,v1)v2.
The simplest equation (114) of the resolving system then reads
(118)-2f′·f′v2·(M+Nv2)+ff′D(M+Nv2)=((ff′′-2f′2)v22+ff′v3)N.
Clearly
(119)D(M+Nv2)=𝒟M+(Mv1+𝒟N)v2+Nv1v22+Nv3,
where the reduced operator
(120)𝒟=∂∂x+u1∂∂u0+F∂∂u1+v1∂∂v0
appears and we obtain three so-called crucial requirements(121)𝒟M=0,2Mf′=(Mv1+𝒟N)f,Nv1f′=Nf′′
for the functions M, N by inspection of the variable v2. Altogether taken, the last resolving equation (114) is equivalent to three requirements (121). We will see with great pleasure in (νιιι) below that requirements (121) ensure even the remaining equations (112) and (113) of the resolving system.(νιι) The Crucial Requirements Simplified. The right-hand equation (121) reads
(122)Qv1=0(Q=Nf′)
and the middle equation (121) reads
(123)f2Pv1+f′𝒟Q=0(P=Mf2),
whence altogether
(124)p=Pf2+Qf′v2=Pf2+QDf.
The left-hand equation (121) does not change much; it may be expressed by 𝒟P=0.Let us summarize our achievements.In order to determine function  p  given by (124), we have three simplified requirements(125)𝒟P=0,f2Pv1+f′𝒟Q=0,Qv1=0for the coefficients  P=P(x,u0,u1,v0,v1) and Q=Q(x,u0,u1,v0,v1).(νιιι) Resolving System is Deleted. Let us recall the primary transcription (108) of the resolving system. We have already seen that (125) implies (114) and hence the equivalent and simplest right-hand equation (108).Let us turn to the middle equation (108) equivalent to (113). One can directly find formulae
(126)Dp=DP·f2+PD(f2)+DQ·Df+QD2f=PD(f2)+QD2f
by using (124) and (125). Moreover
(127)a=Pf,a0=Q,a1=Da0=𝒟Q,b0=a1-af=𝒟Qf-P
by using (124) and right-hand formulae (110). Substitution into middle equation (108) with
(128)pv1=∂∂v1(Pf2+Qf′v2)=Pv1f′+2Pff′+Qf′′v2
gives the identity.As the right-hand equation (108) equivalent to (112) is concerned, we may use
(129)b1=Db0=𝒟2Qf+∂∂v1(𝒟Qf)v2-Pv1v2,𝒟2Q=-𝒟(f2f′Pv1),
where
(130)(𝒟P)v1=𝒟(Pv1)+Pu1f+Pv0=0,(𝒟Q)v1=Qu1f+Qv0.
Moreover
(131)fpu1+pv0=f(Pu1f2+Qu1f′v2)+Pv0f2+Qv0f′v2
and (108) again becomes the identity.(ικ) Back to the Crucial Requirements. Passing to the final part of this example, let us eventually solve (125) with the unknown functions P,Q and given function f. This is already a task of classical mathematical analysis. We abbreviate v=v1 from now on since this variable v frequently occurs in our formulae.Let us begin with middle equation (125) which reads
(132)Pv=(1f)′·(q+FQu1+vQv0)(q=Qx+u1Qu0)
whence
(133)P=1fq+∫‍(1f)′Fdv·Qu1+∫‍(1f)′vdv·Qv0+P-(P-=P-(x,u0,u1,v0))
since Q is independent of variable v due to the right-hand equation (125). We may insert
(134)∫‍(1f)′Fdv=Ff-v,∫(1f)′vdv=vf-∫v-v‍dvf(fixedv-∈ℝ)
and the remaining left-hand equation (125) is expressed by the identity
(135)1·(P-x+u1P-u0)+F·P-u1+v·P-v0+1f·(qx+u1qu0+F·qu1+v·qv0)+(Ff-v)·(Qu1x+u1Qu1u0+F·Qu1u1+v·Qu1v0)+(vf-∫‍dvf)·(qv0+F·Qv0u1+v·Qv0v0)=0.
Functions P-, q, Q are independent of v and thereby subjected to very strong conditions by the inspection of the coefficients of functions
(136)1,F,v,1f,1fF,1fv,(Ff-v)F,(Ff-v)v,vf-∫‍dvf,(vf-∫‍dvf)F,(vf-∫‍dvf)v
in identity (135). The final result depends on the properties of function F and we mention only a few instructive subcases here.(κ) The Generic Subcase. Functions (136) are in general linearly independent over ℝ and identity (135) implies
(137)P-x+u1P-u0=P-u1=P-v0-Qu1x-u1Qu1u0=0,(138)(qx+u1qu0)=Qxx+2u1Qxu0+u12Qu0u0=0,(139)(qu1+Qu1x+u1Qu1u0)=Qu0+2(Qu1x+u1Qu1u0)=0,(140)(qv0)=Qxv0+u1Qu0v0=Qu1u1=Qu1v0=Qv0v0=0.
The unknown functions P- and Q can be easily found as follows. We may suppose that
(141)Q=a(x,u0)u1+b(x,u0)v0+c(x,u0),b(x,u0)=B∈ℝ,
by using (140). Then P-x=P-u0=P-u1=0; hence P-=P-(v0) due to (139). Moreover P-′=ax+u1a0 which implies P-′=ax∈ℝ,a0=0; hence a=a(x) and altogether
(142)a=Ax+A-,P-=Av0+C(A,A-,C∈ℝ).
Then
(143)c=C1x+C2u0+C3(C1,C2,C3∈ℝ)
follows from (138). Hence, C2+2A=0 due to (139) and altogether
(144)Q=(Ax+A-)u1+Bv0+C1x-2Au0+C3,P-=Av0+C.Recalling moreover (133), we have explicit formulae for the solutions  P, Q  of crucial requirements (125) and the symmetry problem is resolved. While P- and Q are mere polynomials, the total coefficient P given by (133) depends on the quadrature ∫(dv/f) and this may be globally rather complicated function. It follows that, in our approach, the elementary and the “transcendental” parts of the solution are in a certain sense separated.(κι) A Special Case of Function
F
. Let us choose F(v)=ev. Then series (136) becomes quite explicit; namely,
(145)1,ev,v,e-v,1,ve-v,(1-v)ev,(1-v)v,v+1,(v+1)ev,(v+1)vev
and these functions are linearly dependent. Identity (135) implies smaller number of requirements; the first term in (137) is combined with (139) into the single equation
(146)P-x+u1P-u0+qu1+Qu1x+u1Qu1u0=0
without any other change. We can state the final solution
(147)Q=(Ax+A-)u1+Bv0+C1x+C2u0+C3,P-=Av0+(C2+2A)x+C
with only one additional parameter C2∈ℝ if compared to the previous formulae (144).(κιι) Another Special Case. Let us eventually mention the very prominent function F(v)=v1/2; see [1, 7, 16]. Then the series
(148)1,v1/2,v,2v1/2,2v,2v3/2,v3/2,v2,23v3/2,23v2,23v5/2
stands for (136) and the relevant identity (135) implies the system of equations
(149)P-x+u1P-u0=0,(150)P-u1+2(qx+u1qu0)=P-u1+2(Qxx+2u1Qxu0+u12Qu0u0)=0,(151)P-v0+2qu1+Qu1x+u1Qu1u0=P-v0+2Qu0+3(Qu1x+u1Qu1u0)=0,(152)6qv0+3Qu1u1+2(Qv0x+u1Qv0u0)=3Qu1u1+8(Qv0x+u1Qv0u0)=0,(153)Qu1v0=0,Qv0v0=0.
We are passing to the solution of the system of (149)–(153) with unknown functions Q=Q(x,u0,u1,v0) and P-=P-(x,u0,u1,v0). Due to (153), we may put
(154)Q=a(x,u0)v0+b(x,u0,u1)
and then (152) is expressed by 3bu1u1+8(ax+u1au0)=0, whence easily
(155)b=-49au0u13-43axu12+b-(x,u0)u1+b~(x,u0).
Moreover (151) reads P-v0+2(au0v0+bu0)+3(bu1x+u1bu1u0)=0, whence
(156)P-=-au0v02-(2bu0+3(bu1x+u1bu1u0))v0+p-(x,u0,u1).
Remaining equations (149) and (150) do not admit such simple discussion. Using (154) and (155), identity (149) is equivalent to the system
(157)au0x+u1au0u0=0(whencea=Au0+a-(x),A∈ℝ),(158)2bu0x+3(bu1xx+u1bu1u0x)+u1(2bu0u0+3(bu1xu0+u1bu1u0u0))=0,(159)p-x+u1p-u0=0
of three equations and identity (150) is equivalent to the system
(160)2bu0u1+3(bu1xu1+bu1u0+u1bu1u0u1)=2a-′′,(161)p-u1+2(bxx+2u1bxu0+u12bu0u0)=0,
if (157) is moreover employed. At the same time, (155) can be improved as
(162)b=-49Au13-43a-′u12+b-(x,u0)u1+b~(x,u0).
With this improvement, (160) reads 2b-u0+3(-(8/4)a-′′+b-u0)=2a-′′ and it follows that
(163)b-=2a-′′u0+b^(x).
Analogously (158) reads
(164)2(b-u0xu1+b~u0x)+3(-83a-′′′u1+b-xx+u1b-u0x)+u1(2b~u0u0+3b-u0x)=0,
which is equivalent to the system
(165)2b~u0x+6a-(4)u0+3b^′′=0,4a-′′′+b~u0u0=0,
if (163) is inserted. Altogether, it follows that (158) is equivalent to
(166)2b~=-3a-′′′u02-3b^′u0+b~0(u0)+b~1(x),a-′′′=-12b~0′′=A3∈ℝ,
whence
(167)a-=A3x36+A2x2+A1x+A0,b~0=-A3u02+B1u0+B0(A2,A1,A0,B1,B0∈ℝ).
At the same time, we have improvements
(168)b-=2(A3+2A2)u0+b^,2b~=-3A3u02-3b^′u0+b~(u0)+b~1(x)
of the above formulae. Let us eventually turn to the remaining equations (159) and (161). We begin with (161) which can be simplified to
(169)p-u1+2163A3u12+2b^′′u1-(3b^′′′u0+b~1′′)=0,
whence
(170)p-=2163A3u133+b^′′u12-(3b^′′′u0+b~1′′)u1+p˘(x,u0).
Then the last requirement (159) is easily simplified as
(171)b^′′′u12-(3b^(4)u0+b~1′′′)u1+p˘x+u1(-3b^′′′u1+p˘u0)=0
and it follows that
(172)p˘x=0,b~1′′′-3b^(4)+p˘u0=0,b^′′′=0,
whence easily
(173)p˘=p˘(u0),b~1′′′=-p˘′=C3∈ℝ,b^=D2x2+D1x+D0,(174)b~1=C3x36+C2x2+C1x+C0,p˘=-C3u0+C(C3,D2,…,C∈ℝ).
The solution is eventually done. It depends on the parameters
(175)A,A3,A2,A1,A0,B1,B0,C3,C2,C1,C0,C,D2,D1,D0∈ℝ
in the total number of 15. This is seemingly in contradiction with [1, 7, 16] where 14-dimensional symmetry group (namely, the exceptional simple Lie group 𝔾2) was declared. However, our final symmetry in fact depends on the sum B0+C0 as follows from (166), (167), and (174) and therefore no contradiction appears. We will not explicitly state the resulting symmetries Z for obvious reason here. Recall that they are given by (99) where z,p are clarified in (109) and (124). Coefficients appearing in (124) are clarified in (133), (156), and (170) and in (154), (157), (162), (168), (173), and (174).It should be moreover noted that our approach is of the universal nature while the method of explicit calculations which provides the infinitesimal transformations in[7]rests on a lucky accident; see [7, Theorem 3.2, and the subsequent discussion].Remark 29.
VariationsZ were easily found in (ιιι). Due to Theorem 26 and Remark 27, infinitesimal symmetries satisfy moreover ℒZπ0=λπ0 or, alternatively saying, they preserve the Pfaffian equation  π0=0, and this property was just employed. We will now prove the converse without use of Theorem 24. The reasoning is as follows. Let a variation  Z  preserve Pfaffian equation  π0=0. Then Z preserves the space of adjoint variables  x, u0, u1, v0, v1 of this Pfaffian equation. In this finite-dimensional space, the variation Z generates a group which can be prolonged to the higher-order jet variables. It follows that Z is indeed an infinitesimal transformation.Remark 30.
Let us briefly mention the caseF(v1)=Av1+B(A,B∈ℝ;A≠0) as yet excluded by condition (90). In this linear case, clearly
(176)ℒDα1=ℒD(du1-(Av1+B)dx)=d(Av1+B)-Av2dx=Aβ1,ℒD(α1-Aβ0)=0,τ=α1-Aβ0=d(u1-Av0-Bx)∈ℛ(Ω)
and we may introduce standard filtration
(177)ℛ(Ω)⊂Ω-0=Ker⁡2Ω1⊂Ω-1=Ker⁡Ω1⊂Ω-0=Ω1⊂Ω-1=Ω2⊂⋯,
where τ is a basis of ℛ(Ω) and the forms
(178)π0=α0,π1=α1,π2=ℒDα1=Aβ1,…,πl=Aβl-1
provide a basis of module Ω-l(l≥1). The symmetries can be easily found. They are the prolonged contact transformations m defined by m*α0=λα0 depending moreover on the parameter t=u1-Av0-Bx. Roughly saying, the geometry of the linear second-order equation u2=Av1+B is identical with the contact geometry of curves in ℝ2. Quite analogous result can be obtained also for the Monge equation F(x,u0,u1,v0,v1)=0 and, in much greater generality, for the system of two Pfaffian equations in four-dimensional space [17].Remark 31.
Let us once more return to the crucial requirement (125) where operators 𝒟 and ∂/∂v1 are applied to unknown functions P and Q. We have employed the simplicity of the second operator ∂/∂v1 in the above solution; see formula (133). However, analogous “complementary” method can be applied to the first operator 𝒟 as follows. Let us introduce new variables
(179)x-=x,u-0=u0-u1x+Fx22,u-1=u1-Fx,v-0=v0-v1x,v-=v1
with the obvious inverse transformation (not stated here). Then
(180)𝒟=∂∂x-,∂∂v1=f2x-2∂∂u-0-x-(f∂∂u-1+∂∂v-0)+∂∂v-(f=f(v1)=f(v-))
in terms of new variables. We again abbreviate v=v-=v1. Passing to new coordinates, the left-hand requirement (125) is simplified as P=P-(u-0,u-1,v-0,v). The middle requirement (125) reads
(181)f2(f2x-2P-u-0-x-(fP-u-1+P-v-0)+P-v)+f′Q-x-=0(Q=Q-(x-,u-0,u-1,v-0,v))
and determines the function Q- in terms of new variables as
(182)Q-=-f2f′(f6x-3P-u-0-x-22(fP-u-1+P-v-0)+x-P-v)+q-,q-=q-(u-0,u-1,v-0,v),
where q- is constant of integration. This is a polynomial in variable  x- and it follows easily that the remaining right-hand requirement (130) applied to function Q- is equivalent to the system
(183)P-u-0u-0=0,𝒫u-0=0,q-v=0,fq-u-1+q-v-0+∂∂v(f2f′P-v)=0,f3f′P-u-0v+f2f′(f𝒫u-1+𝒫v-0)+13∂∂v(f3f′P-u-0)=0,fq-u-0+2f2f′𝒫v+∂∂v(f2f′𝒫)=0(𝒫=fP-u-1+P-v-0).
We will not discuss this alternative approach here in more detail.Remark 32.
Though the symmetries of (77) can be completely determined by applying the common methods, several formally quite different ways of the calculation are possible. It would certainly be of practical interest which of them is the “most economical” one. Let us mention such an alternative way for better clarity. We start with the “opposite” transcription
(184)dvdx=G(d2udx2)(G=F-1,theinversefunction)
of (77). The primary concepts are retained, the same underlying space M, diffiety Ω, and contact forms αr, βr(r=0,1,…). However, we choose x,u0,u1,…,v0 for new coordinates on M from now on and the forms
(185)αr=dur-ur+1dx(r=0,1,…),β0=dv0-G(u2)dx
for new basis of Ω. We have moreover
(186)D=∂∂x+∑‍ur+1∂∂ur+G∂∂v0
in terms of new coordinates. The standard filtration is formally simplified. The forms
(187)πr=ℒDrπ0(r=0,1,…;π0=β0-G′α1+DG′α0)
may be taken for new standard basis if the inequality D2G′≠0 is supposed. This follows from the obvious formulae
(188)π1=ℒDπ0=D2G′α0,π2=ℒDπ1=D3G′α0+D2G′α1,…,
simplifying the analogous left-hand side (93). Then, analogously to (95) and (99), we introduce the variations(189)Z=z∂∂x+∑‍zr1∂∂ur+z02∂∂v0=z∂∂dx+∑‍Drp∂∂πr
of diffiety Ω where z=Zx and p=π0(Z) may be arbitrary functions. Recall that we have even infinitesimal symmetry of Ω if and only if the requirement (100) is satisfied. However clearly
(190)dπ0=dx∧π1+G′′α1∧α2+(G′′′u3α2+G′′α3)∧α0
and one can obtain the resolving equations as follows. First of all, we obtain equations
(191)zD2G′+G′′u3a2+G′′a3+pu0=λDG′,pv0=λ(ar=αr(Z))
which determine coefficients z and λ analogously to (107). Moreover
(192)G′′a2-pu1-G′pv0=G′′a1-G′′′u3a0+pu2=G′′a0-pu3=pur=0(r>3)
are conditions for the unknown function p=p(x,u0,…,u3,v0) analogous to (108). The “vertical” coefficients ar can be expressed in terms of functions p,Dp,D2p and D3p, by using the equation
(193)D2G′·a0=D2G′·α0(Z)=π1(Z)=Dπ0(Z)=Dp
and the recurrence ar+1=Dar. As yet the calculations are much easier then for the above case of formulae (110); however, the resulting resolving system of three equations analogous to (112)–(114) is again complicated and will not be explicitly stated here. Remarkable task appears when we investigate the corresponding crucial requirements and try to determine the structure of function p in terms of new coordinates. For instance, the “very prominent” and seemingly rather artificial case (κιι) turns into the “simplest possible” and quite natural equation dv/dx=(d2u/dx2)2 in new coordinates.


## 7. Brief Digression to the Calculus of Variations

The classicalLagrange problem of the calculus of variations deals with an underdetermined system of differential equations (better with a diffiety) together with a variational integral. We are interested in internal symmetries of this variational problem.Let us start with a diffietyΩ⊂Φ(M). We choose a standard filtration Ω-* and the corresponding standard basis πrj(j=1,…,μ(Ω);r=0,1,…). For better clarity, we suppose the controllable case ℛ(Ω)=0. Let x∈ℱ(M) be an independent variable. Let us consider x-parametrized solutions p of diffiety Ω in the sense
(194)p:I⟶M(I⊂ℝ),p*ω=0(ω∈Ω),p*x=x∈I⊂ℝ.
Here I⊂ℝ is a closed interval a≤x≤b with a little confusion: letter x denotes both a function on M and the common coordinate (that is, a point) in ℝ.Definition 33.
A vector fieldV∈𝒯(M) is called a variation of solution p of diffiety Ω if p*ℒVω=0(ω∈Ω). This is a mere slight adaptation of the familiar classical concept.Lemma 34.
A vector fieldV∈𝒯(M) is a variation of p if and only if
(195)p*πr+1j(V)=p*Dπrj(V)=ddxp*πrj(V)(j=1,…,μ(Ω);r=0,1,…).Proof.
A variationV satisfies p*ℒVπrj=0, where
(196)p*ℒVπrj=p*(V⌋dπrj+dπrj(V))=p*(-πr+1j(V)+Dπrj(V))dx
by virtue of (55).Remark 35.
It follows easily that a vector fieldZ∈𝒯(M) is a variation of diffiety Ω in the sense of Definition 8 if and only if Z is a variation of every solution p of Ω; see Lemma 23. Conversely, if V is a variation of a solution p then there exist many variations Z of Ω such that Z=V at every point of p, and they are characterized by the identities p*π0j(Z)=p*π0j(V)(j=1,…,μ(Ω)) along the curve p; see formula (72). We conclude that the concepts “variation Z of Ω” and “variations V of p” are closely related. Roughly saying, variations V of p are “restrictions” of variations Z of Ω to the curve p.Definition 36.
A couple{Ω,φ} where Ω⊂Φ(M) is a diffiety and φ∈Φ(M) is a differential form will be identified with a variational problem in the (common) sense that diffiety Ω represents the differential constraints to the variational integral  ∫φ. A solution p of Ω is called an extremal of this variational problem, if
(197)∫p*ℒVφ=∫p*V⌋dφ=0(specialvariationsV)
for every variation V of p which is vanishing at the endpoints p(a),p(b)∈M. This definition provides the common classical extremals; see Remark 43.Remark 37.
The phrase “variationV of p” can be replaced with “variation V of Ω”. The form φ can be replaced with arbitrary form φ+ω(ω∈Ω). The extremals do not change.Theorem 38.
To every standard basis ofΩ and given φ∈Φ(M) there exists unique form φ˘∈Φ(M) such that
(198)φ˘≅φ(mod⁡Ω),dφ˘≅0(mod⁡Ω∧Ωandallinitialformsπ0j).
In accordance with (198) we assume that
(199)dφ˘≅∑‍ejπ0j∧dx(mod⁡Ω∧Ω).
Then a solution p of Ω is extremal if and only if p*ej=0(j=1,…,μ(Ω)) and therefore if and only if
(200)p*Z⌋dφ˘=0(Z∈𝒯(M))
for all vector fields Z∈𝒯(M).Proof (see [9]).
For a givenφ∈Φ(M), let us look at a top-order summand
(201)dφ≅∑‍arjπrj∧dx=⋯+aRJπRJ∧dx(mod⁡Ω∧Ω).
If R>0, the summand can be deleted if the primary differential form φ is replaced with the new form φ+aRJπR-1J. The extremals do not change. The procedure is unique and terminates in form φ˘ satisfying (198). Then (200) follows from the identity
(202)p*Z⌋dφ˘=p*Z⌋∑‍ejπ0j∧dx=p*∑‍ejπ0j(Z)dx,
where the functions π0j(Z) may be quite arbitrary if Z is a variation,  see Lemma 34.Definition 39.
The differential formφ˘ can be regarded for the internal Poincaré-Cartan form of our variational problem and equations ej=0(j=1,…,μ(Ω)) for the Euler-Lagrange system.We turn to the symmetries.Definition 40.
A symmetrym of diffiety Ω is called a symmetryof variational problem  {Ω,φ}, if m*φ≅φ(mod⁡Ω).  A variation (infinitesimal symmetry) Z of Ω is called a variation (infinitesimal symmetry, resp.) of variational problem  {Ω,φ}, if ℒZφ∈Ω. Let V∈𝒯(M) be a variation of a solution  p of diffiety Ω. Then V is called a Jacobi vector field of p, if moreover p*ℒVφ=0. Roughly saying, variations Z of variational problem {Ω,φ} are “universal” Jacobi vector fields for all solutions p of Ω. In classical theory, Jacobi vector fields are introduced only for the particular case when p is an extremal.We will see in the following example that Poincaré-Cartan formsφ˘ simplify the calculation of symmetries and variations. On this occasion, we also recall the following admirable result.Theorem 41 (E. Noether).
IfZ is a variation of variational problem {Ω,φ} and φ˘ is a Poincaré-Cartan form then p*φ˘(Z)=const. for every extremal p.Proof.
We haveℒZφ˘∈Ω,p*ω=0(ω∈Ω), and therefore
(203)0=p*ℒZφ˘=p*(Z⌋dφ˘+dφ˘(Z))=p*dφ˘(Z)=dp*φ˘(Z)
by virtue of (200).Remark 42.
Many concepts of the classical calculus of variations lose the geometrical meaning if the higher-order symmetries are accepted; for example, this concerns the common concept of a nondegenerate variational problem and even the order of a variational integral. On the other hand, the most important concepts can be appropriately modified; for example, the Hilbert-Weierstrass extremality theory together with the Hamilton-Jacobi equations [18–21] since the Poincaré-Cartan forms φ˘ make “absolute sense” along the extremals.Remark 43.
In the common classical calculus of variations, extremalsp are defined by the property ∫‍p*ℒVφ=0, where variations V satisfy certain weak boundary conditions at the endpoints (“fixed ends” or transversality) in order to delete some “boundary effects” of the variational integral. Much stronger conditions appear in Definition 36. Therefore
(204)classicalextremals⊂ourextremals.
However, φ can be replaced by the form φ˘. Then
(205)∫‍p*ℒVφ˘=∫‍p*V⌋dφ˘+boundaryterm.
For the above special variations V, the boundary term vanishes. If p is extremal in the sense of Definition 36, then (200) and Remark 15 may be applied and it follows that
(206)classicalextremals⊃ourextremals.
In topical Griffiths’ theory [22], extremals are defined by the property
(207)p*Z⌋d(φ+ω)=0(allZ∈𝒯(M),appropriateω∈Ωdependingon⁡p)
which is clearly equivalent to the condition
(208)∫‍p*Z⌋d(φ+ω)=∫‍p*ℒZ(φ+ω)=0(specialvectorfieldsZ,appropriateω∈Ω),
where Z are vector fields vanishing at the endpoints. This condition trivially implies
(209)∫‍p*ℒV(φ+ω)=0hence∫‍p*ℒVφ=0(specialvariationsV)
with variations  V vanishing at the endpoints; see Remark 35. Therefore
(210)Griffithsextremals⊂ourextremals.
The converse inclusion
(211)Griffithsextremals⊃ourextremals
is, however, trivial since the universal form  φ˘=φ+ω˘(ω˘∈Ω) satisfies p*Z⌋dφ˘=0 even for every extremal in the sense of Definition 36. We conclude that all the mentioned concepts of extremals are identical. (We apologize for this hasty exposition. Roughly saying, the Griffiths’ theory and our approach are almost identical. The Griffiths’ correction  ω∈Ω  depending on  p is made universal here. The classical approach rests on a special choice of boundary conditions for the variations V. However, such a special choice is misleading since it does not affect the resulting family of extremals and we prefer a universal choice here as well.)


## 8. Particular Example of a Variational Integral

A simple illustrative example is necessary at this place. Let us again deal with diffietyΩ of Section 6. So we recall coordinates x,u0,u1,v0,v1,… of the underlying space M, the contact forms αr, βr(r=0,1,…) generating Ω, the vector field
(212)D=∂∂x+u1∂∂u0+F∂∂u1+∑‍vr+1∂∂vr=∂∂x+∑ω∈Ω‍0·∂∂ω∈ℋ,
and the standard basis π0,π1,… of Ω. We moreover introduce variational integrals
(213)∫φ(φ=gdx,g∈ℱ(M)).
Assuming ∂g/∂πr=0(r>R) and therefore
(214)dg=Dgdx+∂g∂u0α0+∂g∂u1α1+∑‍∂g∂vrβr=Dgdx+∂g∂π0π0+⋯+∂g∂πRπR,
we introduce the functions
(215)gR=∂g∂πR,gr-1=∂g∂πr-1-Dgr(r=R,…,1).
Then
(216)φ˘=gdx+g1π0+⋯+gRπR-1
is the Poincaré-Cartan form since the identity
(217)dφ˘=g0·π0∧dx(mod⁡Ω∧Ω)
can be directly verified. In accordance with formula (199) where e=e1,π0=π01 is abbreviated, we have e=g0. Let us denote
(218)e=e[g]=g0=∂g∂π0-D∂g∂π1+⋯+(-1)RDR∂g∂πR,
for better clarity. The following simple result will be needed.Lemma 44.
Identitye[g]=0 is equivalent to the equation g=DG with appropriate G∈ℱ(M).Proof.
By virtue of (200), the identity is equivalent to the congruence dφ˘≅0(mod⁡Ω∧Ω). However, if the rule d(dφ˘)=0 is applied to the congruence, it follows easily that dφ˘=0 identically. Therefore φ˘=dG≅DGdx(mod⁡Ω) by using the Poincaré lemma.Let us mention symmetriesm and variations Z of our variational problem in more detail. In the favourable case μ(Ω)=1, the task is not difficult.The symmetrym of our variational problem {Ω,φ} clearly preserves the unique Poincaré-Cartan form φ˘ and therefore also the vector field 𝒟=D/g∈ℋ determined by the condition φ˘(𝒟)=1. We suppose g≠0 here. It follows that all differential forms
(219)π˘0=ℒ𝒟φ˘=𝒟⌋dφ˘+dφ˘(𝒟)=eπ0·𝒟x=egπ0,π˘r+1=ℒ𝒟π˘r(r=0,1,…)
are preserved, too. Let us moreover suppose e=e[g]≠0. Clearly
(220)π˘1=1gℒDπ0=1g(Degπ0+egπ1),π˘r+1=1gℒ𝒟πr≅egr+1πr(mod⁡dx,π0,…,πr).
Therefore φ˘,π˘0,π˘1,… is invariant basis of module Φ(M) in the sense
(221)m*φ˘=φ˘,m*π˘r=π˘r(r=0,1,…).
It follows that the symmetries  m  of our variational problem  {Ω,φ}can be comfortably determined. Quite analogous conclusion can be made for the infinitesimal symmetries, of course.Passing to the variationsZ of the variational problem, we have explicit formula (99) for the variations of Ω and moreover condition ℒZφ∈Ω equivalent to ℒZφ˘∈Ω. However,
(222)ℒZφ˘=Z⌋dφ˘+dφ˘(Z)≅Z⌋(eπ0∧dx)+Dφ˘(Z)dx=(ep+Dφ˘(Z))dx(mod⁡Ω)
and therefore
(223)0=ep+Dφ˘(Z)=ep+DG(G=gz+g0p+g1Dp+⋯+gRDRp).
Assume g≠0. We obtain condition e[ep]=0 for the unknown function p. In more precise notation and in full detail
(224)e[e[g]p]=(∂∂π0-D∂∂π1+⋯)(∂g∂π0-D∂g∂π1+⋯)p=0.
This is formally a very simple condition concerning the unknown function p; alas, it is not easy to be resolved. Paradoxically, variationsZcause serious difficulties.For better clarity, we continue this example with particular choice of the variational integral. Let us consider variational integral∫‍g(x,u0,v0)dx. Equation (214) reads
(225)dg=Dgdx+∂g∂u0α0+∂g∂v0β0=Dgdx+∂g∂π0π0+∂g∂π1π1+∂g∂π2π2
and it follows that
(226)∂g∂π0=-2gu0DfC+gv0AC2,∂g∂π1=gu0fC+gv0BC2,∂g∂π2=gv0C
by using (93). We have R=2 and therefore
(227)φ˘=gdx+(∂g∂π1-D∂g∂π2)π0+∂g∂π2π1,e=∂g∂π0-D∂g∂π1+D2∂g∂π2
by virtue of (215)–(218). Both the Poincaré-Cartan form φ˘ and the Euler-Lagrange equation e=0 can be expressed in terms of common coordinates, if derivatives (226) are inserted. We omit the final formulae here. Passing to the symmetries m, we may simulate the moving frames method and express the differential
(228)dφ˘=∑‍Crπ˘r∧φ˘+∑r<s‍Crsπ˘r∧π˘s=π˘0∧φ˘+∑r<s‍Crsπ˘r∧π˘s
in terms of the invariant basis (221). Then all coefficientsCrsare invariants of symmetry  m; that is,
(229)m*Crs=Crshencem*𝒟kCrs=𝒟kCrs(𝒟=Dg).
In fact we have obtained all invariants. (Hint: for instance, differential
(230)dπ˘0=dℒ𝒟φ˘=ℒ𝒟dφ˘=∑‍𝒟Crsπ˘r∧π˘s+ℒ𝒟(π˘0∧φ˘)+∑‍Crsℒ𝒟(π˘r∧π˘s)
does not provide any novelty.) It follows that the symmetry problem is resolved. Compatibility of the system of (229) ensures the existence of symmetries  m  of the variational problem  {Ω,φ}  since the Frobenius theorem can be applied to the Pfaffian system (221). In the most favourable case, Crs are even constants. Explicit calculation of invariants Crs is a lengthy but routine procedure. First of all
(231)dφ˘≅dg1∧π0+dg2∧π1+g1dπ0+g2dπ1(mod⁡dx)
by using the primary formula (216). Then
(232)dg1≅∑‍∂g1∂πrπr,dg2≅∑‍∂g2∂πrπr(mod⁡dx)
may be substituted where the coefficient can be determined analogously as in (226). As the differential
(233)dπ0≅β1∧(f′α-ff′β0+f′′v2α0)+f′β2∧α0(mod⁡dx)
is concerned, we refer to formula in Section 6. The contact forms must be replaced with the standard basis by using the right-hand formulae (93). Then we may use the lucky identity
(234)dπ1=dℒDπ0=ℒDdπ0
in order to determine the last summand in (231). In the end, the standard basis πr in (231) can be easily replaced by the invariant forms π˘r(r=0,1,…) and we are done.


## 9. The Order-Increasing Case

Let us eventually return to the main topic, the differential equations. We will finish this paper with decisive examples of higher-order symmetries, namely, with symmetries of the Monge equation(235)dwdx=F(x,u,v,w,dudx,dvdx)
involving three unknown functions u=u(x),v=v(x), and w=w(x). Let us directly turn to the internal theory carried out by using the underlying space M with coordinates
(236)x,ur,vr,w0(r=0,1,…),
diffiety Ω⊂Φ(M) with the basis
(237)αr=dur-ur+1dx,βr=dvr-vr+1dx(r=0,1,…),γ0=dw0-F(x,u0,v0,w0,u1,v1)dx
and the total derivative
(238)D=∂∂x+∑‍ur+1∂∂ur+∑‍vr+1∂∂vr+F∂∂w0∈ℋ.
We also introduce functions and differential forms
(239)wr=Drw0∈ℱ(M),γr=ℒDrγ0=dwr-wr+1dx∈Ω(r=0,1,…)
for the formal reasons. The natural filtration Ω* in accordance with the order is such that the forms α0,…,αl,β0,…,βl,γ0 are taken for the basis of submodule Ωl⊂Ω(l=0,1,…). Let us determine the corresponding standard filtration Ω-*. Clearly
(240)ℒDγ0=γ1=Fu0α0+Fv0β0+Fw0γ0+Fu1α1+Fv1β1
and therefore
(241)ℒD(γ0-Fu1α0-Fv1β0)=(Fu0-DFu1)α0+(Fv0-DFv1)β0+Fw0γ0∈Ω0.
Denoting π=γ0-Fu1α0-Fv1β0, we obtain
(242)ℒDπ=(Fu0-DFu1)α0+(Fv0-DFv1)β0+Fw0γ0=Aα0+Bβ0+Fw0π∈Ω0(A=Fu0-DFu1+Fw0Fu1,B=Fv0-DFv1+Fw0Fv1).
We will not deal with the case when A=B=0 identically. Let us instead suppose that A≠0 from now on. Then ℛ(Ω)=0 and we may introduce standard filtration Ω-* of diffiety Ω where the form
(243)π01=π=γ0-Fu1α0-Fv1β0
generates Ω-0 and in general the forms
(244)πr1=ℒDrπ(r=0,…,l),πr2=βr(r=0,…,l-1)
generate module Ω-l(l≥1). Notation (53) with indices is retained here. With this preparation, we are passing to the symmetries of diffiety Ω. Theorem 26 and Remark 27 fail since μ(Ω)=2 in our case. There exist many standard filtrations of Ω and we may also expect the existence of the order-destroying symmetries.The preparation is done; however, before passing to quite explicit examples, certain general aspects are worth mentioning. We recall Figure3 which can be transparently illustrated just at this place for the first time.First of all, everyorder-preserving symmetry  m on scheme (a) of Figure 3 obviously satisfies certain formulae
(245)m*α0=a1α0+a2β0+a3γ0,m*β0=b1α0+b2β0+b3γ0,m*γ0=c1α0+c2β0+c3γ0,det⁡(a1a2a3b1b2b3c1c2c3)≠0,
where the coefficients cannot be in fact arbitrary since they are subjected to identity (240). In more detail, we have
(246)DW·m*γ1=ℒDm*γ0=Dc1α0+Dc2β0+Dc3γ0+c1α1+c2β1+c3γ1(W=m*x)
in accordance with (63). Alternatively (240) implies
(247)m*γ1=m*Fu0·m*α0+⋯+m*Fv1·m*β1,
where the forms m*α0,…,m*β1 can be expressed in terms of forms α0,…,γ1. The comparison provides many unpleasant interrelations among coefficients a1,…,c3.However, by using the standard basis, the same symmetry satisfies shorter formulae(248)m*π01=aπ01(hence,automaticallyDWm*π11=Daπ01+aπ11),m*π02=b01π01+b02π02+b11π11
with coefficients subjected only to the inequalities a≠0 and b02≠0 at this place. We employ the fact that both triples α0,β0,γ0 and π01,π02,π11 are bases of module Ω0. Moreover m* preserves the natural filtration Ω* and therefore also the corresponding standard filtration Ω-*. Especially, the initial term Ω-0 is preserved and m*π01 is a mere multiple of π01.Theorder-preserving infinitesimal symmetry  Z corresponding to scheme (a) satisfies either the system
(249)ℒZα0=λ1α0+λ2β0+λ3γ0,ℒZβ0=μ1α0+μ2β0+μ3γ0,ℒZγ0=ν1α0+ν2β0+ν3γ0
with coefficients subjected to many identities analogous as above or, alternatively, the equivalent and shorter system
(250)ℒZπ01=μπ01(henceℒZπ11=Dμπ01+(μ-DZx)π11),ℒZπ02=λ01π01+λ02π02+λ11π11
with arbitrary coefficients in terms of the standard basis. For the middle equation use the identity
(251)ℒZπ11=ℒZℒDπ01=ℒDℒZπ01-Dzπ11(z=Zx).
This follows from the Lie bracket formula [D,Z]=Dz·D which is true if and only if Z is a variation of diffiety Ω. The Cartan’s general equivalence method [23] can be applied to this order-preserving symmetry problem; however, we will mention the Lie approach later on.With this result, the simplest possibleorder-increasing symmetry  m on scheme (c) of Figure 3 can be introduced by the equations
(252)m*π01=a1π01+a2π02,m*π02=b1π01+b2π02+b(a1π11+a2π12),det⁡(a1b1-bDa1a2b2-bDa2)≠0.
Let us prove the invertibility of m. Clearly
(253)DWm*π11=ℒDm*π01=Da1π01+Da2π02+a1π11+a2π12
and it follows that
(254)m*π02-bDWm*π11=(b1-bDa1)π01+(b2-bDa2)π02∈m*Ω.
Inclusions π01,π02∈m*Ω therefore hold true and Lemma 21 can be applied.In order to state another example to scheme (c), let us consider the equations(255)m*π01=a01π01+a02π02+a11π11+a12π12,m*π02=b01π01+b02π02+b11π11+b12π12.
Invertibility of such morphism m is ensured if b1i=ba1i,b0i-ba0i=aa1i(i=1,2) for appropriate factors a≠0,b and if moreover
(256)det⁡(a11a01-Da11a12a02-Da12)≠0.
For the proof of invertibility, apply ℒD to the inclusion
(257)1a(m*π02-bm*π01)=a11π01+a12π02∈m*Ω
and verify that π01,π02∈m*Ω.In both examples,the common general equivalence method [23] fails. The corresponding variations  Z can be introduced and are rather interesting though they do not generate any symmetry groups. See Remark 46 below.It is also easy to illustrate scheme (b) of Figure3 by using the symmetries m and infinitesimal symmetries Z such that
(258)m*π01=a1π01+a2π02,m*π02=b1π01+b2π02,det⁡(a1a2b1b2)≠0,ℒZπ01=λ1π01+λ2π02,ℒZπ02=μ1π01+μ2π02.
(Hint: Theorem 24 can be trivially applied and the natural filtration is not preserved, if a2≠0 and λ2≠0.) Another example is provided by the equations
(259)m*π01=a01π01+a02π02+a12π12,m*π02=bπ02,ℒZπ01=μ01π01+μ02π02+μ12π12,ℒZπ02=μπ02
“symmetrical” to the order-preserving case. The classical Lie’s infinitesimal symmetries and the Cartan’s equivalence method can be both applied without any change.We have briefly indicated only the simplest devices here and refer to [2, Section 4] for the universal construction. A complete overview of all possible higher-order symmetries of (235) is lying beyond any actual imagination. For instance, the composition m1∘m2 of symmetries and the conjugate groups m∘m(λ)∘m-1 to a given group provide much more complicated examples than the original components m1,m2,m, and m(λ). The definition equations for such composition of symmetries can be directly found and they look rather depressively for the time being.


## 10. Concluding Examples on Infinitesimal Symmetries

We deal only with a simplified equation (235), namely, with the equation
(260)dwdx=F(dudx,dvdx)
for good reasons to be clarified in the Appendix. Let us abbreviate
(261)F1=Fu1,F1=Fv1,F11=Fu1u1,F11=Fu1v1,F11=Fv1v1
from now on. The crucial identity (240) then reads
(262)ℒDγ0=γ1=F1α1+F1β1
and we recall the standard basis πr1=ℒDrπ01,πr2=ℒDrπ02=βr(r=0,1,…), where
(263)π01=γ0-F1α0-F1β0,π11=-DF1α0-DF1β0
in terms of the simplified notation. The formulae
(264)dπ01=dx∧π11+(F11α0+F11β0)∧α1+(F11α0+F11β0)∧β1,dπ02=dx∧β1
easily follow. On this occasion, we also recall more general adjustments
(265)λm*ℒDω=ℒDm*ω(ω∈Ω,λ=DW,W=m*x),(266)(ℒDω)(Z)=D(ω(Z))(ω∈Ω)
of Lemmas 17 and 23. The factor λ≠0 appearing here can be defined by the congruence m*dx≅λdx(mod⁡Ω) as well.Several symmetry problems for (260) will be mentioned. We start with examples on infinitesimal symmetries  Z and demonstrate our approach both using the traditional order-preserving case and then employing two technically quite analogous order-increasing symmetry problems. The calculations are elementary but not of a mere mechanical nature and the concise form of the final results is worth attention. That is, by using the series (268) with the standard basis, the unknown functions z,p, and q satisfy quite reasonable and explicitly solvable conditions. Denoting
(267)a0=α0(Z),a1=α1(Z)=Da0,p=π01(Z),q=π02(Z)=β0(Z),z=Zx,
we simulate the procedure of Section 6 and our method again rests on the explicit formula
(268)Z=z∂∂x+∑‍Drp∂∂πr1+∑‍Drq∂∂πr2
for all variations  Z. We recall that infinitesimal symmetries  Z moreover satisfy certain additional requirements in order to ensure the conditions of Theorem 24. The choice of such requirements which is arbitrary to a large extent (dotted lines in Figure 3(b)) strongly affects the final result, the resulting symmetries Z. Altogether taken, reasonings of this Section 10 belong to the Lie’s theory appropriately adapted to the infinite-dimensional spaces. On the contrary, we will conclude this paper with only few remarks on the true (not group-like) higher-order symmetries  m in subsequent Section 11. The reasonings can be related to the E. Cartan’s general equivalence method [16, 23] and they would deserve more space than it is possible here.Let us turn to proper examples.(ι) The Order-Preserving Symmetry Problem. We again intentionally start with a mere “traditional” case. Let us deal with infinitesimal symmetries Z satisfying
(269)ℒZπ01=μπ01=μ(γ0-F1α0-F1β0),ℒZπ02=λ01π01+λ02π02+λ11π11=μ1α0+μ2β0+μ3γ0.
We use the “hybrid” equations involving both the standard basis and the contact forms. Let us recall the explicit formula (268) for all variations. We have moreover the above equations (269) in order to obtain the true infinitesimal symmetries. In more detail
(270)Z⌋dπ01+dp=μ(γ0-F1α0-F1β0),Z⌋dπ02+dq=μ1α0+μ2β0+μ3γ0
should be satisfied. Analogously as in Section 6, this is expressed by the resolving system(271)zDF1+F11a1+F11Dq=pu0+μF1,zDF1+F11a1+F11Dq=pv0+μF1,F11a0+F11q+pu1=0,F11a0+F11q+pv1=0,(272)pw0=μ,pur=pvr=0(r≥2),qu0=μ1,qv0=μ2,qw0=μ3,qur=0(r≥1),z+qv1=0,qvr=0(r≥2)
by using (264) and β1(Z)=Dβ0(Z)=Dq. It follows that only (271) with μ=pw0,z=-qv1 inserted and coefficients a0,a1 given by
(273)a0DF1+qDF1+Dp=0,a1=Da0
are the most important.Let us denoteΔ=(F11)2-F11F11 and assume  Δ≠0  from now on. Equations (271) are equivalent to
(274)ΔDq=det⁡(pu0+pw0F1+qv1DF1F11pv0+pw0F1+qv1DF1F11),Δa1=det⁡(F11pu0+pw0F1+qv1DF1F11pv0+pw0F1+qv1DF1,),(275)Δq=det⁡(F11pu1F11pv1),Δa0=det⁡(pu1F11pv1F11).
We have unknown functions
(276)p=p(x,u0,v0,w0,u1,v1),q=q(x,u0,v0,w0,v1)
and let us pass to the solution of (273), (274), and (275).The first equation (273) multiplied by function Δ reads
(277)(F11u2+F11v2)Δa0+(F11u2+F11v2)Δq+(𝒟p+pu1u2+pv1v2)Δ=0
and therefore implies only the identity
(278)𝒟p=0(𝒟=∂∂x+u1∂∂u0+v1∂∂v0+F∂∂w0),
if both equations (275) are accepted (direct verification). Alternatively saying, second equation (275) can be regarded for a definition of function  a0 if (278) is taken into account. Let us denote G=Δa0 for a moment. Then
(279)DG=DΔ·a0+Δ·Da0=DΔ·GΔ+Δ·a1,Δ·DG-G·DΔ=a1·(Δ)2
and the second equation (274) reads
(280)Δ·DG-G·DΔ=Δ·det⁡(⋯)
with the same determinant. Lower-order terms clearly provide the equation
(281)𝒟G=det⁡(F11pu0+pw0F1F11pv0+pw0F1)
and coefficients of v2 give
(282)Δ·Gv1-G·Δv1=0,henceG=g(x,u0,v0,w0)Δ
for appropriate function g; however trivially g=a0. With this result, we obtain
(283)Δ(gu1Δ+gΔu1)-gΔ·Δu1=Δdet⁡(F11qv1F11F11qv1F11)=(Δ)2qv1
by inspection of coefficients of v2. It follows that gu1=qv1, whence
(284)g=R(x,u0,v0,w0)u1+S(x,u0,v0,w0),q=R(x,u0,v0,w0)v1+T(x,u0,v0,w0).
Analogously the lower-order terms of the first equation (274) give
(285)Δ·𝒟q=det⁡(pu0+pw0F1F11pv0+pw0F1F11),
while the second-order terms do not provide any new requirements.Let us finally recall (275) with a0=g and q given by (284) inserted. These equations turn into the compatible system
(286)pu1+(Ru1+S)F11+(Rv1+T)F11=0,pv1+(Ru1+S)F11+(Rv1+T)F11=0
for the function p with the solution
(287)p=(F-F1u1-F1v1)R-F1S-F1T+C(C=C(x,u0,v0,w0)).
Then (278) is expressed by the crucial requirement(288)(F-F1u1-F1v1)·𝒟R-F1·𝒟S-F1·𝒟T+𝒟C=0
for the functions R, S, T and C. One can moreover verify with the help of
(289)0=(𝒟p)u1=𝒟(pu1)+pu0+pw0F1,0=(𝒟p)v1=𝒟(pv1)+pv0+pw0F1
that the remaining equations (281) and (285) become identities.Let us summarize our achievements.Assuming  (F11)2≠F11F11,  all infinitesimal symmetries (268) are determined by formula (287), the second equation (284) and z=-qv1=-R  with functions  R,S,T,C  of variables  x0,u0,v0,w0  satisfying (288).Traditional methods are sufficient to analyze thoroughly (288). Passing to more details, we have
(290)(F-F1u1-F1v1)(u1Ru0+v1Rv0+FRw0)-F1(Sx+u1(Su0+Rx)+v1Sv0+FSw0)-F1(Tx+u1Tu0+v1(Tv0+Rx)+FTw0)+Cx+u1Cu0+v1Cv0+F(Cw0+Rx)=0.
Analogously as in Section 6, the large series of coefficients
(291)Fu1,F1(u1)2,F1v1u1,Fv1,F1u1v1,…,1,u1,v1,F
appears. If these functions are ℝ-linearly independent, only the solution R,S,T,C such that
(292)Ru0=Rv0=Rw0=Sx=Su0+Rx=⋯=Cv0=Cw0+Rx=0
is possible. It follows that
(293)R=a1x+a2,S=-a1u0+a3,T=-a1v0+a4,C=-a1w0+a5,
where a1,…,a5∈ℝ are arbitrary constants. This result provides the obvious symmetries which are self-evident at a first glance, the coordinate shifts and the similarity.For a special choice of functionF, the symmetry group may be very large and less trivial. We can mention the case F=u1v1. Then the arising system of five equations
(294)Rx+Su0+Tv0-Cw0=Ru0+Tw0=Rv0+Sw0=Tx-Cu0=Sx-Cv0=0
for the unknown functions
(295)R=R(x,u0,v0),S=S(x,u0,w0),T=T(x,v0,w0)=C(u0,v0,w0),
can be resolved by
(296)R=a1x+a2u0+a3v0+a4,S=a5x+a6u0-a3w0+a7,T=a8x+a9v0-a2w0+a10,C=a8u0+a6v0+(a1+a6+a9)w0+a11,
where a1,…,a11∈ℝ are arbitrary constants.We omit more examples, in particular the interesting cases (withℝ-linear dependence of functions F-F1u1-F1v1,F1,F1,1) where the infinitesimal symmetries depend on arbitrary functions and the “degenerate” cases when either Δ=0 or DF1=DF1=0 identically.(ιι) The Order-Increasing Infinitesimal Symmetry. Let us mention variations (268) satisfying moreover the equations
(297)ℒZπ01=μ01π01+μ02π02+μ12π12,ℒZπ02=μπ02
which provide the order-increasing case, if μ12≠0. One can then obtain the resolving system(298)zDF1+F11a1+F11Dq-pu0-pw0F1=F11a0+F11q+pu1=z+qv1=0
for the unknown functions
(299)p=p(x,u0,v0,w0,u1,v1),q=q(x,v0,v1),z=z(x,v0,v1)
and moreover formula
(300)μ12=F11a0+F11q+pv1=-F11qDF1+DpDF1+F11q+pv1
for the coefficient μ12. We mention only the particular case F=u1v1. Then the resolving system reads -qv1v2+Dq-pu0-pw0v1=q+pu1=0 and admits the solution
(301)p=-qu1+(qx+qv0v1)u0+P(x,v1u0-w0,v0,v1),
where the functions q=q(x,v0,v1) and P=P(x,v1u0-w0,v0,v1) may be arbitrarily chosen. Since the above coefficient
(302)μ12=-1v2(qu2+Dp)+pv1
does not in general vanish, we have a large family of order-increasing infinitesimal symmetries.(ιιι) Another Order-Increasing Case. Let us mention variations (268) satisfying the equations
(303)ℒZπ01=λ1π01+λ2π02,ℒZπ02=μ1π01+μ2π02
which provide an order-increasing case if λ2≠0. The resolving system(304)zDF1+F11a1+F11Dq-pu0-pw0F1=0,F11a0+F11q+pu1=F11a0+F11q+pv1=0,qu0+qw0F1=qv1+z=0
for the unknown functions
(305)p=p(x,u0,v0,w0,u1,v1),q=q(x,u0,v0,w0,v1),z=z(x,u0,v0,w0,v1)
looks more complicated. One can also obtain the formula
(306)λ2=qv1DF1-F11a1-F11Dq+pv0+pw0F1
for the important coefficient λ2. Let us again mention only the particular case F=u1v1. Then the resolving system is simplified as
(307)qv1v2-Dq+pu0+pw0v1=q+pu1=qu2+Dp-pv1v2=qu0+qw0v1=0.
It follows immediately that p=-qu1+P and the resolving system is reduced to the equations
(308)𝒟q=Pu0+Pw0v1,u1𝒟q=𝒟P,qu0+qw0v1=0(𝒟=∂∂x+u1∂∂u0+v1∂∂v0+F∂∂w0)
for the unknown functions P and q of variables x,u0,v0,w0,v1. This implies that q=Q(x,w,v0,v1), where w=w0-v1u0 and we obtain two equations
(309)Qx+Qv0v1=Pu0+Pw0v1,Px+Pv0v1=0
with the solution P=P-(v0-xv1,w0-u0v1,v1)+Q-, where P- may be arbitrary function while Q-=Q-(x,u0,v0,w0,v1) is a fixed particular solution of differential equation
(310)Qx+Qv0v1=Q-u0+Q-w0v1
satisfying moreover the identity Q-x+Q-v0v1=0. We may choose the particular solution Q-=(Qx+Qv0v1)u0. Then the identity turns into the requirement (∂/∂x+v1∂/∂v0)2Q=0 which is satisfied if
(311)Q=Q1(w,v1)x+Q0(w,v1)(w=w0-v1u0).
Altogether taken, we have obtained the final solution
(312)p=-qu1+P-+(Qx+Qv0v1)u0,q=Q=Q1x+Q0,
where
(313)P-=P-(v0-xv1,w,v1),Q1=Q1(w,v1),Q0=Q0(w,v1),w=w0-v1u0
and P-, Q1, Q0 are quite arbitrary functions. The above-mentioned coefficient λ2 does not in general vanish. (Indeed, look at the top-order summands
(314)λ2=⋯-F11a1+⋯=⋯-a1+⋯,
where v2a0=⋯+Dp by virtue of (263); hence, v2a1=⋯+D2p=⋯-D2(qu1)=⋯-qu3 may be substituted.) We again have an order-increasing infinitesimal symmetry.Remark 45.
VariationsZ satisfying (269) preserve the Pfaffian system π01=π02=π11=0 and therefore generate a group for analogous reasons as in Remark 29. Variations Z satisfying (297) preserve the Pfaffian system π01=π02=π12=0 and the case of requirements (303) is quite trivial in this respect. It follows that we have indeed obtained the infinitesimal symmetries  Z.


## 11. Concluding Example on Order-Increasing Symmetries

Passing from infinitesimal symmetriesZ to the true symmetries m, the linear theory is replaced with highly nonlinear area of Pfaffian equations and the prolongation into involutiveness. In accordance with E. Cartan’s notice, nobody should expect such easily available results as in the Lie’s infinitesimal theory. Our modest aim is twofold: to perform an economical reduction of the symmetry problem to finite dimension and to point out a useful interrelation between appropriate variations  Z and one-parameter families  m(t) of higher-order symmetries. We again deal only with (260).(ι) Setting the Problem. Let us deal with symmetries m such that
(315)m*π0i=aiα0+biβ0+ciγ0=(ai+F1ci)α0+(bi+F1ci)β0+ciπ01(i=1,2).
Invertibility of m is obviously ensured if
(316)det⁡(a1+F1c1a2+F1c2b1+F1c1b2+F1c2)=0,det⁡(a1a2Da1-uDa2b1b2Db1-uDb2c1c2Dc1-uDc2)≠0,
where
(317)u=a1+F1c1a2+F1c2=b1+F1c1b2+F1c2.
We tacitly suppose a2+F1c2≠0,b2+F1c2≠0 and one can observe that the particular case u=0 provides the traditional order-preserving symmetries. Equations (315) can be simplified to the equivalent system of equations
(318)m*π01-um*π02=vπ01,m*π02=aα0+bβ0+cγ0,
where v=c1-uc2 and a=a2,b=b2,c=c2. The invertibility is ensured by the inequalities
(319)v≠0,det⁡(aF1DF1bF1DF1c-10)≠0.
Equations (318) will be represented by a Pfaffian system in a certain finite-dimensional space; however, let us again simplify the notation by bars; for example,
(320)x-=m*x,π-ri=m*πri,α-r=m*αr,β-r=π-r2=m*βr=m*πr2,
and so like. Then we have the system
(321)π-01-uπ-02=vπ01,π-02=aα0+bβ0+cγ0
which should be completed by the exterior derivatives
(322)dπ-01-udπ-02-du∧π-02=dv∧π01+vdπ01,dπ-02=da∧α0+db∧β0+dc∧γ0+dx∧((a+F1c)α1+(b+F1c)β1).
We refer to (264) for terms dπ01, dπ-01 appearing here. We have obtained the compatibility problem of (322). The familiar prolongation criterion can be shortly expressed as follows. All coefficients and variables with bars are functions of the primary jet variables. So we may suppose, for example,
(323)du=Udx+∑‍ur1αr+∑‍ur2βr+u03γ0(U=Du)
(with summands of uncertain lengths) and analogously for dv,da,db,dc with a little adjustment for the differential
(324)dx-=λ(dx+∑‍λr1αr+∑‍λr2βr+λ03γ0)(λ=Dx-=Dm*x=DW).
Such substitutions into (322) should give identities. However, a short inspection of the summand F-11α-0∧α-1 in differential dπ-01 implies that then necessarily either u=0 (the group case) or F11=0 identically.(ιι) A Particular Case. It follows that the assumption F=f(v1)u1+g(v1) is necessary; however, let us again suppose F=u1v1 from now on. Then (321) may be retained and (322) become more explicit
(325)dx-∧(π-11-uπ-12)+β-0∧α-1+α-0∧β-1-du∧π-02=dv∧π01+v(dx∧π11+β0∧α1+α0∧β1),(326)dx-∧π-12=da∧α0+db∧β0+dc∧γ0+dx∧((a+v1c)α1+(b+u1c)β1).
We turn to the prolongation procedure in more detail.(ιιι) On the Equation (326). The prolongation should satisfy the identity
(327)λ(dx+∑‍λr1αr+∑‍λr2βr+λ03γ0)∧1λ(Aα0+Bβ0+Cγ0+(a+v1c)α1+(b+u1c)β1)=(Adx+∑‍ar1αr+∑‍ar2βr+a03γ0)∧α0+⋯+(Cdx+∑‍cr1αr+∑‍cr2βr+c03γ0)∧γ0+dx∧((a+v1c)α1+(b+u1c)β1),
where A=Da,B=Db,C=Dc. All summands with factor dx∧ mutually cancel. Then we conclude that necessarily λr1=λr2=0(r>1) and also ar1=⋯=cr2=0(r>1). It follows that the problem is reduced to finite dimension:  x-,a,b,c are functions only of coordinates x,u0,v0,u1,v1. Even the explicit formulae can be easily obtained as follows
(328)a11=λ11A-λ01(a+v1c),…,b12=λ12B-λ02(b+u1c),b01-a02=λ01B-λ02A,c01-a03=λ01C-λ03A,c02-b03=λ02C-λ03B
for the prolongation where moreover λ11(b+u1c)=λ12(a+v1c) is supposed.(ιν) On the Equation (325). Calculations modulo dx are also sufficient here. The prolongation should satisfy
(329)(λ01α0+λ02β0+λ03γ0+λ11α1+λ12β1)∧(Vπ01+vπ11+Uπ-02)-(∑‍ur1αr+∑‍ur2βr+u03γ0)∧π-02-(∑‍vr1αr+∑‍vr2βr+v03γ0)∧π01=-β-0∧α-1-α-0∧β-1+v(β0∧α1+α0∧β1),
where U=Du,V=Dv. We have used the identity
(330)π-11-uπ-12=1λℒD(π-01-uπ-02)+Uλπ-02,
where π-01-uπ-02=vπ01 is moreover substituted. In order to prove the existence of prolongation, the right-hand side terms should be made more explicit. We recall the identity (263) which gives
(331)π11+v2α0+u2π02=0,π21+v2α0+v1α1+u3π02+u2π12=0
and uniquely determines the forms α-0 and α-1 in terms of forms π-11,π-02,π-21,π-12 and therefore in terms of contact forms αr,βr, and γ0, if the rule (265) is applied to the primary equations (321). The result is that
(332)β-0∧α-1=π-02∧(acertainsumofformsα0,β0,γ0,α1,β1,α2,β2)α-0∧β-1=-1v-2(u-2π-02+π-11)∧(acertainsumofformsα0,β0,γ0,α1,β1).
On the other hand, inequalities (266) imply that the factors
(333)Vπ01+vπ11+Uπ-02,π-02,π11
on the left-hand side of (329) are linearly independent and we conclude that the prolongation can be realized. Moreover u becomes a function of second-order coordinates while x and v are functions of first-order coordinates x,u0,v0,w0,u1,v as before. The problem is again reduced to finite-dimension; however, we do not state explicit formula for the prolongation here.Remark 46.
In accordance with Lie’s classical theory, the existence ofinfinitesimal symmetries  Z (Figure 5(a)) is equivalent to the existence of a one-parameter group  m(λ) of symmetries (Figures 3(a) and 3(b)) due to the solvability of the Lie system ensured by Theorem 24. Alas, the “genuine” higher-order symmetries (Figure 3(c)) cannot be obtained in this way and they rest on the toilsome mechanisms of Pfaffian systems. We nevertheless propose a hopeful conjecture as follows. Every one-parameter family  m(t) of symmetries ensures the existence of many variations  Z(t) depending on parameter t (Figure 5(b)). We believe that the converse can be proved as well: one-parameter families of symmetries can be reconstructed from a “sufficiently large” supply of variations. Indeed, if t is regarded as additional variable of the underlying space, then the family Z(t) turns into a single vector field.In any case, the existence of many variations is anecessary condition for the existence of “genuine” higher-order symmetries and the following point (ν) will be instructive in this respect.(ν) On the Variations. If a one-parameter family m(t)=m (abbreviation) satisfies (318) then the corresponding family Z(t)=Z (abbreviation) of variations clearly satisfies the system
(334)ℒZπ01-uℒZπ02-u′π02=v′π01,ℒZπ02=a′α0+b′β0+c′γ0,
where u′=Zu,…,c′=Zc may be regarded as new parameters. Assuming formula (268), one can obtain the resolving system
(335)zv2+Dq-pu0-a′v1+uqu0=qu0+a′v1+c′v2=0,(336)zu2+a1-pv0-a′u1+u′+uqv0=qv0+a′u1-b′+c′u2=0,(337)q+pu1=qu1=a0+pv1=z+qv1=pw0-v′-uqw0=qw0-a′=0.
It follows from right-hand equations (337) that p=-qu1+q-, where q,q- do not depend on u1. Recalling the identity
(338)a0v2+qu2+Dp=0,
then the middle equations (337) yield the conditions
(339)qu0+qw0v1=q-x+q-v0v1=qx+qv0v1-q-u0-q-w0v1=0
and q=q(x,u0,v0,v1,w0),q-=q-(x,u0,v0,v1,w0). With this result, (335) turns into identity and (336) reduces to the equation
(340)u′=u(qw0u1-qv0)+u2qv1+D(qv1u1-q-v1)+q-v0
for the parameter u′. Equations (339) are trivially satisfied if
(341)q=Q(xv1-v0,u0v1-w0,v1),q-=Q-(xv1-v0,u0v1-w0,v1).There exist many variations corresponding to (318). The necessary condition for the existence of higher-order symmetries is satisfied.Remark 47.
Let us briefly sketch the connection to the general equivalence method [23] by using slightly adapted Cartan’s notation. We consider space ℝn (and its counterpart ℝ-n) with coordinates (x)=(x1,…,xn)(or(x-)=(x-1,…,x-n), resp.) and linearly independent 1-forms ω1,…,ωn(andω-1,…,ω-n). In the classical equivalence problem, a mapping m should be determined such that
(342)m*ω-i=∑‍aij(u)ωj(i,j=1,…,n;(x)=m*(x-),u=u(x)),
where (aij(u)) is a matrix of a linear group with parameters (u)=(u1,…,ur). In Cartan’s approach, this requirement is made symmetrical:
(343)m*∑‍aij(u-)ω-i=∑‍aij(u)ωj((x)=m*(x-),u-=u-(x,u)).
This provides the invariant differential forms by appropriate simultaneous adjustments of both sides (343). Such procedure fails, if (aij(x)) is not a matrix of a linear group which happens just in the case of higher-order symmetries on Figure 3(c). Then the corresponding total system (342) with i,j=1,2,… is invertible only in the infinite-dimensional underlying space M and (aij(u)) need not be even a square matrix in any finite portion of the system (342). On the other hand, such a finite portion is quite sufficient since Lemma 17 ensures the extension on the total space M. The “symmetrization” procedure cannot be applied, invariant differential forms need not exist, and only the common prolongation procedure is available, if the problem is reduced to a finite-dimensional subspace of M.


## 12. Concluding Survey

Our approach to differential equations and our methods differ from the common traditional use. For better clarity, let us briefly report the main novelties as follows: clear interrelation between the external and internal concepts in Remarks1 and 2; introduction and frequent use of “nonholonomic” series (18); the “absolute” and coordinate-free Definition 4 of ordinary differential equations; the distinction between variations and infinitesimal symmetries in Definition 8; the main tool, the standard bases generalizing the common contact forms in jet spaces; the invariance of constants K=K(Ω) and μ=μ(Ω), the controllability concept related to the Mayer problem; the distinction between order-preserving, group-like, and true higher-order symmetries in Figure 1; technical Lemmas 17, 19, and 23 and Theorem 24 which provide new universal method of solution of the higher-order symmetry problem; new explicit formula as (136) for the famous and “well-known” symmetry problem of a Monge equation with two unknown functions; the Lagrange variational problem without Lagrange multipliers and with easy proofs; see Theorem 41; particular results of new kind for the Monge equation with three unknown functions; a note on the insufficience of G-structures in Remark 47.Figure 1(a)
Non-contrillable case
(b)
Mayer extermalFigure 3Figure 4(a)
Vector fieldZ:ZP(λ)=(d/dλ)m(λ)P
(b)
VariationsZ(t):Z(t)P(t)=(d/dt)m(t)PAll these achievements can be carried over the partial differential equations.On this occasion, the actual extensive theory of thecontrol systems(344)dxdt=f(x,u)(t∈ℝ,x∈ℝn,u∈ℝm)
is worth mentioning. It may be regarded as a mere formally adapted individual subcase of the theory of underdetermined systems of ordinary differential equations. However, the exceptional role of theindependent variable  t (the change of notation), the state variables  x, and the control  u is emphasized in applications; see [24–26] and references therein. In particular, only the t-preserving and moreover t-independent symmetries of the system (344) are accepted. So in our notation (1), such restriction means that we suppose x-=W=x and functions Wi are independent of x. This is a fatal restriction of the impact of the theory of control systems. It follows that the results of this theory do not imply the classical results by Lie and Cartan; they are of rather special nature. The lack of new effective methods adapted to the control systems theory should be moreover noted. The absence of explicit solutions of particular examples is also symptomatic. Last but not least, unlike our diffieties, the control systems cannot be reasonably generalized for the partial differential equations.We believe that the internal and higher-order approach to some nonholonomic theories are possible, for instance, in the case of the higher-order subriemannian geometry [12]. It seems that the advanced results [27] in the theory of geodesics can be appropriately adapted and rephrased in terms of invariants (as in [28]) instead of adjoint tensor fields.



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