Applications to the Economy of Axioms

We now come back to the end of Chap. 3, Sect. 3.9, where the three standard group axioms (composition; identity element; existence of inverses) were superseded by the hypothesis of existence of differential equations. We remind that in the proof of Lemma 3.3, a relocalization was needed to assure that detψk j(a)6=0. For the sake of clarity and of rigor, we will, in the hypotheses, explicitly mention the subdomain A¹⊂A where the determinant of theψk j(a)does not vanish.

The following (apparently technical) theorem which is a mild modification of the Theorem 9 on p. 72 of [2], will be used in an essential way by Lie to derive his famous three fundamental theorems in Chap. 9 below.

Theorem 9.If, in the transformation equations defined for(x,a)∈X ×A:

(1) x⁰_i=f_i(x₁, . . . ,x_n;a₁, . . . ,a_r) (i=1···n),

the r parameters a₁, . . . ,a_rare all essential and if in addition, certain differential equations of the form:

(2) ∂^x0i

∂^ak

=

∑

r j=1

ψk j(a₁, . . . ,a_r)ξji(x⁰₁, . . . ,x⁰_n) (i=1···n;k=1···r)

are identically satisfied by x⁰₁= f₁(x;a), . . . ,x⁰_n= f_n(x;a), where the matrix ψk j(a)is holomorphic and invertible in some nonempty subdomainA¹⊂A, and where the functionsξji(x⁰)are holomorphic inX, then by introducing the r infinitesimal transformations:

X_k:=

∑

n i=1

ξki(x) ∂

∂^xi

,

it holds true that every transformation x⁰_i=f_i(x;a)whose parameters a₁, . . . ,a_r lie in a small neighborhood of some fixed a⁰∈A¹can be obtained by firstly performing the transformation:

x_i=f_i(x₁, . . . ,x_n;a⁰₁, . . . ,a⁰_r) (i=1···n), and then secondly, by performing a certain transformation:

x⁰_i=exp tλ1X₁+···+tλrX_r

(x_i) (i=1···n)

4.6 Applications to the Economy of Axioms 83

of the one-term group generated by some suitable linear combination of the X_k, where t andλ1, . . . ,λrare small complex numbers.

Especially, this technical statement will be useful later to show that wheneverr infinitesimal transformationsX₁, . . . ,X_rform a Lie algebra, the composition of two transformations of the formx⁰=exp tλ1X₁+···+tλrX_r

is again of the same form, hence the totality of these transformations truly constitutes a group.

Proof. The arguments are essentially the same as the ones developed at the end of the previous section (p. 81) for a genuinely local continuous transformation group, except that the identity parametere(which does not necessarily exist here) should be replaced bya⁰.

For the sake of completeness, let us perform the proof. On the first hand, we fixa⁰∈A¹and we introduce the solutionsa_k=a_k(t,λ1, . . . ,λr)of the following system of ordinary differential equations:

da_k dt =

∑

r j=1

λjαjk(a) (k=1···r),

with initial conditiona_k(0,λ1, . . . ,λr) =a⁰_k, whereλ1, . . . ,λrare small complex pa-rameters and where, as before,αjk(a)denotes the inverse matrix ofψjk(a), which is holomorphic in the whole ofA1.

On the second hand, we introduce the local flow:

exp tλ1X₁+···+tλrX_r

(x) =:h x;t,λ

of the general linear combinationλ1X₁+···+λrX_r of therinfinitesimal transfor-mations X_k=∑ⁿi=1ξki(x)_∂^∂_x

i, wherexis assumed to run inA¹. Thus by its very definition, this flow integrates the ordinary differential equations:

dh_i dt =

∑

r j=1

λjξji(h₁, . . . ,h_n) (i=1···n)

with the initial conditionh x; 0,λ^=x.

On the third hand, we solve first the ξji in the fundamental differential equa-tions (2) using the inverse matrixα^:

ξji(f₁, . . . ,f_n) =

∑

r k=1

αjk(a)∂^fi

∂^ak

(i=1···n; j=1···r).

Then we multiply byλj, we sum and we recognize^da_dt^k, which we then substitute:

∑

r addition, if we set xequal to f(x;a⁰), both collections of solutions will have the sameinitial value fort=0, namely f(x;a⁰). In conclusion, by observing that thef_i and theh_isatisfy the same equations, the uniqueness property enjoyed by first order ordinary differential equations yields the identity:

f x;a(t,λ⁾≡exp t₁λ1X₁+···+tλrX_r

f(x;a⁰)

expressing that every transformationx⁰= f(x;a)for ain a neighbourhood of a⁰ appears to be the composition of the fixed transformationx=f(x;a⁰)followed by a certain transformation of the one-term group exp tλ1X1+···+tλrXr

(x). ut

§ 18.

We now apply the preceding general developments to the peculiar case where the∞^rtransformationsx⁰_i=f_i(x₁, . . . ,x_n,a₁, . . . ,a_r)consitute anr-term group.

If the equations (1) represent anr-term group, then according to Theorem 3 p. 40, there always are differential equations of the form (2); so we do not need to specially enunciate this requirement.

Moreover, we observe that all infinitesimal transformations of the form

∑ⁿi=1

n∑^rj=1λjξji(x) o∂f

∂xi can be linearly expressed by means of the following rinfinitesimal transformations:

since all the infinitesimal transformations (9) are contained in the expression:

∑

r k=1

λkX_k(f).

Here, as we have underscored already in the introduction of this chapter, the infinitesimal transformationsX₁(f), . . . ,X_r(f)are independent of each other.

Consequently, we can state the following theorem about arbitrary r-term groups:

Theorem 10.To every r-term group:

4.6 Applications to the Economy of Axioms 85

x⁰_i= f_i(x₁, . . . ,x_n,a₁, . . . ,a_r) (i=1···n)

are associated r independent infinitesimal transformations:

X_k(f) =

∑

n i=1

ξki(x₁, . . . ,x_n)∂^f

∂^xi

(i=1···n),

which stand in the following relationship to the finite transformations of the group: if x⁰_i= f_i(x₁, . . . ,x_n;a⁰₁, . . . ,a⁰_r)is any transformation of the group, then every transformation x⁰_i = f_i(x,a) whose parameter lies in a certain neigh-bourhood of a⁰₁, . . . ,a⁰_r can be obtained by firstly executing the transforma-tion x_i= f_i(x₁, . . . ,x_n,a⁰₁, . . . ,a⁰_r)and secondly a certain transformation x⁰_i=

ωi(x₁, . . . ,x_n) of a one-term group, the infinitesimal transformation of which

has the formλ1X₁(f) +···+λrX_r(f), whereλ1, . . . ,λr denote certain suitably chosen constants.

If we not only know that the equationsx⁰_i=f_i(x,a)represent anr-term group, but also that this group contains the identity transformation, and lastly also that the parameters a⁰_k represent a system of values of the identity transformation in the domain ((a_k)), then we can still say more. Indeed, if we in particular choose for the transformationx_i=f_i(x,a⁰)the identity transformation, we then realize immediately that the transformations of our group are nothing but the transformations of those one-term groups that are generated by the infinitesimal transformations:

λ1X1(f) +···+λrXr(f).

Thus, if for abbreviation we set:

∑

r k=1

λkX_k(f) =C_k(f),

then the equations:

x⁰_i=x_i+t

1C(x_i) + t²

1·2C C(x_i)

+··· ^(i=1···n)

represent the∞^rtransformations of the group. The fact that ther+1 parameters:

λ1, . . . ,λr,tappear is just fictitious here, for they are indeed only found in ther

combinationsλ1t, . . . ,λrt. We can therefore quietly settequal to 1. In addition, if we remember the representation of a one-term group by a single equation given in eq. (3) on p. 67, then we realize that the equations of ourr-term group may be condensed in the single equation:

f(x⁰₁, . . . ,x⁰_n) =f(x₁, . . . ,x_n) +C(f) + 1

1·2C C(f) +···.

That it is possible to order the transformations of the group as inverses in pairs thus requires hardly any mention.

We can briefly state as follows the above result about ther-term groups with identity transformation.

Theorem 11.If an r-term group:

x⁰_i= f_i(x₁, . . . ,x_n,a₁, . . . ,a_r) (i=1···n)

contains the identity transformation, then its∞^r transformations can be orga-nized in∞^r−1 families of∞¹ transformations in such a way that each family amongst these ∞^r−1 families consists of all the transformations of a certain one-term group with the identity transformation. In order to find these one-term groups, one forms the known equations:

∂^x0i

∂^ak

=

∑

r j=1

ψk j(a₁, . . . ,a_r)ξji(x⁰₁, . . . ,x⁰_n) (i=1···n;k=1···r),

which are identically satisfied after substituting x⁰_i=f_i(x,a), one further sets:

∑

n i=1

ξki(x₁, . . . ,xn)∂^f

∂^xi

=X_k(f) (k=1···r), hence the expression:

∑

r k=1

λkX_k(f)

with the r arbitrary parametersλ1, . . . ,λr represents the infinitesimal transfor-mations of these∞^r−1one-term groups, and their finite equations have the form:

x⁰_i=x_i+

∑

r k=1

λkξki(x) +

1...r

∑

k,j

λkλj

1·2 X_k(ξji) +··· (i=1···n). The totality of all these finite transformations is identical with the totality of all transformations of the group x_i⁰= f_i(x,a). Besides, the transformations of this group can be ordered as inverses in pairs.

§ 19.

In general, if anr-term group contains all transformations of some one-term group and if, in the sense discussed earlier, this one-term group is generated by the infinitesimal transformationX(f), then we say that ther-term group con-tains the infinitesimal transformationX(f). Now, we have just seen that every r-term group with the identity transformation can be brought to the form:

4.6 Applications to the Economy of Axioms 87 mutually independent infinitesimal transformations. So we can say that every such r-term group contains r independent infinitesimal transformations.

One is very close to presume that anr-term group cannot contain more than rindependent infinitesimal transformations.

In order to clarify this point, we want to consider directly the question of when the infinitesimal transformation:

is contained in ther-term group with therindependent infinitesimal transfor-mationsX₁(f), . . . ,X_r(f).

IfY(f)belongs to ther-term group in question, then the same is also true of the transformations:

x⁰_i=x_i+τ

1ηi(x) + τ²

1·2Y(ηi) +··· (i=1···n)

of the one-term group generated byY(f). Hence if we execute first an arbitrary transformation of this one-term group and after an arbitrary transformation:

x⁰⁰_i =x⁰_i+

of ther-term group, we then must obtain a transformation which also belongs to ther-term group. By a calculation, we find that this new transformation has the form:

x⁰⁰_i =x_i+τ ηi(x) +

∑

r k=1

λkξki(x) +··· (i=1···n),

where all the left out terms are of second and of higher order with re-spect to λ1, . . . ,λr,τ. For arbitrary λ1, . . . ,λr,τ, this transformation must be-long to the r-term group. Now, if the r+1 infinitesimal transformations X₁(f), . . . ,X_r(f),Y(f) were independent of each other, then according to the Proposition 4 p. 80, the last written equations would represent∞^r+1 transfor-mations; but this is impossible, for ther-term group contains in general only

∞^r transformations. Consequently,X₁(f), . . . ,Xr(f),Y(f)are not independent

of each other, but sinceX₁(f), . . . ,X_r(f)are so, thenY(f)must be linearly

wherel₁, . . . ,l_rdenote appropriate constants. That is to say, the following holds:

Proposition 2.If an r-term group contains the identity transformation, then it contains r independent infinitesimal transformations X₁(f), . . . ,X_r(f)and every infinitesimal transformation contained in it possesses the formλ1X₁(f) +···+ λrX_r(f), whereλ1, . . . ,λrdenote constants.

We even saw above that every infinitesimal transformation of the form λ1X₁(f) +···+λrX_r(f) belongs to the group; hence in future, we shall call the expressionλ1X₁(f) +···+λrX_r(f)with therarbitrary constantsλ1, . . . ,λr

the general infinitesimal transformationof ther-term group in question.

From the preceding considerations, it also comes the following certainly spe-cial, but nevertheless important:

Proposition 2.If an r-term group contains the m6r mutually independent infinitesimal transformations X1(f), . . . ,Xr(f), then it also contains every in-finitesimal transformation of the following form: λ1X1(f) +···+λmXm(f), whereλ1, . . . ,λmdenote completely arbitrary constants.

Of course, the researches done so far give the means to determine the in-finitesimal transformations of anr-term groupx⁰_i= f_i(x,a) with the identity transformation. But it is possible to reach the objective more rapidly.

Let the identity transformation of our group go with the parameters:

a⁰₁, . . . ,a⁰_r, and leta⁰₁, . . . ,a⁰_r lie in the domain ((a)), so that the determinant

4.6 Applications to the Economy of Axioms 89

Now, sinceX₁(f), . . . ,X_r(f)are independent infinitesimal transformations and since in addition the determinant of theψk j(a⁰)is different from zero, the right-hand sides of the latter equations representrindependent infinitesimal transfor-mations of our group.

The following method is even somewhat simpler.

One setsa_k=a⁰_k+δ^tk, where it is understood thatδ^t1, . . . ,δ^trare infinitely small quantities. Then it comes:

x⁰_i=f_i x₁, . . . ,x_n,a⁰₁+δ^t1, . . . ,a⁰_r+δ^tr

=x_i+

∑

r k=1

∂

∂^ak

f_i(x,a)

a=a⁰

δ^tk+···,

where the left out terms are of second and of higher order with respect to theδ^tk. Here, it is now immediately apparent that our group contains therinfinitesimal transformations:

x⁰_i=xi+ ∂

∂^ak

fi(x,a)

a=a₀

δ^tk (i=1···n) (k=1···r)

However, the question whether theserinfinitesimal transformations are inde-pendent of each other requires in each individual case yet a specific examina-tion, if one does not know from the beginning that the determinant of theψ^does not vanish fora_k=a⁰_k.

Example.We consider the general projective group:

x⁰= x+a1

a₂x+a₃ of the once-extended manifold.

The infinitesimal transformations of this group are obtained very easily by means of the method shown right now. Indeed, one hasa⁰₁=0,a⁰₂=0,a⁰₃=1, hence we have:

x⁰= x+δ^t1

xδ^t2+1+δ^t3

=x+δ^t1−xδ^t3−x²δ^t2+···,

that is to say, our group contains the three mutually independent infinitesimal transformations:

X₁(f) =df

dx, X₂(f) =xdf

dx, X₃(f) =x²df dx. The general infinitesimal transformation of our group has the form:

λ1+λ2x+λ3x²df dx,

hence we obtain its finite transformations by integrating the ordinary differential equation:

dx⁰

λ1+λ2x⁰+λ3x⁰²=dt, adding the initial condition:x⁰=xfort=0.

In order to carry out this integration, we bring the differential equation to the form:

dx⁰ x⁰−α ⁻

dx⁰

x⁰−β ⁼γ^dt, by setting:

λ1= αβγ

α−β^, λ2=−α⁺β

α−βγ^, λ3= γ α−β^, whence:

2α⁼−λ2

λ3

+

qλ₂²−4λ1λ3

λ3

, 2β ⁼−λ2

λ3−

qλ₂²−4λ1λ3

λ3

,

γ^=qλ2²−4λ1λ3. By integration, we find:

l(x⁰−α⁾−l(x⁰−β^{) =}γ^t+l(x−α⁾−l(x−β^), or:

x⁰−α x⁰−β ^=eγt

x−α x−β^,

and now there is absolutely no difficulty to expressα^,β^,γin terms ofλ1,λ2,λ3, in order to receive in this way the∞³transformations of our three-term group arranged in∞²one-term groups, exactly as is enunciated in Theorem 11.

Besides, a simple known form of our group is obtained if one keeps the two parametersα ^andβ, while one introduces the new parameterγ ^{instead ofeγt;} then our group appears under the form:

x⁰−α

x⁰−β ⁼γ^x−α x−β^.

References 91

References

1. Arnol’d, V.I.: Ordinary differential equations. Translated from the Russian and edited by R.A. Silverman, MIT Press, Cambridge, Mass.-London (1978)

2. Engel, F., Lie, S.: Theorie der Transformationsgruppen. Erster Abschnitt. Unter Mitwirkung von Prof. Dr. Friedrich Engel, bearbeitet von Sophus Lie, Verlag und Druck von B.G. Teubner, Leipzig und Berlin, xii+638 pp. (1888). Reprinted by Chelsea Publishing Co., New York, N.Y.

(1970)

3. Gröbner, W.: Die Lie-Reihen und ihre Anwendungen. Math. Monog. Veb Deutschen Verlag der Wissenschaften (1960)

4. Rao, M.R.M.: Ordinary differential equations, theory and applications. Edward Arnold, Lon-don (1981)

—————–

Part II

English Translation

Chapter 5

Complete Systems of Partial Differential Equations

Abstract Any infinitesimal transformationX =∑ⁿi=1ξi(x)_∂^∂_x

i can be consid-ered as the first order analytic partial differential equationXω⁼0 with the un-knownω. After a relocalization, a renumbering and a rescaling, one may supp-poseξn(x)≡1. Then the general solutionωhappens to be any (local, analytic) functionΩ ω1, . . . ,ωn−1

of the(n−1)functionally independent solutions de-fined by the formula:

ωk(x):=exp −x_nX

(x_k) (k=1···n−1).

What about first ordersystems X₁ω⁼···=X_qω⁼0 of such differential equa-tions? Any solutionω trivially satisfies alsoX_i X_k(ω⁾−X_k X_i(ω^{)=0. But} it appears that the subtraction in the Jacobi commutatorX_i X_k(·)

−X_k X_i(·) kills all the second-order differentiation terms, so that one may freely add such supplementary first-order differential equations to the original system, contin-uing again and again, until the system, still denoted byX₁ω⁼···=X_qω^=0, becomescompletein the sense of Clebsch, namely satisfies, locally in a neigh-borhood of a generic pointx⁰:

(i)for all indicesi,k=1, . . . ,q, there are appropriate functionsχikµ(x)so that X_i X_k(f)

−X_k X_i(f)

=χik1(x)X₁(f) +···+χikq(x)X_q(f);

(ii)the rank of the vector space generated by theqvectorsX₁

x, . . . ,X_q

x is constant equal toqfor allxnear the central pointx⁰.

Under these assumptions, it is shown in this chapter that there aren−q func-tionally independent solutionsx^(q)₁ , . . . ,x^(q)_n−qof the system that are analytic near x₀such that any other solution is a suitable function of thesen−qfundamental solutions.

95

First Order Scalar Partial Differential Equation

As a prologue, we ask what are the general solutionsω of a first order partial differential equationXω⁼0 naturally associated to a local analytic vector field X =∑ⁿi=1ξi(x)_∂^∂_x

i. Free relocalization being always allowed in the theory of Lie, we may assume, after possibly renumbering the variables, thatξndoes not vanish in a small neighborhood of some point at which we center the origin of the coordinates. Dividing then byξn(x), it is equivalent to seek functionsω^that are annihilated by the differential operator:

X=

still denoted byX and which now satisfiesX(x_n)≡1. We recall that the corre-sponding system of ordinary differential equations which defines curves that are everywhere tangent toX, namely the system:

dx₁ with initial condition fort=0 being an arbitrary point of the hyperplane{xn= 0}:

x₁(0) =x₁, . . . ,x_n−1(0) =x_n−1, x_n(0) =0

issolvableand has a unique vectorial solution(x₁, . . . ,x_n−1,x_n)which is ana-lytic in a neighborhood of the origin. In fact,x_n(t) =tby an obvious integration, and the(n−1)otherx_k(t)are given by the marvelous exponential formula al-ready shown on p. 67:

x_k(t) =exp(tX)(x_k) =

∑

l>0

t^l

l!X^l(x_k) (k=1···n−1).

We then sett:=−x_nin this formula (minus sign will be crucial) and we define the(n−1)functions that are relevant to us:

ωk(x₁, . . . ,x_n):=x_k(−x_n) =exp −x_nX independent solutions of the partial differential equation Xω⁼⁰with the rank of their Jacobian matrix ^∂ω_∂x^k

i

_16k6n−1

16i6n being equal to n−1at the origin. Fur-thermore, for every other solutionω ^{of X}ω⁼0, there exists a local analytic functionΩ⁼Ω ω1, . . . ,ωn−1

defined in a neighborhood of the origin inKⁿ⁻¹

§ 21. 97

such that:

ω^(x)≡Ω ω1(x), . . . ,ωn−1(x) .

Proof. Indeed, when applyingXto the above series defining theωk, we see that all terms do cancel out, just thanks to an application Leibniz’ formula developed in the form:

X

(x_n)^lX^l(x_k)

=l(x_n)^l−1X^l(x_k) + (x_n)^lX^l+1(x_k).

Next, the assertion that the mapx7−→ ω1(x), . . . ,ωn−1(x)

has rankn−1 is clear, forωk(x₁, . . . ,x_n−1,0)≡x_kby construction. Finally, after straighteningX toX⁰:=_∂^∂_x0

n in some new coordinates(x⁰₁, . . . ,x⁰_n)thanks to the theorem on p. 64, the general solutionω^0(x0)toX0ω⁰⁼0 happens trivially to just be any function

Ω^0(x01, . . . ,x⁰_n−1)ofx⁰₁≡ω1⁰, . . . ,x⁰_n−1≡ω_n−1^{0 .} ut

C h a p t e r 5.

The Complete Systems.

We assume that the theory of the integration of an individual first order linear partial differential equation:

X(f) =

∑

n i=1

ξi(x₁, . . . ,x_n)∂^f

∂^xi

=0,

or of the equivalent simultaneous system of ordinary differential equations:

dx₁ ξ1

=···=dx_n ξn

,

is known; nonetheless, as an introduction, we compile without demonstration a few related propositions. Based on these propositions, we shall very briefly derive the theory of the integration of simultaneous linear partial differential equations of the first order. In the next chapter, we shall place in a new light [IN EIN NEUESLICHT SETZEN] this theory due in the main whole to JACOBIand to CLEBSCH, by ex-plaining more closely the connection between the concepts [BEGRIFFEN] of “linear partial differential equation” and of “infinitesimal transformation”, a connection that we have already mentioned earlier (Chap. 4, p. 69).

§ 21.

One can suppose thatξ1, . . . ,ξn behave regularly in the neighborhood of a deter-minate system of valuesx⁰₁, . . . ,x⁰_n, and as well thatξn(x⁰₁, . . . ,x⁰_n)is different from zero. Under these assumptions, one can determinex₁, . . . ,x_n−1as analytic functions ofx_nin such a way that by sustitution of these functions, the simultaneous system:

dx₁ dx_n=ξ1

ξn

, . . . ,dx_n−1 dx_n =ξn−1

ξn

is identically satisfied, and that in additionx₁, . . . ,x_n−1forx_n=x⁰_ntake certain pre-scribedinitial values[ANFANGSWERTHE]x⁰₁, . . . ,x⁰_n−1. These initial values have to be interpreted as the integration constants.

The equations which, in the concerned way, representx₁, . . . ,x_n−1as functions of x_nare called thecomplete integral equations of the simultaneous system; they can receive the form:

x_k=x⁰_k+ (x⁰_n−x_n)P_k x⁰₁−x⁰₁, . . . ,x⁰_n−1−x⁰_n−1,x⁰_n−x_n

(k=1···n−1),

where the P_k denote ordinary power series in their arguments. By inversing the relation betweenx₁, . . . ,x_n−1,x_nandx⁰₁, . . . ,x⁰_n−1,x⁰_n, one again obtains the integral equations, resolved with respect to only the initial valuesx⁰₁, . . . ,x⁰_n−1:

x⁰_k=x_k+ (x_n−x⁰_n)P_k x1−x⁰₁, . . . ,xn−x⁰_n

=ωk(x₁, . . . ,xn)

(k=1···n−1).

Here, the functionsωkare the so-calledintegral functionsof the simultaneous sys-tem, since the differentials of these functions:

dωk=

∑

n i=1

∂ωk

∂^xi

dx_i (k=1···n−1)

all vanish identically by virtue of the simultaneous system, and every function of this sort is called an integral function of the simultaneous system. But every such integral function is at the same time a solution of the linear partial differential equa-tion X(f) =0, whenceω1, . . . ,ωn−1 are solutions ofX(f) =0, and in fact, they are obviously independent. In a certain neighborhood ofx⁰₁, . . . ,x⁰_nthese solutions behave regularly; in addition, they reduce forx_n=x⁰_n tox₁, . . . ,x_n−1respectively;

that is why they are called thegeneral solutions of the equationX(f) =0relative to x_n=x⁰_n.

If one knows altogethern−1 independent solutions:

ψ1(x₁, . . . ,x_n), . . . ,ψn−1(x₁, . . . ,x_n)

of the equation X(f) =0, then the most general solution of it has the form

Ω⁽ψ1, . . . ,ψn−1), whereΩ denotes an arbitrary analytic function of its arguments.

§ 22.

If a functionψ^(x1, . . . ,x_n)satisfies the two equations:

X₁(f) =0, X₂(f) =0,

§ 22. 99 then it naturally also satisfies the two differential equations of second order:

X₁ X₂(f)

=0, X₂ X₁(f)

=0, and in consequence of that, also the equation:

X₁ X₂(f)

−X₂ X₁(f)

=0, which is obtained by subtraction from the last two written ones.

If now one lets:

because all terms which contain second order differential quotients are cancelled.

Thus, the following holds: then it also satisfies the equation:

X₁ X₂(f) which, likewise, is of first order.

It is of great importance to know how the expression X₁ X₂(f)

−X₂ X₁(f) behaves when, in place ofx₁, . . . ,x_n, new independent variablesy₁, . . . ,y_nare intro-duced.

We agree that by introduction of they, it arises:

X_k(f) =

Since f denotes here a completely arbitrary function ofx₁, . . . ,x_n, we can substitute X₁(f)orX₂(f)in place of f, so we have:

and consequently:

Proposition 2.If, by the introduction of a new independent variable, the expres-sions X₁(f)and X₂(f)are transferred to Y₁(f)and respectively to Y₂(f), then the in the course of our study. The same proposition can be stated more briefly:the expression X₁ X₂(f)

−X₂ X₁(f)

behaves invariantly through the introduction of a new variable.

We now consider theqequations:

(1) X₁(f) =0, . . . ,X_q(f) =0, and we ask about its possible joint solutions.

It is thinkable that between the expressionsX_k(f), there are relations of the form:

(2)

∑

q k=1

χk(x₁, . . . ,x_n)X_k(f)≡0.

If this would be the case, then certain amongst our equations would be a conse-quence of the remaining ones, and they could easily be left out while taking for granted the solution of the stated problem. Therefore it is completely legitimate to make the assumption that there are no relations of the form (2), hence that the equa-tions (1) are solvable with respect to theqof the differential quotients ^∂_∂x^f

i. It is to be understood in this sense, when we refer to theequations(1) asindependentof each other¹.

According to what has been said above about the two equationsX₁(f) =0 and X2(f) =0, it is clear that the possible joint solutions of ourq equations do also satisfy all equations of the form:

X_i X_k(f)

−X_k X_i(f)

=0.

And now, two cases can occur.

Firstly, the equations obtained this way can be a consequence of the former, when for everyiandk6q, a relation of the following form:

X_i X_k(f)

−X_k X_i(f)

=

=χik1(x₁, . . . ,x_n)X₁(f) +···+χikq(x₁, . . . ,x_n)X_q(f).

1 This is a typical place where a relocalization is in general required in order to insure that the

1 This is a typical place where a relocalization is in general required in order to insure that the

Dans le document Lie's Theory of Transformation Groups I, Contemporary Approach and Translation (Page 96-0)