Changes of Coordinates and of Parameters

done automatically, without emphasizing it.

3.2 Changes of Coordinates and of Parameters

In the variablesx₁, . . . ,x_n, let the equations:

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

of anr-term group be presented. Then there are various means to derive from these equations other equations which represent again anr-term group.

On the first hand, in place of thea, we can introducerarbitrary independent functions of them:

a_k=βk(a₁, . . . ,a_r) (k=1···r)

as new parameters. By resolution with respect toa₁, . . . ,a_r, one can obtain:

a_k=γk(a₁, . . . ,a_r) (k=1···r)

and by substitution of these values, one may set:

f_i(x₁, . . . ,x_n,a₁, . . . ,a_r) =f_i(x₁, . . . ,x_n,a₁, . . . ,a_r).

Then if we yet set:

βk(b₁, . . . ,b_r) =b_k, βk(c₁, . . . ,c_r) =c_k (k=1···r),

the composition equations:

f_i f₁(x,a), . . . , f_n(x,a), b₁, . . . ,b_r

=f_i(x₁, . . . ,x_n,c₁, . . . ,c_r) take, without effort, the form:

f_i f₁(x,a), . . . ,f_n(x,a), b₁, . . . ,b_r

=f_i(x₁, . . . ,x_n,c₁, . . . ,c_r), from which it results that the equations:

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

with theressential parametersa1, . . . ,ar represent in the same way anr-term group.

Certainly, the equations of this new group are different from the ones of the original group, but these equations obviously represent exactly the same

transformations as the original equationsx⁰_i= fi(x,a). Consequently, the new group is fundamentally identical to the old one.

On the other hand, we can also introduce new independent variables y₁, . . . ,y_nin place of thex:

y_i=ωi(x₁, . . . ,x_n) (i=1···n), or, if resolved:

x_i=w_i(y₁, . . . ,y_n) (i=1···n).

Afterwards, we have to set:

x⁰_i=w_i(y⁰₁, . . . ,y⁰_n) =w⁰_i, x⁰⁰_i =w_i(y⁰⁰₁, . . . ,y⁰⁰_n) =w⁰⁰_i, and we hence obtain, in place of the transformation equations:

x⁰_i=f_i(x₁, . . . ,x_n,a₁, . . . ,a_r), the following ones:

w_i(y⁰₁, . . . ,y⁰_n) =f_i(w₁, . . . ,w_n,a₁, . . . ,a_r), or, by resolution:

y⁰_i=ωi f₁(w,a), . . . ,f_n(w,a)

=F_i(y₁, . . . ,y_n,a₁, . . . ,a_r).

It is easy to prove that the equations:

y⁰_i=F_i(y₁, . . . ,y_n,a₁, . . . ,a_r) (i=1···n)

with the ressential parameters a₁, . . . ,a_r again represent anr-term group. In fact, the known equations:

f_i f₁(x,a), . . . , f_n(x,a), b₁, . . . ,b_r

=f_i(x₁, . . . ,x_n,c₁, . . . ,c_r) are transferred, after the introduction of the new variables, to:

fi f(w,a),b

= fi(w₁, . . . ,wn,c₁, . . . ,cr), which can also be written:

f_i(w⁰₁, . . . ,w⁰_n,b₁, . . . ,b_r) = f_i(w₁, . . . ,w_n,c₁, . . . ,c_r) =w⁰⁰_i; but from this, it comes by resolution with respect toy⁰⁰₁, . . . ,y⁰⁰_n:

y⁰⁰_ν=ων f₁(w⁰,b), . . . , f_n(w⁰,b)

=ων f₁(w,c), . . . ,f_n(w,c) ,

3.2 Changes of Coordinates and of Parameters 29

or, what is the same:

y⁰⁰_ν=F_ν(y⁰₁, . . . ,y⁰_n,b₁, . . . ,b_r) =F_ν(y₁, . . . ,y_n,c₁, . . . ,c_r), that is to say: there exist the equations:

F_ν F₁(y,a), . . . ,F_n(y,a),b₁, . . . ,b_r

=F_ν(y₁, . . . ,y_n,c₁, . . . ,c_r), whence it is indeed proved that the equationsy⁰_i=F_i(y,a)represent a group.

Lastly, we can naturally introduce at the same time new parameters and new variables in a given group; it is clear that in this way, we likewise obtain a new group from the original group.

At present, we set up the following definition:

Definition.Twor-term groups:

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

y⁰_i=f_i(y₁, . . . ,y_n,b₁, . . . ,b_r) (i=1···n)

in the same number of variables aresimilar[ÄHNLICH] to each other as soon as the one converts into the other by the introduction of appropriate new variables and of appropriate new parameters.

Obviously, there is an unbounded number of groups which are similar to a given one; but all these unboundedly numerous groups are known simultane-ously with the given one. For this reason, as it shall also happen in the sequel, we can consider that two mutually similar groups are not essentially distinct from each other.

Above, we spoke about the introduction of new parameters and of new vari-ables without dealing with the assumptions by which we can ascertain that all group-theoretic properties essential for us are preserved here. Yet a few words about this point.

For it to be permitted to introduce, in the groupx⁰_i=f_i(x₁, . . . ,x_n,a₁, . . . ,a_r), the new parameters a_k =βk(a₁, . . . ,a_r) in place of the a, the a_k must be univalent functions of the a in the complete region (a) defined earlier on, and they must behave regularly everwhere in it; the functional determinant

∑±∂β1/∂^a1···∂βr/∂^arshould vanish nowhere in the region(a), and lastly, to two distinct systems of valuesa1, . . . ,arof this region, there must always be associated two distinct systems of valuesa₁, . . . ,a_r. In other words: in the re-gion of thea_k, one must be able to delimit a region(a)on which the systems of values of the region(a)are represented in a univalent way by the equations a_k=βk(a₁, . . . ,a_r).

On the other hand, for the introduction of the new variables y_i =

ωi(x₁, . . . ,x_n) to be allowed, the y must be univalent and regular

func-tions of the x for all sytems of values x₁, . . . ,x_n which come into con-sideration after establishing the group-theoretic properties of the equations

x⁰_i = fi(x₁, . . . ,xn,a1, . . . ,ar); inside this region, the functional determinant

∑±∂ω1/∂^x1···∂ωn/∂^xn should vanish nowhere, and lastly, to two distinct systems of valuesx₁, . . . ,x_nof this region, there must always be associated two distinct systems of valuesy₁, . . . ,y_n. The concerned system of values of thex must therefore be represented univalently onto a certain region of systems of values of they.

If one would introduce, in the groupx⁰_i= f_i(x,a), new parameters or new variables without the requirements just explained being satisfied, then it would be thinkable in any case that important properties of the group, for instance the group composition property itself, would be lost; a group with the identity transformation could convert into a group which does not contain the identity transformation, and conversely.

But it certain circumstances, the matter is only to study the family of trans-formations x⁰_i= f_i(x,a) in the neighbourhood of a single point a₁, . . . ,a_r or

x₁, . . . ,x_n. This study will often be facilitated by introducing new variables or

new parameters which satisfy the requirements mentioned above in the neigh-bourhood of the concerned points.

In such a case, one does not need at all to deal with the question whether the concerned requirements are satisfied in the whole extension of the regions(x) and(a).

Now, if we have a family of∞^rtransformations:

x⁰_i= f_i(x₁, . . . ,x_n,a₁, . . . ,a_r)

which forms an r-term group, then there corresponds to this family a family of∞^r operations by which the points of the spacex₁, . . . ,x_nare permuted. Evi-dently, any two of these∞^roperations, when executed one after the other, always produce an operation which again belongs to the family.

Thus, if we actually call a family of operations of this sort a group operation [OPERATIONSGRUPPE], or shortly, a group, then we can say:every given r-term transformation group can be interpreted as the analytic representation of a certain group of∞^rpermutations of the points x₁, . . . ,x_n.

Conversely, if a group of∞^r permutations of the points x₁, . . . ,x_nis given, and if it is possible to represent these permutations byanalytictransformation equations, then the corresponding∞^rtransformations naturally form a transfor-mation group.

Now, if one imagines that a determined group-operation is given, and in ad-dition, that an analytic representation of it is given — hence if one has a trans-formation group —, then this representation has in itself two obvious incidental characters [ZUFÄLLIGKEIT].

The first incidental character is the choice of the parameters a₁, . . . ,a_r. It stands to reason that this choice has in itself absolutely no influence on the group-operation, when we introduce in place of the a the new parameters