One thus has ρ linearly-independent couplings of the Lagrangian expressions with divergences; the linear independence follows from the fact that, from (9), it would follow that δ u = 0, ∆u = 0, ∆x = 0, so there would be a dependency between the infinitesimal transformations.

However, by assumption, such a thing is not fulfilled for any parameter values, since otherwise the Gρ that further arises from the infinitesimal transformations by integration would depend upon less than ρ essential parameters. The further possibility that δ u = 0, Div(f ∆x) = 0 was, however, excluded.

These conclusions are also still true in the limiting case of infinitely many parameters. Now, let G be an infinite, continuous group; δ u and its derivatives, and therefore also B, will be linear in the arbitrary functions p(x) and their derivatives 1). By substituting the values of δ u , still independently of (12), let:

One may now, analogously to the formula for partial integration, replace the derivatives of p with p itself and divergences that are linear in p and its derivatives using the identity: φ(x, u, …)

One thus gets:

The fact that it is no restriction to assume that the p are free of the u, ∂u / ∂x, … shows the converse.Noether – Invariant variational problems

I now construct the n-fold integral of (15), taken over any domain, and choose the p(x) such that they vanish on the boundary of (B – Γ), along with all of the derivatives that appear. Since the integral of a divergence reduces to a boundary integral, the integral of the left-hand side of (15) thus also vanishes for arbitrary p(x) that only vanish on the boundary, along with sufficiently many derivatives, and from this, it follows, by a well- known argument, that the integrands vanish for any p(x), so one has the ρ relations:

These are the desired dependencies between the Lagrangian expressions and their derivatives for the invariance of I under G∞ρ ; the linear independence is clear, as above, since the inverse leads back to (12), and since one can again go from the infinitesimal transformations back to the finites ones, as will be done more thoroughly in § 4. Thus, ρ arbitrary transformations already appear in the infinitesimal transformations for a G∞ρ . From (15) and (16), it then follows that Div(B – Γ) = 0. If one correspondingly assumes a “mixed group” of ∆x and ∆u that are linear in the ε and the p(x) then one sees, when one sets the p(x) equal to zero and then the ε, that the divergence relations (13) exist, as well as the dependencies (16). § 3. Converse in the case of the finite group. In order to show the converse, one must essentially follow through the foregoing argument in the opposite sequence. The validity of (12) follows from the validity of (13) upon multiplication by ε and addition, and by means of the identity (3), this implies a relation: δ f + Div(A – B) = 0. If one then sets: ∆x = 1/f ⋅ (A – B) then one arrives at (11) as a result of this. From this, (7) finally follows by integration: ∆I = 0, and thus the invariance of I under the infinitesimal transformation that is determined by ∆x, ∆u, where the ∆u is to be determined from ∆x and δ u by means of (9), and ∆x and ∆u become linear in the parameters. However, ∆I = 0 implies, in a well-known way, the invariance of I under the finite transformations that arise by integrating the simultaneous system: (17) dxi = ∆xi, dt dui = ∆ui dt  xi = y  for t = 0  .   ui = vi  These finite transformations include ρ parameters a1, …, aρ, namely, the couplings tε1, …, tερ . From the assumption that there should be ρ and only ρ linearly independent divergence relations (13), it follows moreover that the finite transformations always define a group, as long as they do not include the derivatives ∂u / ∂x. In the opposite case − namely, at least one infinitesimal transformation arises from the Lie bracket process − there would be no linear coupling of the ρ remaining divergence relations, and since I also admits this transformation, there would be more than ρ linearly independentNoether – Invariant variational problems 9 divergence relations, or else this infinitesimal transformation would be of the special form in which δ u = 0, Div(f ⋅ ∆x) = 0, but then ∆x or ∆u would depend upon derivatives, contrary to assumption. Whether or not this case can occur when derivatives appear in ∆x or ∆u must remain undecided. One then adds all functions ∆x for which Div(f ⋅ ∆x) = 0 to the ∆x that was determined above in order to once more preserve the group property. By convention, the parameters that are thus added shall not, however, be counted. The converse is thus proved. From this converse, it then follows that, in fact, ∆x and ∆u can be assumed to be linear in the parameters. Namely, if ∆u and ∆x were of higher degree in ε then, due to the linear independence of the products of powers of ε, entirely analogous relations to (18) would follow, only in a greater number, from which, by the converse, one infers the invariance of I under a group whose infinitesimal transformations include the parameters linearly. Should this group contain precisely ρ parameters, then there would have to exist linear dependencies between the original divergence relations due to the terms of higher order in ε. Let it be remarked that in the case where ∆x and ∆u also contain derivatives of the u the finite transformations can depend upon infinitely many derivatives of the u. In this  ∂u  d 2 xi d 2ui case, the integration of (17) then leads from the determination of , to ∆   2 2 dt dt  ∂xκ  ∂∆u ∂u ∂∆uλ −∑ , such that the number of derivatives of u generally increases at

∂xκ κ ∂xλ ∂xκ each step. Perhaps the following will serve as an example: f = 12 u′2 , ψ = − u′′, ∆x = −2u ε, u ′λ ψ⋅x= d (u − u′x), dx δ u = x ⋅ ε, 2u   ∆u =  x −  ⋅ ε . u′   Since the Lagrangian expression of a divergence vanishes identically, the converse ultimately shows the following: If I admits a Gρ then any integral that differs from I only by a boundary integral – i.e., an integral of a divergence – will likewise admit a Gρ with the same δ u , whose infinitesimal transformation will generally contain derivatives of the d  u 2    u. Thus, perhaps referring to the example above, f* = 12 u ′2 −    admits the dx  x    infinitesimal transformation ∆u = xε, ∆x = 0, while derivatives of the u appear in the infinitesimal transformations that correspond to f.Noether – Invariant variational problems 10 If one goes on to the variational problem − i.e., if one sets ψi = 0 1) − then (18) goes to the equations: Div B(λ) = 0, …, Div B(ρ) = 0, which are often referred to as “conservation laws.” In the one-dimensional case, it follows from this that B(1) = const., …, B(ρ) = const., and therefore the B contain at most (2κ – 1)th derivatives of the u (from (6)), as long as ∆u and ∆x include no higher derivatives than κth that appear in f. Since 2κth derivatives appear in ψ, in general 2), one thus has the existence of ρ first integrals. The f above once more shows that nonlinear dependencies can exist between them. The linearly independent ∆u = ε1, ∆x = ε2 correspond to the linearly independent relations: u′′ d 1 d

u′ , u′′⋅⋅ u′ = (u′)2 , while a nonlinear dependency exists between the first dx 2 dx integrals u′ = const., u′2 = const. Thus, one is dealing with the elementary case in which ∆u, ∆x contain no derivatives of the u 3). § 4. Converse in the case of infinite groups. First, let us show that the assumption of the linearity of ∆x and ∆u presents no restriction, which one deduces here without the converse from the fact that G∞ρ formally depends upon ρ and only ρ arbitrary functions. Namely, it shows that in the nonlinear case the number of arbitrary functions would increase under the composition of transformations in which the terms of lowest order would add together. In fact, let, say: ∂u ∂p   y = A  x, u , ,⋯ ; p  = x + ∑ a(x, u, …) pν + b(x, u, …) pν−1 ∂x ∂x   ν  ∂p   ∂p 

• cpν−1   + … + d   + …  ∂x   ∂x  2 and analogously v = ∂u   B  x , u , ,⋯ ; p  , ∂x   (pν = ( p(1) )ν , …, ( p ( ρ ) )ν ), so under 1ρ composition withz =   ∂v A  y, v, ,⋯ ; q  , one gets, for the terms of lowest order: ∂y   ) ψi = 0, or, more generally, ψi = Ti ψi , where Ti are new functions that are to be added to the others, are referred to as “field equations” in physics. In the case ψi = Ti , the identities (13) go to identities: Div B(λ) = ∑ Tiδ ui( λ ) , which are also referred to as conservation laws in physics. 2 ) As long as f is nonlinear in the κth derivatives. 3 ) Otherwise, one would have u′λ = const. for any λ, corresponding to: 1 u′′ ⋅ (u′)λ−1 = 1 d λ dx λ (u ′) .Noether – Invariant variational problems 11 2 2  ∂p ∂q    ∂p   ∂q   z = x + ∑ a(pν + qν) + b  pν −1 + qν −1  + c  pν − 2   + qν − 2    + … ∂x ∂x    ∂x   ∂x    If a coefficient that is different from a and b here is different from zero then a term σ σ  ∂p   ∂q  p   + qν −σ   actually appears for σ > 1, so one cannot write this as the  ∂x   ∂x  differential quotient of a single function or products of powers of them; the number of arbitrary functions thus has increased, contrary to assumption. If all of the coefficients that are different from a and b vanish then each of the values of the exponents v1, …, vρ will be the second term of the differential quotient of the first one (as is always the case for, e.g., a G∞1), such that linearity actually enters in, or else the number of arbitrary functions would also increase here. Due to the linearity of the p(x), the infinitesimal transformations thus satisfy a system of linear partial differential equations, and since the group property is fulfilled, they define an “infinite group of infinitesimal transformations,” by Lie’s definition (Grundlagen, § 10). One deduces the converse now in a manner that is similar to the one in the case of finite groups. The existence of the dependencies (16) leads, upon multiplication by p(λ)(x) and addition, using the identity conversion (14), to ∑ψ iδ ui = Div Γ, and from ν −σ this, as in § 3, one infers the determination of ∆x and ∆u and the invariance of I under these infinitesimal transformations, which, in fact, depend linearly upon ρ arbitrary functions and their derivatives up to order σ. The fact that these infinitesimal transformations, when they include no derivatives ∂u / ∂x, …, certainly define a group follows, as in § 3, from the fact that otherwise more arbitrary functions would appear by composition, while, by assumption, there shall be only ρ dependencies (16); they thus define an “infinite group of infinitesimal transformations.” However, such a thing consists (Grundlagen, Theorem VII, pp. 391) of the most general infinitesimal transformations of a certain “infinite group G of finite transformations,” in Lie’s sense. Every finite transformation will then be generated by infinitesimal ones (Grundlagen, §

7. 1), and thus arise from the integration of the simultaneous system: dxi = ∆xi, dt dui = ∆ui, dt  xi = yi  for t = 0  ,   ui = v  in which, it can, however, be necessary to choose the arbitrary p(x) to be independent of t. G thus depends, in fact, on ρ arbitrary functions; in particular, if it suffices to choose p(x) to be free of t then this dependency will be analytic in the arbitrary functions q(x) = t 1 ) From this, it follows, in particular, that the group G that is generated by the infinitesimal transformations ∆x, ∆u of a G∞ρ again leads back to G∞ρ . G∞ρ then includes no infinitesimal transformations that are different from ∆x, ∆u that depend upon arbitrary functions, and can also contain none that are independent of them that depend upon parameters, since it would then be a mixed group. However, from the above, the finite transformations are determined by means of the infinitesimal ones.

p(x) 1).

If derivatives ∂u / ∂x, … appear then it can be necessary to add infinitesimal transformations δ u = 0, Div(f ⋅ ∆x) = 0 before one can reach the same conclusion.

In connection with an example of Lie (Grundlagen, § 7), let a somewhat more general case be given, where one can advance to explicit formulas that likewise show that the derivatives of the arbitrary functions up to order σ appear, from which the converse is then complete.

It is the example of those groups of infinitesimal transformations that correspond to the group of all transformations of the x and the transformations of the u that are “induced” by them; i.e., those transformations of the u for which ∆u, and consequently u, depend upon only the arbitrary functions that appear in ∆x, whereby let it be assumed that the derivatives ∂u / ∂x, … do not appear in ∆u. One thus has:

Since the infinitesimal transformation ∆x = p(x) generates any transformation x = y + g(y) with arbitrary g(y), in particular, p(x) can be determined to be independent of t, such that the following one-parameter group will be generated:

which goes to the identity for t = 0 and to the desired x = y + g(y) for t = 1. In fact, it follows by differentiation of (18) that:

where p(x, t) is determined from g(y) by inversion, and conversely, (18) arises from (19) by means of the auxiliary condition that xi = yi for t = 0, by which, the integral is established uniquely. By means of (18), the x in ∆u can be replaced with the “integration constants” y and t; thus, the g(y) appear up to precisely the σth derivatives when one

and, in general, replaces expresses the ∂y / ∂x in terms of ∂x / ∂y in

For the determination of the u,

one then gets the system of equations:

The question of whether this latter case always occurs was posed by Lie in a different formulation (Grundlagen, § 7 and § 13, conclusion).Noether – Invariant variational problems in which only t and u are variable, but the g(y), … belong to the coefficient domain, such that the integration yields:

and therefore transformations that depend upon precisely σ derivatives of the arbitrary functions. From (18), the identity is included in this for g(y) = 0, and the group property follows from the fact that the chosen process produces any transformation x = y + g(y), from which the one that is induced on the u is established uniquely, so the group G will be exhausted.

Incidentally, it then follows from the converse that it is no restriction to choose the arbitrary functions to depend upon only the x, but not on the u, ∂u / ∂x, … In the latter

enter into the identity transformation (14), as well as into

(15), in addition to the p(λ). If one now chooses the p(λ) to be successively of degree zero, one, … in u, ∂u / ∂x, …, with arbitrary functions of x as coefficients, then the dependencies (16) emerge again, but in greater numbers, which, however, from the converse above, lead back to previous case under composition with arbitrary functions that depend upon only x. One likewise shows that the simultaneous appearance of dependencies and divergence relations that are independent of them corresponds to mixed groups 1).

As in § 3, it also follows from the converse here that, along with I, also any integral I* that differs by a divergence likewise admits an infinite group with the same δ u , in which, however, ∆x and ∆u will generally include derivatives of the u. Einstein has introduced such an integral into the general theory of relativity in order to obtain a simpler statement of the energy theorem; I shall give the infinitesimal transformations that this I* admits, for which I preserve the notation of Klein’s second note precisely. The integral I = ∫…∫ K dw = ∫…∫ K dS admits the group of all transformations of the w and the one that it induces on gμν ; they correspond to the dependencies ((30), in Klein):

Now, one has: I* = ∫…∫ K* dS, where K* = K + Div, and consequently, one will have: K μν = Kμν , where ∗ K μν , Kμν mean the Lagrangian expressions in each case. Therefore, the dependencies that were given are also true for K μν , and after multiplying by pτ and adding, one gets by the reverse conversion of the product differentiation:

Comparing this with Lie’s differential equation: δK* + Div(K* ∆w) = 0, it then follows that:Noether – Invariant variational problems

§ 5. Invariance of the individual components of the relations.

If one specializes the group G to the simplest case that is ordinarily considered by specifying that one allows no derivatives of the u in the transformations and that the transformed independent variables depend upon only the x, but not the u then one can deduce the invariance of the individual components in the formulas.

First of all, this yields, from known reasons, the invariance of ∫…∫ (∑ψi δui) dx; thus, one infers the relative invariance of ∑ψi δui 1), where we understand δ to mean any variation. In fact, one has, on the one hand:

, … that vanish on the boundary, due to the linear, ∂x ∂v ∂u homogeneous nature of the transformation of the δu, δ , …, the δv, δ , … also ∂x ∂y vanish on the boundary, so one has, correspondingly: and, on the other hand, for δu, δ

and it follows that for δu, δ

i ∂u

that vanish on the boundary:

If one expresses y, v, δv in the third integral in terms of x, u, δu and one sets it equal to the first one then one has a relation:

are infinitesimal transformations that I* admits. These infinitesimal transformations thus depend upon the first and second derivatives of the gμν, and include the arbitrary p up to the first derivatives. 1 ) I.e., under transformation, ∑ψi δui takes on a factor, which is always referred to as relative invariance in the algebraic theory of invariants.Noether – Invariant variational problems 15 ∫ ⋯ ∫ ( ∑ χ (u,…) δ u ) dx = 0 i i for a du that vanishes on the boundary, but is otherwise arbitrary, and, as is known, the vanishing of the integrands for arbitrary δu follows from this; one thus has the following relation identically in δu: ∂yi (∑ ψi (v, …) δvi ), ∑ ψi (u, …) δui = ∂xκ which expresses the relative invariance of ∑ ψi δui , and consequently, the invariance of ∫…∫ (∑ ψi δui) 1). In order to apply this to the derived divergence relations and the dependencies, one must first confirm that the δ u that is derived from the ∆u, ∆x actually satisfies the transformation laws for the variation δu, as long as only the parameter (arbitrary functions, resp.) in δ v are determined in a way that corresponds to the way that they are determined for the similar group of infinitesimal transformations in y, v. Let Tq denote the transformation that takes x, u to y, v; since Tq is an infinitesimal in x, u, the one that is similar to it in y, v is given by T = TqTp Tq−1 , where the parameters (arbitrary functions r, resp.) are therefore determined from p and q. One expresses this in formulas as: Tp :ξ = x + ∆x(x, p),u* = u + ∆u(x, u, p), Tq :y = A(x, q),v = B(x, u, q), TqTp : η = A(x + ∆x(x, p), q), v* = B(x + ∆x(p), u + ∆u(p), q). From this, one has, however, Tr = TqTp Tq−1 , so: 1 )  ∂v ∂y  This conclusion breaks down when y also depends upon the u, since then δ f  y ,v , ,…  also   ∂f includes terms like ∑ δ y , so the divergence conversion does not lead to the Lagrangian expressions, just ∂y as when one allows derivatives of the u; then, in fact, the δv will lead to linear combinations of δu, δ ∂u , ∂x …, so after a further divergence conversion this will lead to an identity ∫…∫ (∑ χi (u, …) δui ) dx = 0, such that the Lagrangian expressions once again do not appear on the right-hand side. The question of whether one can also already conclude the existence of divergence relations from the invariance of ∫…∫ (∑ ψi δui ) dx is, from the converse, equivalent to the question of whether one can conclude that from the invariance of I under a group that does not necessarily lead to the same ∆u, ∆x, but still leads to the same δ u . In the special case of simple integrals and only first derivatives in f, one can deduce the existence of first integrals from the invariance of the Lagrangian expressions for finite groups (cf., e.g., Engel, Gött. Nachr. (1916), pp. 270.).Noether – Invariant variational problems η = y + ∆y(r) = y + ∑ v* = v + ∆v(r) = v + ∑ 16 ∂A( x, q ) ∆x(p), ∂x ∂B( x, u , q ) ∂B( x, u , q ) ∆x ( p ) + ∑ ∆u ( p ) . ∂x ∂u One replaces x = x + ∆x in this with ξ − ∆ξ, from which, x again goes to x, so ∆x vanishes; thus, from the first formula in (20), η also again goes to y = η − ∆η. If ∆u(p) goes to δ u ( p) then ∆v(r) also goes to δ v(r ) , and the second formula in (20) gives: v + δ v ( y , v ,… , r ) = v + ∑ δ v ( y , v ,… , r ) = ∑ ∂B( x, u , q ) δ u( p) , ∂u ∂B δ uκ ( x, u , p) , ∂uκ such that the transformation formulas for variations are, in fact, therefore fulfilled, as long as δ v is assumed to depend only on the parameters (arbitrary functions r, resp.) 1). In particular, the relative invariance of ∑ψ iδ ui then follows; thus, the relative invariance of Div B also follows, since the divergence relations are also fulfilled in y, v, and furthermore, from (14) and (13), one also has the relative invariance of Div Γ and that of the left-hand side of the dependencies, when composed with the p(λ), where the arbitrary p(x) (the parameters, resp.) are always replaced with the r in the transformation formulas. This then yields the relative invariance of Div(B − Γ), and therefore that of a divergence of a non-vanishing system of functions B – Γ whose divergence vanishes identically. From the relative invariance of Div B, one may, in the one-dimensional case and for finite groups, draw a conclusion about the invariance of the first integrals. The parameter transformation that corresponds to the infinitesimal transformation will, from (20), be linear and homogeneous, and due to the invertibility of all transformations, the ε will also be linear and homogeneous in the transformed parameters ε*. This invertibility certainly remains preserved when one sets ψ = 0, since no derivatives of the u enter into (20). By equating the coefficients of the ε* in: Div B(x, u, …, ε) = 1 dy ⋅ Div B(y, v, …, ε*), dx ) This again shows that y must be assumed to independent of u, etc., in order for the conclusion to be valid. As an example, let us, perhaps, mention the δgμν and δqρ that were given by Klein, which satisfy the transformations for variations, as long as p is subject to a vector transformation.Noether – Invariant variational problems 17 d (λ) d (λ ) B (y, v, …) will then be linear, homogeneous functions of the B (x, u, …), dy dx d (λ ) d (λ ) such that from B (x, u, …) = 0 or B(λ)(x, u) = const. it also follows that: B (y, v, dx dy …) = 0 or B(λ)(y, v) = const. The first ρ integrals that correspond to a Gρ thus admit the group in any case, such that the further integration is also simplified. The simplest example of this is the one in which f is free of x or one u, which corresponds to the du transformation ∆x = ε, ∆u = 0 (∆x = 0, ∆u = ε, resp.). One has δ u = − ε (ε, resp.), and dx since B can be derived from f and δ u by differentiation and rational couplings, it is then also free of x (u, resp.) and admits the corresponding groups 1). the § 6. An assertion of Hilbert. From the foregoing, one ultimately finds the proof of an assertion of Hilbert about the connection between the break-down of the proper energy theorem and “general relativity” (Klein’s first note, Göttinger Nachr. (1917), answer, first passage), and indeed, in a generalized group-theoretic context. Let the integral I admit a G∞ρ , and let Gσ be any finite group that arises from specializing the arbitrary functions, so it is a subgroup of G∞ρ . The infinite group G∞ρ then corresponds to dependencies (16), and the finite one Gσ , to divergence relations (13), and conversely, it follows from the existence of any sort of divergence relations that I is invariant under a finite group that is identical to Gσ when and only when the δ u are linear combinations of the ones obtained from Gσ . The invariance under Gσ can thus lead to no divergence relations that differ from (13). However, since the invariance of I under the infinitesimal transformations ∆u, ∆x of G∞ρ for arbitrary p(x) follows from the validity of (16), it already follows from this, in particular, that it is invariant under the infinitesimal transformations of Gσ that arise by specialization, and consequently, under ∑ψ δ u λ = Div B(λ) must then must then be consequences of the dependencies (16), which can also be written: ∑ψ a λ = Div χ(λ), Gσ . The divergence relations ∗ i ( ) i ( ) i i

where the χ(λ) are linear couplings of the Lagrangian expressions and their derivatives.

Since the ψ enter into (13), as well as (16), linearly, the divergence relations must then be linear combinations of the dependencies (16), in particular, and the B(λ) themselves are ) In the case where the existence of first integrals already follows from the invariance of ∫ (∑ ψi δui ) dx, they do not admit the complete group Gρ ; e.g., ∫ (u′′ δu) dx admits the infinitesimal transformation: ∆x = ε2 , ∆u = ε1 + xε2 , while the first integral u − u′x = const., which corresponds to ∆x = 0, ∆u = xε3 , does not admit the other two infinitesimal transformations, since it includes u, as well as x, explicitly. This first integral corresponds simply to infinitesimal transformations of f that include derivatives. One then sees that, in any case, the invariance of ∫…∫ (∑ ψi δui ) dx is achieved less often than the invariance of I, which responds to a question that was posed in a previous remark.

thus composed linearly from the χ – i.e., from the Lagrangian expressions and their derivatives, and from functions whose divergences vanish identically, like perhaps the B – Γ that appeared in the conclusion to § 2, for which Div (B – Γ) = 0 and the divergence likewise has the invariant property. I will refer to divergence relations for which B() of the given kind can be composed from the Lagrangian expressions and their derivatives as “unreal,” and all others as “real.”

Conversely, if the divergence relations are linear couplings of the dependencies (16) – hence, “unreal” – then the invariance under Gσ follows from the invariance under G∞ρ ; Gσ becomes a subgroup of G∞ρ . The divergence relations that correspond to a finite group Gσ will then be unreal when and only when Gσ is a subgroup of an infinite group that I is invariant under.

The original Hilbert assertion is obtained from this by specializing the group. Let the term “translation group” mean the finite group:

As is known, invariance under the translation group expresses the idea that the x do not

enter into I = ∫ ⋯ ∫ f  x, u, ,…  dx explicitly. Let the associated n divergence

relations:

be referred to as “energy relations,” since the “conservation law” Div B(λ) = 0 that corresponds to the variational problem corresponds to the “energy law,” while the B(λ) correspond to the “energy components.” One then has: If I admits the translation group then the energy relations become unreal when and only when I is invariant under an infinite group that includes the translation group as a subgroup 1).

An example of such an infinite group is given by the group of all transformations of the x, along with those induced transformations of the u(x) in which only derivatives of the arbitrary functions p(x) appear; the translation group then arises by specializing p(i)(x) = εi.

Therefore, it must remain undecided whether the most general of these groups is therefore already given – along with the groups that arise from altering I by a boundary integral.

Induced transformations of the given sort arise perhaps when one subjects the u to the coefficient transformations of a “total differential form;” i.e., a form ∑ a dλxi + ∑ b dλ−1xi dxκ + … that includes higher differentials, in addition to the dx. Special induced transformations for which the p(x) only appear in the first derivatives are given by the) The energy law in classical mechanics, and likewise in the older “relativity theory” (where ∑ dx2 goes to itself), are “unreal,” since no infinite groups appear there.

1Noether – Invariant variational problems coefficient transformations of ordinary differential forms ∑ c dxi1 … dxiλ , and ordinarily one has considered only these.

Another group of the given kind that cannot be a coefficient transformation, due to the appearance of logarithmic terms, is perhaps the following one:

Here, the dependencies (16) become:

while the unreal energy relations become:

The simplest invariant integral for the group is:

The most general I is determined by integrating Lie’s differential equation (11):

which goes to:

(identically in p(x), p′(x), p′′(x)) by substituting the values of ∆x and δ u , as long as one assumes that f depends upon only first derivatives of the u. This system of equations already possesses solutions that actually include the derivatives for two functions u(x), namely:

One computes the finite transformations from these infinitesimal ones backwards from the method that was given in the conclusion of § 4.Noether – Invariant variational problems where Φ means an arbitrary function of the given arguments.

Hilbert expressed his assertion in such a way that the break-down of the proper energy law was a characteristic feature of the “general theory of relativity.” In order for this assertion to be literally true, the term “general relativity” must then be further regarded as it usually is, and also extended to the previous groups that depend upon n arbitrary functions 1).

With this, the validity is again confirmed of a remark of Klein that the usual terminology “relativity” in physics should be replaced with “invariance under a group.” (“Über die geometrischen Grundlagen der Lorentzgruppe,” Jber. d. deutsch. Math. Verein. 19 (1910), pp. 287; printed in Phys. Zeit.)