Invariant variational problems

Transcription

1 Invarante Varatonsprobleme, Kgl. Ges. d. Wss. Nachrchten, Math.-phys. Klasse (1918), Invarant varatonal problems (F. Klen on hs ffty-year Doctoral Jublee) By Emmy Noether n Göttngen. (Presented by F. Klen at the sesson on 26 July ) Translated by D. H. Delphench We shall deal wth varatonal problems that admt a contnuous group (n the Le sense); the results that ths yelds for the assocated dfferental equatons fnd ther most general expresson n the theorems that are formulated n 1 and are proved n the followng paragraphs. One can make more precse statements about these dfferental equatons that arse from varatonal problems than one can about arbtrary dfferental equatons that admt a group, whch defnes the context of Le s nvestgatons. Thus, what follows rests upon a couplng of the methods of the formal calculus of varatons wth those of the theory of Le groups. For specal groups and varatonal problems, ths couplng s not new; I menton Hamel and Herglotz for specal fnte groups, and Lorentz and hs school (e.g., Fokker), Weyl, and Klen for specal nfnte groups 2 ). In partcular, Klen s second note and the present efforts were mutually nfluenced by each other, so I wll refer to the concludng remarks of Klen s note. 1. Prefatory remarks and the formulaton of the theorem. All of the functons that enter nto what follows shall be assumed to be analytc, or at least contnuous and contnuously dfferentable fntely often, and sngle-valued n the doman n queston. As one knows, one understands the term transformaton group to mean a system of transformatons such that to every transformaton ncluded n the system there exsts an nverse, and the composton of any two transformatons of the systems agan belongs to the system. The group s called a fnte, contnuous group G ρ when ts transformatons 1 ) The fnal verson of the manuscrpt was frst submtted at the end of September. 2 ) Hamel: Math. Ann, Bd. 59 and Zet. f. Math. u. Phys., Bd. 50. Herglotz: Ann. d. Phys. (4) Bd. 36, esp. 9, pp Fokker, Verslag d. Amsterdamer Akad., 27/1, For further lterature, cf., the second note of Klen: Göttnger Nachrchten, 19 July In a recently-appearng paper of Kneser (Math. Zet., Bd. 2), he treated the constructonof nvarants by smlar methods.

2 Noether Invarant varatonal problems 2 are ncluded n the most general group that depends analytcally upon ρ essental parameters s (.e., the ρ parameters shall not be representable as ρ functons of fewer parameters). Correspondngly, one understands an nfnte contnuous group G ρ to mean a group whose most general transformatons depend analytcally upon ρ essental, arbtrary functons p(x) and ther dervatves, or at least, one that are contnuous and contnuously dfferentable fntely often. As an ntermedate step between the two, one fnds the groups that depend upon nfntely many parameters, but not on arbtrary functons. Fnally, one refers to the mxed groups as the ones that depend upon arbtrary functons, as well as parameters 1 ). Let x 1,, x n be ndependent varables, and let u 1 (x),, u µ (x) be functons that depend upon them. If one subjects the x and u to the transformatons of a group then, due to the assumed nvertblty of the transformatons, among the transformed quanttes, there must agan be found precsely n ndependent ones: y 1,, y n ; let the remanng ones that are ndependent of them be denoted by v 1 (y),, v µ (y). The dervatves of the u wth respect to the x vz., u /, 2 u / 2, can also enter nto the transformatons 2 ). A functon s called an nvarant of the group when there exsts a relaton: 2 u u P x, u,,, 2 = 2 v v P y, v,,, 2. y y In partcular, an ntegral I becomes an nvarant of the group when there exsts a relaton: (1) I = = 2 u u f x, u,,, dx 2 2 v v f y, v,,, dy 2 y y 3 ) when one ntegrates over an arbtrary real x-doman and the correspondng y-doman 4 ). 1 ) In Grundlagen für de Theore der unendlchen kontnuerlchen Transformatonsgruppen (Ber. d. K. Sächs. Ges. der Wss. 1891) [cted as Grundlagen ], Le defned the nfnte, contnuous groups to be transformaton groups whose transformatons are gven by the most general solutons of a system of partal dfferental equatons, as long as these solutons do not depend upon only a fnte number of parameters. In ths way, one thus obtans one of the aforementoned types that are dfferent from the fnte groups, whle, conversely, the lmtng case of nfntely many parameters does not necessarly need to satsfy a system of dfferental equatons. 2 ) To the greatest extent possble, I wll omt ndces, as well as summatons; hence, one mght have 2 u / 2 for 2 u α / β γ, etc. 3 ) To abbrevate, I wrte dx, dy for dx 1,, dx n, dy 1,, dy n. 4 ) All of the arguments x, u, ε, p(x) that enter nto the transformatons shall be assumed to be real, whle the coeffcents mght be complex. However, snce one deals wth denttes n the x, u parameters and arbtrary functons n the fnal results, they are also true for complex values, as long as all of the functons that appear n them are assumed to be analytc. A greater part of the results can be establshed wthout ntegrals, moreover, such that the restrcton to the reals s not necessary for the proof here ether. On the other hand, the consderatons at the concluson of 2 and the begnnng of 5 do not seem to be practcable wthout ntegrals.

3 Noether Invarant varatonal problems 3 On the other hand, I defne the frst varaton δi for an arbtrary not necessarly nvarant ntegral I, and convert t accordng to the rules of the calculus of varatons by partal ntegraton. As s known, as long as one assumes that δu, along wth all of the dervatves that appear, vansh at the boundary, but are otherwse arbtrary, t becomes: (2) δi = δ f dx = u ψ x, u,, δu dx, where ψ means the Lagrangan expressons;.e., the left-hand sdes of the Lagrange equatons for the assocated varatonal problem δi = 0. These ntegral relatons correspond to an ntegral-free dentty n du and ts dervatves, whch arses when one wrtes down the boundary terms. As partal ntegraton shows, these boundary terms are ntegrals over dvergences.e., over expressons: A1 An Dv A = + +, where A s lnear n δu and ts dervatves. One thus comes to: (3) ψ δu = δf + Dv A. 1 In partcular, f f contans only frst dervatves of u then n the case of a smple ntegral the dentty (3) s dentcal wth the one that Heun called the central Lagrange equaton : n d f (4) ψ δu = δf dx u δu du u = dx, whle for a n-fold ntegral, (3) goes to: f f (5) ψ δu = δf δ u u δ u. 1 u n 1 n For a smple ntegral and κ dervatves wth respect to u, (3) s gven by: (6) ψ δu = δf d 1 f 2 f (1) κ f ( κ 1) δu (1) δu (2) δu ( κ ) dx u 1 u 1 + u

4 Noether Invarant varatonal problems d 2 f 3 f (1) κ f ( κ 2) u 2 δ (2) δu (3) δu ( κ ) dx u 2 u 2 + u + ( 1) n κ d κ f u κ δ ( κ ), dx κ u and a correspondng dentty s true for n-fold ntegrals; n partcular, A ncludes δu up to ts (k 1) th dervatve. The fact that the Lagrangan expressons ψ are, n fact, defned by (4), (5), (6) follows from the fact that all hgher dervatves of the δu are elmnated by way of the combnatons on the rght-hand sde, whle, on the other hand, relaton (2) s fulflled, whch leads to the partal ntegraton unquely. One now deals wth the two theorems n what follows: I. If the ntegral I s nvarant under a G ρ then there wll be ρ lnearly ndependent couplngs of the Lagrangan expressons wth dvergences; conversely, the nvarance of I under a G ρ follows from the latter. The theorem s also true n the lmtng case of nfntely many parameters. II. If the ntegral I s nvarant under a G ρ, n whch the arbtrary functons appear up to ther σ th dervatves then there exst r dentty relatons between the Lagrangan expressons and ther dervatves up to σ th order; the converse s also true here 1 ). The statement of both theorems s true for the mxed groups, so dependent, as well as ndependent dvergence relatons wll appear. If one goes from these denttes to the assocated varatonal problem so one sets ψ = 0 2 ) then n the one-dmensonal case, for whch the dvergence goes to a total dfferental, Theorem I expresses the exstence of ρ frst ntegrals, between whch, nonlnear dependences can generally exst 3 ); n the mult-dmensonal case, one obtans dvergence equatons that are often referred to recently as conservaton theorems ; Theorem II expresses the dea that ρ of the Lagrangan equatons are consequences of the remanng ones. The smplest example of Theorem II wthout the converse s defned by the Weerstrass parametrc representaton. For t, the ntegral s known to be nvarant due to ts homogenety of frst order when one replaces the ndependent varable x by an arbtrary functon of x that leaves u unchanged (y = p(x); v (y) = u (x)). Thus, an arbtrary functon enters n, but wthout any of ts dervatves, and ths corresponds to the wellknown lnear relaton between the Lagrangan expressons themselves: ψ du = 0. dx The general theory of relatvty of the physcsts wll serve as a further example. Here, one deals wth the group of all transformatons of the x: y = p (x), whle the u (whch are 1 ) Wth certan trval exceptons; cf., 2, remark 2. 2 ) Somewhat more generally, one can also set ψ = T ; cf., 3, frst remark. 3 ) Cf., the concluson of 3.

5 Noether Invarant varatonal problems 5 denoted by g µν and q) wll be subjected to transformatons that are nduced by the coeffcents of a quadratc and lnear dfferental form that ncludes the frst dervatves of the arbtrary functons p(x). They correspond to the well-known n dependences between the Lagrangan expressons and ther frst dervatves 1 ). In partcular, f one specalzes the group by sayng that one allows no dervatves of the u(x) n the transformatons, and, n addton, the transformed ndependent quanttes may depend upon only the x, but not the u, then that the relatve nvarance 2 ) of ψ δu follows (as wll be shown n 5) from the nvarance of I, and lkewse for the dvergences that appear n Theorem I, as long as the parameters are subjected to certan transformatons. From ths, t follows that the aforementoned frst ntegrals also admt the group. For Theorem II, one lkewse deduces the relatve nvarance of the left-hand sdes of the dependences that are composed by means of arbtrary functons, and as a consequence of ths, one has a functon whose dvergence vanshes dentcally and admts the group that medates the connecton between dependences and the energy theorem n the relatvty theory of the physcsts 3 ). Theorem II ultmately gves a group-theoretc proof of an asserton of Hlbert that s connected wth ths concernng the breakdown of the proper energy theorems for general relatvty. Wth these extra remarks, Theorem I ncludes all of the known theorems on frst ntegrals n mechancs, etc., whle Theorem II can be regarded as the greatest possble group-theoretc generalzaton of general relatvty theory. 2. Dvergence relatons and dependences. Let G be a fnte or nfnte group; one may then always arrange that the dentty transformaton corresponds to the value zero for the parameter s (the arbtrary functons p(x), resp.) 4 ). The most general transformaton then takes the form: u y = A x, u,, x = x + x + u v (y) = B x, u,, x = u + u +, where x, u mean the terms of lower dmenson n ε (p(x), resp.) and ts dervatves; ndeed, they shall be assumed to be lnear n them. As we wll show later, ths s no loss of generalty. Now, let the ntegral I be an nvarant under G, so relaton (1) s fulflled. In partcular, I s then also nvarant under the nfntesmal transformaton: 1 ) Cf., perhaps, Klen s presentaton. 2 ) I. e, ψ δu takes on a factor under transformaton. 3 ) Cf., Klen s second note. 4 ) Cf., perhaps, Le: Grundlagen, pp If one s dealng wth an arbtrary functon then the specal values a σ of the parameter must be replaced wth fxed functons p σ, p σ /,, and correspondngly the values a σ + ε must be replaced wth p σ + p(x), p σ / + p/, etc.

6 Noether Invarant varatonal problems 6 y = x + x, v (y) = u + u that s ncluded n G, and under t relaton (1) goes to: v u (7) 0 = I = f y, v( y),, dy f x, u( x),, dx y, where the frst ntegral s taken over the x+ x-doman that corresponds to the x-doman. However, ths ntegraton can be converted nto an ntegraton over the x-doman by means of the converson that s true for nfntesmal x: v v (8) f y, v( y),, dy = f x, v( x),, dx Dv( f x) dx y +. If one then ntroduces the varaton: u (9) δ u = v (x) u (x) = u λ x n place of the nfntesmal transformaton x then (7) and (8) go to: (10) 0 = { δ f + Dv( f x)} dx. The rght-hand sde s the well-known formula for the smultaneous varaton of the dependent and ndependent varables. Snce the relaton (10) s fulflled under ntegraton over an arbtrary doman, the ntegrand must vansh dentcally; Le s dfferental equatons for the nvarance of I then go to the relaton: (11) δ f + Dv(f x) = 0. λ If one expresses δ f from (3) n ths usng the Lagrangan expressons then one gets: (12) ψ δ u = Dv B (B = A f x), and ths relaton thus represents an dentty for any nvarant ntegral I n all of the arguments that appear; ths s the desred form for Le s dfferental equatons for I 1 ). 1 ) (12) goes to 0 = 0 for the trval case when Dv(f x) = 0, δu = 0 whch can come about only when x, u also depend upon the dervatves of u; these nfntesmal transformatons are therefore always separate from the group, and only the number of the remanng parameters or arbtrary functons are to be counted n the formulaton of the theorems. Whether or not the remanng nfntesmal transformatons stll defne a group must reman undecded.

7 Noether Invarant varatonal problems 7 Now let G be assumed to be a fnte, contnuous group, to begn wth. Snce, by assumpton, u and x are lnear n the parameters ε 1,, ε ρ, from (9), the same thng s true for δ u and ts dervatves; therefore, A and B are lnear n the ε. I therefore set: B = B (1) ε B (ρ) ε ρ, (1) ( ) δ u = δ u ε1 + + δ u ρ ε ρ, (1) where the δ u, are thus functons of x, u, u /,, so the desred dvergence relatons follow from (12): (13) (1) ψ δ u = Dv B (1),, ( ) ψ δ u ρ = Dv B (ρ). One thus has ρ lnearly-ndependent couplngs of the Lagrangan expressons wth dvergences; the lnear ndependence follows from the fact that, from (9), t would follow that δ u = 0, u = 0, x = 0, so there would be a dependency between the nfntesmal transformatons. However, by assumpton, such a thng s not fulflled for any parameter values, snce otherwse the G ρ that further arses from the nfntesmal transformatons by ntegraton would depend upon less than ρ essental parameters. The further possblty that δ u = 0, Dv(f x) = 0 was, however, excluded. These conclusons are also stll true n the lmtng case of nfntely many parameters. Now, let G be an nfnte, contnuous group; δ u and ts dervatves, and therefore also B, wll be lnear n the arbtrary functons p(x) and ther dervatves 1 ). By substtutng the values of δ u, stll ndependently of (12), let: ψ δ u = ( λ ) σ ( λ ) ( λ ) ( λ) ( λ ) p ( λ ) p ψ a ( x, u, ) p ( x) + b ( x, u, ) + + c ( x, u, ) σ λ,. One may now, analogously to the formula for partal ntegraton, replace the dervatves of p wth p tself and dvergences that are lnear n p and ts dervatves usng the dentty: τ p( x) ϕ(x, u, ) = ( 1) τ τ ϕ p(x) τ τ One thus gets: (14) ψ δ u = mod dvergences. σ ( λ) ( λ ) σ ( λ) ( λ ) ( a ψ ) ( b ψ ) + + ( 1) ( c ψ ) p σ λ = Dv(B Γ). 1 ) The fact that t s no restrcton to assume that the p are free of the u, u /, shows the converse.

8 Noether Invarant varatonal problems 8 I now construct the n-fold ntegral of (15), taken over any doman, and choose the p(x) such that they vansh on the boundary of (B Γ), along wth all of the dervatves that appear. Snce the ntegral of a dvergence reduces to a boundary ntegral, the ntegral of the left-hand sde of (15) thus also vanshes for arbtrary p(x) that only vansh on the boundary, along wth suffcently many dervatves, and from ths, t follows, by a wellknown argument, that the ntegrands vansh for any p(x), so one has the ρ relatons: (16) σ ( λ) ( λ ) σ ( λ) ( a ψ ) ( b ψ ) + + ( 1) ( c ψ ) σ = 0 (λ = 1, 2,, ρ). These are the desred dependences between the Lagrangan expressons and ther dervatves for the nvarance of I under G ρ ; the lnear ndependence s clear, as above, snce the nverse leads back to (12), and snce one can agan go from the nfntesmal transformatons back to the fntes ones, as wll be done more thoroughly n 4. Thus, ρ arbtrary transformatons already appear n the nfntesmal transformatons for a G ρ. From (15) and (16), t then follows that Dv(B Γ) = 0. If one correspondngly assumes a mxed group of x and u that are lnear n the ε and the p(x) then one sees, when one sets the p(x) equal to zero and then the ε, that the dvergence relatons (13) exst, as well as the dependences (16). 3. Converse n the case of the fnte group. In order to show the converse, one must essentally follow through the foregong argument n the opposte sequence. The valdty of (12) follows from the valdty of (13) upon multplcaton by ε and addton, and by means of the dentty (3), ths mples a relaton: δ f + Dv(A B) = 0. If one then sets: x = 1/f (A B) then one arrves at (11) as a result of ths. From ths, (7) fnally follows by ntegraton: I = 0, and thus the nvarance of I under the nfntesmal transformaton that s determned by x, u, where the u s to be determned from x and δ u by means of (9), and x and u become lnear n the parameters. However, I = 0 mples, n a well-known way, the nvarance of I under the fnte transformatons that arse by ntegratng the smultaneous system: dx (17) dt du = x, dt x = y = u for t = 0. u = v These fnte transformatons nclude ρ parameters a 1,, a ρ, namely, the couplngs tε 1,, tε ρ. From the assumpton that there should be ρ and only ρ lnearly ndependent dvergence relatons (13), t follows moreover that the fnte transformatons always defne a group, as long as they do not nclude the dervatves u /. In the opposte case namely, at least one nfntesmal transformaton arses from the Le bracket process there would be no lnear couplng of the ρ remanng dvergence relatons, and snce I also admts ths transformaton, there would be more than ρ lnearly ndependent

9 Noether Invarant varatonal problems 9 dvergence relatons, or else ths nfntesmal transformaton would be of the specal form n whch δ u = 0, Dv(f x) = 0, but then x or u would depend upon dervatves, contrary to assumpton. Whether or not ths case can occur when dervatves appear n x or u must reman undecded. One then adds all functons x for whch Dv(f x) = 0 to the x that was determned above n order to once more preserve the group property. By conventon, the parameters that are thus added shall not, however, be counted. The converse s thus proved. From ths converse, t then follows that, n fact, x and u can be assumed to be lnear n the parameters. Namely, f u and x were of hgher degree n ε then, due to the lnear ndependence of the products of powers of ε, entrely analogous relatons to (18) would follow, only n a greater number, from whch, by the converse, one nfers the nvarance of I under a group whose nfntesmal transformatons nclude the parameters lnearly. Should ths group contan precsely ρ parameters, then there would have to exst lnear dependences between the orgnal dvergence relatons due to the terms of hgher order n ε. Let t be remarked that n the case where x and u also contan dervatves of the u the fnte transformatons can depend upon nfntely many dervatves of the u. In ths 2 2 d x case, the ntegraton of (17) then leads from the determnaton of d u, u to 2 2 dt dt κ u u uλ =, such that the number of dervatves of u generally ncreases at κ κ λ κ each step. Perhaps the followng wll serve as an example: f = u, ψ = u, ψ x = d dx (u u x), δ u = x ε, x = 2u ε, u = u λ 2u x ε. u Snce the Lagrangan expresson of a dvergence vanshes dentcally, the converse ultmately shows the followng: If I admts a G ρ then any ntegral that dffers from I only by a boundary ntegral.e., an ntegral of a dvergence wll lkewse admt a G ρ wth the same δ u, whose nfntesmal transformaton wll generally contan dervatves of the u. Thus, perhaps referrng to the example above, f * d u = 2 u admts the dx x nfntesmal transformaton u = xε, x = 0, whle dervatves of the u appear n the nfntesmal transformatons that correspond to f.

10 Noether Invarant varatonal problems 10 If one goes on to the varatonal problem.e., f one sets ψ = 0 1 ) then (18) goes to the equatons: Dv B (λ) = 0,, Dv B (ρ) = 0, whch are often referred to as conservaton laws. In the one-dmensonal case, t follows from ths that B (1) = const.,, B (ρ) = const., and therefore the B contan at most (2κ 1) th dervatves of the u (from (6)), as long as u and x nclude no hgher dervatves than κ th that appear n f. Snce 2κ th dervatves appear n ψ, n general 2 ), one thus has the exstence of ρ frst ntegrals. The f above once more shows that nonlnear dependences can exst between them. The lnearly ndependent u = ε 1, x = ε 2 correspond to the lnearly ndependent relatons: u d 1 d = u 2, u u = ( u ), whle a nonlnear dependency exsts between the frst dx 2 dx ntegrals u = const., u 2 = const. Thus, one s dealng wth the elementary case n whch u, x contan no dervatves of the u 3 ). 4. Converse n the case of nfnte groups. Frst, let us show that the assumpton of the lnearty of x and u presents no restrcton, whch one deduces here wthout the converse from the fact that G ρ formally depends upon ρ and only ρ arbtrary functons. Namely, t shows that n the nonlnear case the number of arbtrary functons would ncrease under the composton of transformatons n whch the terms of lowest order would add together. In fact, let, say: u y = A x, u,, ; p = x + a(x, u, ) p ν + b(x, u, ) p ν 1 p 2 + cp ν 1 p + + d ν p + (pν (1) 1 = ( p ) ν,, ( ) ν ρ ( p ) ρ ), u and analogously v = B x, u,, ; p, so under composton wth z = v A y, v,, ; q y, one gets, for the terms of lowest order: 1 ) ψ = 0, or, more generally, ψ = T ψ, where T are new functons that are to be added to the others, are referred to as feld equatons n physcs. In the case ψ = T, the denttes (13) go to denttes: Dv B (λ) ( ) = Tδu λ, whch are also referred to as conservaton laws n physcs. 2 ) As long as f s nonlnear n the κ th dervatves. 3 ) Otherwse, one would have u λ = const. for any λ, correspondng to: u (u ) λ 1 = 1 d λ ( u ). λ dx

11 Noether Invarant varatonal problems 11 z = x + a(p ν + q ν ) + b p p + q ν 1 ν 1 q + c p 2 2 ν 2 p ν 2 q + q + If a coeffcent that s dfferent from a and b here s dfferent from zero then a term σ σ ν σ p p + ν σ q q actually appears for σ > 1, so one cannot wrte ths as the dfferental quotent of a sngle functon or products of powers of them; the number of arbtrary functons thus has ncreased, contrary to assumpton. If all of the coeffcents that are dfferent from a and b vansh then each of the values of the exponents v 1,, v ρ wll be the second term of the dfferental quotent of the frst one (as s always the case for, e.g., a G 1 ), such that lnearty actually enters n, or else the number of arbtrary functons would also ncrease here. Due to the lnearty of the p(x), the nfntesmal transformatons thus satsfy a system of lnear partal dfferental equatons, and snce the group property s fulflled, they defne an nfnte group of nfntesmal transformatons, by Le s defnton (Grundlagen, 10). One deduces the converse now n a manner that s smlar to the one n the case of fnte groups. The exstence of the dependences (16) leads, upon multplcaton by p (λ) (x) and addton, usng the dentty converson (14), to ψ δ u = Dv Γ, and from ths, as n 3, one nfers the determnaton of x and u and the nvarance of I under these nfntesmal transformatons, whch, n fact, depend lnearly upon ρ arbtrary functons and ther dervatves up to order σ. The fact that these nfntesmal transformatons, when they nclude no dervatves u /,, certanly defne a group follows, as n 3, from the fact that otherwse more arbtrary functons would appear by composton, whle, by assumpton, there shall be only ρ dependences (16); they thus defne an nfnte group of nfntesmal transformatons. However, such a thng conssts (Grundlagen, Theorem VII, pp. 391) of the most general nfntesmal transformatons of a certan nfnte group G of fnte transformatons, n Le s sense. Every fnte transformaton wll then be generated by nfntesmal ones (Grundlagen, 7) 1 ), and thus arse from the ntegraton of the smultaneous system: dx dt = x, du dt = u x = y, for t = 0, u = v n whch, t can, however, be necessary to choose the arbtrary p(x) to be ndependent of t. G thus depends, n fact, on ρ arbtrary functons; n partcular, f t suffces to choose p(x) to be free of t then ths dependency wll be analytc n the arbtrary functons q(x) = t 1 ) From ths, t follows, n partcular, that the group G that s generated by the nfntesmal transformatons x, u of a G ρ agan leads back to G ρ. G ρ then ncludes no nfntesmal transformatons that are dfferent from x, u that depend upon arbtrary functons, and can also contan none that are ndependent of them that depend upon parameters, snce t would then be a mxed group. However, from the above, the fnte transformatons are determned by means of the nfntesmal ones.

12 Noether Invarant varatonal problems 12 p(x) 1 ). If dervatves u /, appear then t can be necessary to add nfntesmal transformatons δ u = 0, Dv(f x) = 0 before one can reach the same concluson. In connecton wth an example of Le (Grundlagen, 7), let a somewhat more general case be gven, where one can advance to explct formulas that lkewse show that the dervatves of the arbtrary functons up to order σ appear, from whch the converse s then complete. It s the example of those groups of nfntesmal transformatons that correspond to the group of all transformatons of the x and the transformatons of the u that are nduced by them;.e., those transformatons of the u for whch u, and consequently u, depend upon only the arbtrary functons that appear n x, whereby let t be assumed that the dervatves u /, do not appear n u. One thus has: x = p () (x), u = n ( λ ) σ ( λ ) ( λ ) ( λ ) ( λ ) p ( λ ) p a ( x, u) p + b + + c σ λ= 1. Snce the nfntesmal transformaton x = p(x) generates any transformaton x = y + g(y) wth arbtrary g(y), n partcular, p(x) can be determned to be ndependent of t, such that the followng one-parameter group wll be generated: (18) x = y + t g (y), whch goes to the dentty for t = 0 and to the desred x = y + g(y) for t = 1. In fact, t follows by dfferentaton of (18) that: dx (19) dt = g (y) = p () (x, t), where p(x, t) s determned from g(y) by nverson, and conversely, (18) arses from (19) by means of the auxlary condton that x = y for t = 0, by whch, the ntegral s establshed unquely. By means of (18), the x n u can be replaced wth the ntegraton constants y and t; thus, the g(y) appear up to precsely the σ th dervatves when one p g yκ expresses the y / n terms of / y n =, and, n general, replaces y σ p g wth ts values n terms of σ y,, y,, one then gets the system of equatons: κ σ x. For the determnaton of the u, σ y du dt σ g g = F g( y),,,, u, t σ y y (u = v for t = 0) 1 ) The queston of whether ths latter case always occurs was posed by Le n a dfferent formulaton (Grundlagen, 7 and 13, concluson).

13 Noether Invarant varatonal problems 13 n whch only t and u are varable, but the g(y), belong to the coeffcent doman, such that the ntegraton yelds: σ g g u = v + B v, g( y),,,, t σ y y, and therefore transformatons that depend upon precsely σ dervatves of the arbtrary functons. From (18), the dentty s ncluded n ths for g(y) = 0, and the group property follows from the fact that the chosen process produces any transformaton x = y + g(y), from whch the one that s nduced on the u s establshed unquely, so the group G wll be exhausted. Incdentally, t then follows from the converse that t s no restrcton to choose the arbtrary functons to depend upon only the x, but not on the u, u /, In the latter ( λ ) ( λ ) p p case, n fact,, enter nto the dentty transformaton (14), as well as nto u u, x (15), n addton to the p (λ). If one now chooses the p (λ) to be successvely of degree zero, one, n u, u /,, wth arbtrary functons of x as coeffcents, then the dependences (16) emerge agan, but n greater numbers, whch, however, from the converse above, lead back to prevous case under composton wth arbtrary functons that depend upon only x. One lkewse shows that the smultaneous appearance of dependences and dvergence relatons that are ndependent of them corresponds to mxed groups 1 ). t = 1 1 ) As n 3, t also follows from the converse here that, along wth I, also any ntegral I * that dffers by a dvergence lkewse admts an nfnte group wth the sameδ u, n whch, however, x and u wll generally nclude dervatves of the u. Ensten has ntroduced such an ntegral nto the general theory of relatvty n order to obtan a smpler statement of the energy theorem; I shall gve the nfntesmal transformatons that ths I * admts, for whch I preserve the notaton of Klen s second note precsely. The ntegral I = K dw = K ds admts the group of all transformatons of the w and the one that t nduces on g µν ; they correspond to the dependences ((30), n Klen): µτ µν g K µτ K µν g + 2 = 0. τ σ w Now, one has: I * = K * ds, where K * = K + Dv, and consequently, one wll have: K = K µν µν, where K, K µν µν mean the Lagrangan expressons n each case. Therefore, the dependences that were gven are also true for K, and after multplyng by p τ and addng, one gets by the reverse converson of the product µν dfferentaton: K µν p µν + 2 Dv ( g µσ K µν p τ ) = 0, δk * + Dv 2g K µτ p K p µν g = 0. σ µσ τ µν Comparng ths wth Le s dfferental equaton: δk * + Dv(K * w) = 0, t then follows that:

14 Noether Invarant varatonal problems Invarance of the ndvdual components of the relatons. If one specalzes the group G to the smplest case that s ordnarly consdered by specfyng that one allows no dervatves of the u n the transformatons and that the transformed ndependent varables depend upon only the x, but not the u then one can deduce the nvarance of the ndvdual components n the formulas. Frst of all, ths yelds, from known reasons, the nvarance of ( ψ δu ) dx; thus, one nfers the relatve nvarance of ψ δu 1 ), where we understand δ to mean any varaton. In fact, one has, on the one hand: u v δi = δ f x, u,, dx = f y, v,, dy δ, y and, on the other hand, for δu, homogeneous nature of the transformaton of the δu, vansh on the boundary, so one has, correspondngly: and t follows that for δu, u δ f x, u,, dx u δ, that vansh on the boundary, due to the lnear, x u δ v,, the δv, x = ( ψ (, ) δ ) v δ f y, v,, dy y u u dx, = ( ψ (, ) δ ) v v dy, u δ, that vansh on the boundary: x ( ψ ( u, ) δu ) dx = ( ψ (, ) δ ) y v v dx. = ( ψ (, ) δ ) n v v dy δ y, also If one expresses y, v, δv n the thrd ntegral n terms of x, u, δu and one sets t equal to the frst one then one has a relaton: w σ = 1 K 2g K µτ p K p µν g, g µν = p µν + gσ w σ µσ τ µν µν σ are nfntesmal transformatons that I * admts. These nfntesmal transformatons thus depend upon the frst and second dervatves of the g µν, and nclude the arbtrary p up to the frst dervatves. 1 ) I.e., under transformaton, ψ δu takes on a factor, whch s always referred to as relatve nvarance n the algebrac theory of nvarants.

15 Noether Invarant varatonal problems 15 ( χ (, ) δ ) u u dx = 0 for a du that vanshes on the boundary, but s otherwse arbtrary, and, as s known, the vanshng of the ntegrands for arbtrary δu follows from ths; one thus has the followng relaton dentcally n δu: ψ (u, ) δu = y x κ ( ψ (v, ) δv ), whch expresses the relatve nvarance of ψ δu, and consequently, the nvarance of ( ψ δu ) 1 ). In order to apply ths to the derved dvergence relatons and the dependences, one must frst confrm that the δ u that s derved from the u, x actually satsfes the transformaton laws for the varaton δu, as long as only the parameter (arbtrary functons, resp.) n δ v are determned n a way that corresponds to the way that they are determned for the smlar group of nfntesmal transformatons n y, v. Let T q denote the transformaton that takes x, u to y, v; snce T q s an nfntesmal n x, u, the one that s 1 smlar to t n y, v s gven by T = T q T p T, where the parameters (arbtrary functons r, resp.) are therefore determned from p and q. One expresses ths n formulas as: T p : ξ = x + x(x, p), u * = u + u(x, u, p), T q : y = A(x, q), v = B(x, u, q), q T q T p : η = A(x + x(x, p), q), v * = B(x + x(p), u + u(p), q). 1 From ths, one has, however, T r = T q T p T, so: q 1 v ) Ths concluson breaks down when y also depends upon the u, snce then δ f y, v,, also y f ncludes terms lke y δ y, so the dvergence converson does not lead to the Lagrangan expressons, just u as when one allows dervatves of the u; then, n fact, the δv wll lead to lnear combnatons of δu, δ x,, so after a further dvergence converson ths wll lead to an dentty ( χ (u, ) δu ) dx = 0, such that the Lagrangan expressons once agan do not appear on the rght-hand sde. The queston of whether one can also already conclude the exstence of dvergence relatons from the nvarance of ( ψ δu ) dx s, from the converse, equvalent to the queston of whether one can conclude that from the nvarance of I under a group that does not necessarly lead to the same u, x, but stll leads to the same δ u. In the specal case of smple ntegrals and only frst dervatves n f, one can deduce the exstence of frst ntegrals from the nvarance of the Lagrangan expressons for fnte groups (cf., e.g., Engel, Gött. Nachr. (1916), pp. 270.).

16 Noether Invarant varatonal problems 16 η = y + y(r) = y + A( x, q) x(p), v * = v + v(r) = v + B( x, u, q) B( x, u, q) x( p) + u( p) u. One replaces x = x + x n ths wth ξ ξ, from whch, x agan goes to x, so x vanshes; thus, from the frst formula n (20), η also agan 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 n (20) gves: v + δ v( y, v,, r) = v + B( x, u, q) δ u( p), u B δ v( y, v,, r) = δ u κ ( x, u, p ), u such that the transformaton formulas for varatons are, n fact, therefore fulflled, as long as δ v s assumed to depend only on the parameters (arbtrary functons r, resp.) 1 ). In partcular, the relatve nvarance of ψ δ u then follows; thus, the relatve nvarance of Dv B also follows, snce the dvergence relatons are also fulflled n y, v, and furthermore, from (14) and (13), one also has the relatve nvarance of Dv Γ and that of the left-hand sde of the dependences, when composed wth the p (λ), where the arbtrary p(x) (the parameters, resp.) are always replaced wth the r n the transformaton formulas. Ths then yelds the relatve nvarance of Dv(B Γ), and therefore that of a dvergence of a non-vanshng system of functons B Γ whose dvergence vanshes dentcally. From the relatve nvarance of Dv B, one may, n the one-dmensonal case and for fnte groups, draw a concluson about the nvarance of the frst ntegrals. The parameter transformaton that corresponds to the nfntesmal transformaton wll, from (20), be lnear and homogeneous, and due to the nvertblty of all transformatons, the ε wll also be lnear and homogeneous n the transformed parameters ε *. Ths nvertblty certanly remans preserved when one sets ψ = 0, snce no dervatves of the u enter nto (20). By equatng the coeffcents of the ε * n: Dv B(x, u,, ε) = dy dx Dv B(y, v,, ε* ), κ 1 ) Ths agan shows that y must be assumed to ndependent of u, etc., n order for the concluson to be vald. As an example, let us, perhaps, menton the δg µν and δq ρ that were gven by Klen, whch satsfy the transformatons for varatons, as long as p s subject to a vector transformaton.

17 Noether Invarant varatonal problems 17 the d dy B(λ) (y, v, ) wll then be lnear, homogeneous functons of the d dx B(λ) (x, u, ), such that from d dx B(λ) (x, u, ) = 0 or B (λ) (x, u) = const. t also follows that: d dy B(λ) (y, v, ) = 0 or B (λ) (y, v) = const. The frst ρ ntegrals that correspond to a G ρ thus admt the group n any case, such that the further ntegraton s also smplfed. The smplest example of ths s the one n whch f s free of x or one u, whch corresponds to the transformaton x = ε, u = 0 ( x = 0, u = ε, resp.). One has δ u = ε du (ε, resp.), and dx snce B can be derved from f and δ u by dfferentaton and ratonal couplngs, t s then also free of x (u, resp.) and admts the correspondng groups 1 ). 6. An asserton of Hlbert. From the foregong, one ultmately fnds the proof of an asserton of Hlbert about the connecton between the break-down of the proper energy theorem and general relatvty (Klen s frst note, Göttnger Nachr. (1917), answer, frst passage), and ndeed, n a generalzed group-theoretc context. Let the ntegral I admt a G ρ, and let G σ be any fnte group that arses from specalzng the arbtrary functons, so t s a subgroup of G ρ. The nfnte group G ρ then corresponds to dependences (16), and the fnte one G σ, to dvergence relatons (13), and conversely, t follows from the exstence of any sort of dvergence relatons that I s nvarant under a fnte group that s dentcal to G σ when and only when the δ u are lnear combnatons of the ones obtaned from G σ. The nvarance under G σ can thus lead to no dvergence relatons that dffer from (13). However, snce the nvarance of I under the nfntesmal transformatons u, x of G ρ for arbtrary p(x) follows from the valdty of (16), t already follows from ths, n partcular, that t s nvarant under the nfntesmal transformatons of G σ that arse by specalzaton, and consequently, under ( ) G σ. The dvergence relatons ψ δ u λ = Dv B (λ) must then must then be ( ) consequences of the dependences (16), whch can also be wrtten: ψ a λ = Dv χ (λ), where the χ (λ) are lnear couplngs of the Lagrangan expressons and ther dervatves. Snce the ψ enter nto (13), as well as (16), lnearly, the dvergence relatons must then be lnear combnatons of the dependences (16), n partcular, and the B (λ) themselves are 1 ) In the case where the exstence of frst ntegrals already follows from the nvarance of ( ψ δu ) dx, they do not admt the complete group G ρ ; e.g., (u δu) dx admts the nfntesmal transformaton: x = ε 2, u = ε 1 + xε 2, whle the frst ntegral u u x = const., whch corresponds to x = 0, u = xε 3, does not admt the other two nfntesmal transformatons, snce t ncludes u, as well as x, explctly. Ths frst ntegral corresponds smply to nfntesmal transformatons of f that nclude dervatves. One then sees that, n any case, the nvarance of ( ψ δu ) dx s acheved less often than the nvarance of I, whch responds to a queston that was posed n a prevous remark.

18 Noether Invarant varatonal problems 18 thus composed lnearly from the χ.e., from the Lagrangan expressons and ther dervatves, and from functons whose dvergences vansh dentcally, lke perhaps the B Γ that appeared n the concluson to 2, for whch Dv (B Γ) = 0 and the dvergence lkewse has the nvarant property. I wll refer to dvergence relatons for whch B () of the gven knd can be composed from the Lagrangan expressons and ther dervatves as unreal, and all others as real. Conversely, f the dvergence relatons are lnear couplngs of the dependences (16) hence, unreal then the nvarance under G σ follows from the nvarance under G ρ ; G σ becomes a subgroup of G ρ. The dvergence relatons that correspond to a fnte group G σ wll then be unreal when and only when G σ s a subgroup of an nfnte group that I s nvarant under. The orgnal Hlbert asserton s obtaned from ths by specalzng the group. Let the term translaton group mean the fnte group: so y = x + ε, v (y) = u (x), u x = ε, u = 0, δ u = ελ. λ λ As s known, nvarance under the translaton group expresses the dea that the x do not u enter nto I = f x, u,, dx explctly. Let the assocated n dvergence relatons: u ψ = Dv B (λ) (λ = 1, 2,, n) λ be referred to as energy relatons, snce the conservaton law Dv B (λ) = 0 that corresponds to the varatonal problem corresponds to the energy law, whle the B (λ) correspond to the energy components. One then has: If I admts the translaton group then the energy relatons become unreal when and only when I s nvarant under an nfnte group that ncludes the translaton group as a subgroup 1 ). An example of such an nfnte group s gven by the group of all transformatons of the x, along wth those nduced transformatons of the u(x) n whch only dervatves of the arbtrary functons p(x) appear; the translaton group then arses by specalzng p () (x) = ε. Therefore, t must reman undecded whether the most general of these groups s therefore already gven along wth the groups that arse from alterng I by a boundary ntegral. Induced transformatons of the gven sort arse perhaps when one subjects the u to the coeffcent transformatons of a total dfferental form;.e., a form a d λ x + b d λ 1 x dx κ + that ncludes hgher dfferentals, n addton to the dx. Specal nduced transformatons for whch the p(x) only appear n the frst dervatves are gven by the 1 ) The energy law n classcal mechancs, and lkewse n the older relatvty theory (where dx 2 goes to tself), are unreal, snce no nfnte groups appear there.

19 Noether Invarant varatonal problems 19 coeffcent transformatons of ordnary dfferental forms c dx dx 1 λ, and ordnarly one has consdered only these. Another group of the gven knd that cannot be a coeffcent transformaton, due to the appearance of logarthmc terms, s perhaps the followng one: y = x+ p(x), v = u + ln(1 + p (x)) = u + ln dy dx, x = p(x), u = p (x) 1 ), δ u = p (x) u p( x). Here, the dependences (16) become: dψ ψ u + dx = 0, whle the unreal energy relatons become: d( ψ + const.) ψ u + dx = 0. The smplest nvarant ntegral for the group s: I = 2u1 e dx. u u 1 2 The most general I s determned by ntegratng Le s dfferental equaton (11): whch goes to: d δ f + ( f x) = 0, dx f f f f p( x) + u + f p ( x) + p ( x) u u u = 0 (dentcally n p(x), p (x), p (x)) by substtutng the values of x and δ u, as long as one assumes that f depends upon only frst dervatves of the u. Ths system of equatons already possesses solutons that actually nclude the dervatves for two functons u(x), namely: u1 e f = ( u 1 u 2) Φ u1 u2,, u 1 u 2 1 ) One computes the fnte transformatons from these nfntesmal ones backwards from the method that was gven n the concluson of 4.

20 Noether Invarant varatonal problems 20 where Φ means an arbtrary functon of the gven arguments. Hlbert expressed hs asserton n such a way that the break-down of the proper energy law was a characterstc feature of the general theory of relatvty. In order for ths asserton to be lterally true, the term general relatvty must then be further regarded as t usually s, and also extended to the prevous groups that depend upon n arbtrary functons 1 ). 1 ) Wth ths, the valdty s agan confrmed of a remark of Klen that the usual termnology relatvty n physcs should be replaced wth nvarance under a group. ( Über de geometrschen Grundlagen der Lorentzgruppe, Jber. d. deutsch. Math. Veren. 19 (1910), pp. 287; prnted n Phys. Zet.)