The Noether theorem. Elisabet Edvardsson. Analytical mechanics - FYGB08 January, 2016

Transcription

1 The Noether theorem Elsabet Evarsson Analytcal mechancs - FYGB08 January,

2 1 Introucton The Noether theorem concerns the connecton between a certan kn of symmetres an conservaton laws n physcs. It was proven by the German mathematcan Emmy Noether, n her artcle Invarante Varatonsprobleme n In ths report we see how ths theorem s use n fel theory as well as n screte mechancal systems. We wll see how symmetres of the Lagrangan or Lagrangan ensty gve rse to conserve quanttes an contnuty equatons, respectvely. Secton 2 gves a short ntroucton to how we escrbe fel theores. We ntrouce the Lagrangan ensty an show how to express the Euler- Lagrange equatons of a fel n terms of t. Then we move on to efnng symmetres of physcal systems n secton 3. In partcular we efne a Noether symmetry as a symmetry uner whch the Lagrangan ensty s nvarant, but the acton ntegral s allowe to change wth an ntegral of a total vergence. After ths, n secton 4, we escrbe Noether s orgnal result from her 1918-artcle, an prove part of t n a specal case that s nterestng n physcs. In secton 5, we use the results from secton 4 to fn the connecton between symmetres an conservaton laws an n secton 6 we look at some examples of how to use the theorem. Fnally, we conclue n secton 7 by scussng some subtletes an lmtatons of the theorem. 2 Fel theory Ths secton wll be a short ntroucton to how we escrbe fel theores n the Lagrangan formulaton, t wll manly follow [2]. A screte mechancal system s escrbe usng tme, t, as an nepenent parameter together wth a number of generalze coornates, q (t, that epen on tme. We efne the Lagrangan an the acton L = L(q, q, t, (2.1 S [q, q, t] := t1 t 0 Lt. (2.2 By conserng the varaton of the acton an by usng Hamlton s prcple, we can now erve the equatons of moton for the system. These are gve by the Euler-Lagrange equatons: q = 0. (2.3 t q Ths escrpton works well for screte systems wth a fnte (or countably nfnte number of egrees of freeom. Conser however a system such as a vbratng ro. To escrbe the system completely, we woul have to conser the vbraton of each nvual pont n the ro. Ths amounts to efnng one generalze coornate for each pont. Clearly ths s not possble. The soluton to ths problem s to conser not only the tme, t, as an nepenent parameter, but also the space coornates, (x, y, z, as such. To smplfy notaton, we wll enote (ct, x, y, z, where c s the spee of lght, by (x 0, x 2, x 2, x 3, (2.4 1

3 an n general we wll refer to one or all of these nepenent parameters by x µ. Now, to escrbe propertes of the system, we efne fels, φ, that epen on the nepenent parameter,.e. φ = φ(x µ. (2.5 The x µ are consere nepenent of each other, so all ervatves of the fels wth respect to some x µ can be taken as total ervatves. To smplfy notaton further, we make the followng efnton: φ,µ := φ x µ. (2.6 We wll also use the Ensten summaton conventon for Greek nces. It turns out that nstea of the Lagrangan, t s convenent to speak of a Lagrangan ensty L = L(φ, φ,µ, x µ. (2.7 Ths can be ntegrate over space to gve the ornary Lagrangan: L = L 3 x. (2.8 As before, we efne the acton as the ntegral of the Lagrangan over tme. Ths gves us S = L 4 x, (2.9 where s a regon n space-tme. From ths we can erve the Euler-Lagrange equatons for fels: φ x µ = 0. (2.10 (φ,µ 3 Symmetry transformatons Before we can scuss Noether s theorem n etal, we nee to scuss what we mean by a symmetry of a system. What we use to escrbe the system are the equatons of moton, so t s natural to say that a symmetry transformaton of a system s a transformaton of the epenent an nepenent varables that leaves the explct form of the equatons of moton unchange. Ths efnton, however, s too general for out purposes, so we wll now ntrouce what by some s calle a Noether symmetry. To arrve at ths efnton, we conser the argument mae n [1]. Suppose we have a coornate transformaton x µ x µ = x µ (x ν, φ φ = φ (x µ. (3.1 Now the Lagrangan ensty wll change accorng to 2

4 L(φ, φ,µ, x µ L (φ, φ,µ, x µ. (3.2 In the Lagrangan formulaton, the equatons of moton are gven by the Euler-Lagrange equatons. Stuyng equaton (2.10, we see that a suffcent conton for the equatons of moton to be nvarant uner ths transformaton s that the Lagrangan ensty takes the same form before an after the transformaton. That s L (φ, φ,µ, x µ = L(φ, φ,µ, x µ. (3.3 However, n the case of a screte system, we know that the Euler-Lagrange equatons reman unchange f a total tme ervatve s ae to the Lagrangan. The corresponng statement n fel theory s that the Euler-Lagrange equatons are unchange f a total vergence s ae to the Lagrangan ensty. So we get a symmetry f for some Λ. L (φ, φ,µ, x µ = L(φ, φ,µ, x µ + x µ Λµ, (3.4 We now turn to the acton. Uner ths same transformaton (3.1, the acton wll be transforme as S [φ, φ,µ, x µ ] S [ φ, φ,µ, x µ ]. (3.5 We can calculate the fference n the acton: S = S [ φ, φ,µ, x µ ] S [φ, φ,µ, x µ ] = L (φ, φ,µ, x µ 4 x L(φ, φ,µ, x µ 4 x. (3.6 Now, nepenently of whether the transformaton s a symmetry transformaton or not, the new Lagrangan must satsfy the Euler-Lagrange equatons. Ths n turn means that the new acton must satsfy Hamlton s prncple. By assumng that S satsfes Hamltons prncple an by conserng the varaton of S t s possble to show that a suffcent conton for ths s for some Λ. S [ φ, φ,µ, x µ ] = S [φ, φ,µ, x µ ] + x µ Λµ 4 x, (3.7 So the acton s only etermne up to an ntegral of a total vergence. Comparng equaton (3.7 wth equaton (3.4, we see that both of these expressons contan a total vergence. Snce the acton s the ntegral of the Lagrangan ensty, we can brng these two terms together an make the followng efnton: Defnton 3.1. A Noether symmetry s a symmetry transformaton of the form (3.1 that satsfes 1. Invarance of the Lagrangan: L (φ, φ,µ, x µ = L(φ, φ,µ, x µ. (3.8 3

5 2. Invarance of the acton up to an ntegral of a total vergence: S [ φ, φ,µ, x µ ] = S [φ, φ,µ, x µ ] + x µ Λµ 4 x. (3.9 So far we have talke about transformatons that can take any form. What we wll manly be ntereste n, are nfntesmal transformatons. These are of the form x µ x µ + x µ, φ φ + φ, (3.10 where x µ an φ are nfntesmal. We wll treat the nfntesmal quanttes as varatons, so we wrte x µ x µ + δx µ, φ(x µ φ(x µ + δφ(x µ. (3.11 Now, the x µ are nepenent varables, so they can be vare nepenently. However, some care s requre when conserng the varaton of φ, snce the φ epen on the x µ. The total varaton of φ(x µ s gven by δφ(x µ = φ (x µ φ(x µ. (3.12 It s natural to splt ths varaton nto two parts [1, 2], so that we separate the varaton of the epenent an nepenent varables. So we efne δ 0 as the varaton only n φ: δ 0 (x µ := φ (x µ φ(x µ. (3.13 Combnng these, we en up wth δφ = φ (x µ φ(x µ = φ (x µ + δ 0 φ φ (x µ = δ 0 φ + φ (x µ φ (x µ. (3.14 Now, by Taylor expanson, to frst orer we have [4] φ (x µ = φ (x µ + (x µ x µ φ,µ, (3.15 whch s the same thng as φ (x µ φ (x µ = δx µ φ,µ. (3.16 By efnton of δ 0, to frst orer we now have φ (x µ φ (x µ = δx µ φ,µ. (3.17 Insertng ths nto (3.14, we get δφ = δ 0 φ + δx µ φ,µ. (3.18 4

6 4 Noether s frst theorem The paper that Noether publshe n 1918, [3], was purely mathematcal wthout any reference to symmetres or conservaton laws n physcs. We wll here escrbe one of her results an prove part of t n a specal case that s nterestng for physcs. Let x 1, x 2,..., x n be nepenent varables an let u 1 (x, u 2 (x,..., u ν (x be functons of these varables. A transformaton group, G ρ s calle a fnte contnuous transformaton group f the transformatons can be expresse n a form that epens on ρ constant parameters ɛ. Smlarly an nfnte contnuous transformaton group, G ρ, s a group whose transformatons epen on ρ arbtrary functons, p(x an ther ervatves n an analytcal or at least contnuously fferentable way. The most general form of a transformaton n G ρ to nepenent varables y 1,..., y n an epenent varables v 1 (y,..., v ν (y can be wrtten as ( y = A x, u, u x,... v = B ( x, u, u x,... where the x an u terms are lnear n ɛ. A functon, f, s sa to be nvarant uner G ρ f ( f x, u, u x,... = f = x + x +... = u + u +... ( y, v, v y,... Conser an ntegral, I, that s nvarant uner G ρ. Then t satsfes ( I = f x, u, u x,... x = f ( y, v, v y,... (4.1 (4.2 y (4.3 where s an arbtrary oman n x an s the corresponng oman n y. The varaton of the ntegral s gven by ( ( δi = Ψ x, u, u x,... δu x, (4.4 where the functons Ψ are referre to as the Lagrangan expressons. We note that f I ha been the acton ntegral, the Ψ woul smply have been the quantty that we set to zero n the Euler-Lagrange equatons. From ths we can now formulate Noether s two theorems Theorem 4.1. An ntegral I s nvarant uner a fnte contnuous transformaton group G ρ f an only f there are ρ lnearly nepenent combnatons among the Lagrangan expressons that become vergences. Theorem 4.2. An ntegral I s nvarant uner an nfnte contnuous transformaton group G ρ epenng on arbtrary functons an ther ervatves up to orer l f an only f there are ρ enttes among the Lagrangan expressons an ther ervatves up to orer l. 5

7 Even though both of these theorems are nterestng n physcs, we wll only conser the frst one, whch we from now on wll refer to as Noether s frst theorem. As mentone, we wll not gve the proof of Noether s frst theorem n full generalty, nstea we wll conser the followng statement, for whch the proof wll use results manly from [1]. Suppose that we have a system n the Lagrangan formulaton escrbe by nepenent coornates x µ wth fels φ. Assume further that the equatons of moton of ths system are nvarant uner the fnte contnuous transformaton group, G ρ, of Noether symmetres. Then there are ρ lnearly nepenent combnatons among the Lagrangan expressons that become vergences. That s to say, we are ntereste n the case where the x an y represent space an tme an the u an v represent fels. In ths case I s the acton ntegral an f s the Lagrangan ensty. Also, the transformatons that are of physcal nterest are those where A an B are nepenent of ervatves. We begn by conserng the group G ρ. We efne t as a group of transformatons that epen on ρ constant parameters ɛ. Now, n partcular ths group contans nfntesmal transformatons on the form (3.10, where the -terms are consere lnear n ɛ, n agreement wth the notaton n equaton (4.1. Conserng the transformatons as varatons, we want to see what happens to the varaton of the acton ntegral. We have δs = L(φ, φ, x µ 4 x L(φ, φ, x µ 4 x. (4.5 To rewrte ths, we make a change of varables so that the regons of ntegraton become the same. We then get ( δs = L(φ, φ, x µ + δl(φ, ( φ, x µ 1 + x µ (δxµ 4 x L(φ, φ, x µ 4 x. (4.6 We are ntereste n nfntesmal transformatons, an therefore n varatons of the frst orer, so ( δs = δl(φ, φ, x µ + L(φ, φ, x µ x µ (δxµ 4 x. (4.7 The frst term n the ntegral contans a varaton of the Lagrangan. It s esrable to rewrte ths n terms of varatons of the nepenent varables an fels. By Taylor-expanng the frst term [4], we get (to frst orer δs = ( φ δ 0 φ + We rewrte the ervatves n the last two terms an get δs = ( φ δ 0 φ + Conser now the varaton δ 0 (φ,µ. We have x µ (δ 0φ = (φ,µ δ 0(φ,µ + L x µ δxµ + L (δxµ x µ 4 x. (4.8 (φ,µ δ 0(φ,µ + x µ (φ (x φ(x = φ x µ φ x µ = δ 0 6 x µ (Lδxµ 4 x. (4.9 ( φ x µ = δ 0 (φ,µ. (4.10

8 Usng ths n equaton (4.9, t gves us ( δs = δ 0 φ + φ (φ,µ x µ (δ 0φ + x µ (Lδxµ 4 x. (4.11 Some algebrac manpulaton now leas to [ δs = δ 0 φ + ( φ x µ (φ,µ δ 0φ ( x µ φ,µ [( = ( φ x µ δ 0 φ + (( (φ,µ x µ φ,µ δ 0 φ + ] x µ (Lδxµ 4 x ] δ 0 φ + (Lδx µ 4 x. (4.12 Now we requre that G ρ shoul be a group of Noether symmetres. ememberng our efnton of a Noether symmetry n secton 3, we have the conton that S = µ Λµ 4 x, (4.13 for the transformaton of the acton ntegral. Ths ffers from Noether s orgnal theorem that requre I = 0. Ths, however, as we wll see oes not make the problem more ffcult t merely as an extra term. Our other conton on a Noether symmetry, was the nvarance of the Lagrangan ensty. Combnng ths wth the expresson (3.6 for S an equaton (4.5, whch escrbes the varaton of S, we see that δs = S. (4.14 Ths means that δs = µ Λµ 4 x. (4.15 Equatng ths wth the above expresson for δs an notng that the regon of ntegraton,, s arbtrary, we get ( φ x µ δ 0 φ = (φ,µ ( x µ (φ,µ δ 0φ + Lδx µ Λ µ. (4.16 It s clear that f we ha use Noether s orgnal conton, S = 0, the only fference woul be the abcence of the Λ-term. Now, we can get r of the varatonal terms. By efnton, a fnte contnuous group of transformatons contans transformatons that can be wrtten n terms of a number of constant parameters, ɛ. The varatons can now be expresse as lnear combnatons of the (nfntesmal ɛ by δ 0 φ = k (δ 0 φ ɛ k = ɛ k k η k ɛ k, (4.17 7

9 an. Smlarly, we wrte δx µ = k Λ µ = k (δx µ ɛ k = ɛ l k Λ µ ɛ k ɛ k = k ξ µ k ɛ k. (4.18 λ µ k (xɛ k. (4.19 Substtutng all of ths nto (4.16, we get ( φ x µ δ 0 φ = ( (φ,µ x µ η k ɛ k + L ( µ φ k k ξ µ k ɛ k k λ µ k (xɛ k. (4.20 Snce the ɛ are constant parameters, they are unaffecte by the ervatve. For each k we thus obtan an equaton ( φ x µ η k = (φ,µ x µ jµ k, (4.21 where j µ k = ( (φ,µ η k + Lξ µ k λµ k. (4.22 Ths s calle the Nother current. Here we see why the argument fals n the case of an nfnte contnuous group of transformatons. When the constant ɛ are swtche for functons, p(x, they are no longer unaffecte by the ervatves. For future convenence, we rewrte the expresson for j µ k slghtly [2]. We know from equaton (3.18 how to express δφ n terms of δ 0 φ an δx µ. Insertng equatons (4.17 an (4.18 nto (3.18 gves us δφ = η k ɛ k + φ,µ ξ µ k ɛ k = (η k + φ,µ ξ µ k ɛ k. (4.23 k k k Lettng we get that ψ k = η k + φ,µ ξ µ k, (4.24 δφ = k ψ k ɛ k. (4.25 Insertng ths nto (4.16 we get the followng expresson for the Noether current: j µ k = [ ( (φ,µ ψ k + Lδ ν µ ] (φ,µ φ,ν ξk ν λµ k. (4.26 8

10 5 Conservaton laws Now we want to relate the prevous scusson to conservaton laws n physcs. A conserve quantty, Q, s a quantty that satsfes Q = 0. (5.1 t We see from equaton (4.21 that f all fels satsfy Hamlton s prncple, the left-han se of (4.21 s zero. We are then left wth x µ jµ k = 0. (5.2 Ths s a contnuty equaton an s not a conserve quantty n the sense that we just efne. However, contnuty equatons are at least as mportant, f not more, n physcs, so ths s a nce result. It turns out that we actually can extract a conserve quantty from the contnuty equaton. Namely, f we ntegrate equaton (5.2 over some volume V n space we get [1] 0 = V x µ jµ k 3 x = V x 0 j0 k 3 x + V x j k 3 x. (5.3 From ths we get 0 = jk 0 t 3 x + ˆn jk S, (5.4 V S where we have converte the secon ntegral nto a surface ntegral usng Gauss theorem. If we let V be all of space an assume that the fels vansh at nfnty, we see that the secon ntegral must be zero. We thus get an we see that s a conserve quantty. jk 0 t x = 0, 3 (5.5 Q k = jk 0 3 x 3 (5.6 So now we have scusse Noether s theorem n the case of fel theores n physcs. From the scusson n secton 2, t s clear that t shoul be of nterest to fn the corresponng result n the case of a screte system. Although the ervaton of equaton (5.2 was long, we note that we o not have to go through all the steps agan [2]. Instea we can mmeately make the followng substtutons n equaton (4.26: Ths gves us the equaton t L L, x µ t, φ q, φ,µ q. (5.7 [( q L ξ k ] ψ k + λ k = 0, (5.8 q q 9

11 where δt = k ξ k ɛ k, (5.9 δq = k ψ k ɛ k, (5.10 an Λ = k λ k ɛ k, (5.11 where S [ q, q, t ] = S [q, q, t] + Λ t. (5.12 t Ths gves us a conserve quantty n the sense of equaton (5.1. We note that here we obtane the conserve quantty mmeately. Ths of course, s a consequence of the fact that we only have one nepenent parameter,.e. tme. We wll now look at some examples of how to use these results. 6 Examples In ths secton we wll see how Noether s theorem can be use to obtan conservaton laws an contnuty equatons. 6.1 The stress-energy tensor [2] Assume that we have a system, escrbe by x µ an φ, wth a Lagrangan ensty that s nvarant uner an nfntesmal transformaton x µ x µ + x µ, (6.1 φ φ, (6.2 such that x µ = a µ an φ = 0, where a s a constant. Ths means that we are conserng systems nvarant uner translatons n space an tme. We can escrbe ths kn of symmetry wth a fnte contnuous transformaton group, G 4. Lettng ɛ α = a α an conserng the nfntesmal transformatons as varatons, we get usng equatons (4.18 an (4.25 that ξ µ α = δ µ α, (6.3 ψ α = 0. (6.4 Assumng that the Lagrangan ensty s not explctly epenng on any of the x µ, we see that the acton ntegral must be nvarant uner ths transformaton. Thus λ µ α = 0. Insertng all of ths nto equaton (4.26, we get j α µ = [( Lδ ν µ ] (φ,µ φ,ν δα ν. (6.5 10

12 So we have [ ] x µ φ,α Lδ α µ = 0. (6.6 φ,µ The quantty nse the parenthess we recognze as the stress-energy tensor, so we have just erve the conservaton of the stress-energy tensor. The stress-energy tensor gves us nformaton about both the energy an the momentum of a system. In fel-theory we get ths conservaton law n one step. However, when we eal wth screte systems, where space an tme are treate fferently, we nee to conser two fferent kns of symmetres to arrve at conservaton laws for both energy an momentum. Ths wll be shown n the followng two examples. 6.2 The Hamltonan [2] Conser a screte system wth generalze coornates q (t. Conser the nvarance uner tmetranslatons. Ths means that we conser a transformaton whch transforms the tme, but leaves the generalze coornates nvarant, t t + δt. (6.7 Ths symmetry can be escrbe by a fnte contnuous transformaton group G 1, wth ɛ = δt. Ths gves us ξ 1 = ξ = 1, (6.8 ψ k = ψ = 0. (6.9 If the Lagrangan s not explctly tme epenent, t s clear that the form of the Lagrangan must be nvarant uner ths transformaton. Ths means that also the acton ntegral s unchange an therefore λ k = 0. Insertng all ths nto (5.8 gves us ( q L = 0. (6.10 t q The quantty q q L, (6.11 we recognze as the Hamltonan, an thus we see that the Hamltonan s conserve f the Lagrangan s not explctly tme epenent. If the Hamltonan s the total energy we then see that ths s just the conservaton of energy. 6.3 Canoncal momentum [1] Conser a one-mensonal system of partcles escrbe by spatal coornates x. Suppose that the Lagrangan, L(x, ẋ, t, s nvarant uner spatal translatons. An nfntesmal spatal translaton can be wrtten as x x + a, (

13 where a s an nfntesmal constant parameter. We requre that t s unchange n the transformaton, so we get the followng: So an thus t ξ k = 0, (6.13 η k = 1. (6.14 ψ k = 1, (6.15 ( + λ k = 0. (6.16 ẋ Assumng that the acton s numercally nvarant uner the transformaton, we get that λ k = 0 an thus ( = 0. (6.17 t ẋ The quantty we recognze as the total lnear canoncal momentum of the system. quantty. ẋ, (6.18 Ths s thus a conserve Even though t s certanly nce to have been able to erve the conservaton of canoncal momentum an the conservaton of the Hamltonan n the last secton n ths way, t shoul be note that these conservaton laws coul easly have been foun by just lookng at the Euler-Lagrange equatons an settng approprate terms to zero. 6.4 The Klen-Goron equaton [2] As a fnal example we wll conser another example from fel theory, namely the Klen-Goron equaton. It s a relatvstc verson of the Schrönger equaton that escrbes a spnless partcle. It looks lke 2 φ + 1 c 2 2 φ t 2 + µ2 0φ = 0. (6.19 A corresponng Lagrangan ensty s gven by L = c 2 φ,ν φ, ν µ 2 0c 2 φφ. (6.20 We see that the Lagrangan ensty s nvarant uner a phase-transformaton of φ, namely one of the form φ φe θ, (6.21 where θ s a constant parameter. Clearly ths transforms φ φ e θ. Assumng that ths s an nfntesmal transformaton, we have to frst orer by Taylor expanson that φ φ + θφ, φ φ θφ. (

14 Now we conser φ an φ as two fferent fels, so we enote them by φ 1 an φ 2 respectvely. The x µ are unaffecte by ths transformaton, so we can escrbe the transformaton usng one constant parameter ɛ = θ. Ths means that ξ µ k = 0, (6.23 ψ 1 = φ, (6.24 ψ 2 = φ. (6.25 Snce both the Lagrangan ensty an the x are nvarant uner the transformaton, the acton ntegral must also be nvarant. Thus λ µ k = 0. Inserte nto (4.26 ths gves us ( j µ = (φ, µ φ (φ, µ φ = (φ, µ φ φφ, µ. (6.26 In contrast to the Schrönger equaton, we cannot nterpret j 0 as a probablty ensty snce t s not postve-efnte. Accorng to [6], the way we shoul nterpret t s that qj 0, where q s the charge, s charge ensty an qj s electrc current. Wth ths nterpretaton, we thus get a statement about the conservaton of charge. The symmetry transformaton, (6.22, escrbe here s a specal case of a bgger class of symmetry transformatons calle gauge transformatons of the frst kn. Those are transformatons of the fel varables only,.e. they leave x µ nvarant, such that φ φ + ɛc φ, (6.27 where c are constants. We then get j µ = c φ. (6.28 φ, µ 7 Some remarks It shoul be clear from the last secton that Noether s theorem s useful. In general, especally n fel theory, we are ntereste n contnuty equatons an conserve quanttes. These can be har to fn by just lookng at the equatons escrbng the system an t can take a lot of work ervng them. Noether s theorem offers a way aroun ths. We smply nee to fn symmetres of the fel equatons that keep the Lagrangan ensty nvarant, somethng that usually s much easer than fnng conservaton laws by lookng at the fel equatons, an then t s straghtforwar to just nsert ths nto equaton (4.26. So even though Noether s frst theorem s state n both rectons, t s most useful when gong from symmetres to conservaton laws an ths s how t generally s use. Therefore t s not a severe lmtaton to only have proven t n one recton. One mportant thng to note s that t s not true that all symmetres correspon to conservatons laws an conversley, not all conservaton laws correspon to symmetres of the system. The frst shoul be mmeately clear from how we erve Noether s theorem. We chose to only conser Noether symmetres, so even though e.g. φ Cφ, for C C, s a symmetry of equaton (6.19, t only gves rse to a conservaton law f C = 1. 13

15 To see how conservaton laws can fal to have a corresponng symmetry, we look at the followng example [2]: Conser the one-mensonal Klen-Goron equaton for real fels. Startng from equatons (6.20 an (6.19, lettng φ = φ an conserng only one spatal menson, we obtan the Lagrangan ensty L = c2 2 wth the corresponng fel equaton [ φ2 c 2 ( ] φ 2 µ 2 x 0φ 2, (7.1 2 φ x φ c 2 t 2 = µ2 0φ. (7.2 We see that L s nepenent of x an t, so the Lagrangan ensty s nvarant uner space-tme translatons. Also there s a symmetry arsng from a Lorentz transformaton. Usng Noether s theorem, we woul therefore expect to get three conserve quanttes. However, t can be shown that there are n fact nfntely many conserve quanttes n ths system (ths s a consequence of the fact that equaton (7.2 has solton solutons. Namely, we can fn nfntely many polynomals P an Q n φ an ts ervatves, such that P t + Q x = 0. (7.3 However, to obtan these conservaton laws t seems lke one has to use a non-lagrangan escrpton of fels, so clearly these are not conservaton laws we woul expect to fn usng the Noether theorem. As a fnal remark, we brefly menton Noether s secon theorem. We mentone t n secton 4, where we saw that t concerns transformatons whch epen on a number of functons, nstea of the constant parameters n the frst theorem. These transformatons are not symmetres n the usual sense, nstea one talks of local gauge transformatons [5], where the transformaton epens on where we are. An easy example that relates to gauge transformatons of the frst kn, escrbe n secton 6.4, s the followng transformaton: φ(x φ (x = e θ(x φ. (7.4 The fference s that we now allow θ to epen on x. We wll not say more about ths than that local gauge transformatons are mportant n physcs, e.g. n quantum electroynamcs. 8 Concluson In ths report we have examne the connecton between Noether symmetres an conservaton laws n physcs usng Noether s theorem. We have seen how to erve part of the theorem n a specal case nterestng n physcs an how to use the theorem n some examples. It s clear that the theorem offers a nce way to fn conservaton laws, snce t s much smpler to fn approprate symmetres of the equatons of moton than to see the conservaton laws by just lookng at the equatons escrbng the system. 14

16 eferences [1] K. Brang, 2001, Symmetres, Conservaton Laws, an Noether s Varatonal Problem, PhD thess, St. Hugh s Collage, Oxfor [2] H. Golsten, C.P. Poole, J.L. Safko, 2014, Classcal Mechancs, 3 eton, Pearson Eucaton Lmte [3] Y. Kosmann-Schwarzbach, 2011, The Noether Theorems - Invarance an Conservaton Laws n the Twenteth Century, Sprnger, New York [4] J.W. Norbury, 2000, Quantum Fel Theory, Physcs Department, Unversty of Wsconsn- Mlwaukee [5] B. e Wt, E. Laenen. 2011, Fel Theory n Partcle Physcs. lecture notes, Unverstet Utrecht [6] Department of Physcs Wk at Flora State Unversty, Klen-Goron equaton, wk.physcs.fsu.eu/wk/nex.php/klen-goron_equaton#contnuty_equaton, accesse