Symmetric Lie Groups and Conservation Laws in Physics
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1 Symmetrc Le Groups and Conservaton Laws n Physcs Audrey Kvam May 1, 1 Abstract Ths paper eamnes how conservaton laws n physcs can be found from analyzng the symmetrc Le groups of certan physcal systems. To begn, I wll gve some background on the physcs of Lagrangan mechancs. Then I wll dscuss Le symmetres, Le groups, and Le algebras. Ths wll nclude an eposton on contnuous groups, tangent vectors, and nfntesmal generators. I wll end wth an eample usng the nfntesmal generators of a Le algebra to derve conservaton of energy for the moton of a free partcle n specal relatvty. 1 Lagrangan Mechancs A common objectve n theoretcal physcs s to eamne a gven physcal system and predct how the system changes qualtatvely and quanttatvely as tme ncreases. One way of dong ths, whch s dscussed n any ntroductory physcs course, s to fnd all the forces actng on the system and use Newton s Thrd Law, F = m a = p, to predct future behavor. Ths s called Newtonan mechancs. In more advanced mechancs courses, another method s ntroduced, called Lagrangan mechancs; nstead of lookng at forces, we look at energy. For more complcated physcal stuatons, workng wth the energes of a system s almost always easer than workng wth forces. However, ths method s not ntroduced n lower dvson courses as t requres a famlarty wth calculus of varatons. The Lagrangan L s gven by L = T V (1) where T s knetc energy and V s potental energy. The acton S s defned as S = 1 L[, ẋ, t] dt () and the equaton of moton for the system s gven by (t) such that S s statonary (an etremum). Ths leads to the Euler-Lagrange equaton: ( ) d L = L for L[, ẋ, t] (3) dt ẋ Note: The Lagrangan may be a functon of several varables, n whch case the Euler-Lagrange equaton becomes a set of equatons: ( ) d L = L for L[q 1, q,..., q, q 1, q,..., q, t] () dt q q Eample 1.1 Consder a mass on a sprng, as seen n Fgure 1. When the mass s dsplaced from ts rest poston, the sprng eerts a force back towards equlbrum. The potental energy assocated wth ths s V = 1 k, where k s the sprng constant and s the dsplacement of the mass from equlbrum. The knetc energy assocated wth the velocty of the mass s T = 1 mẋ, where m s mass and ẋ s the frst tme dervatve of the mass s dsplacement,.e. ts velocty. Thus the Lagrangan for ths system s L[, ẋ, t] = 1 mẋ 1 k (5) 1
2 Fgure 1: A mass m on a sprng wth sprng constant k. Fgure : An equlateral trangle has a dscrete set of symmetres. and the Euler-Lagrange gves ( ) d L = L dt ẋ d (mẋ) = k dt mẍ = k ẍ = Ths s obvously an equaton for smple harmonc moton, so our mass on a sprng s a smple harmonc ( k oscllator wth the soluton (t) = A cos m ). t δ ( k m ) (6) From Eample 1.1 we see that the Euler-Lagrange equaton results n a dfferental equaton, the soluton of whch gves our system s equaton(s) of moton. Wth smple systems the dfferental equaton may be easy to solve, but a more complcated system may result n a much more obscure soluton. In some stuatons, Le symmetres may be appled to solve more dffcult equatons. Le Symmetres An academc ntroducton to group theory wll generally nclude a dscusson on the symmetres of geometrcal objects. In a smple case, t s easy to see that an equlateral trangle, such as n Fgure, wll have the trval symmetry, two non-trval rotatons, and three reflectons n ts symmetrc group. In the case of the equlateral trangle we are eamnng dscrete symmetres, because only a certan fnte set of dscrete rotatons and reflectons est wthn the symmetrc group. For eample, a rotaton of π 3 s a symmetry of the equlateral trangle, but rotatons of π or π 6 or are not n the symmetrc group. On the other hand, consder a crcle. The set of symmetres of a crcle s nfnte. Any rotaton by an angle ɛ may be represented n Cartesan coordnates as the mappng Γ ɛ : (, y) ( cos ɛ y sn ɛ, sn ɛ + y cos ɛ) (7) as can be seen n Fgure 3. In ths case, ɛ s a contnuous parameter. Smlarly the set of reflectons of a crcle can be represented by the mappng Γ ρ : (, y) (, y) (8) followed by a rotaton Γ ɛ. [] Therefore, n the case of a crcle we are eamnng a symmetrc group whch s not dscrete; these knd of symmetres are known as Le symmetres, and they form a Le group, whch wll be eplored n the net secton. Restrctng ourselves to ODEs of the form any symmetry mappng must look lke y (n) = ω(, y, y,..., y (n 1) ), y (k) dk y d k, (9) Γ: (, y, y,..., y (n) ) (ˆ, ŷ, ŷ,..., ŷ (n) ), ŷ (k) = dk ŷ dˆ k, k N (1)
3 Fgure 3: Rotaton of a crcle by. and must fulfll the symmetry condton that y (n) = ω(, y, y,..., y (n 1) ), y (k) = dy (k 1) D y (k 1) = d D (11) holds when (1) holds, where D = + y y + y y +... and y () y. [] Eample.1 The usefulness of symmetry mappngs can be seen vsually. In Fgure, the vector feld of some system clearly has no dependence, so t s nvarant n. In another system, whose vector feld can be seen n Fgure 5, there s clearly symmetry, but t s more dffcult to defne n terms of and y than Fgure s. However we can apply the symmetry mappng y p + y, arctan (, y) 7 = (r, θ) (1) and we see that Fgure 5 maps to Fgure 6. Fgure 6 has a much more obvous symmetry we ve straghtened out the vector feld so that t s obvously nvarant n θ. We call r and θ the canoncal coordnates of the system, and ths symmetry s a canoncal transformaton. 5 Θ y y Fgure : Vector feld wth nvarance. Fgure 5: Vector feld wth rotatonal symmetry r Fgure 6: Vector feld wth θ nvarance. The ODE that results from the Euler-Lagrange equatons n Eample 1.1 s easly ntegrable, so Le symmetry analyss appears to be superfluous n the soluton. The followng eample from [] s more complcated. Eample. Consder the followng ODE: dy y 3 + y y = d y y (13) Ths s a nasty lookng equaton. The solutons are plotted n Fgure 7, and there appears to be rotatonal symmetry, meanng the symmetres (, y) 7 ( cos y sn, sn + y cos ) 3 (1)
4 Fgure 7: The solutons to (13), from Symmetry Methods for Dfferental Equatons by Peter E. Hydon. form a Le group for the dfferental equaton. Verfyng ths would be etremely tedous. However, we can rewrte the dfferental equaton n terms of polar coordnates where = r cos θ and y = r sn θ. Then (13) becomes dy dy d = d whch we can rearrange to solve for dr : = = d d dr dr [r sn θ] [r cos θ] sn θ + r cos θ cos θ r sn θ (15) dy dr = d r sn θ r cos θ sn θ dy d cos θ dy d = y = r y r dy d ( dy r d y y dy d = r(1 r ) ) (16) Ths ODE s easly ntegrated, and the symmetres (1) smplfy to (r, θ) (r, θ + ɛ) (17) whch s easly verfable by applyng the symmetry condton (11). Ths s just a translaton n θ, so a vector feld for a system wth ths dfferental equaton may look somethng lke Fgure 6; clearly there s no θ dependence so the system s nvarant wth respect to θ. 3 Le Groups Defnton 3.1 A topologcal space s a set X together wth a collecton of open subsets T that satsfes the followng aoms [3]:. The empty set s n T.. X s n T.. The ntersecton of a fnte number of sets n T s also n T. v. The unon of an arbtrary number of sets n T s also n T.
5 Defnton 3. A manfold s a topologcal space whch s locally Eucldean. A three-dmensonal sphere s an eample of a manfold. Defnton 3.3 Let U R k, V R n be open sets. A map f : U V s called smooth f ts every component (and there are n) s an nfntely dfferentable functon. Our defnton of smooth need not always be as strct as to have nfnte dervatves; nstead, we only need dervatves to est up to a desred pont. For eample, f we have a thrd-order dfferental equaton, we only need dervatves to est up to fourth order for the functon to qualfy (for our purposes) as smooth. Defnton 3. A Le group s a group G, whch s also a manfold, such that the group operatons are smooth maps from G G and G, respectvely, nto G, y G. (, y) y, 1 (18) So, based on our defntons, an element n a Le group s a contnuous dfferentable transformaton and has an nverse wth the same propertes. [1] Because a Le group s a manfold, t s locally Eucldean. Ths means that any symmetry parametrzed by some suffcently small ɛ (n ts canoncal coordnate system) can be appromated as a translaton. Ths can be done wth a Taylor epanson, and leads to the concept of a Le algebra. Le Algebras Defnton.1 A Le algebra g s a vector space over a feld F on whch a product [, ], called the Le bracket, s defned, wth the propertes I. X, Y g mply [X, Y ] g II. [X, αy + βz] = α[x, Y ] + β[x, Z] for α, β F and X, Y, Z g III. [X, Y ] = [Y, X] IV. [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = [] Propertes III. and IV are known as skew-symmetry and the Jacob dentty, respectvely. A Le algebra s equvalently defned as the tangent space at the dentty of a Le group, whch s much easer to vsualze than Defnton.1. The Le bracket may be defned n varous ways, as long as t fulflls the condtons of Defnton.1. A lnear Le algebra has matr elements, where the Le bracket s defned as the commutator [X, Y ] = XY Y X. Lnear Le algebras are frequently used n quantum mechancs. Another common eample of a Le algebra s the vector space R 3 wth the Le bracket defned as the cross product, [X, Y ] = X Y []. The bass of a Le algebra forms the nfntesmal generators of ts assocated Le group, whch are etremely useful n physcal applcatons of fndng dfferental symmetres. Let s go ahead and fnd the nfntesmal generators of a Le group; we begn by fndng the tangent vectors at any element n the Le group. For the Le group acton (, y) (ˆ, ŷ), the tangent vector to (ˆ, ŷ) s (ξ(, y), η(, y)) where dˆ dɛ = ξ(ˆ, ŷ), dŷ dɛ = η(ˆ, ŷ). The tangent vector at a pont (, y) s ( dˆ (ξ(, y), η(, y)) = dɛ, dŷ ) ɛ= dɛ. (19) ɛ= 5
6 So, f we Taylor epand, our Le group acton becomes, to frst order n ɛ: ˆ = + ɛξ(, y) + O(ɛ ) ŷ = y + ɛη(, y) + O(ɛ ) () and our nfntesmal generator X s []. X = ξ(, y) + η(, y) y (1) For ths to make more sense, let s do a smple eample nspred by [1]. Eample.1 Consder a Le group acton (, y) (ˆ, ŷ) where ˆ = e ɛ, ŷ = e ɛ y. () Then we can fnd our tangent vectors ξ(ˆ, ŷ), η(ˆ, ŷ): ξ(ˆ, ŷ) = dˆ dɛ = ɛ= η(ˆ, ŷ) = dŷ dɛ = y ɛ= so our Taylor epanson becomes (3) and our nfntesmal generator s ˆ = + ɛ, ŷ = y ɛy () X = y y. (5) 5 Noether s Theorem Theorem 5.1 Noether s Theorem states that any dfferentable symmetry of the acton of a physcal system has a correspondng conservaton law. In other words, f the Lagrangan functon for a physcal system s not affected by changes n the coordnate system used to descrbe them, then there wll be a correspondng conservaton law. [5] Clam 5.1 If the Lagrangan s nvarant under a translaton of any spatal coordnate, then lnear momentum s conserved. Clam 5. If the Lagrangan s nvarant under an angular rotaton, then angular momentum s conserved. Clam 5.3 If the Lagrangan s nvarant under a translaton n tme, then energy s conserved. Here I wll show a proof, from [5], of Noether s Theorem for conservaton of energy. Proof. Consder a Lagrangan L[, v, t] = T [v ] V [ ] (6) where knetc energy T s proportonal to velocty squared v = ẋ and potental energy V s a functon only of poston. Consder the frst tme dervatve of (6): 6
7 dl dt = [ L d dt + L ] dv v dt and wth the Euler-Lagrange equaton (3), ths can be wrtten as dl dt = [ ( )] d L v dt v [ = d ( ) ] (8) L v L dt v The tme dervatve of the epresson n square brackets s, so that quantty call t H must be ndependent of tme; H s conserved n tme. The frst term of H s ( ) L v (9) v but we know that only the knetc energy term n the Lagrangan s dependent on v. So (9) s equvalent to ( ) T v. (3) v Now we can apply Euler s Theorem for Homogeneous Functons [5], whch states that for a homogeneous functon f(a ) of degree k, kf(a ) = ( ) f(a ) a (31) a T s a homogeneous functon of order, so by (31), Whch means we can wrte H as T = H = T L ( ) T v v = T (T V ) = T + V. So H s equal to the knetc plus potental energes of the system, whch s the total energy of the system. Therefore, when the Lagrangan s ndependent of tme, total energy of the system s conserved. (7) (3) (33) Eample 5.1 Consder Eample.. We found that the system descrbed by the ODE has rotatonal symmetry, wth the set of symmetres dy d = y3 + y y y y (r, θ) (r, θ + ɛ) formng a Le group. By ths symmetry, θ s nvarant, so by Noether s Theorem, angular momentum s conserved n the system. Theorem 5. Let a Le group have generators of the form where the acton () s statonary. Then the vector X = ξ (, t) + η α (, t) t α (3) T = Lξ + (η j ξ t j ) L t j (35) 7
8 s a conserved vector on the Euler-Lagrange solutons,.e. D [T ] = where D s defned as n (11) The proof of ths s found n [1]. Now let s te everythng together. In ths last eample from [1], we wll use the generators of a specfc Le algebra called the Lorentz group to analyze the Lagrangan of a free partcle n specal relatvty and conclude the conservaton of energy of the system. Eample 5. The Lagrangan of a free partcle n specal relatvty s L = mc 1 β (36) where β = ẋ c and ẋ = ẋ 1 + ẋ + ẋ 3, assumng the partcle s spatal coordnates are ( 1,, 3 ). The generators of the Lorentz group are as follows for {1,, 3} X = t X = X j = j j X = t + 1 c t (37) Let s focus on X. By Theorem 5., T = L ẋ L ẋ = mc 1 β m ẋ 1 β (38) = mc 1 β T s an arbtrarly defned constant, so we can set T = E and conclude that total energy E = conserved. 6 Eercses mc 1 β 1. Fnd the tangent vector at ( 1, 3 ) for the Le group acton (, y) ( cos φ y sn φ, sn φ + y cos φ).. Consder the followng Lagrangan. Whch quanttes are conserved for ths system? Eplan. 7 Solutons L = 1 mẏ mgy 1. We have the mappng (, y) ( cos φ y sn φ, sn φ + y cos φ) = (ˆ, ŷ) dˆ = ξ(ˆ, ŷ) = ( cos φ y sn φ) dφ dŷ = η(ˆ, ŷ) = ( cos φ y sn φ) dφ So the tangent vector (ξ(, y), η(, y)) s ( dˆ dɛ, dŷ ) ɛ= dɛ = ( y, ) ɛ= 8 s
9 and at the pont ( 1, 3 ) the tangent vector s evaluated to be ( 3, 1 ).. The Euler-Lagrange equaton for ths Lagrangan s d [mẏ] = mg dt mÿ = mg ÿ = g Ths ODE doesn t depend on tme, poston, OR angle. So, energy, lnear momentum, and angular momentum are all conserved. 8 Acknowledgements A huge thank you to Professors Greg Ellott and Davd Latmer for takng the tme to dscuss Le groups and conservaton laws wth me, and for pontng me n the drecton of some fantastc sources. 9 Bblography [1] Erksson, Marcus. Symmetres and Conservaton Laws Obtaned by Le Group Analyss for Certan Physcal Systems. Thess. Uppsala School of Engneerng/Uppsala Unversty, 8. Web. [] Hydon, Peter E. Symmetry Methods for Dfferental Equatons: A Begnner s Gude. New York: Cambrdge UP,. Prnt. [3] Lpp, Johannes and Wessten, Erc W. Topologcal Space. From MathWorld A Wolfram Web Resource. [] Sattnger, Davd H., and O. L. Weaver. Le Groups and Algebras wth Applcatons to Physcs, Geometry, and Mechancs. New York: Sprnger-Verlag, Prnt. [5] Watkns, Thayer. Noether s Theorem: Its Eplanaton and Proof. Noether s Theorem: Its Eplanaton and Proof. San Jose State Unversty, n.d. Web. Mar. 1. 9
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