The Classical Particle Coupled to External Electromagnetic Field Symmetries and Conserved Quantities. Resumo. Abstract


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1 The Classical Particle Couple to External Electromagnetic Fiel Symmetries an Conserve Quantities G. D. Barbosa R. Thibes, Universiae Estaual o Suoeste a Bahia Departamento e Estuos Básicos e Instrumentais R. R. Ferreira Universiae Estaual o Suoeste a Bahia Departamento e Ciências Exatas Resumo Consieramos uma partícula clássica acoplaa a um campo eletromagnético externo, em ambos os regimes relativístico e nãorelativístico. O acoplamento é construío através o potencial eletromagnético satisfazeno as equações e Maxwell clássicas. Revemos o teorema e Noether a nível clássico associano simetrias infinitesimais a quantiaes conservaas. As simetrias funamentais o espaçotempo são investigaas consierano uma ação nãorelativística e outra relativística em um referencial particular. Consierano algumas hipóteses e simetria sobre a variação a ação calculamos em etalhes as quantiaes conservaas. Palavraschave Simetrias, teorema e Noether, ação clássica, constantes e movimento, grupo e Lorentz. Abstract We consier a classical particle minimally couple to an external electromagnetic fiel, in both nonrelativistic an relativistic regimes. The coupling is constructe via the electromagnetic potential which is assume to satisfy the classical Maxwell equations. We review Noether s theorem at classical level associating infinitesimal symmetries to conserve quantities. The funamental spacetime symmetries are investigate consiering a nonrelativistic an a relativistic action in a particular reference frame. Consiering some symmetry hypothesis on the action variation we work out in etail the corresponing conserve quantities. Keywors Simmetries, Noether s theorem, classical action, conserve quantities, Lorentz group. 1 Introuction The celebrate Noether s theorem [1, 2, 3], originally state an prove by the German mathematician Emmy Noether in 1915 an first publishe in 1918 [1], relates continuous ifferentiable symmetries of the classical action to conserve quantities, i.e., constants of motion. Such connection is achieve through Hamilton s principle of least action [2, 3, 4]. That is a central result in theoretical physics, with eep consequences not only to Classical Physics but also to Quantum Fiel Theory, for example in the stuy of Noether currents an
2 anomalies at quantum level [5]. The latter ones, for instance, concern classically conserve quantities which are not quantum mechanically conserve; i.e., anomalies are relate to the breaking of Noether s theorem at quantum level [5, 6]. Besies its practical applications, Noether s theorem has also a certain philosophical appeal. It relates the more funamental conservation laws of nature, such as energy an momentum, to simple symmetries of the very spacetime continuum we live in, such as time an space translations. It also plays an important role in relating the invariance of physical laws uner symmetry operations, incluing for instance the equivalence of ifferent inertial frames, with corresponing conserve quantities [2, 3]. The present paper is organize as follows. In section 2 below, we iscuss symmetries of the action in classical mechanics. We introuce an prove Noether s theorem using the Lagrangian formalism. In section 3 we iscuss the nonrelativistic particle couple to external electromagnetic fiel, applying Noether s theorem an calculating the conserve quantities. In section 4 we generalize the previous results to the relativistic particle, particularly consiering Lorentz symmetries. We close in the last section with some perspectives an concluing remarks. 2 Noether s Theorem Consier a classical system escribe by n generalize coorinates q i, i = 1,...,n, with time evolution governe by the Lagrangian function L = L(q i, q i,t). The action S of the system is given by the integral S = t2 L(q i (t), q i (t),t). (1) The real parameter t represents the physical time. Loosely speaking, Noether s theorem states that to each continuous symmetry of action (1) correspons a constant of motion. Suppose we perform an infinitesimal transformation in the coorinates q i (t) an time t; if the action (1) remains invariant then there is a conserve quantity for each real parameter of the transformation. We shall be more explicit in this statement an proof below. Hamilton s principle of least action applie to (1) leas to the EulerLagrange equations of motion [2, 3] L L = 0, (2) q i q i whose solution q i = q i (t) gives us the temporal evolution of the system. Associate to the system (1) we efine the energy function h as h(q i, q i,t) = q i L q i L, (3) also known as Jacobi integral. A simple calculation, using (2), leas to h = L t. (4) Thus we see that if the Lagrangian of the system oes not epen explicitly on time, then the Jacobi integral (3) is a constant of motion. Consier now the q i an t coorinates transformation t t = t+ǫx(q,t), q i q i = q i +ǫψ i (q,t), (5)
3 where ǫ represents an infinitesimal real parameter an X an Ψ i are given real functions of the generalize coorinates. The corresponing variation of the action (1) reas t 2 S = L(q i(t ), q i (t ) t2,t ) L(q i (t), q i(t),t). (6) t 1 At this point we may finally state an prove Noether s theorem: Theorem (Noether) Given a classical physical system governe by action S in (1), an an infinitesimal transformation of variables (5), if the corresponing variation of the action (6) assumes the form S = ǫg(q(t),t) t 2 t1 for some (possibly zero) function G(q,t), then C L ( q i X Ψ i ) LX +G (7) q i isaconstantofmotion. Inotherwors, ifthetransformation(5)isasymmetryofthelagrangian up to a total erivative then C = 0, (8) with C given by (7). proof: Applying the infinitesimal transformation (5) to (1) we may rewrite the action variation (6) as = t2 t2 where we have efine S = L(q i +ǫψ i, q i +ǫξ i,t+ǫx)(1+ǫẋ) L(q i, q i,t), { } L(q i, q i,t)+ [ L ǫψ i + L ǫξ i ]+ L t2 q i q i t ǫx (1+ǫẊ) L(q i, q i,t), t2 ξ i Ψ i q i Ẋ. (9) Reagrouping an keeping only terms up to first orer in ǫ we obtain { t2 [ L S = ǫ Ψ i + L ] } ξ i + L q i q i t X +LẊ. (10) Now the S = ǫg(q(t),t) t 2 t1 hypothesis leas immeiately to Noether s conition [ L Ψ i +( q Ψ i q i Ẋ) L ] i q i +LẊ + L X = Ġ. (11) t Finally using the EulerLagrange equations (2) an relations (3) an (4), Noether s conition (11) can be rewritten as [ ] Ψ i ( L )+ q Ψ L i hẋ i q h i X = { } L Ψ i hx = Ġ, (12) q i
4 or simply concluing thus the proof of the theorem. [ ] L ( q i X Ψ i ) LX +G = 0, (13) q i When the function G(q, t) ientically vanishes, the transformation is a full symmetry of the Lagrangian, otherwise the transforme Lagrangian iffers from the original one by the total time erivative ǫġ(q,t) generating thus the same equations of motion. Naturally, for a given action S, it is possible to have more than one conserve quantity C; for each inepenent symmetry (5) of the same action we have a ifferent conserve quantity (7). 3 Non Relativistic Point Particle in External Electromagnetic Fiel Let us consier a classical nonrelativistic particle embee in an external electromagnetic fiel. The particle has mass m an electrical charge q an is escribe by the position vector r(t) with respect to a inertial frame. The Lagrangian L(r,ṙ,t) of this system can be constructe from the particle kinetic energy minus the interacting potential energy between the particle an the electromagnetic fiel. In Gaussian units, we may thus write it as L(r,ṙ,t) = mṙ2 2 qϕ(r,t)+ q cṙ A(r,t), (14) where ϕ(r,t) an A(r,t) represent respectively the scalar an vector potentials, an c is the vacuum spee of light. We are tacitly assuming orinary threeimensional space. The potentials ϕ(r,t) an A(r,t) are external fiels in the sense that they represent given functions of spacetime. Different functions ϕ an A lea to istinct electromagnetic fiel configurations E an B which are given by E(r,t) = ϕ 1 A c t, B(r,t) = A. (15) The EulerLagrange equations erive from (14) rea m r+ q ( cȧ = q ϕ ṙ ) c A, (16) an can be rewritten using (15) as q m r = qe+ (17) cṙ B, which can be promptly recognize as Newton s secon law for a particle moving uner the Lorentz electromagnetic force. Summarizing, the Lagrangian (14) consistently escribes a nonrelativistic particle couple to a given external electromagnetic fiel, with equations of motion (17). Now we turn our attention to the symmetries an conserve quantities of (14). What are the symmetries of the action associate to Lagrangian (14)? The answer obviously epens on the functional form of the potentials ϕ an A. Let us analyse in the following the possibility of realization of the funamental spacetime symmetries in (14), namely, time an spatial translations, spatial rotations an boosts.
5 Time an Spatial Translations The energy function (3) corresponing to Lagrangian (14) reas h(r,ṙ,t) = 1 2 mṙ2 +qϕ(r,t), (18) an inee represents the total energy of the system, being the sum of the particle kinetic energy an the electric potential energy. Thus, from (4), the conition for the energy (18) to represent a constant of motion amounts to 0 = L t = q ϕ q A + t cṙ t. (19) If the potentials o not epen explicitly on time, then the Lagrangian (14) is trivially invariant uner time translation an (5) gives us a symmetry with X 1 an Ψ i 0 whose conserve quantity (7) is precisely (18). On the other han, if the potentials carry an explicitly time epenence out, then (18) oes not represent a constant of motion. However, still in this case, we may have a conserve quantity for (14) associate to time translational symmetry. Performing an infinitesimal time translation t t = t+ǫ in (14), it changes by the amount δl = L t δt = ǫq t Using the first equation of (15) it is possible to rewrite (20) as ( ϕ+ ṙ ) c A. (20) δl = ǫq[ ϕ+e ṙ]. (21) Thus if the electric power qe.ṙ transmitte by the fiel to the particle is null, we conclue from (7), with G = qϕ, that the particle kinetic energy mṙ 2 /2 is a conserve quantity. This comprises, in other wors, the wellknown fact that the magnetic fiel cannot change the particle s spee. Next we iscuss spatial translational symmetry. Let n be a constant unit vector in threeimensional space. Performing an infinitesimal spatial translation along the fixe n irection r = r+ǫn, t = t. (22) the Lagrangian (14) changes to [ L = L+q ϕ+ ṙ ] c.a nǫ. (23) Once again we see we have two possibilities. First, if the potentials themselves enjoy translational invariance in the n irection, which means ϕ n = 0, an A i n = 0, (24) then L = L an (7) gives us C ST = [mṙ+ q c A ] n (25) as a conserve quantity. We interpret (25) as the ncomponent of the total linear momentum of the system, mae up from the sum of the particle an fiel linear momenta along the n irection.
6 Notice that (25) is also the n projection of the Hamiltonian canonical momentum associate to (14). The secon possibility allows for L an L to iffer by a total time erivative. In this case we write L = L+δL in (23) an rearrange δl using (15) in orer to obtain δl = q ( E+ ṙ c B ) nǫ+ ( q c A n ) ǫ. (26) Consequently, if the ncomponent of the Lorentz force applie to the particle is null, the transforme Lagrangian iffers from the original one by a total time erivative Ġǫ with G = q A n, (27) c an (7) gives us the familiar result of particle linear momentum conservation (mṙ n) = 0. (28) We see that uner a translational symmetry operation, if both the particle an the fiel remain invariant, then we have the system total linear momentum conservation as in (25); however, if the fiel oes not remain invariant, then the particle linear momentum is conserve, as in (28), only if the applie electromagnetic force is null. Spatial Rotations Let us focus now on spatial rotations. Performing an infinitesimal rotation ǫ aroun a fixe axis n, the particle coorinates transform as r = r+ǫn r, t = t, (29) an the corresponing Lagrangian variation reas [ ( δl = q φ ṙ )] c A (n r)ǫ+ q A (n ṙ)ǫ. (30) c If it so happens that the system configuration is fully symmetric uner arbitrary spatial rotations aroun n, then δl in (30) is null an we have, as consequence from (7)(8), the conservation of the total angular momentum projection [ C SR = r (mṙ+ q )] c A n. (31) If it is not the case that (30) vanishes ientically, we may rewrite it as δl = ǫr (qe+ q ) cṙ B n+ǫ [ q ] c (r A) n. (32) Once more we have use the electromagnetic fiel equations (15) to obtain (32) from (30). From (32) we infer that, if the torque ue to the external electromagnetic fiel on the particle is null, the variation of the Lagrangian is a total time erivative. In this case, with G = q c (r A) n, Noether s theorem gives us [(r mṙ) n] = 0. (33) So we see that null external torque leas to the particle angular momentum conservation even if the fiel configuration oes not respect rotational symmetry.
7 Galilean Boosts The previous transformations all involve only rest reference frames. Now let us consier a change of coorinates to a inertial frame moving with constant velocity relative to the first, along a fixe n irection. Since we are not yet consiering the relativistic case we escribe it by a Galilean infinitesimal transformation r = r+ǫnt, t = t. (34) The infinitesimal parameter ǫ represents the relative velocity between the two inertial frames. The system Lagrangian (14) transforms uner (34) as δl = ǫ [( q ) ] c At+mr n +ǫ (qe+ q ) cṙ B nt. (35) If the ncomponent of the external force is zero, the Lagrangian variation (35) is a total time erivative. Noether s theorem then leas, from (7), to the center of mass position conservation C GB = ( mṙt+mr) n, (36) which in turn gives an immeiate solution for the particle vector position projection along n (r n) = (ṙ n)t+(r 0 n) (37) for some constant r 0. This simply signals consistently the lack of acceleration for the particle along the n irection. 4 Relativistic Particle in External Electromagnetic Fiel In this section we generalize the previous analyses to the relativistic case. Obviously the kinetic energy term in Lagrangian (14) is not invariant uner a general Lorentz transformation an so is not consistent with the special theory of relativity. We can write a Lagrangian for the relativistic particle embee in an external electromagnetic fiel as L(r,ṙ,t) = m 1 ṙ2 q qϕ+ (38) c2 cṙ A, where once more ϕ(r,t) an A(r,t) are the scalar an vector potentials. The EulerLagrange equations corresponing to (38) rea mṙ + q ( 1 ṙ2 cȧ = q ϕ ṙ ) c A, (39) or simply mṙ q = qe+ (40) 1 ṙ2 cṙ B, by means of (15). Equation (40) is the relativistic generalization of (17) an can be interprete as the relativistic Newton s secon law of motion justifying thus the use of (38).
8 In fact (38) is written for a particular reference frame an is not Lorentz invariant. However, incluing integration along time, the corresponing action can be written as S = L = τ { m q } c Uµ A µ (41) being τ the proper time an U µ = (γc,γṙ) the relativistic quarivelocity. Action (41) is manifestly Lorentz invariant. We are now reay to look for other conserve quantities associate to (38) through Noether s Theorem. Time an Spatial Translations A irect calculation of the energy function (3) for (38) leas to the expression h(r,ṙ,t) = mc2 +qϕ. (42) 1 ṙ2 As we have alreay pointe out, if the Lagrangian oes not epen explicitly on time, the energy function prouces a constant of motion. In the present situation this happens if the potentials φ an A in (38) o not exhibit an explicit time epenence; in this case the total energy of the system (42) is conserve. If the potentials o have an explicitly time epenence then an infinitesimal translation in time t t + ǫ leas again to equations (20) an (21). Following the same line of reasoning, we conclue that if the transmitte electric power E ṙ is null then we have, from (7), the conservation of the quantity E = mc2 (43) 1 ṙ2 which represents the relativistic particle s energy. Concerning spatial translations, if (38) remains invariant uner an infinitesimal isplacement along a fixe n irection r = r+ǫn, (44) the corresponing Noether conserve quantity (7) reas C RST = mṙ q + 1 ṙ2 c A n. (45) That means that the total relativistic linear momentum of the particleplusfiel system is conserve along the n irection. Similarly to the nonrelativistic case, even if (38) is not fully invariant uner spatial translations, equations (23) an (26) hol. Assuming the ncomponent of the Lorentz force to be null we are le through (7), with G given by (27), to the conservation of the particle relativistic momentum along the nirection mṙ 1 ṙ2 n = 0. (46) Equations (45) an (46) are respectively the relativistic generalizations of (25) an (28).
9 Spatial Rotations Let us perform a spatial rotation (29) in the relativistic system (38). If the Lagrangian (38) enjoys full rotational invariance it immeiately follows from (7) that the generalization of (31) reas C RSR = r mṙ q + 1 ṙ2 c A n, (47) which means the nprojection of the total relativistic momentum of the particle an fiel is conserve. Furthermore, if (38) is not rotational invariant, a simple calculation shows that (30) an (32) rea exactly the same an express the variation of (38) uner (29). Then, if the electromagnetic torque applie by the fiel on the particle is null, the Lagrangian variation (32) is a total time erivative an (33) generalizes to r mṙ 1 ṙ2 n = 0, (48) expressing now the conservation of only the particle angular momentum along the nirection. Lorentz Boosts A Lorentz boost is a homogeneous Lorentz transformation connecting two inertial frames with constant relative velocity an no spatial rotation. In usual relativistic notation an homogeneous infinitesimal Lorentz transformation can be written as x α x α = (δ α β +ω α β)x β, (49) being ω α β antisymmetric an representing six real inepenent infinitesimal parameters corresponing to three egrees of freeom escribing orinary spatial rotations an three escribing Lorentz boosts. In the stanar vectorial notation we have been using, an infinitesimal Lorentz boost along the spatial n irection can be written as r = r+ǫnt, t = t+ ǫ c2(n r), (50) where the infinitesimal parameter ǫ represents the relative velocity between the two frames. Clearly (50) is the relativistic generalization of the infinitesimal Galilean boost (34). Since the transformation (50) explicitly changes the time, it is not enough to check the Lagrangian variation  rather we shoul stuy the action variation (6) uner (50) which reas t2 [( ) ( S = ǫ qe+qṙ c B nt+q(ṙ E) n.r )+ [ q ( n r )]] c A nt qφ (51) In case of null external nprojection force an null transmitte electric power the variation S is a total time erivative an (7) gives us the conserve quantity which generalizes (36). C LB = m ) 1/2 (1 ṙ2 (r ṙt) n (52)
10 5 Conclusion Noether s theorem connects continuous symmetries of the classical action to conserve quantities. We have reviewe an prove Noether s theorem at classical level. We have applie Noether s theorem to a system mae up of a particle interacting with an external electromagnetic fiel, consiering both relativistic an nonrelativistic cases. We have iscusse the consequences of all spacetime symmetries of the action, calculating the corresponing conserve quantities. We have seen that when a spacetime transformation is performe in the particle s spacetime coorinates, the fiel s behavior uner the transformation may lea to ifferent results, namely to conservation of quantities either relate solely to the particle or to the whole configuration set, particle plus fiel. Acknowlegment The authors gratefully acknowlege the Brazilian research agencies Conselho Nacional e Desenvolvimento Científico e Tcnológico (CNPq) an Funação e Amparo à Pesquisa o Estao a Bahia (FAPESB) for finacial support. Referências [1] E. Noether, Invariante Variationsprobleme, Nachr. Gesell. Wissenchaft. Göttinger 2, 235, [2] H. Golstein, C. Poole, an J. Safko, Classical Mechanics, AisonWesley, San Francisco, [3] N. A. Lemos, Mecânica Analítica, E. Livraria a Física, São Paulo, [4] A. O. Lopes, Introução à Mecânica Clássica, Eusp, São Paulo, [5] C. Itzykson an J. B. Zuber, Quantum Fiel Theory, Dover Publications, Mineola, [6] K. Fujikawa an H. Suzuki, Path Integral an Quantum Anomalies, Oxfor University Press, Oxfor, 2004.
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