Chapter 2 Review of Classical Action Principles


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1 Chapter Review of Classical Action Principles This section grew out of lectures given by Schwinger at UCLA aroun 1974, which were substantially transforme into Chap. 8 of Classical Electroynamics (Schwinger 1998). (Remarkably, consiering his work on waveguie theory uring Worl War II, now partially recore in Ref. (Milton 006), he never gave lectures on this subject at Harvar after 1947.) We start by reviewing an generalizing the LagrangeHamilton principle for a single particle. The action, W 1, is efine as the time integral of the Lagrangian, L, where the integration extens from an initial configuration or state at time t to a final state at time t 1 : t1 W 1 = L. (.1) t The integral refers to any path, any line of time evelopment, from the initial to the final state, as shown in Fig..1. The actual time evolution of the system is selecte by the principle of stationary action: In response to infinitesimal variations of the integration path, the action W 1 is stationary oes not have a corresponing infinitesimal change for variations about the correct path, provie the initial an final configurations are hel fixe, δw 1 = 0. (.) This means that, if we allow infinitesimal changes at the initial an final times, incluing alterations of those times, the only contribution to δw 1 then comes from the enpoint variations, or δw 1 = G 1 G, (.3) where G a, a = 1 or, is a function, calle the generator, epening on ynamical variables only at time t a. In the following, we will consier three ifferent realizations The Author(s) 015 K.A. Milton, Schwinger s Quantum Action Principle, SpringerBriefs in Physics, DOI / _ 3
2 4 Review of Classical Action Principles Fig..1 A possible path from initial state to final state t 1 time t state variables of the action principle, where, for simplicity, we will restrict our attention to a single particle..1 Lagrangian Viewpoint The nonrelativistic motion of a particle of mass m moving in a potential V (r, t) is escribe by the Lagrangian L = 1 ( ) r m V (r, t). (.4) Here, the inepenent variables are r an t, so that two kins of variations can be consiere. First, a particular motion is altere infinitesimally, that is, the path is change by an amount δr: r(t) r(t) + δr(t). (.5) Secon, the final an initial times can be altere infinitesimally, by δt 1 an δt, respectively. It is more convenient, however, to think of these time isplacements as prouce by a continuous variation of the time parameter, δt(t), so chosen that, at the enpoints, t t + δt(t), (.6) δt(t 1 ) = δt 1, δt(t ) = δt. (.7) The corresponing change in the time ifferential is ( (t + δt) = 1 + δt ), (.8)
3 .1 Lagrangian Viewpoint 5 which implies the transformation of the time erivative, ( 1 δt ). (.9) Because of this reefinition of the time variable, the limits of integration in the action, [ ] 1 W 1 = m (r) V, (.10) are not change, the time isplacement being prouce through δt(t) subject to (.7). The resulting variation in the action is now δw 1 = = { [ m r δr δr V δt ( ) ] } 1 r m + V δt t V { [ ( m r ( ) ) ] δr 1 r m + V δt + δr [ ] ( [ ( ) ] )} m r V 1 r + δt m + V t V, (.11) where, in the last form, we have integrate by parts in orer to isolate δr an δt. Because δr an δt are inepenent variations, the principle of stationary action implies that the actual motion is governe by [ 1 m m r = V, ( ) r ] + V while the total time erivative gives the change at the enpoints, (.1a) = V, (.1b) t G = p δr Eδt, (.1c) with momentum = p = m r, energy = E = 1 m ( ) r + V. (.1) Therefore, we have erive Newton s secon law [the equation of motion in seconorer form], (.1a), an, for a static potential, V/ t = 0, the conservation of energy, (.1b). The significance of (.1c) will be iscusse later in Sect..4.
4 6 Review of Classical Action Principles. Hamiltonian Viewpoint Using the above efinition of the momentum, we can rewrite the Lagrangian as L = p r H(r, p, t), (.13) where we have introuce the Hamiltonian H = p + V (r, t). (.14) m We are here to regar r, p, an t as inepenent variables in W 1 = The change in the action, when r, p, an t are all varie, is [ δw 1 = p H δr δr r [ = (p δr Hδt) + δr ( p H ) r + δp (r H ) ( H + δt H p t The action principle then implies [p r H]. (.15) r + δp δp H p δt H δt H t )]. (.16) ] r = H p = p m, p = H r = V, H = H t, G = p δr Hδt. (.17a) (.17b) (.17c) (.17) In contrast with the Lagrangian ifferential equations of motion, which involve secon erivatives, these Hamiltonian equations contain only first erivatives; they are calle firstorer equations. They escribe the same physical system, because when (.17a) is substitute into (.17b), we recover the LagrangianNewtonian equation (.1a). Furthermore, if we insert (.17a) into the Hamiltonian (.14), we ientify H with E. The thir equation (.17c) is then ientical with (.1b). We also note the equivalence of the two versions of G.
5 . Hamiltonian Viewpoint 7 But probably the most irect way of seeing that the same physical system is involve comes by writing the Lagrangian in the Hamiltonian viewpoint as L = m ( ) r V 1 ( p m r ). (.18) m The result of varying p in the stationary action principle is to prouce p = m r. (.19) But, if we accept this as the efinition of p, the corresponing term in L isappears an we explicitly regain the Lagrangian escription. We are justifie in completely omitting the last term on the right sie of (.18), espite its epenence on the variables r an t, because of its quaratic structure. Its explicit contribution to δl is 1 m ( p m r ) ( δp m r δr + m ) δt, (.0) an the equation supplie by the stationary action principle for p variations, (.19), also guarantees that there is no contribution here to the results of r an t variations..3 A Thir, Schwingerian, Viewpoint Here we take r, p, an the velocity, v, as inepenent variables, so that the Lagrangian is written in the form L = p (r ) v + 1 mv V (r, t) p r H(r, p, v, t), (.1) where H(r, p, v, t) = p v 1 mv + V (r, t). (.) The variation of the action is now δw 1 =δ = [p r H] [ δp r δt H t + p H δt ] H δr δr r δp H p δv H v
6 8 Review of Classical Action Principles [ = (p δr Hδt) δr (p + H ) r + δp (r H ) δv H ( H p v + δt H )], (.3) t so that the action principle implies p = H r = V, r = H p = v, 0 = H = p + mv, (.4c) H v = H t, G = p δr Hδt. (.4a) (.4b) (.4) (.4e) Notice that there is no equation of motion for v since v/ oes not occur in the Lagrangian, nor is it multiplie by a time erivative. Consequently, (.4c) refers to a single time an is an equation of constraint. From this thir approach, we have the option of returning to either of the other two viewpoints by imposing an appropriate restriction. Thus, if we write (.) as H(r, p, v, t) = p 1 + V (r, t) m m (p mv), (.5) an we aopt v = 1 m p (.6) as the efinition of v, we recover the Hamiltonian escription, (.13) an (.14). Alternatively, we can present the Lagrangian (.1) as L = m ( ) r V + (p mv) (r ) v m ( ) r v. (.7) Then, if we aopt the following as efinitions, v = r, p = mv, (.8) the resultant form of L is that of the Lagrangian viewpoint, (.4). It might seem that only the efinition v = r/, inserte in (.7), suffices to regain the Lagrangian escription. But then the next to last term in (.7) woul give the following aitional
7 .3 A Thir, Schwingerian, Viewpoint 9 contribution to δl, associate with the variation δr: (p mv) δr. (.9) In the next Chapter, where the action formulation of electroynamics is consiere, we will see the avantage of aopting this thir approach, which is characterize by the introuction of aitional variables, similar to v, for which there are no equations of motion..4 Invariance an Conservation Laws There is more content to the principle of stationary action than equations of motion. Suppose one consiers a variation such that δw 1 = 0, (.30) inepenently of the choice of initial an final times. We say that the action, which is left unchange, is invariant uner this alteration of path. Then the stationary action principle (.3)assertsthat δw 1 = G 1 G = 0, (.31) or, there is a quantity G(t) that has the same value for any choice of time t; itis conserve in time. A ifferential statement of that is G(t) = 0. (.3) The G functions, which are usually referre to as generators, express the interrelation between conservation laws an invariances of the system. Invariance implies conservation, an vice versa. A more precise statement is the following: If there is a conservation law, the action is stationary uner an infinitesimal transformation in an appropriate variable. The converse of this statement is also true. If the action W is invariant uner an infinitesimal transformation (that is, δw = 0), then there is a corresponing conservation law. This is the celebrate theorem of Amalie Emmy Noether (Noether 1918). Here are some examples. Suppose the Hamiltonian of (.13) oes not epen explicitly on time, or W 1 = [p r H(r, p)]. (.33)
8 10 Review of Classical Action Principles Fig.. δω r is perpenicular to δω an r, an represents an infinitesimal rotation of r about the δω axis δω δω r r Then the variation (which as a rigi isplacement in time, amounts to a shift in the time origin) δt = constant (.34) will give δw 1 = 0 [see the first line of (.16), with δr = 0, δp = 0, δt/ = 0, H/ t = 0]. The conclusion is that G in (.17), which here is just G t = Hδt, (.35) is a conserve quantity, or that H = 0. (.36) This inference, that the Hamiltonian the energy is conserve, if there is no explicit time epenence in H, is alreay present in (.17c). But now a more general principle is at work. Next, consier an infinitesimal, rigi rotation, one that maintains the lengths an scalar proucts of all vectors. Written explicitly for the position vector r, itis δr = δω r, (.37) where the constant vector δω gives the irection an magnitue of the rotation (see Fig..). Now specialize (.14) to H = p + V (r), (.38) m where r = r, a rotationally invariant structure. Then W 1 = [p r H] (.39) is also invariant uner the rigi rotation, implying the conservation of
9 .4 Invariance an Conservation Laws 11 This is the conservation of angular momentum, G δω = p δr = δω r p. (.40) L = r p, L = 0. (.41) Of course, this is also containe within the equation of motion, V L = r V = r ˆr = 0, (.4) r since V epens only on r. Conservation of linear momentum appears analogously when there is invariance uner a rigi translation. For a single particle, (.17b) tells us immeiately that p is conserve if V is a constant, say zero. Then, inee, the action ] W 1 = [p r p m (.43) is invariant uner the isplacement δr = δɛ = constant, (.44) an G δ = p δɛ (.45) is conserve. But the general principle acts just as easily for, say, a system of two particles, a an b, with Hamiltonian H = This Hamiltonian an the associate action W 1 = p a + p b + V (r a r b ). (.46) m a m b are invariant uner the rigi translation with the implication that [p a r a + p b r b H] (.47) δr a = δr b = δɛ, (.48) G δɛ = p a δr a + p b δr b = (p a + p b ) δɛ (.49)
10 1 Review of Classical Action Principles is conserve. This is the conservation of the total linear momentum, P = p a + p b, P = 0. (.50) Something a bit more general appears when we consier a rigi translation that grows linearly in time: δr a = δr b = δv t, (.51) using the example of two particles. This gives each particle the common aitional velocity δv, an therefore must also change their momenta, The response of the action (.47) to this variation is δw 1 = = δp a = m a δv, δp b = m b δv. (.5) [(p a + p b ) δv + δv (m a r a + m b r b ) (p a + p b ) δv ] [(m a r a + m b r b ) δv]. (.53) The action is not invariant; its variation has enpoint contributions. But there is still a conservation law, not of G = P δvt, but of N δv, where Written in terms of the centerofmass position vector N = Pt (m a r a + m b r b ). (.54) R = m ar a + m b r b, M M = m a + m b, (.55) the statement of conservation of N = Pt MR, (.56) namely 0 = N = P M R, (.57) is the familiar fact that the center of mass of an isolate system moves at the constant velocity given by the ratio of the total momentum to the total mass of that system.
11 .5 Nonconservation Laws: The Virial Theorem 13.5 Nonconservation Laws: The Virial Theorem The action principle also supplies useful nonconservation laws. Consier, for constant δλ, δr = δλr, δp = δλp, (.58) which leaves p r invariant, But the response of the Hamiltonian δ(p r) = ( δλp) r + p (δλr) = 0. (.59) H = T (p) + V (r), T (p) = p m, (.60) is given by the noninvariant form δh = δλ( T + r V ). (.61) Therefore we have, for an arbitrary time interval, for the variation of the action (.15), δw 1 = [δλ(t r V )] =G 1 G = (p δλr) (.6) or, the theorem r p = T r V. (.63) For the particular situation of the Coulomb potential between charges, V = constant/r, where r V = r V = V, (.64) r the virial theorem asserts that (r p) = T + V. (.65) We apply this to a boun system prouce by a force of attraction. On taking the time average of (.65) the time erivative term isappears. That is because, over an arbitrarily long time interval τ = t 1 t,thevalueofr p(t 1 ) can iffer by only a finite amount from r p(t ), an (r p) 1 t1 τ t r p = r p(t 1 ) r p(t ) 0, (.66) τ
12 14 Review of Classical Action Principles as τ. The conclusion, for time averages, T = V, (.67) is familiar in elementary iscussions of motion in a 1/r potential. Here is one more example of a nonconservation law: Consier the variations δr =δλ r r, (.68a) ( p δp = δλ r rp r ) r (r p) r 3 = δλ r 3. (.68b) Again p r is invariant: ( p δ(p r) = δλ r rp r ) r r 3 + p ( δλ r ) r r δλr r r 3 = 0, (.69) an the change of the Hamiltonian (.60) isnow The resulting theorem, for V = V (r), is δh = δλ [ L mr 3 + r ] r V. (.70) ( r p) = L r mr 3 V r, (.71) which, when applie to the Coulomb potential, gives the bounstate time average relation L ( ) ( ) 1 V m r 3 =. (.7) r This relation is significant in hyrogen finestructure calculations (for example, see (Schwinger 001)).
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