Generalized Coordinates. The Kepler-Coulomb Problem

Transcription

1 Chter Generlized Coordintes. The Keler-Coulomb Problem.1 Generlized coordintes In generl let us suose we describe system with generlized coordintes, {q } N =1, nd generlized velocities, { q } N =1 ; for exmle, they might be the Crtesin coordintes nd velocities of N/3 rticles moving in 3 dimensions, or the shericl olr coordintes nd velocities of rticle: {q } = {r, θ, φ}, { q } = {ṙ, θ, φ}. Let us suose lthough this my not lwys be true tht the Lgrngin deends only on q nd q, s well s, erhs, on the time exlicitly, The ction rincile sttes W 1 = t1 t dt L{q }, { q }, t..1 δw 1 = G 1 G [ t1 L L = dt δq + δ q + δt L t q q t + dδt L dt ] L q, q. where the lst term rises from the chnge in the time coordinte; see, for exmle, Eq Becuse δ q = d dt δq, this vrition my be rewritten s { t1 [ d L d L δw 1 = dt δq δq L ] t dt q dt q q [ + d δt L ] [ [ L L q + δt dt q t d L ]]} L q. dt q 15 Version of Setember 9, 015

2 16 Version of Setember 9, 015CHAPTER. GENERALIZED COORDINATES. THE KEPLER-COUL Defining the generlized momentum by.3 = L q,.4 nd the energy by E = q L,.5 we hve [ δw 1 = δq δte t1 t dt [ ] 1 δq ṗ L d δt q dt E + L ]..6 t By the ction rincile, we red off the genertors nd the equtions of motion: G = δq Eδt,.7 ṗ = d L = L, dt q q d dt E = L t = E..8 t. Hmiltonin The Hmiltonin is defined by the Legendre trnsformtion: H{q, } = q L{q, q },.9 which coincides, numericlly, with the energy.5. However, the oint here is tht the vribles re chnged from {q, q } to {q, }. Here the generlized cnonicl momentum is obtined from the Lgrngin by Eq..4. Therefore, The equtions of motion for q nd re H = q, H = L = q q H = L = ṗ,.11 q q where the lst follows from the Euler-Lgrnge equtions.8. The Hmiltonin is time deendent only if exlicitly so: dh dt = H t + H H q + ṗ,.1 q

3 .3. MOTION OF A PARTICLE IN A 1/R POTENTIAL17 Version of Setember 9, 015 where the summnd vnishes by virtue of the eqution of motion.11. Thus, the comlete set of Hmilton s equtions re: q = H, ṗ = H q, dh dt = H t..13 As sketched in Chter 1, these results cn be directly obtined from the ction rincile, which we rewrite s 1 1 W 1 = dt L = dt q H,.14 which hs s its vrition [ 1 δw 1 = dt = 1 d δ q + dt δq H H δq δ q + δq ṗ H q [ d δq Hδt dt dh + δt dt H t δt H ] t dδt dt H + δ q H ]..15 The ction rincile 1.3 yields the Hmilton equtions.13 s well s the exected form of the genertor, G = δq Hδt Motion of rticle in 1/r otentil For rticle described in shericl olr coordintes, r, θ, φ, which re relted to the Crtesin coordintes by x = r sin θ cosφ, y = r sin θ sin φ, z = r cosθ,.17 the velocity vector is given in terms of the generlized coordintes nd velocities by ṙ = ˆxṙ sin θ cosφ + r θ cosθ cosφ r φ sin θ sin φ + ŷṙ sin θ sin φ + r θ cosθ sin φ + r φ sin θ cosφ + ẑṙ cosθ r θ sin θ..18 When this is squred, there is considerble cncelltion: which is reflective of the geometricl line element, ṙ = ṙ + r θ + r sin θ φ,.19 ds = dr + r dθ + r sin θdφ..0

4 18 Version of Setember 9, 015CHAPTER. GENERALIZED COORDINATES. THE KEPLER-COUL We will consider the Lgrngin of single rticle in centrl otentil, L = 1 mṙ V r,.1 nd, in fct, we will secilize to the Newtonin or Coulomb otentil, V r = α r,. where in the Newtonin cse of light lnet of mss m orbiting very hevy sun of mss M, α = GMm in terms of Newton s constnt G, while for the interction between hevy nucleus, of electric chrge +Ze nd light electron of chrge e, α = Ze Gussin units. The mthemtics is identicl in the two cses. The generlized moment re r = L ṙ = mṙ, The equtions of motion re θ = L θ = mr θ, φ = L φ = mr sin θ φ..3 ṗ r = m r = L r = V r + mr θ + mr sin θ φ, ṗ θ = L θ = mr sin θ cosθ φ, ṗ φ = L φ = b.4c The lst eqution is evidently conservtion lw; in fct, it is esy to check tht φ = L z, L = r,.5 so this is just the sttement of the conservtion of the z comonent of ngulr momentum. The remining equtions cn be gretly simlified by recognizing tht the motion is confined to lne, which we my, without loss of generlity, tke to be the z = 0 lne, in which cse θ = π/. This is consistent with the second eqution of motion.4b, since both sides vnish in tht cse. The first, rdil, eqution.4 then simlifies to m r = α 1 r + L z mr 3..6 We cn obtin first integrl of this differentil eqution by multilying it by ṙ: d 1 dt mṙ + L z mr α = 0;.7 r the quntity in rentheses is simly the energy, which we lredy know is conserved, since there is no exlicit time deendence in this roblem: E = q L = m ṙ + r φ α r = m ṙ + L z m r α r..8

5 .3. MOTION OF A PARTICLE IN A 1/R POTENTIAL19 Version of Setember 9, 015 The solution of the roblem is now given by qudrtures: Tht is, we solve for ṙ: ṙ = E + α L z m r m r,.9 or dt = m dr E + α r L z m r.30 Since ngulr momentum is conserved, we cn mke this n eqution for φ, becuse dφ = Lz mr dt: φ = L z dr/r me + α/r L z /r..31 This solution in fct holds for generl centrl otentil, with α/r V r. For the 1/r otentil, the solution is simle. Let s = 1/r nd then we cn write this s 1 φ = ds,.3 m E + αs s where one cn comlete the squre, nd in terms of s = s αm/l z find φ = rcsin L z s + constnt..33 me/l z + α m /L 4 z Therefore, s = 1 r = αm L z + αm L z 1 + EL z mα sinφ φ φ 0 is n rbitrry constnt; choose it to be φ 0 = π/, so the bove ers in stndrd form = 1 + e cosφ, r.35 where = L z αm, e = 1 + EL z mα..36 This is the eqution of n ellise, if e < 1, tht is, for negtive energies bound sttes E < 0. In tht cse, e is the eccentricity, nd is the ltus rectum. The sun or nucleus lies t one of the foci of the ellise. The distnce of closest roch of the lnet to the sun is clled the erihelion. Tht distnce occurs when φ = 0, so r = 1 + e ;.37 the helion, the furthest distnce from the sun occurs for φ = π, or r = 1 e..38

6 0 Version of Setember 9, 015CHAPTER. GENERALIZED COORDINATES. THE KEPLER-COUL The semimjor xis is therefore = 1 e so r = 1 e;.39 smll clcultion homework revels the semiminor xis is In hysicl terms, then, b = 1 e..40 = α E, E < 0, or E = α..41 Wht is the eriod of this closed orbit? The constncy of the ngulr momentum imlies tht of the rel velocity. Evidently, for smll ngulr chnge, the re swet out by the rdius vector is ds = 1 r dφ,.4 so ds dt = 1 L z r φ = m..43 This is Keler s second lw. Therefore, so the eriod is dt = m L z ds,.44 T = m L z S,.45 where S is the re enclosed by the orbit, here S = πb. Therefore, inserting the vrious quntities we hve m α T = π..46 E 3/ Note tht the ngulr momentum cncels out. But becuse E = α/, this cn be written in the more fmilir form T = 4π m3 α = 4π GM 3,.47 which sys tht the squre of the eriod is roortionl to the cube of the semimjor xis, Keler s third lw. Note the crucil oint, in the lnetry cse, the mss of the lnet dros out. The sme orbit eqution.35 lies for E > 0, or e > 1. Now the erihelion distnce is given by r = = e 1,.48 e + 1

7 .4. AXIAL VECTOR 1 Version of Setember 9, 015 where = e 1 = α E..49 The orbit is hyerbol rbol if E = 0, nd the distnce of the comet from the sun, sy, becomes infinite when φ = rccos 1 e. A similr orbit eqution to.35 lies for reulsive field, with α < 0. Then the energy is necessrily ositive, e > 1, nd = 1 + e cosφ..50 r Agin the orbit is hyerbol. The erihelion distnce is given by r = e 1,.51 but now the focus is outside of the hyerbol. The symtotes of the hyerbol occurs t φ = rccos 1 e. See figures in Lndu nd Lifshitz,. 37 nd Axil Vector The fct tht the orbit closes is remrkble feture of the 1/r otentils. It does not close for other centrl otentils excet r. This reflects hidden symmetry of the Keler roblem. In ddition to the vector L, which is erendiculr to the lne of the orbit, there is nother vector, in the lne of the orbit, A = v L α r r,.5 which is conserved. Indeed, d dt A = v r mv αv r + αrv r r 3 = rv vr m v + α r = 0,.53 r3 by virtue of the eqution of motion. Thus A is constnt in direction nd mgnitude. At the erihelion it is esy to evlute it, becuse there v nd r re erendiculr. We see tht A oints long the semimjor xis from the sun to the erihelion oint nd hs mgnitude A = mv r α = r T α = r α + α α = α r r + 1 = αe, so.54 A = ˆr αe..55 Becuse it oints in the direction of the xis of the ellise, it might be dubbed the xil vector lthough it is olr vector!. It is often referred to s the Llce-Runge-Lenz vector, or by some subset of those nmes, but it ws

8 Version of Setember 9, 015CHAPTER. GENERALIZED COORDINATES. THE KEPLER-COUL ctully first discovered by Hermnn in 1710, nd generlized by Bernoulli in the sme yer. Wht is the corresonding symmetry? It turns out to be the 4-dimensionl rottion grou, O4, equivlent to SU SU, but since this is of most interest in quntum mechnics, we will not sto to elucidte this oint now..5 Reduced mss In fct, the mss of the sun is not infinite, nd it moves somewht s lnet orbits round it. We hve then two-body roblem, but with the otentil deending only on the distnce between the bodies. Let us stte this roblem generlly, which is governed by the Lgrngin L = 1 m 1ṙ m ṙ V r 1 r..56 Let us introduce the center of mss nd reltive coordintes, with M = m 1 + m. This mens tht MR = m 1 r 1 + m r, r = r 1 r,.57 r 1 = R + m M r, r = R m 1 r..58 M When this is substituted into the kinetic energy, the cross term between ṙ Ṙ cncels out, nd we re left with where the reduced mss ers, µ = m 1m m 1 + m, T = 1 MṘ + 1 µṙ,.59 or 1 µ = 1 m m..60 Thus, the kinetic energy breks u into the energy of the center of mss nd tht of the reltive motion. The equtions for the reltive motion re the sme s before, but with the mss m relced by the reduced mss µ = mm/m+m m if m M. The sme kind of breku occurs for the liner nd ngulr momentum. For the former, P = m 1 ṙ 1 + m ṙ = MṘ,.61 s we lredy know, see Eq The ngulr momentum breks u like the energy, L = R MṘ + r µṙ = L CM + L R,.6 into the ngulr momentum of the center of mss, nd tht of the reltive motion. So the modifiction entiled by the finite mss of the sun is trivil. This is not the cse when the effect of forces due to third bodies, e.g. Juiter, re considered!

9 .6. PROBLEMS FOR CHAPTER 3 Version of Setember 9, Problems for Chter 1. Prove tht φ = L z, Eq..5. Prove tht the semiminor xis of the elliticl orbit is given by Eq Prove the sttements mde bout the hyerbolic orbits in the lst two rgrhs of Sec..3.