Chapter 6. Hamilton s Equations. 6.1 Legendre transforms 156 CHAPTER 6. HAMILTON S EQUATIONS

156 CHAPTER 6 HAMILTON S EQUATIONS Chapter 6 Hamlton s Equatons We dscussed the generalzed momenta p = L(q, q, t) q, and how the canoncal varables {q,p j } descrbe phase space One can use phase space rather than {q, q j } to descrbe the state of a system at any moment In ths chapter we wll explore the tools whch stem from ths phase space approach to dynamcs 61 Legendre transforms The mportant object for determnng the moton of a system usng the Lagrangan approach s not the Lagrangan tself but ts varaton, under arbtrary changes n the varables q and q, treated as ndependent varables It s the vanshng of the varaton of the acton under such varatons whch determnes the dynamcal equatons In the phase space approach, we want to change varables q p, where the p are components of the gradent of the Lagrangan wth respect to the veloctes Ths s an example of a general procedure called the Legendre transformaton We wll dscuss t n terms of the mathematcal concept of a dfferental form Because t s the varaton of L whch s mportant, we need to focus our attenton on the dfferental dl rather than on L tself We frst want to gve a formal defnton of the dfferental, whch we wll do frst for a functon f(x 1,, x n )ofn varables, although for the Lagrangan we wll later subdvde these nto coordnates and veloctes We wll take the space n whch x takes values to be some general n-dmensonal space we call M, whch mght be ordnary Eucldean space but mght be somethng else, lke the surface of a sphere 1 Gven a dfferentable functon f of n ndependent varables x, the dfferental s df = n =1 f x dx (61) What does that mean? As an approxmate statement, ths can be regarded as sayng df f f(x + x ) f(x )= n =1 f x x + O( x x j ), wth some statement about the x beng small, followed by the droppng of the order ( x) 2 terms Notce that df s a functon not only of the pont x M, but also of the small dsplacements x A very useful mathematcal language emerges f we formalze the defnton of df, extendng ts defnton to arbtrary x, even when the x are not small Of course, for large x they can no longer be thought of as the dfference of two postons n M and df no longer has the meanng of the dfference of values of f at two dfferent ponts Our formal df s now defned as a lnear functon of these x varables, whch we therefore consder to be a vector v lyng n an n-dmensonal vector space R n Thus df : M R n R s a real-valued functon wth two arguments, one n M and one n a vector space The dx whch appear n (61) can be thought of as operators actng on ths vector space argument to extract the th component, and the acton of df on the argument (x, v) sdf (x, v) = ( f/ x )v Ths dfferental s a specal case of a 1-form, as s each of the operators dx All n of these dx form a bass of 1-forms, whch are more generally ω = ω (x)dx, where the ω (x) are n functons on the manfold M If there exsts an ordnary functon f(x) such that ω = df, then ω s sad to be an exact 1-form 1 Mathematcally, M s a manfold, but we wll not carefully defne that here The precse defnton s avalable n Ref [16] 155

61 LEGENDRE TRANSFORMS 157 Consder L(q,v j,t), where v = q At a gven tme we consder q and v as ndependant varables The dfferental of L on the space of coordnates and veloctes, at a fxed tme, s dl = L q dq + L v dv = L q dq + p dv If we wsh to descrbe physcs n phase space (q,p ), we are makng a change of varables from v to the gradent wth respect to these varables, p = L/ v, where we focus now on the varables beng transformed and gnore the fxed q varables So dl = p dv, and the p are functons of the v j determned by the functon L(v ) Is there a functon g(p ) whch reverses the roles of v and p, for whch dg = v dp? If we can nvert the functons p(v), we can defne g(p )= p v (p j ) L(v (p j )), whch has a dfferental dg = = p dv + v dp v dp dl = p dv + v dp p dv as requested, and whch also determnes the relatonshp between v and p, v = g = v (p j ), p gvng the nverse relaton to p k (v l ) Ths partcular form of changng varables s called a Legendre transformaton In the case of nterest here, the functon g s called H(q,p j,t), the Hamltonan, H(q,p j,t)= k Then for fxed tme, p k q k (q,p j,t) L(q, q j (q l,p m,t),t) (62) dh = (dp k q k + p k d q k ) dl = ( q k dp k p k dq k ), k k H H = q p k k, = p q,t q k k (63) p,t Other examples of Legendre transformatons occur n thermodynamcs The energy change of a gas n a varable contaner wth heat flow s sometmes wrtten de =d Q pdv, 158 CHAPTER 6 HAMILTON S EQUATIONS where d Q s not an exact dfferental, and the heat Q s not a well defned system varable Though Q s not a well defned state functon, the dfferental d Q s a well defned 1-form on the manfold of possble states of the system It s not, however, an exact 1-form, whch s why Q s not a functon on that manfold We can express d Q by defnng the entropy and temperature, n terms of whch d Q = TdS, and the entropy S and temperature T are well defned state functons Thus the state of the gas can be descrbed by the two varables S and V, and changes nvolve an energy change de = TdS pdv We see that the temperature s T = E/ S V If we wsh to fnd quanttes approprate for descrbng the gas as a functon of T rather than S, we defne the free energy F by F = TS E so df = SdT pdv, and we treat F as a functon F (T,V ) Alternatvely, to use the pressure p rather than V, we defne the enthalpy X(p, S) =Vp+ E, dx = Vdp+ TdS To make both changes, and use (T,p) to descrbe the state of the gas, we use the Gbbs free energy G(T,p)=X TS = E + Vp TS, dg = Vdp SdT Each of these nvolves a Legendre transformaton startng wth E(S, V ) Unlke Q, E s a well defned property of the gas when t s n a volume V f ts entropy s S, soe = E(S, V ), and T = E, p = E S V V S 2 E As S V = 2 E T we can conclude that = p V S V S S V consder the state of the gas to be descrbed by T and V,so de = E T ds = 1 T de + p T dv = 1 T from whch we can conclude ( ) 1 E = V T T T V dt + E dv V V T E T dt + V [ ( 1 T T [ ( 1 T p + E V p + E V )], T V We may also )] dv, T

61 LEGENDRE TRANSFORMS 159 and therefore T p p = E T V V T Ths s a useful relaton n thermodynamcs Let us get back to mechancs Most Lagrangans we encounter have the decomposton L = L 2 +L 1 +L 0 nto terms quadratc, lnear, and ndependent of veloctes, as consdered n 242 Then the momenta are lnear n veloctes, p = j M j q j + a, or n matrx form p = M q + a, whch has the nverse relaton q = M 1 (p a) As H = L 2 L 0, H = 1(p a) M 1 (p a) L 2 0 As a smple example, wth a = 0 and a dagonal matrx M, consder sphercal coordnates, n whch the knetc energy s T = m 2 (ṙ2 + r 2 θ2 + r 2 sn 2 θ φ 2) = 1 ( ) p 2 r + p2 θ 2m r + p2 φ 2 r 2 sn 2 θ Note that the generalzed momenta are not normalzed components of the ordnary momentum, as p θ p ê θ, n fact t doesn t even have the same unts The equatons of moton n Hamltonan form (63), q k = H, ṗ p k k = H, q,t q k p,t are almost symmetrc n ther treatment of q and p dmensonal coordnate η for phase space, } η = q for 1 N, η N+ = p If we defne a 2N we can wrte Hamlton s equaton n terms of a partcular matrx J, η j = 2N k=1 ( H 0 1I J jk, where J = N N η k 1I N N 0 J s lke a multdmensonal verson of the σ y whch we meet n quantummechancal descrptons of spn 1/2 partcles It s real, antsymmetrc, and because J 2 = 1I, t s orthogonal Mathematcans would say that J descrbes the complex structure on phase space, also called the symplectc structure ) 160 CHAPTER 6 HAMILTON S EQUATIONS In Secton 21 we dscussed how the Lagrangan s unchanged by a change of generalzed coordnates used to descrbe the physcal stuaton More precsely, the Lagrangan transforms as a scalar under such pont transformatons, takng on the same value at the same physcal pont, descrbed n the new coordnates There s no unque set of generalzed coordnates whch descrbes the physcs But n transformng to the Hamltonan language, dfferent generalzed coordnates may gve dfferent momenta and dfferent Hamltonans An nce example s gven n Goldsten, a mass on a sprng attached to a fxed pont whch s on a truck movng at unform velocty v T, relatve to the Earth If we use the Earth coordnate x to descrbe the mass, the equlbrum poston of the sprng s movng n tme, x eq = v T t, gnorng a neglgble ntal poston Thus U = 1k(x v 2 T t) 2, whle T = 1 2 mẋ2 as usual, and L = 1 2 mẋ2 1k(x v 2 T t) 2, p = mẋ, H = p 2 /2m + 1k(x v 2 T t) 2 The equaton of moton s ṗ = mẍ = H/ x = k(x v T t), of course Ths shows that H s not conserved, dh/dt =(p/m)dp/dt + k(ẋ v T )(x v T t)= (kp/m)(x v T t)+(kp/m kv T )(x v T t)= kv T (x v T t) 0 Alternatvely, dh/dt = L/ t = kv T (x v T t) 0 Ths s not surprsng; the sprng exerts a force on the truck and the truck s dong work to keep the fxed pont movng at constant velocty On the other hand, f we use the truck coordnate x = x v T t, we may descrbe the moton n ths frame wth T = 1 2 mẋ 2, U = 1 2 kx 2, L = 1 2 mẋ 2 1 2 kx 2, gvng the correct equatons of moton p = mẋ, ṗ = mẍ = L / x = kx Wth ths set of coordnates, the Hamltonan s H = ẋ p L = p 2 /2m + 1 2 kx 2, whch s conserved From the correspondence between the two sets of varables, x = x v T t, and p = p mv T, we see that the Hamltonans at correspondng ponts n phase space dffer, H(x, p) H (x,p )=(p 2 p 2 )/2m =2mv T p 1 2 mv2 T 0 Thus the Hamltonan s not nvarant, or a scalar, under change of generalzed coordnates, or pont transformatons We shall see, however, that t s nvarant under tme ndependent transformatons that are more general than pont transformatons, mxng coordnates and momenta 62 Varatons on phase curves In applyng Hamlton s Prncple to derve Lagrange s Equatons, we consdered varatons n whch δq (t) was arbtrary except at the ntal and fnal tmes, but the veloctes were fxed n terms of these, δ q (t) =(d/dt)δq (t)

63 CANONICAL TRANSFORMATIONS 161 In dscussng dynamcs n terms of phase space, ths s not the most natural varaton, because ths means that the momenta are not vared ndependently Here we wll show that Hamlton s equatons follow from a modfed Hamlton s Prncple, n whch the momenta are freely vared We wrte the acton n terms of the Hamltonan, [ ] tf I = p q H(q j,p j,t) dt, t and consder ts varaton under arbtrary varaton of the path n phase space, (q (t),p (t)) The q (t) s stll dq /dt, but the momentum s vared free of any connecton to q Then δi = tf t [ ( δp q H ) p ( δq ṗ + H )] dt + q p δq t f where we have ntegrated the p dδq /dt term by parts Note that n order to relate statonarty of the acton to Hamlton Equatons of Moton, t s necessary only to constran the q (t) at the ntal and fnal tmes, wthout mposng any lmtatons on the varaton of p (t), ether at the endponts, as we dd for q (t), or n the nteror (t,t f ), where we had prevously related p and q j The relaton between q and p j emerges nstead among the equatons of moton The q seems a bt out of place n a varatonal prncple over phase space, and ndeed we can rewrte the acton ntegral as an ntegral of a 1-form over a path n extended phase space, ( ) I = p dq H(q, p, t)dt We wll see, n secton 66, that the frst term of the ntegrand leads to a very mportant form on phase space, and that the whole ntegrand s an mportant 1-form on extended phase space 63 Canoncal transformatons We have seen that t s often useful to swtch from the orgnal set of coordnates n whch a problem appeared to a dfferent set n whch the problem t, 162 CHAPTER 6 HAMILTON S EQUATIONS became smpler We swtched from cartesan to center-of-mass sphercal coordnates to dscuss planetary moton, for example, or from the Earth frame to the truck frame n the example n whch we found how Hamltonans depend on coordnate choces In all these cases we consdered a change of coordnates q Q, where each Q s a functon of all the q j and possbly tme, but not of the momenta or veloctes Ths s called a pont transformaton But we have seen that we can work n phase space where coordnates and momenta enter together n smlar ways, and we mght ask ourselves what happens f we make a change of varables on phase space, to new varables Q (q, p, t), P (q, p, t), whch also cover phase space, so there s a relaton (q, p) (Q, P ) gven by the two descrptons of the same pont n phase space We should not expect the Hamltonan to be the same ether n form or n value, as we saw even for pont transformatons, but there must be a new Hamltonan K(Q, P, t) from whch we can derve the correct equatons of moton, Q = K P, P = K Q ( ) Q The analog of η for our new varables wll be ζ =, and the relaton P η ζ means each can be vewed as a functon of the other Hamlton s equatons for ζ are ζ = J K ζ If ths exsts, we say the new varables (Q, P ) are canoncal varables and the transformaton (q, p) (Q, P )sa canoncal transformaton Note that the functons Q and P may depend on tme as well as on q and p These new Hamltonan equatons are related to the old ones, η = J H/ η, by the functon whch gves the new coordnates and momenta n terms of the old, ζ = ζ(η, t) Then ζ = dζ dt = j ζ η j η j + ζ t Let us wrte the Jacoban matrx M j := ζ / η j In general, M wll not be a constant but a functon on phase space The above relaton for the veloctes now reads ζ = M η + ζ t η

63 CANONICAL TRANSFORMATIONS 163 The gradents n phase space are also related, = ζ j, or η t,η j η ζ j η = M T ζ t,ζ Thus we have t,η ζ = M η + ζ t = M J ηh + ζ t = M J M T ζ H + ζ t = J ζ K Let us frst consder a canoncal transformaton whch does not depend on tme, so ζ/ t η = 0 We see that we can choose the new Hamltonan to be the same as the old, K = H (e K(ζ,t):=H(η(ζ),t)), and get correct mechancs, provded M J M T = J (64) We wll requre ths condton even when ζ does depend on t, but then we need to revst the queston of fndng K The condton (64) on M s smlar to, and a generalzaton of, the condton for orthogonalty of a matrx, OO T = 1I, whch s of the same form wth J replaced by 1I Another example of ths knd of relaton n physcs occurs n specal relatvty, where a Lorentz transformaton L µν gves the relaton between two coordnates, x µ = ν L µν x ν, wth x ν a four dmensonal vector wth x 4 = ct Then the condton whch makes L a Lorentz transformaton s 1 0 0 0 L g L T 0 1 0 0 = g, wth g = 0 0 1 0 0 0 0 1 The matrx g n relatvty s known as the ndefnte metrc, and the condton on L s known as pseudo-orthogonalty In our current dscusson, however, J s not a metrc, as t s antsymmetrc rather than symmetrc, and the word whch descrbes M s symplectc So far the only examples of canoncal transformaton whch we have dscussed have been pont transformatons Before we contnue, let us recall that such transformatons can be dscrete, eg (x, y, z) (r, θ, φ), but they can 164 CHAPTER 6 HAMILTON S EQUATIONS also be contnuous, or dependng on a parameter For example, wth rotatng coordnates we consdered x = r cos(θ + φ),y = r sn(θ + φ), whch can be vewed as a set of possble dscrete transformatons (r, θ) (x, y) dependng on a parameter φ, whch mght even be a functon of tme, φ(t) =Ωt For each fxed φ ths transformaton P φ :(x, y) (r, θ) s an acceptble pont transformaton, whch can be used to descrbe phase space at any tme t Thuswe may also consder a composton, the pont tranformaton P φ(t+ t) P 1 φ(t) from the polar coordnate system at tme t to that at t + t Ths s an nfntesmal pont transformaton, and s what we used n dscussng the angular velocty n rgd body moton Just as for orthogonal transformatons, symplectc transformatons can be dvded nto those whch can be generated by nfntesmal transformatons (whch are connected to the dentty) and those whch can not Consder a transformaton M whch s almost the dentty, M j = δ j + ɛg j,orm = 1I + ɛg, where ɛ s consdered some nfntesmal parameter whle G s a fnte matrx As M s symplectc, (1 + ɛg) J (1 + ɛg T )=J, whch tells us that to lowest order n ɛ, GJ + JG T = 0 Comparng ths to the condton for the generator of an nfntesmal rotaton, Ω = Ω T, we see that t s smlar except for the appearence of J on opposte sdes, changng orthogonalty to symplectcty The new varables under such a canoncal transformaton are ζ = η + ɛg η The condton (64) for a transformaton η ζ to be canoncal does not nvolve tme each canoncal transformaton s a fxed map of phase-space onto tself, and could be used at any t We mght consder a set of such maps, one for each tme, gvng a tme dependant map g(t) :η ζ Each such map could be used to transform the trajectory of the system at any tme In partcular, consder the set of maps g(t, t 0 ) whch maps each pont η at whch a system can be at tme t 0 nto the pont to whch t wll evolve at tme t That s, g(t, t 0 ):η(t 0 ) η(t) If we consder t = t 0 + t for nfntesmal t, ths s an nfntesmal transformaton As ζ = η + t η = η + t k J k H/ η k,wehavem j = ζ / η j = δ j + t k J k 2 H/ η j η k, so G j = k J k 2 H/ η j η k, (GJ + JG T ) j = kl = kl ( J k 2 H 2 ) H J lj + J l J jk η l η k η l η k (J k J lj + J l J jk ) 2 H η l η k

64 POISSON BRACKETS 165 The factor n parentheses n the last lne s ( J k J jl + J l J jk ) whch s antsymmetrc under k l, and as t s contracted nto the second dervatve, whch s symmetrc under k l, we see that (GJ + JG T ) j = 0 and we have an nfntesmal canoncal transformaton Thus the nfntesmal flow of phase space ponts by the velocty functon s canoncal As compostons of canoncal transformatons are also canoncal 2, the map g(t, t 0 ) whch takes η(t 0 )ntoη(t), the pont t wll evolve nto after a fnte tme ncrement t t 0, s also a canoncal transformaton Notce that the relatonshp ensurng Hamlton s equatons exst, M J M T ζ H + ζ t = J ζk, wth the symplectc condton M J M T = J, mples ζ (K H) = J ζ/ t, so K dffers from H here Ths dscusson holds as long as M s symplectc, even f t s not an nfntesmal transformaton 64 Posson Brackets Suppose I have some functon f(q, p, t) on phase space and I want to ask how f, evaluated on a dynamcal system, changes as the system evolves through phase space wth tme Then df dt = = f q q + f H q p f ṗ + f p t f H + f p q t (65) The structure of the frst two terms s that of a Posson bracket, a blnear operaton of functons on phase space defned by [u, v] := u v q p Thus Eq 65 may be rewrtten as u v (66) p q df dt =[f,h]+ f t (67) 2 If M = M 1 M 2 and M 1 J M T 1 = J, M 2 J M T 2 = J, then M J M T = (M 1 M 2 ) J( M T 2 M T 1 )=M 1 (M 2 J M T 2 ) M T 1 = M 1 J M T 1 = J, som s canoncal 166 CHAPTER 6 HAMILTON S EQUATIONS The Posson bracket s a fundamental property of the phase space In symplectc language, [u, v] = j u v J j =( η u) T J η v (68) η η j If we descrbe the system n terms of a dfferent set of canoncal varables ζ, we should stll fnd the functon f(t) changng at the same rate We may thnk of u and v as functons of ζ as easly as of η Really we are thnkng of u and v as functons of ponts n phase space, represented by u(η) =ũ(ζ) and we may ask whether [ũ, ṽ] ζ s the same as [u, v] η Usng η = M T ζ, we have [u, v] η = ( M T ζ ũ ) T J M T ζ ṽ =( ζ ũ) T M J M T ζ ṽ = ( ζ ũ) T J ζ ṽ =[ũ, ṽ] ζ, so we see that the Posson bracket s ndependent of the coordnatzaton used to descrbe phase space, as long as t s canoncal The Posson bracket plays such an mportant role n classcal mechancs, and an even more mportant role n quantum mechancs, that t s worthwhle to dscuss some of ts abstract propertes Frst of all, from the defnton t s obvous that t s antsymmetrc: [u, v] = [v, u] (69) It s a lnear operator on each functon over constant lnear combnatons, but s satsfes a Lebntz rule for non-constant multples, [uv, w] =[u, w]v + u[v, w], (610) whch follows mmedately from the defnton, usng Lebntz rule on the partal dervatves A very specal relaton s the Jacob dentty, [u, [v, w]]+[v, [w, u]]+[w, [u, v]] = 0 (611) We need to prove that ths s true To smplfy the presentaton, we ntroduce some abbrevated notaton We use a subscrpt, to ndcate partal dervatve wth respect to η,sou, means u/ η, and u,,j means ( u/ η )/ η j We wll assume all our functons on phase space are sutably dfferentable, so u,,j = u,j, We wll also use the summaton conventon, that any ndex

64 POISSON BRACKETS 167 whch appears twce n a term s assumed to be summed over 3 Then [v, w] = v, J j w,j, and [u, [v, w]] = [u, v, J j w,j ] = [u, v, ]J j w,j + v, J j [u, w,j ] = u,k J kl v,,l J j w,j + v, J j u,k J kl w,j,l In the Jacob dentty, there are two other terms lke ths, one wth the substtuton u v w u and the other wth u w v u, gvng a sum of sx terms The only ones nvolvng second dervatves of v are the frst term above and the one found from applyng u w v u to the second, u, J j w,k J kl v,j,l The ndces are all dummy ndces, summed over, so ther names can be changed, by k j l, convertng ths second term to u,k J kl w,j J j v,l, Addng the orgnal term u,k J kl v,,l J j w,j, and usng v,l, = v,,l, gves u,k J kl w,j (J j +J j )v,l, = 0 because J s antsymmetrc Thus the terms n the Jacob dentty nvolvng second dervatves of v vansh, but the same argument apples n pars to the other terms, nvolvng second dervatves of u or of w, so they all vansh, and the Jacob dentty s proven Ths argument can be made more elegantly f we recognze that for each functon f on phase space, we may vew [f, ] as a dfferental operator on functons g on phase space, mappng g [f,g] Callng ths operator D f, we see that D f = ( ) f J j, j η η j whch s of the general form that a dfferental operator has, D f = f j, j η j where f j are an arbtrary set of functons on phase space For the Posson bracket, the functons f j are lnear combnatons of the f,j, but f j f,j Wth ths nterpretaton, [f,g] =D f g, and [h, [f,g]] = D h D f gthus [h, [f,g]]+[f,[g, h]] = [h, [f,g]] [f,[h, g]] = D h D f g D f D h g = (D h D f D f D h )g, (612) 3 Ths conventon of understood summaton was nvented by Ensten, who called t the greatest contrbuton of my lfe 168 CHAPTER 6 HAMILTON S EQUATIONS and we see that ths combnaton of Posson brackets nvolves the commutator of dfferental operators But such a commutator s always a lnear dfferental operator tself, D h D f = j D f D h = j h f j f j = η η j j h = η j η j h f j η f j h η j + η j j + η j 2 h f j η η j 2 h f j η η j so n the commutator, the second dervatve terms cancel, and D h D f D f D h = f j h h f j j η η j j η j η = ( ) f j h j h f j η η η j Ths s just another frst order dfferental operator, so there are no second dervatves of g left n (612) In fact, the dentty tells us that ths combnaton s D h D f D f D h = D [h,f] (613) An antsymmetrc product whch obeys the Jacob dentty s what makes a Le algebra Le algebras are the nfntesmal generators of Le groups, or contnuous groups, one example of whch s the group of rotatons SO(3) whch we have already consdered Notce that the product here s not assocatve, [u, [v, w]] [[u, v],w] In fact, the dfference [u, [v, w]] [[u, v],w]= [u, [v, w]] + [w, [u, v]] = [v, [w, u]] by the Jacob dentty, so the Jacob dentty replaces the law of assocatvty n a Le algebra Le groups play a major role n quantum mechancs and quantum feld theory, and ther more extensve study s hghly recommended for any physcst Here we wll only menton that nfntesmal rotatons, represented ether by the ω t or Ω t of Chapter 4, consttute the three dmensonal Le algebra of the rotaton group (n three dmensons) Recall that the rate at whch a functon on phase space, evaluated on the system as t evolves, changes wth tme s df = [H, f]+ f dt t, (614)

64 POISSON BRACKETS 169 where H s the Hamltonan The functon [f,g] on phase space also evolves that way, of course, so d[f,g] dt = [H, [f,g]] + [f,g] t [ ] [ f = [f,[g, H]]+[g, [H, f]] + t,g + f, g ] t [ ( = f, [H, g]+ g )] [ ( + g, [H, f] f )] t t [ = f, dg ] [ g, df ] dt dt If f and g are conserved quanttes, df/dt = dg/dt = 0, and we have the mportant consequence that d[f,g]/dt = 0 Ths proves Posson s theorem: The Posson bracket of two conserved quanttes s a conserved quantty We wll now show an mportant theorem, known as Louvlle s theorem, that the volume of a regon of phase space s nvarant under canoncal transformatons Ths s not a volume n ordnary space, but a 2n dmensonal volume, gven by ntegratng the volume element 2n =1 dη n the old coordnates, and by 2n =1 dζ = det ζ η j 2n =1 dη = det M n the new, where we have used the fact that the change of varables requres a Jacoban n the volume element But because J = M J M T, det J = det M det J det M T = (det M) 2 det J, and J s nonsngular, so det M = ±1, and the volume element s unchanged In statstcal mechancs, we generally do not know the actual state of a system, but know somethng about the probablty that the system s n a partcular regon of phase space As the transformaton whch maps possble values of η(t 1 ) to the values nto whch they wll evolve at tme t 2 s a canoncal transformaton, ths means that the volume of a regon n phase space does not change wth tme, although the regon tself changes Thus the probablty densty, specfyng the lkelhood that the system s near a partcular pont of phase space, s nvarant as we move along wth the system 2n =1 dη 170 CHAPTER 6 HAMILTON S EQUATIONS 65 Hgher Dfferental Forms In secton 61 we dscussed a renterpretaton of the dfferental df as an example of a more general dfferental 1-form, a map ω : M R n R We saw that the {dx } provde a bass for these forms, so the general 1-form can be wrtten as ω = ω (x) dx The dfferental df gave an example We defned an exact 1-form as one whch s a dfferental of some well-defned functon f What s the condton for a 1-form to be exact? If ω = ω dx s df, then ω = f/ x = f,, and ω,j = ω x j = 2 f x x j = 2 f x j x = ω j, Thus one necessary condton for ω to be exact s that the combnaton ω j, ω,j = 0 We wll defne a 2-form to be the set of these objects whch must vansh In fact, we defne a dfferental k-form to be a map ω (k) : M R n R n R }{{} k tmes whch s lnear n ts acton on each of the R n and totally antsymmetrc n ts acton on the k copes, and s a smooth functon of x M At a gven pont, a bass of the k-forms s 4 dx 1 dx 2 dx k := P S k ( 1) P dx P 1 dx P 2 dx Pk For example, n three dmensons there are three ndependent 2-forms at a pont, dx 1 dx 2, dx 1 dx 3, and dx 2 dx 3, where dx 1 dx 2 = dx 1 dx 2 dx 2 dx 1, whch means that, actng on u and v, dx 1 dx 2 ( u, v) =u 1 v 2 u 2 v 1 The product s called the wedge product or exteror product, and can be extended to act between k 1 - and k 2 -forms so that t becomes an assocatve dstrbutve product Note that ths defnton of a k-form agrees, for k = 1, 4 Some explanaton of the mathematcal symbols mght be n order here S k s the group of permutatons on k objects, and ( 1) P s the sgn of the permutaton P, whch s plus or mnus one f the permutaton can be bult from an even or an odd number, respectvely, of transpostons of two of the elements The tensor product of two lnear operators nto a feld s a lnear operator whch acts on the product space, or n other words a blnear operator wth two arguments Here dx dx j s an operator on R n R n whch maps the par of vectors ( u, v) tou v j

65 HIGHER DIFFERENTIAL FORMS 171 wth our prevous defnton, and for k = 0 tells us a 0-form s smply a functon on M The general expresson for a k-form s ω (k) = ω 1k (x)dx 1 dx k 1<< k Let us consder some examples n three dmensonal Eucldean 5 space E 3, where there s a correspondance we can make between vectors and 1- and 2-forms In ths dscusson we wll not be consderng how the objects change under changes n the coordnates of E 3, to whch we wll return later k =0: As always, 0-forms are smply functons, f(x), x E 3 k =1: A 1-form ω = ω dx can be thought of, or assocated wth, a vector feld A(x) = ω (x)ê Note that f ω = df, ω = f/ x, so A = f The 1-form s not actually the vector feld, as t s a functon that depends on two arguments, x and v If the 1-form ω s assocated wth the vector feld A = A (x)ê, actng on the vector feld B = B (x)ê, we have ω(x, B)= A B = A B k =2: A general two form s a sum over the three ndependent wedge products wth ndependent functons B 12 (x),b 13 (x),b 23 (x) Let us extend the defnton of B j to make t an antsymmetrc matrx, so B = B j dx dx j = B j dx dx j <j,j As we dd for the angular velocty matrx Ω n (42), we can condense the nformaton n the antsymmetrc matrx B j nto a vector feld B = B ê, wth B j = ɛ jk B k Note that ths step requres that we are workng n E 3 rather than some other dmenson Thus B = jk ɛ jk B k dx dx j Also note 1 2 j ɛ jk B j = 1 2 jl ɛ jk ɛ jl B l = B k The 2-form B takes two vector arguments, so actng on vectors A = A (x)ê and C = C (x)ê, we have B(x, A, B) = B j A C j = ɛjk B k A C j =( A(x) C(x)) B(x) 5 Forms are especally useful n dscussng more general manfolds, such as occur n general relatvty Then one must dstngush between covarant and contravarant vectors, a complcaton we avod here by treatng only Eucldean space 172 CHAPTER 6 HAMILTON S EQUATIONS k =3: There s only one bass 3-form avalable n three dmensons, dx 1 dx 2 dx 3 Any other 3-form s proportonal to ths one, though the proportonalty can be a functon of {x } In partcular dx dx j dx k = ɛ jk dx 1 dx 2 dx 3 The most general 3-form C s smply specfed by an ordnary functon C(x), whch multples dx 1 dx 2 dx 3 Havng establshed, n three dmensons, a correspondance between vectors and 1- and 2-forms, and between functons and 0- and 3-forms, we can ask to what the wedge product corresponds n terms of these vectors If A and C are two vectors correspondng to the 1-forms A = A dx and C = C dx, and f B = A C, then B = A C j dx dx j = (A C j A j C )dx dx j = B j dx dx j, j j j so B j = A C j A j C, and B k = 1 ɛkj B j = 1 ɛkj A C j 1 ɛkj A j C = ɛ kj A C j, 2 2 2 so B = A C, and the wedge product of two 1-forms s the cross product of ther vectors If A s a 1-form and B s a 2-form, the wedge product C = A B = C(x)dx 1 dx 2 dx 3 s gven by C = A B = A B jk dx dx j dx k }{{}}{{} j<k ɛ jkl B l ɛ jk dx 1 dx 2 dx 3 = A B l ɛ jkl ɛ jk dx 1 dx 2 dx 3 }{{} l j<k symmetrc under j k = 1 A B l ɛ jkl ɛ jk dx 1 dx 2 dx 3 = A B l δ l dx 1 dx 2 dx 3 2 l jk l = A Bdx 1 dx 2 dx 3, so we see that the wedge product of a 1-form and a 2-form gves the dot product of ther vectors If A and B are both 2-forms, the wedge product C = A B must be a 4-form, but there cannot be an antsymmetrc functon of four dx s n three dmensons, so C =0

65 HIGHER DIFFERENTIAL FORMS 173 The exteror dervatve We defned the dfferental of a functon f, whch we now call a 0-form, gvng a 1-form df = f, dx Now we want to generalze the noton of dfferental so that d can act on k-forms for arbtrary k Ths generalzed dfferental d : k-forms (k + 1)-forms s called the exteror dervatve It s defned to be lnear and to act on one term n the sum over bass elements by d (f 1k (x)dx 1 dx k )=(df 1k (x)) dx 1 dx k = f 1k,jdx j dx 1 dx k j Clearly some examples are called for, so let us look agan at three dmensonal Eucldean space k =0: For a 0-form f, df = f, dx, as we defned earler In terms of vectors, df f k =1: For a 1-form ω = ω dx, dω = dω dx = j ω,j dx j dx = j (ω j, ω,j ) dx dx j, correspondng to a two form wth B j = ω j, ω,j These B j are exactly the thngs whch must vansh f ω s to be exact In three dmensonal Eucldean space, we have a vector B wth components B k = 1 2 ɛkj (ω j, ω,j )= ɛ kj ω j =( ω) k, so here the exteror dervatve of a 1-form gves a curl, B = ω k =2: On a two form B = <j B j dx dx j, the exteror dervatve gves a 3-form C = db = k <j B j,k dx k dx dx j In three-dmensonal Eucldean space, ths reduces to C = ( k ɛ jl B l ) ɛ kj dx 1 dx 2 dx 3 = k B k dx 1 dx 2 dx 3, kl <j k so C(x) = B, and the exteror dervatve on a 2-form gves the dvergence of the correspondng vector k =3: If C s a 3-form, dc s a 4-form In three dmensons there cannot be any 4-forms, so dc = 0 for all such forms 174 CHAPTER 6 HAMILTON S EQUATIONS We can summarze the acton of the exteror dervatve n three dmensons n ths dagram: d f ω (1) A d ω (2) B d ω (3) f A B Now that we have d operatng on all k-forms, we can ask what happens f we apply t twce Lookng frst n three dmenons, on a 0-form we get d 2 f = da for A f, and da A, sod 2 f f But the curl of a gradent s zero, so d 2 = 0 n ths case On a one form d 2 A = db, B A and db B = ( A) Now we have the dvergence of a curl, whch s also zero For hgher forms n three dmensons we can only get zero because the degree of the form would be greater than three Thus we have a strong hnt that d 2 mght vansh n general To verfy ths, we apply d 2 to ω (k) = ω 1k dx 1 dx k Then dω = ( j ω 1k ) dx j dx 1 dx k j 1< 2< < k d(dω) = lj = 0 d ( l j ω 1 }{{} k ) dx l dx j dx 1 dx k }{{} 1< 2< < k symmetrc antsymmetrc Ths s a very mportant result A k-form whch s the exteror dervatve of some (k 1)-form s called exact, whle a k-form whose exteror dervatve vanshes s called closed, and we have just proven that all exact k-forms are closed The converse s a more subtle queston In general, there are k-forms whch are closed but not exact, gven by harmonc functons on the manfold M, whch form what s known as the cohomology of M Ths has to do wth global propertes of the space, however, and locally every closed form can be wrtten as an exact one 6 The precsely stated theorem, known as Poncaré s 6 An example may be useful In two dmensons, rrotatonal vortex flow can be represented by the 1-form ω = yr 2 dx + xr 2 dy, whch satsfes dω = 0 wherever t s well defned, but t s not well defned at the orgn Locally, we can wrte ω = dθ, where θ s the polar coordnate But θ s not, strctly speakng, a functon on the plane, even on the plane wth the orgn removed, because t s not sngle-valued It s a well defned functon on the plane wth a half axs removed, whch leaves a smply-connected regon, a regon wth no holes In fact, ths s the general condton for the exactness of a 1-form a closed 1-form on a smply connected manfold s exact 0

65 HIGHER DIFFERENTIAL FORMS 175 Lemma, s that f ω s a closed k-form on a coordnate neghborhood U of a manfold M, and f U s contractble to a pont, then ω s exact on U We wll gnore the possblty of global obstructons and assume that we can wrte closed k-forms n terms of an exteror dervatve actng on a (k 1)-form Coordnate ndependence of k-forms We have ntroduced forms n a way whch makes them appear dependent on the coordnates x used to descrbe the space M Ths s not what we want at all 7 We want to be able to descrbe physcal quanttes that have ntrnsc meanng ndependent of a coordnate system If we are presented wth another set of coordnates y j descrbng the same physcal space, the ponts n ths space set up a mappng, deally an somorphsm, from one coordnate system to the other, y = y( x) If a functon represents a physcal feld ndependent of coordnates, the actual functon f(x) used wth the x coordnates must be replaced by another functon f(y) when usng the y coordnates That they both descrbe the physcal value at a gven physcal pont requres f(x) = f(y) when y = y(x), or more precsely 8 f(x) = f(y(x)) Ths assocated functon and coordnate system s called a scalar feld If we thnk of the dfferental df as the change n f correspondng to an nfntesmal change dx, then clearly d f s the same thng n dfferent coordnates, provded we understand the dy to represent the same physcal dsplacement as dx does That means dy k = j y k x j dx j As f(x) = f(y(x)) and f(y) =f(x(y)), the chan rule gves f x = j f y j y j x, f y j = f x x y j, 7 Indeed, most mathematcal texts wll frst defne an abstract noton of a vector n the tangent space as a drectonal dervatve operator, specfed by equvalence classes of parameterzed paths on M Then 1-forms are defned as duals to these vectors In the frst step any coordnatzaton of M s ted to the correspondng bass of the vector space R n Whle ths provdes an elegant coordnate-ndependent way of defnng the forms, the abstract nature of ths defnton of vectors can be unsettlng to a physcst 8 More elegantly, gvng the map x y the name φ, soy = φ(x), we can state the relaton as f = f φ 176 CHAPTER 6 HAMILTON S EQUATIONS so d f = k f y k dy k = jk f x x y k y k x j dx j = f δ j dx j = f, dx = df j x We mpose ths transformaton law n general on the coeffcents n our k- forms, to make the k-form nvarant, whch means that the coeffcents are covarant, ω j = ω j1j k = Integraton of k-forms x y j ω 1, 2,, k ( k l=1 x l y jl ) ω 1 k Suppose we have a k-dmensonal smooth surface S n M, parameterzed by coordnates (u 1,,u k ) We defne the ntegral of a k-form ω (k) = ω 1k dx 1 dx k 1<< k over S by S ( ω (k) k = ω 1k (x(u)) 1, 2,, k l=1 x l u l ) du 1 du 2 du k We had better gve some examples For k = 1, the surface s actually a path Γ : u x(u), and Γ ω dx = umax u mn ω (x(u)) x u du, whch seems obvous In vector notaton ths s Γ A d r, the path ntegral of the vector A For k =2, ω (2) x x j = B j S u v dudv

65 HIGHER DIFFERENTIAL FORMS 177 In three dmensons, the parallelogram whch s the mage of the rectangle [u, u + du] [v, v + dv] has edges ( x/ u)du and ( x/ v)dv, whch has an area equal to the magntude of d S = ( x u x ) dudv v and a normal n the drecton of d S Wrtng B j n terms of the correspondng vector B, B j = ɛ jk B k,so S ω (2) = = S S ( ) x ɛ jk B k u ( x B k u x v ( ) x dudv v j ) dudv = k so ω (2) gves the flux of B through the surface Smlarly for k = 3 n three dmensons, ( ) x ɛjk u ( ) x v j S ( ) x dudvdw w k B d S, s the volume of the parallelopped whch s the mage of [u, u + du] [v, v + dv] [w, w + dw] As ω jk = ω 123 ɛ jk, ths s exactly what appears: ω (3) x x j x k = ɛjk ω 123 u v w dudvdw = ω 123 (x)dv Notce that we have only defned the ntegraton of k-forms over submanfolds of dmenson k, not over other-dmensonal submanfolds These are the only ntegrals whch have coordnate nvarant meanngs Also note that the ntegrals do not depend on how the surface s coordnatzed We state 9 a marvelous theorem, specal cases of whch you have seen often before, known as Stokes Theorem Let C be a k-dmensonal submanfold 9 For a proof and for a more precse explanaton of ts meanng, we refer the reader to the mathematcal lterature In partcular [14] and [3] are advanced calculus texts whch gve elementary dscussons n Eucldean 3-dmensonal space A more general treatment s (possbly???) gven n [16] u v 178 CHAPTER 6 HAMILTON S EQUATIONS of M, wth C ts boundary Let ω bea(k 1)-form Then Stokes theorem says dω = ω (615) C Ths elegant jewel s actually famlar n several contexts n three dmensons If k =2,C s a surface, usually called S, bounded by a closed path Γ= S Ifω s a 1-form assocated wth A, then Γ ω = Γ A d lnowdω s the 2-form A, and S dω = ( ) S A ds, so we see that ths Stokes theorem ncludes the one we frst learned by that name But t also ncludes other possbltes We can try k = 3, where C = V s a volume wth surface S = V Then f ω B s a two form, S ω = S B ds, whle dω B, so V dω = BdV, so here Stokes general theorem gves Gauss s theorem Fnally, we could consder k =1,C = Γ, whch has a boundary C consstng of two ponts, say A and B Our 0-form ω = f s a functon, and Stokes theorem gves 10 Γ df = f(b) f(a), the fundamental theorem of calculus 66 The natural symplectc 2-form We now turn our attenton back to phase space, wth a set of canoncal coordnates (q,p ) Usng these coordnates we can defne a partcular 1- form ω 1 = p dq For a pont transformaton Q = Q (q 1,,q n,t)we may use the same Lagrangan, reexpressed n the new varables, of course Here the Q are ndependent of the veloctes q j, so on phase space 11 dq = j( Q / q j )dq j The new veloctes are gven by Q = j C Q q j q j + Q t, so Q q j = Q q j 10 Note that there s a drecton assocated wth the boundary, whch s nduced by a drecton assocated wth C tself Ths gves an ambguty n what we have stated, for example how the drecton of an open surface nduces a drecton on the closed loop whch bounds t Changng ths drecton would clearly reverse the sgn of A d lwehavenot worred about ths ambguty, but we cannot avod notcng the appearence of the sgn n ths last example 11 We have not ncluded a term Q t dt whch would be necessary f we were consderng a form n the 2n + 1 dmensonal extended phase space whch ncludes tme as one of ts coordnates

66 THE NATURAL SYMPLECTIC 2-FORM 179 Thus the old canoncal momenta, p = L(q, q, t) q = q,t j Thus the form ω 1 may be wrtten L(Q, Q, t) Q j Q = j q,t q q,t j P j Q j q ω 1 = Q j P j dq = P j dq j, j q j so the form of ω 1 s nvarant under pont transformatons Ths s too lmted, however, for our current goals of consderng general canoncal transformatons on phase space, under whch ω 1 wll not be nvarant However, ts exteror dervatve ω 2 := dω 1 = dp dq s nvarant under all canoncal transformatons, as we shall show momentarly Ths makes t specal, the natural symplectc structure on phase space We can reexpress ω 2 n terms of our combned coordnate notaton η, because J j dη dη j = dq dp = dp dq = ω 2 <j We must now show that the natural symplectc structure s ndeed form nvarant under canoncal transformaton Thus f Q,P are a new set of canoncal coordnates, combned nto ζ j, we expect the correspondng object formed from them, ω 2 = j J j dζ dζ j, to reduce to the same 2-form, ω 2 We frst note that dζ = ζ dη j = M j dη j, j η j j wth the same Jacoban matrx M we met n secton 63 Thus ω 2 = j = kl J j dζ dζ j = J j M k dη k j k l ( M T J M ) dη k dη l kl M jl dη l Thngs wll work out f we can show M T J M = J, whereas what we know for canoncal transformatons from Eq (64) s that M J M T = J We 180 CHAPTER 6 HAMILTON S EQUATIONS also know M s nvertble and that J 2 = 1, so f we multply ths known equaton from the left by J M 1 and from the rght by J M, we learn that J M 1 M J M T J M = J M 1 J J M = J M 1 M = J = J J M T J M = M T J M, whch s what we wanted to prove Thus we have shown that the 2-form ω 2 s form-nvarant under canoncal transformatons, and deserves ts name One mportant property of the 2-form ω 2 on phase space s that t s non-degenerate A 2-form has two slots to nsert vectors nsertng one leaves a 1-form Non-degenerate means there s no non-zero vector v on phase space such that ω 2 (, v) = 0, that s, such that ω 2 ( u, v) = 0, for all u on phase space Ths follows smply from the fact that the matrx J j s non-sngular Extended phase space One way of lookng at the evoluton of a system s n phase space, where a gven system corresponds to a pont movng wth tme, and the general equatons of moton corresponds to a velocty feld Another way s to consder extended phase space, a2n + 1 dmensonal space wth coordnates (q,p,t), for whch a system s moton s a path, monotone n t By the modfed Hamlton s prncple, the path of a system n ths space s an extremum of the acton I = t f t p dq H(q, p, t)dt, whch s the ntegral of the one-form ω 3 = p dq H(q, p, t)dt The exteror dervatve of ths form nvolves the symplectc structure, ω 2, as dω 3 = ω 2 dh dt The 2-form ω 2 on phase space s nondegenerate, and every vector n phase space s also n extended phase space On such a vector, on whch dt gves zero, the extra term gves only somethng n the dt drecton, so there are stll no vectors n ths subspace whch are annhlated by dω 3 Thus there s at most one drecton n extended phase space whch s annhlated by dω 3 But any 2-form n an odd number of dmensons must annhlate some vector, because n a gven bass t corresponds to an antsymmetrc matrx B j, and n an odd number of dmensons det B = det B T = det( B) =( 1) 2n+1 det B = det B, so det B = 0 and the matrx

66 THE NATURAL SYMPLECTIC 2-FORM 181 s sngular, annhlatng some vector ξ In fact, for dω 3 ths annhlated vector ξ s the tangent to the path the system takes through extended phase space One way to see ths s to smply work out what dω 3 s and apply t to the vector ξ, whch s proportonal to v =( q, ṗ, 1) So we wsh to show dω 3 (, v) = 0 Evaluatng dp dq (, v) = dp dq ( v) dq dp ( v) = dp q dq ṗ dh dt(, v) = dh dt( v) dt dh( v) ( H = dq + H dp + H ) q p t dt 1 ( H dt q + H ṗ + H ) q p t = H dq + H dp dt ( ) H H q +ṗ q p q p dω 3 (, v) = ( q H ) ( dp ṗ + H ) dq p q + ( ) H H q +ṗ dt q p = 0 where the vanshng s due to the Hamlton equatons of moton There s a more abstract way of understandng why dω 3 (, v) vanshes, from the modfed Hamlton s prncple, whch states that f the path taken were nfntesmally vared from the physcal path, there would be no change n the acton But ths change s the ntegral of ω 3 along a loop, forwards n tme along the frst trajectory and backwards along the second From Stokes theorem ths means the ntegral of dω 3 over a surface connectng δη v dt these two paths vanshes But ths surface s a sum over nfntesmal parallelograms one sde of whch s v t and the other sde of whch 12 s (δ q(t),δ p(t), 0) As ths latter vector s an arbtrary functon of t, each parallelogram must ndependently gve 0, so that ts contrbuton to the ntegral, dω 3 ((δ q, δ p, 0), v) t = 12 It s slghtly more elegant to consder the path parameterzed ndependently of tme, and consder arbtrary varatons (δq, δp, δt), because the ntegral nvolved n the acton, beng the ntegral of a 1-form, s ndependent of the parameterzaton Wth ths approach we fnd mmedately that dω 3 (, v) vanshes on all vectors 182 CHAPTER 6 HAMILTON S EQUATIONS 0 In addton, dω 3 ( v, v) = 0, of course, so dω 3 (, v) vanshes on a complete bass of vectors and s therefore zero 661 Generatng Functons Consder a canoncal transformaton (q, p) (Q, P ), and the two 1-forms ω 1 = p dq and ω 1 = P dq We have mentoned that the dfference of these wll not vansh n general, but the exteror dervatve of ths dfference, d(ω 1 ω 1)=ω 2 ω 2 =0,soω 1 ω 1 s an closed 1-form Thus t s exact 13, and there must be a functon F on phase space such that ω 1 ω 1 = df We call F the generatng functon of the canoncal transformaton 14 If the transformaton (q, p) (Q, P ) s such that the old q s alone, wthout nformaton about the old p s, do not mpose any restrctons on the new Q s, then the dq and dq are ndependent, and we can use q and Q to parameterze phase space 15 Then knowledge of the functon F (q, Q) determnes the transformaton, as ω 1 ω 1 = (p dq P dq )=df = F dq q + F dq Q Q q = p = F, P q = F Q Q q If the canoncal transformaton depends on tme, the functon F wll also depend on tme Now f we consder the moton n extended phase space, we know the phase trajectory that the system takes through extended phase space s determned by Hamlton s equatons, whch could be wrtten n any set of canoncal coordnates, so n partcular there s some Hamltonan K(Q, P, t) such that the tangent to the phase trajectory, v, s annhlated by dω 3, where ω 3 = P dq K(Q, P, t)dt Now n general knowng that two 2-forms both annhlate the same vector would not be suffcent to dentfy them, but n ths case we also know that restrctng dω 3 and dω 3 to ther acton on the dt = 0 subspace gves the same 2-form ω 2 That s to say, f 13 We are assumng phase space s smply connected, or else we are gnorng any complcatons whch mght ensue from F not beng globally well defned 14 Ths s not an nfntesmal generator n the sense we have n Le algebras ths generates a fnte canoncal transformaton for fnte F 15 Note that ths s the opposte extreme from a pont transformaton, whch s a canoncal transformaton for whch the Q s depend only on the q s, ndependent of the p s

66 THE NATURAL SYMPLECTIC 2-FORM 183 u and u are two vectors wth tme components zero, we know that (dω 3 dω 3)( u, u ) = 0 Any vector can be expressed as a multple of v and some vector u wth tme component zero, and as both dω 3 and dω 3 annhlate v, we see that dω 3 dω 3 vanshes on all pars of vectors, and s therefore zero Thus ω 3 ω 3 s a closed 1-form, whch must be at least locally exact, and ndeed ω 3 ω 3 = df, where F s the generatng functon we found above 16 Thus df = pdq PdQ+(K H)dt, or K = H + F t The functon F (q, Q, t) s what Goldsten calls F 1 The exstence of F as a functon on extended phase space holds even f the Q and q are not ndependent, but n ths case F wll need to be expressed as a functon of other coordnates Suppose the new P s and the old q s are ndependent, so we can wrte F (q, P, t) Then defne F 2 = Q P + F Then df 2 = Q dp + P dq + p dq P dq +(K H)dt = Q dp + p dq +(K H)dt, so Q = F 2, p = F 2, K(Q, P, t) =H(q, p, t)+ F 2 P q t The generatng functon can be a functon of old momenta rather than the old coordnates Makng one choce for the old coordnates and one for the new, there are four knds of generatng functons as descrbed by Goldsten Let us consder some examples The functon F 1 = q Q generates an nterchange of p and q, Q = p, P = q, whch leaves the Hamltonan unchanged We saw ths clearly leaves the form of Hamlton s equatons unchanged An nterestng generator of the second type s F 2 = λ q P, whch gves Q = λ q, P = λ 1 p, a smple change n scale of the coordnates wth a correspondng nverse scale change 16 From ts defnton n that context, we found that n phase space, df = ω 1 ω 1, whch s the part of ω 3 ω 3 not n the tme drecton Thus f ω 3 ω 3 = df for some other functon F, we know df df =(K K)dt for some new Hamltonan functon K (Q, P, t), so ths corresponds to an ambguty n K 184 CHAPTER 6 HAMILTON S EQUATIONS n momenta to allow [Q,P j ]=δ j to reman unchanged Ths also doesn t change H For λ = 1, ths s the dentty transformaton, for whch F =0, of course Placng pont transformatons n ths language provdes another example For a pont transformaton, Q = f (q 1,,q n,t), whch s what one gets wth a generatng functon Note that F 2 = p = F 2 q f (q 1,,q n,t)p = j f j q P j s at any pont q a lnear transformaton of the momenta, requred to preserve the canoncal Posson bracket, but ths transformaton s q dependent, so whle Q s a functon of q and t only, ndependent of p, P (q, p, t) wll n general have a nontrval dependence on coordnates as well as a lnear dependence on the old momenta For a harmonc oscllator, a smple scalng gves H = p2 2m + k 2 q2 = 1 k/m ( P 2 + Q 2), 2 where Q =(km) 1/4 q, P =(km) 1/4 p In ths form, thnkng of phase space as just some two-dmensonal space, we seem to be encouraged to consder a second canoncal transformaton Q, P θ, P, generated by F 1 (Q, θ), to a new, polar, coordnate system wth θ = tan 1 Q/P as the new coordnate, and F1 we mght hope to have the radal coordnate related to the new momentum, P = F 1 / θ Q AsP = F 1 / Q θ s also Q cot θ, we can take F 1 = 1 2 Q2 cot θ, so P = 1 2 Q2 ( csc 2 θ)= 1 2 Q2 (1 + P 2 /Q 2 )= 1 2 (Q2 + P 2 )=H/ω Note as F 1 s not tme dependent, K = H and s ndependent of θ, whch s therefore an gnorable coordnate, so ts conjugate momentum P s conserved Of course P dffers from the conserved Hamltonan H only by the factor ω = k/m, so ths s not unexpected Wth H now lnear n the new momentum P, the conjugate coordnate θ grows lnearly wth tme at the fxed rate θ = H/ P = ω P θ Q