Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons, χ A ξ B whch are symplectc transformatons at each pont are called canoncal. Specfcally, those functons χ A ξ satsfyng Ω CD χc χd ΩAB ξa ξ B are canoncal transformatons. Canoncal transformatons preserve Hamlton s equatons. 1 Posson brackets We may also wrte Hamlton s equatons n terms of Posson brackets between dynamcal varables. By a dynamcal varable, we mean any functon f f ξ A of the canoncal coordnates used to descrbe a physcal system. We defne the Posson bracket of any two dynamcal varables f and g by AB f g f, g} Ω The mportance of ths product s that t s preserved by canoncal transformatons. We see ths as follows. Let ξ A be any set of phase space coordnates n whch Hamlton s equatons take the form dξ A AB H Ω ξ B 1 and let f and g be any two dynamcal varables. Denote the Posson bracket of f and g n the coordnates ξ A be denoted by f, g} ξ. In a dfferent set of coordnates, χ A ξ, we have AB f g f, g} χ Ω χ A χ B ξ Ω AB C f ξ D χ A ξ C χ B ξ C ξd f ΩAB χa χ B ξ C Therefore, f the coordnate transformaton s canoncal so that ξ C ξd ΩAB χa χ B ΩCD g ξ D g ξ D 1
we have AB f g f, g} χ Ω ξ C ξ D f, g} ξ and the Posson bracket s unchanged. We conclude that canoncal transformatons preserve all Posson brackets. Conversely, a transformaton whch preserves all Posson brackets satsfes ξ C ξd ΩAB χa χ B f ξ C f, g} χ f, g} ξ g f ΩCD ξd ξ C for all f, g and must therefore be canoncal. An mportant specal case of the Posson bracket occurs when one of the functons s the Hamltonan. In that case, we have AB f H f, H} Ω f H x f H p p x f x dx f p df f dp or smply, df f f, H} + Ths shows that as the system evolves classcally, the total tme rate of change of any dynamcal varable s the sum of the Posson bracket wth the Hamltonan and the partal tme dervatve. If a dynamcal varable has no explct tme dependence, f 0, then the total tme dervatve s just the Posson bracket wth the Hamltonan. The coordnates provde another mportant specal case. Snce nether x nor p has any explct tme dependence, we have g ξ D dx dp H, x } H, p } 2 or smply ξ A H, ξ A}, and we can check ths drectly that ths reproduces Hamlton s equatons, dq H, x } j1 j1 H p x H x j x H p j p j x j δ j H p j 2
and dp H, p } j1 H p H p H q j p j p j q j Notce that snce q, p and are all ndependent, and do not depend explctly on tme, p. We also have the commutator of the Hamltonan wth the Hamltonan tself, p j p q j 0 dh H, H} + H H so f the Hamltonan s not explctly tme-dependent, then t s a constant of the moton. f More generally, a dynamcal varable wth no explct tme dependence, 0, s a constant of the moton f and only f t has vanshng Posson bracket wth the Hamltonan, H, f} 0. 2 Canoncal transformatons We now defne the fundamental Posson brackets. Suppose x and p j are a set of coordnates on phase space such that Hamlton s equatons hold. Snce they themselves are functons of x m, p n they are dynamcal varables and we may compute ther Posson brackets wth one another. Wth ξ A x m, p n we have for x wth x j, for x wth p j and fnally x, x j} AB x x j Ω ξ x x j x m x x j 0 m1 x, p j }ξ p j, x } AB x p j Ω ξ x p j x m x p j δ j m1 δmδ j m m1 p, p j } ξ Ω AB p p j p p j x m p p j 0 m1 3
for p wth p j. The subscrpt ξ on the bracket ndcates that the partal dervatves are taken wth respect to the coordnates ξ A x, p j. We summarze these relatons as ξ A, ξ B} ξ ΩAB However, snce Posson brackets are preserved by canoncal transformatons, ths wll hold n any canoncal coordnates, ξ A, ξ B} χ ΩAB. We summarze the results of ths subsecton wth a theorem: Let the coordnates ξ A be canoncal. Then a coordnate transformaton χ A ξ s canoncal f and only f t satsfes the fundamental bracket relaton χ A, χ B} ξ ΩAB For proof, note that the bracket on the left s defned by χ A, χ B} ξ χa χ B ΩCD ξ C ξ D so n order for χ A to satsfy the canoncal bracket relaton we must have CD χa χ B Ω ξ C ξ D ΩAB 3 whch s just the condton shown above for the coordnate transformaton χ A ξ to be canoncal. Conversely, suppose the transformaton χ A ξ s canoncal, so that eq.3 holds. Then, computng the Posson bracket χ A, χ B} ξ ΩCD χa ξ C χ B ξ D ΩAB so χ A satsfes the fundamental bracked relaton. In summary, each of the followng statements s equvalent: 1. χ A ξ s a canoncal transformaton. 2. χ A ξ s a coordnate transformaton of phase space that preserves Hamlton s equatons. 3. χ A ξ preserves the symplectc form, accordng to AB ξc ξ D Ω χ A χ B ΩCD 4. χ A ξ satsfes the fundamental bracket relatons χ A, χ B} ξ ΩAB These bracket relatons represent a set of ntegrablty condtons that must be satsfed by any new set of canoncal coordnates. When we formulate the problem of canoncal transformatons n these terms, t s not obvous what functons q x j, p j and π x j, p j wll be allowed. Fortunately there s a smple procedure for generatng canoncal transformatons, whch we develop n the next secton. We end ths secton wth three examples of canoncal transformatons. 4
2.1 Example 1: Coordnate transformatons Let x, p j be one set of canoncal varables. Suppose we defne new confguraton space varables, q, be an arbtrary nvertble functon of the spatal coordnates: q q x j We seek a set of momentum varables π j such that q, π j are canoncal. For ths they must satsfy the fundamental Posson bracket relatons: q, q j} x,p 0 q, π j } x,p δ j π, π j } x,p 0 Check each: q, q j} x,p 0 m1 q q j x m q q j snce qj p m 0. For the second bracket, δj q, π j }x,p q π j x m q π j m1 m1 x m π j p m Snce q s ndependent of p m, we can satsfy ths only f Integratng gves π j xm p m q j π j xn q j p n + c j x wth the c j an arbtrary functons of x. Choosng c j 0, we compute the fnal bracket: π, π j } x,p π π j x m π π j p m p m x m x n x s x m p n p m q j p s x n x s p m p n x m q j p s xm x n q j x m p n xm x n x m q j p n 2 x n q j p n 2 x n q j p n 0 Exercse: Show that the fnal bracket, π, π j } x,p stll vanshes provded c f for some functon f q. 5
Therefore, the transformatons q j q j x π j xn q j p n + f q j s a canoncal transformaton for any functons q x. Ths means that the symmetry group of Hamlton s equatons s at least as bg as the symmetry group of the Euler-Lagrange equatons. 2.2 Example 2: Interchange of x and p. The transformaton q p π x s canoncal. We easly check the fundamental brackets: q, q j} x,p p, p j } x,p 0 q, π j } x,p p, x j} x,p x j, p }x,p δ j π, π j } x,p x, x j} x,p 0 Interchange of x and p j, wth a sgn, s therefore canoncal. The use of generalzed coordnates n Lagrangan mechancs does not nclude such a possblty, so Hamltonan dynamcs has a larger symmetry group than Lagrangan dynamcs. For our next example, we frst show that the composton of two canoncal transformatons s also canoncal. Let ψ χ and χ ξ both be canoncal. Defnng the composton transformaton, ψ ξ ψ χ ξ, we compute CD ψa ψ B ψ A Ω ξ C ξ D χ E ψ B χ F ΩCD χ E ξ C χ F ξ D χ E χ F ψ A ψ B ξ C ξ D ΩCD χ E χ F so that ψ ξ s canoncal. Ω EF ψ A χ E Ω AB 2.3 Example 3: Momentum transformatons ψ B By the prevous results, the composton of an arbtratry coordnate change wth x, p nterchanges s canoncal. Consder the effect of composng a an nterchange, b a coordnate transformaton, and c an nterchange. For a, let q p π x χ F 6
Then for b we choose an arbtrary functon of q : Fnally, for c, another nterchange: Combnng all three, we have Q F q j P qn Q π n q P π Q q P qn Q π n pn π x n π Q F q j F p j so that π s replaced by an arbtrary functon of the orgnal momenta. Ths establshes that replacng the momenta by any ndependent functons of the momenta, preserves Hamlton s equatons as long as we choose the proper coordnates q. 3 Generatng functons There s a systematc approach to canoncal transformatons usng generatng functons. We wll gve a smple example of the technque. Gven a system descrbed by a Hamltonan Hx, p j, we seek another Hamltonan H q, π j such that the equatons of moton have the same form, namely n the orgnal system and dx dp dq dπ H p H x H π H n the transformed varables. The prncple of least acton must hold for each par: S S ˆ p dx H ˆ π dq H where S and S dffer by at most a constant. Correspondngly, the ntegrands may dffer by the addton of a total dfferental, df df, snce ths wll ntegrate to a surface term and therefore wll not contrbute to the varaton. In general we may therefore wrte p dx H π dq H + df 7
and solve for the dfferental df df p dx π dq + H H For the dfferental of f to take ths form, t must be a functon of x, q and t, f fx, q, t. Therefore, the dfferental of f s df f x dx + f dq + f Equatng the expressons for df we match up terms to requre The frst equaton p f x 4 π f 5 H H + f p fxj, q j, t x 7 gves q mplctly n terms of the orgnal varables, whle the second determnes π. Ths choce fxes the form of π by eq.5, whle eq.6 gves the new Hamltonan n terms of the old one. The functon f s the generatng functon of the transformaton. There are other types of generatng functons. By makng a Legendre transformaton, we can change the ndependent varables. For example, settng we have f p x + f 2 p, q, t p dx H π dq H + df π dq H + dp x + p dx + df 2 p, q, t H π dq H + dp x + df 2 p, q, t so that the ndependent varables are now p, q, satsfyng 6 We may also defne x f p π f H H + f f π q + f 3 x, π j, t f p x π q + f 4 p, π j, t so that the ndependent varables may be taken as ether of the new coordnates wth ether of the old coordnates. 8
3.1 Example 1 Let f 2 be a general quadratc, f 2 p, q j, t 1 aj t q q j + b j t p q j + c j t p p j 2 Then x 1 aj q q j + 2b p 2 jp q j + c j p p j b jq j + c j p j π 1 aj q q j + 2b 2 jp q j + c j p p j a j q j + b jp H H + 1 ȧ j t q q j + 2 ḃ j t p q j + ċ j t p p j 3.2 Example 2 Let f 2 p, q j, t g p, t + g p q + 1 2 f j p q q j + 1 3! f jk p q q j q k Then x g p, t g p q 12 p f j p q q j 13! f jk p q q j q k π g p f j p q j 1 2 f jk p q j q k H H + g p, t 9