10. Canonical Transformations Michael Fowler

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1 10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst choce q= ( q, 1 q ). n Now transform to a new set, maybe even tme dependent, Q = Q( qt, ). The dervaton of Lagrange s equatons by mnmzng the acton stll works, so Hamlton s equatons must stll also be OK too. Ths s called a pont transformaton: we ve just moved to a dfferent coordnate system, we re relabelng the ponts n confguraton space (but possbly n a tme-dependent way). General and Canoncal Transformatons In the Hamltonan approach, we re n phase space wth a coordnate system havng postons and momenta on an equal footng. It s therefore possble to thnk of more general transformatons than the pont transformaton (whch was restrcted to the poston coordnates). We can have transformatons that mx up poston and momentum varables: where ( p, q ) Q = Q p, q, t, P = P p, q, t, means the whole set of the orgnal varables. In those orgnal varables, the equatons of moton had the nce canoncal Hamlton form, q = H / p = H / q. Thngs won t usually be that smple n the new varables, but t does turn out that many of the natural transformatons that arse n dynamcs, such as that correspondng to gong forward n tme, do preserve the form of Hamlton s canoncal equatons, that s to say Q = H / P, P = H / Q, for the new H P, Q. A transformaton that retans the canoncal form of Hamlton s equatons s sad to be canoncal. (Jargon note: these transformatons are occasonally referred to as contact transformatons.) Generatng Functons for Canoncal Transformatons In ths secton, we go back to consderng the full acton (not the abbrevated--fxed energy--acton used earler). Now, we've establshed that Hamlton s equatons n the orgnal parameterzaton follow from mnmzng the acton n the form

2 2 δ pdq Hdt = 0. For a canoncal transformaton, by defnton the new varables must also satsfy Hamlton's equatons, so, workng backwards, acton mnmzaton must be expressble n the new varables exactly as n the old ones: δ PdQ H dt = 0. Now, we ve prevously stated that two actons lead to the same equatons of moton f the ntegrands dffer by the total dfferental of some functon F of coordnates, momenta and tme. (That s because n addng such a functon to the ntegrand, the functon s contrbuton to the ntegral s just the dfference between ts values at the two (fxed) ends, so n varyng the path between the ends to mnmze the total ntegral and so generate the equatons of moton, ths exact dfferental df makes no contrbuton.) That s to say, the two acton ntegrals wll be mnmzed on the same path through phase space provded the ntegrands dffer by an exact dfferental: p dq Hdt = PdQ H dt + df. F s called the generatng functon of the transformaton. Rearrangng the equaton above, ( ). df = p dq PdQ + H H dt Notce that the dfferentals here are dq, dq, dt, so these are the natural varables for expressng the generatng functon. We wll therefore wrte t as F( qqt,, ), and from the expresson for df above, (,, ) (,, ) (,, ) F qqt F qqt F qqt p =, P =, H = H +. q Q t Let s reemphasze here that a canoncal transformaton wll n general mx up coordnates and momenta they are the same knd of varable, from ths Hamltonan perspectve. They can even be exchanged: for a system wth one degree of freedom, for example, the transformaton Q = p, P = q s a perfectly good canoncal transformaton (check out Hamlton s equatons n the new varables), even though t turns a poston nto a momentum and vce versa! If ths partcular transformaton s appled to a smple harmonc oscllator, the Hamltonan remans the same (we re takng H = 2 ( p + q ) ) so the dfferental df of the generatng functon (gven above) has no H H term, t s just

3 3 df q, Q = pdq PdQ. The generatng functon for ths transformaton s easly found to be from whch as requred. F q, Q = Qq, df = Qdq + qdq = pdq PdQ, Another canoncal transformaton for a smple harmonc oscllator s q= 2Psn Q = 2Pcos Q. You wll nvestgate ths n homework. Generatng Functons n Dfferent Varables Ths F( qqt,, ) s only one example of a generatng functon n dscussng Louvlle s theorem later, we ll fnd t convenent to have a generatng functon expressed n the q 's and P 's. We get that generatng functon, often labeled Φ ( qpt,, ), from (,, ) F qqt by a Legendre transformaton: ( ) ( ) dφ q, P, t = d F + PQ = p dq + Q dp + H H dt. Then, for ths new generatng functon ( qpt,, ) ( qpt,, ) ( qpt,, ) Φ Φ Φ p =, Q =, H = H +.. q P t Evdently, we can smlarly use the Legendre transform to fnd generatng functons dependng on the other possble mxes of old and new varables: pq,, and pp., What s the Pont of These Canoncal Transformatons? It wll become evdent wth a few examples: t s often possble to transform to a set of varables where the equatons of moton are a lot smpler, and, for some varables, trval. The canoncal approach also gves a neat proof of Louvlle s theorem, whch we ll look at shortly. Posson Brackets under Canoncal Transformatons Frst, note that f Hamlton s equatons have the standard canoncal form H H q = [ Hq, ] = = [ Hp, ] = q p wth respect to a par of varables pqthen, those varables are sad to be canoncally conjugate. The Posson bracket s nvarant under a canoncal transformaton, meanng [ f g] = [ f g],,. pq, PQ,

4 4 Let's begn by establshng that [ ] [ ] [ ] Q, Q = 0, PP, = 0, PQ, = δ. k pq, k pq, k pq, k We'll show the method by takng just one par of varables,, Then P Q P Q, =. pq p [ PQ], pq and a generatng functon F( qq ) q p p q Wth the generatng functon F( qq, ), we have p( qq, ) ( F/ q), P( qq, ) ( F/ Q) P P Q = Q p q q q Q,. = =, so q and Puttng these results nto the Posson bracket, [ QP] P P P Q = + Q q p Q q p 2 Q P Q F Q p, = = = = 1. p qq Q q Q q q q These basc results can then be used to prove the general Posson bracket s ndependent of the parametrzaton of phase space, detals n Goldsten and elsewhere. Landau, on the other hand, offers a one-lne proof of the nvarance of the Posson bracket of two f p, q, g p, q under a canoncal transformaton: magne a fcttous system dynamcal functons havng g as ts Hamltonan. Then [, ] pq, system used, so must equal [ f, g ] PQ,. f g s just df / dt, and cannot depend on the coordnate Tme Development s a Canoncal Transformaton Generated by the Acton ( 1) ( 1) The transformaton from the varables q at tme t 1 to q ( 2), ( 2) p at a later tme t 2 has to be canoncal, snce the system obeys Hamlton s (canoncal!) equatons at all tmes. 1 2 In fact, the varaton of the acton along the true path from q at tme t 1 to q at t 2 wth respect to fnal and ntal coordnates and tmes was found earler to be ( 1) ( 2) ( 2) ( 2) ( 1) ( 1) ( 2) ( 1) ( ) = + ds q, q, t, t p dq p dq H dt H dt,

5 5 and, comparng that expresson wth the dfferental form of a canoncal transformaton correspondng to F( q Q, p P) n the dscusson above, whch was ( 1) ( 2) df = p dq PdQ + H H dt, we see that the acton tself s the generatng functon for the canoncal transformaton from the varables q at tme t 1 to the set q at the later tme t 2, actually S generates the forward moton n tme, the equvalent varables n the two equatons above beng ( 1) ( 1) ( 2) ( 2) p p, dq dq P, dq dq.