The generating function of a canonical transformation

Transcription

1 ENSEÑANZA Revsta Mexcana de Físca E DICIEMBRE 2011 The generatng functon of a canoncal transformaton G.F. Torres del Castllo Departamento de Físca Matemátca Insttuto de Cencas Unversdad Autónoma de Puebla Puebla Pue. Méxco. Recbdo el 27 de juno de 2011; aceptado el 3 de octubre de 2011 An elementary proof of the exstence of the generatng functon of a canoncal transformaton s gven. A shorter proof mang use of the formalsm of dfferental forms s also gven. Keywords: Canoncal transformatons; generatng functon. Se da una prueba elemental de la exstenca de una funcón generatrz de una transformacón canónca. Se da tambén una prueba más corta usando el formalsmo de formas dferencales. Descrptores: Transformacones canóncas; funcón generatrz. PACS: Jj 1. Introducton One of the man reasons why the Hamltonan formalsm s more useful than the Lagrangan formalsm s that the set of coordnate transformatons that leave nvarant the form of the Hamlton equatons s much wder than the set of coordnate transformatons that leave nvarant the form of the Lagrange equatons. Furthermore each of the so-called canoncal transformatons leaves nvarant the form of the Hamlton equatons and can be obtaned from a sngle real-valued functon of 2n + 1 varables where n s the number of degrees of freedom of the system whch s therefore called the generatng functon of the transformaton. The proof of the exstence of a generatng functon for an arbtrary canoncal transformaton gven n most standard textboos s usually based on the calculus of varatons see e.g. Refs. 1 to 6 whch allows one to obtan the basc relatons qucly. The am of ths paper s to gve a straghtforward elementary dervaton of the exstence of the generatng functon of a canoncal transformaton not based on the calculus of varatons. One of the advantages of the proof gven here s that t allows one to see clearly the assumptons nvolved by contrast wth the more dffuse proof usually gven n the textboos and to realze that the canoncal transformatons are not the most general transformatons that leave nvarant the form of the Hamlton equatons. In Sec. 2 the defnton of a canoncal transformaton s brefly revewed n order to derve the basc equatons that lead to the exstence of the generatng functon of the transformaton. In Sec. 3 we pont out some of the frequent errors contaned n the proofs gven n some of the standard textboos. For those readers acquanted wth the formalsm of exteror dfferental forms a consderably shorter proof s gven n the appendx. The smplcty of ths latter proof may serve as an nvtaton to learn the language of dfferental forms for those not already famlar wth t. 2. Canoncal transformatons In order to present the deas n a smple way t s convenent to consder frstly the case where there s only one degree of freedom whch greatly smplfes the dervatons Systems wth one degree of freedom We shall consder a system wth one degree of freedom descrbed by a Hamltonan functon Hq p t. Ths means that the tme evoluton of the phase space coordnates q and p s determned by the Hamlton equatons dq dt = H p dp dt = H. 1 We want to fnd the coordnate transformatons Q=Qq p t P =P q p t that mantan the form of the Hamlton equatons 1. That s we want that Eqs. 1 be equvalent to dq dt = K dp dt = K 2 where K may be the orgnal Hamltonan H expressed n terms of the new coordnates or another functon. The last possblty s relevant snce t turns out that the new Hamltonan can be made equal to zero by means of a sutable transformaton. Assumng that the transformaton Q = Qq p t P = P q p t s dfferentable and can be nverted that s t s possble to fnd q and p n terms of Q P and t and therefore H can be vewed also as a functon of Q P and t mang use repeatedly of the chan rule and of Eqs. 1 and 2 we fnd that K = dq dt = H p H p + 3 = H p + H p 4

2 THE GENERATING FUNCTION OF A CANONICAL TRANSFORMATION 159 H p = H + H p p = H {Q P } + 6 where we have made use of the defnton of the Posson bracets {f g} f g p f g p. 7 In a smlar way we obtan K = H {Q P }. 8 Now we have two possbltes: ether the Hamltonan K s essentally the orgnal Hamltonan H expressed n terms of the new varables [that s K Qq p t P q p t t =Hq p t] or K dffers from H. In the frst case Eqs. 6 and 8 wll hold ndependent of the Hamltonan H f and only f {Q P } = 1 9 and the coordnate transformaton does not nvolve the tme Q = Qq p P = P q p. Then Eq. 9 s the necessary and suffcent condton for the local exstence of a functon F such that P dq pdq = df. 10 That s the functon F may not be defned n all the phase space we can only ensure ts exstence n some neghborhood of each pont of the phase space. In fact wrtng the left-hand sde of Eq. 10 n the equvalent form P p one fnds that the condton P = p p dq + P p dp P p s equvalent to Eq. 9 [1]. Even though more general transformatons are also possble see below attenton s restrcted to the transformatons satsfyng Eq. 9 also when the coordnate transformaton nvolves the tme explctly. The coordnate transformatons satsfyng Eq. 9 are called canoncal transformatons. One good reason to consder only canoncal transformatons s that the Posson bracets 7 are nvarant under these transformatons n the sense that f g p f g p = f g f g for any par of functons f g f and only f Eq. 9 holds. In fact mang use of the chan rule one can readly show that f g p f g f p = {Q P } g f g. 11 Thus restrctng ourselves to coordnate transformatons satsfyng Eq. 9 but allowng them to nvolve the tme explctly Eqs. 6 and 8 yeld K = H H K =. 12 Now t turns out that Eqs. 9 and 12 are necessary and suffcent condtons for the local exstence of a functon F such that P dq Kdt pdq + Hdt = df 13 as can be seen wrtng the left-hand sde of the last equaton as P p dq + P p dp + P + H K dt and applyng the standard crteron for a lnear or Pfaffan dfferental form to be exact. For nstance by consderng the coeffcents of dq and dt recallng that q p and t are treated as three ndependent varables we have P + H K = H K + = H K P p H K + H K =0. If q and Q are functonally ndependent then the functon F appearng n Eq. 13 can be expressed n terms of q Q and t n a unque way and from Eq. 13 t follows that P = F p = F H K = F 14 and necessarly 2 F/ 0 otherwse q and p would not be ndependent. Conversely gven a functon F q Q t such that 2 F/ 0 Eqs. 14 can be locally nverted to fnd Q and P n terms of q p and t. In ths way F s a generatng functon of a canoncal transformaton. If q and Q are functonally dependent that s Q can be expressed as a functon of q and t only or q can be expressed as a functon of Q and t only the functon F appearng n Eq. 13 can be wrtten n nfntely many ways n terms of q Q and t and the frst two equatons n 14 mae no sense snce e.g. eepng q and t constant n the partal dfferentaton wth respect to Q would mae Q also constant. In such a case the varables p and Q as well as P and q are necessarly functonally ndependent otherwse q and p would be dependent. Then wrtng Eq. 14 n the equvalent form P dq Kdt + qdp + Hdt = df 15 Rev. Mex. Fs. E

3 160 G.F. TORRES DEL CASTILLO where F F + pq t follows that the generatng functon F can be expressed n a unque way as a functon of Q p and t and the canoncal transformaton s determned by P = F q = F p H K = F 16 and necessarly 2 F /p 0. Conversely a gven functon F p Q t such that 2 F /p 0 defnes a canoncal transformaton by means of the frst two equatons n 16. In a smlar way one can consder generatng functons dependng on q P t or p P t see e.g. Refs. 1 to 6. It should be clear from the dervaton above that the coordnate transformatons satsfyng condton 9 are not the most general coordnate transformatons that leave nvarant the form of the Hamlton equatons and by contrast to what s clamed n some textboos e.g. Refs. 3 and 4 the Posson bracet {Q P } needs not be a trval constant. By a trval constant we mean a functon whose value s the same at all ponts of ts doman or equvalently a functon whose partal dervatves are all dentcally equal to zero. A smple example s gven by the transformaton Q = arctan q p P = p 2 + q 2. One readly fnds that the Posson bracet {Q P } s equal to p 2 + q 2 1/2 whch s not a trval constant but s a constant of the moton f the Hamltonan s for nstance H = 1/2p 2 + q 2 correspondng to a harmonc oscllator. Then the Hamlton equatons 1 yeld dq/dt = p and dp/dt = q; therefore we have dq/dt = 1 and dp/dt = 0 whch can be expressed as the Hamlton equatons 2 f the transformed Hamltonan s chosen as K = P. {Q P } = In place of an equaton of the form 13 n ths case one fnds the relaton P dq Kdt = 2p 2 +q 2 1/2[ pdq Hdt dpq/2 ]. 17 A second example related to the prevous one s gven by the coordnate transformaton Q = p + p p p = H p + H K {Q P } + H K {Q P } p p = {Q P } H p + H {Q P } H p + H p H + {Q P } p H p H p + H p + {Q P } t arctan q 2 P = 1 p 2 p2 + q 2. Now {Q P } = 2t arctan q/p whch s also a constant of moton f H = 1/2p 2 + q 2 as above. Furthermore dq/dt = 0 dp/dt = 0 whch can be wrtten n the form 2 wth a new Hamltonan K = 0. Ths s not strange snce n the Hamlton Jacob method one fnds a transformaton leadng to a new Hamltonan equal to zero but ths s usually done wth the ad of canoncal transformatons the soluton of the Hamlton Jacob equaton s the generatng functon of a canoncal transformaton to a new set of varables correspondng to a Hamltonan equal to zero. For ths transformaton we obtan the relaton P dq = 2 t arctan q [pdq ] Hdt dpq/2 p [cf. Eqs. 13 and 17]. The most general coordnate transformaton that preserves the form of the Hamlton equatons 1 corresponds to {Q P } beng a constant of the moton. Indeed mang use of the defnton of the Posson bracet 7 Eqs. 6 8 the chan rule and Eqs. 1 K {Q P } p K p + K p + K p K p = {Q P } dp p dt {Q P } { Q H } { + P H } { Q K } { P K }. Now accordng to Eq. 11 we have for nstance { Q H } H = {Q P } H = {Q P } H and { P H } = {Q P } H H = {Q P } H ; H K {Q P } + dq dt Rev. Mex. Fs. E

4 THE GENERATING FUNCTION OF A CANONICAL TRANSFORMATION 161 therefore { Q H } { + P H } = 0 and smlarly { Q K } { + P K } = 0 thus showng that {Q P } s a constant of moton cf. Ref. 1. A shorter proof s gven n the appendx Systems wth an arbtrary number of degrees of freedom When the number of degrees of freedom s greater than 1 the exstence of a generatng functon of any canoncal transformaton can be demonstrated followng essentally the same steps as n the precedng subsecton. We start assumng that the set of Hamlton equatons dq dt = H p = n s equvalent to the set dq dt = K dp dt = H 18 dp dt = K 19 where the new coordnates Q and P are functons of q p and possbly also of the tme. Then by vrtue of the chan rule and Eqs. 18 and 19 we obtan here and n what follows there s summaton over repeated ndces K = dq H dt j H p j p j j + = H j + H p j p j H p j j + H j + = H j p j p j j + H j p j p j j + = H {Q Q } + H {Q P } + 20 wth the Posson bracets beng now defned by and smlarly {f g} f g p f p g 21 K = H {P Q } + H {P P } + [cf. Eqs. 6 and 8]. 22 By analogy wth the case where the number of degrees of freedom s 1 the canoncal transformatons are defned by the condtons {Q Q } = 0 {P P } = 0 {Q P } = δ. 23 Then Eqs. 20 and 22 yeld H K = = H K 24 [cf. Eqs. 12]. As s well nown Eqs. 23 mply that m j m j = 0 m p j m = δ p m j 25 j = 0 p m p j p m p j as a matter of fact Eqs. 25 are equvalent to Eqs. 23 [1 4]. Indeed assumng that Eqs. 23 hold we have m = m {Q P } m {Q Q } = m j p j p j j m = j p j and n a smlar manner = p m j p m p j p j p m j m = p j j = p m p j j j p j p j j m p j m j m j m j m p j p m j m p j p m p j p m j m j m p j p m j p m p j p m p j and ths set of relatons mples Eqs. 25. Equatons 24 and 25 are necessary and suffcent condtons for the local exstence of a functon F such that P dq Kdt p dq + Hdt = df 26 Rev. Mex. Fs. E

5 162 G.F. TORRES DEL CASTILLO as can be readly verfed wrtng the left-hand sde of the last equaton n terms of the orgnal varables P j j p dq j +P j dp + P +H K dt p and applyng agan the standard crteron for the local exactness of a lnear dfferental form. If the 2n varables q Q are functonally ndependent whch s not necessarly the case Eq. 26 mples that F can be expressed as a functon of q Q and t n a unque way and P = F p = F H K = F. 27 The ndependence of the 2n varables q p requres that det 2 F/ j 0. Conversely gven a functon F q Q t satsfyng ths condton Eqs. 27 defne a local canoncal transformaton. For the canoncal transformatons such that the set q Q s functonally dependent one can employ generatng functons that depend on other combnatons of old and new varables; some or all of the q can be replaced by ther conjugates p and smlarly some or all of the Q can be replaced by ther conjugates P gvng a total of 2 2n possbltes not only the four cases consdered e.g. n Ref Comparson wth other treatments The presence of the combnatons p dq Hdt and P dq Kdt n Eq. 26 s not accdental. It s related to the fact that one obtans the Hamlton equatons loong for the path n phase space q = q t p = p t along whch the ntegral t 2 t 1 p dq Hdt 28 has a statonary value usually a mnmum when compared wth neghborng paths wth the same end ponts n phase space for t = t 1 and t = t 2. Snce the addton of the dfferental of any dfferentable functon F q p t to the ntegrand n 28 changes the value of the ntegral by a term that s the same for all the paths wth the same end ponts n phase space for t = t 1 and t = t 2 t s rght to say that f P dq Kdt = p dq Hdt + df [whch s Eq. 26] then the Hamlton equatons 18 wll be equvalent to Eqs. 19. What s wrong to say s that the converse s also true see e.g. Refs. 2 5 and 6 or that P dq Kdt and p dq Hdt can only dffer by a trval constant factor and the dfferental of a functon see e.g. Refs. 3 and 4 f Eqs. 18 are equvalent to 19. Even though Eq. 26 mples that there exsts a functonal relaton among F Q q and t another frequent error s to conclude that ths mples that the 2n varables q Q are functonally ndependent see e.g. Refs. 4 to 6. Snce Eq. 26 does not necessarly hold [see e.g. Eq. 17] n the case of a non-canoncal transformaton that preserves the form of the Hamlton equatons the ntegrals and t 2 t 1 t 2 t 1 p dq Hdt P dq Kdt do not concde nor are smply related. However the actual path followed by the system n phase space corresponds to statonary values of both functonals ths s analogous for nstance to the fact that the pont x = 0 s a local mnmum for the functons fx = x 4 and gx = 1 cos x despte the fact that these functons are not the same. Acnowledgements The author would le to than the referees for helpful comments. Appendx A. Dervaton usng exteror forms Mang use of the propertes of the contracton or nteror product of a vector feld wth a dfferental form see e.g. Refs. 7 to 11 one fnds that there s only one vector feld of the form X = + A + B A.1 p whose contracton wth the 2-form ω dp dq dh dt A.2 s equal to zero that s X ω = 0. In fact mang use of the expressons A.1 and A.2 one fnds that X ω = 0 s equvalent to A = H p B = H. Hence the ntegral curves of X correspond to the solutons of the Hamlton equatons 18. Equatons 19 are then equvalent to the condton X Ω = 0 where Ω dp dq dk dt. A.3 If we restrct ourselves to canoncal transformatons then Ω = ω or equvalently dp dq Kdt p dq +Hdt = 0 whch mples the local exstence of a functon F such that Eq. 26 holds. However there s an nfnte number of 2-forms Ω of the form A.3 that do not dffer by a trval multplcatve constant from ω such that smultaneously X ω = 0 and X Ω = 0 [11]. Rev. Mex. Fs. E

6 THE GENERATING FUNCTION OF A CANONICAL TRANSFORMATION 163 Only n the case of systems wth one degree of freedom any two such 2-forms must be related by Ω = fω where f s some nowhere vanshng real-valued functon [11]. Among other thngs from Ω = fω t follows that {Q P } = f [wth the Posson bracets defned by Eq. 7]. Snce ω and Ω are both closed that s ther exteror dervatves are equal to zero equaton Ω = fω mples that f must obey the condton df ω = 0 A.4 that s 0 = f f f dq + dp + p dt dp dq H H f H dq dt dp dt = p p f H p + f dp dq dt. By vrtue of the Hamlton equatons 1 ths equaton holds f and only f f s a constant of the moton that s Xf = 0 see the examples at the end of Sec M.G. Caln Lagrangan and Hamltonan Mechancs World Scentfc Sngapore Chap. VII. 2. H.C. Corben and P. Stehle Classcal Mechancs 2nd ed. Wley New Yor Sec H. Goldsten C. Poole and J. Safo Classcal Mechancs 3rd ed. Addson-Wesley San Francsco Chap D.T. Greenwood Classcal Dynamcs Prentce-Hall Englewood Clffs NJ Chap C. Lanczos The Varatonal Prncples of Mechancs 4th ed. Unversty of Toronto Press Toronto Chap. VII. 6. D. ter Haar Elements of Hamltonan Mechancs 2nd ed. Pergamon Oxford Chap V.I. Arnold Mathematcal Methods of Classcal Mechancs 2nd ed. Sprnger New Yor M. Crampn and F.A.E. Pran Applcable Dfferental Geometry Cambrdge Unversty Press Cambrdge L.H. Looms and S. Sternberg Advanced Calculus Addson- Wesley Readng MA S. Sternberg Lectures on Dfferental Geometry Chelsea New Yor G.F. Torres del Castllo Dfferentable Manfolds: A Theoretcal Physcs Approach Brhäuser Scence New Yor Rev. Mex. Fs. E