PHYS 705: Classcal Mechancs Canoncal Transformaon
Canoncal Varables and Hamlonan Formalsm As we have seen, n he Hamlonan Formulaon of Mechancs,, are ndeenden varables n hase sace on eual foong The Hamlon s Euaon for are symmerc (symlecc, laer) H and, H Ths elegan formal srucure of mechancs affords us he freedom n selecng oher arorae canoncal varables as our hase sace coordnaes and momena - As long as he new varables formally sasfy hs absrac srucure (he form of he Hamlon s Euaons.
3 Canoncal Transformaon Recall (from hw) ha he Euler-Lagrange Euaon s nvaran for a on ransformaon: (, ) L d L d.e., f we have, 0, L d L d hen, 0, Now, he dea s o fnd a generalzed (canoncal) ransformaon n hase sace (no confg. sace) such ha he Hamlon s Euaons are nvaran! P (,, ) P (,, ) (In general, we look for ransformaons whch are nverble.)
Invarance of EL euaon for Pon Transformaon Gven: From he nverse on ransformaon euaon, we have, (, ) and a on ransformaon: 0, d L L d Formally, calculae: L L L L L L (chan rule) (, ) 0 and k k k Frs look a he suaon n confg. sace frs: Need o show: 0 d L L d 4
5 Invarance of EL euaon for Pon Transformaon d L L LHS d d L L L d Formng he LHS of EL euaon wh : LHS L d d L L L d d d L L L d L d d
6 Invarance of EL euaon for Pon Transformaon (exchange order of dff) LHS d L L L d d d 0 (Snce ha s wha gven!) d L L LHS 0 0 d d d 0
7 Canoncal Transformaon Now, back o hase sace, we need o fnd he arorae (canoncal) ransformaon (,, ) and P P (,, ) such ha here exs a ransformed Hamlonan KP (,, ) wh whch he Hamlon s Euaons are sasfed: K and P P K (The form of he EOM mus be nvaran n he new coordnaes.) ** I s moran o furher saed ha he ransformaon consdered ( P, ) mus also be roblem-ndeenden meanng ha mus be canoncal coordnaes for all sysem wh he same number of dofs.
8 Canoncal Transformaon To see wha hs condon mgh say abou our canoncal ransformaon, we need o go back o he Hamlon s Prncle: Hamlon s Prncle: The moon of he sysem n confguraon sace s such ha he acon I has a saonary value for he acual ah,..e., I Ld 0 Now, we need o exend hs o he n-dmensonal hase sace. The negran n he acon negral mus now be a funcon of he ndeenden conugae varable, and her dervaves,. We wll consder varaons n all n hase sace coordnaes
9 Hamlon s Prncle n Phase Sace. To rewre he negran n erms of,,,, we wll ulze he defnon for he Hamlonan (or he nverse Legendre Transform): H L L H(,, ) (Ensen s sum rule) Subsung hs no our varaon euaon, we have I Ld H (,, ) d 0. The varaons are now for n ' s and n ' s : (all s and s are ndeenden) agan, we wll reured he varaons for he o be zero a ends (,, ) H (,, ) The rewren negran s formally a funcon of,,, bu n fac does no deend on,.e. 0 Ths fac wll roved o be useful laer on.
0 Hamlon s Prncle n Phase Sace Affecng he varaons on all n varables,, we have, ' s I d d d d d d 0 ' s As n revous dscusson, he second erm n he sum for ' s can be rewren usng negraon by ars: d d d d 0 0
Hamlon s Prncle n Phase Sace Prevously, we have reured he varaons for he o be zero a end s so ha,, 0 So, he frs sum wh ' s can be wren as: d d d d d d d where d d d d d d
Hamlon s Prncle n Phase Sace Now, erform he same negraon by ars o he corresondng erm for we have, d d d d Noe, snce, whou enforcng he 0 varaons for o be zero a end ons. 0 ' s Ths gves he resul for he nd sum n he varaon euaon for ' s : d d where d d
3 Hamlon s Prncle n Phase Sace Pung boh erms back ogeher, we have: I d d d d d d 0 Snce boh varaons are ndeenden, and mus vansh ndeendenly! d d 0 H 0 (,, ) H (,, ) H and H (one of he Hamlon s euaons)
4 Hamlon s Prncle n Phase Sace I d d d d d d 0 d d 0 H 0 0 (,, ) H (,, ) H 0 and H ( nd Hamlon s euaons)
5 Hamlon s Prncle n Phase Sace So, we have us shown ha alyng he Hamlon s Prncle n Phase Sace, he resulng dynamcal euaon s he Hamlon s Euaons. H H
6 Hamlon s Prncle n Phase Sace Noce ha a full me dervave of an arbrary funcon F of can be u no he negrand of he acon negral whou affecng he varaons: df H(,, ) d d df H(,, ) d d d (,, ) df F cons Thus, when varaon s aken, hs consan erm wll no conrbue!
7 Canoncal Transformaon Now, we come back o he ueson: When s a ransformaon o canoncal? We need Hamlon s Euaons o hold n boh sysems H(,, ) d0 P, Ths means ha we need o have he followng varaonal condons: AND P K(, P, ) d 0 For hs o be rue smulaneously, he negrands mus eual And, from our revous slde, hs s also rue f hey are dffered by a full me dervave of a funcon of any of he hase sace varables nvolved + me: df H(,,) P K(, P,),,, P, d
8 Canoncal Transformaon df H(,,) P KP (,,), P,,, ()( G9.) d F s called he Generang Funcon for he canoncal ransformaon: (, ), P : P (,, ) F s useful n secfyng he exac form of he ransformaon f conans half of he old varables and half of he new varables. I, hen, acs as a brdge beween he wo ses of canoncal varables. Deendng on he form of he generang funcons (whch ar of canoncal varables beng consdered as he ndeenden varables), we can classfy canoncal ransformaons no four basc yes. P (,, )
9 Canoncal Transformaon: 4 Tyes df H(,,) P K(, P,) old, new, d Tye : F F(,, ) Tye : F F (,, ) P P F,, P,, F P,,, P, F F K H F F K H P Tye 3: F F (,, ) 3 Tye 4: F F (,, ) 4 P P F 3 3,, P,, F 4 4, P,, P, F F3 K H F F4 K H P
0 Canoncal Transformaon: Tye Tye : F F(,, ) (old) (new) F s a funcon of and + me Wrng ou he full me dervave for F, E () becomes: H P K F F F (agan E s sum rule) Or, we can wre he euaon n dfferenal form:, and (wre ou and mully he euaon by ) d d d d F d F P d F K H d 0 d
Canoncal Transformaon: Tye, Snce all he are ndeenden, her coeffcens mus vansh ndeendenly. Ths gves he followng se of euaons: F,, ( ) C F P,, ( ) F C K H ( C3) F F For a gven secfc exresson for, e.g.,,,, E.( C),, are n relaons defnng n erms of and hey can be nvered o ge he s se of he canoncal ransformaon: In he secfc examle, we have: F,,
Canoncal Transformaon: Tye E.( C) are n relaons defnng n erms of. Togeher wh our resuls for he, he nd se of he canoncal ransformaon can be obaned. F P,, P K Agan, n he secfc examle, we have: F H K H P,, E.( C3) gves he connecon beween K and H: KP (,, ) (noe: s a funcon of he new varables so ha he RHS needs, P o be re-exress n erms of usng he canoncal ransformaon.)
3 Canoncal Transformaon: Tye In summary, for he secfc examle of a Tye generang funcon: We have he followng: F,, P Canoncal Transformaon and K H Transformed Hamlonan Noe: hs examle resuls n bascally swang he generalzed coordnaes wh her conugae momena n her dynamcal role and hs exercse demonsraes ha hs swang bascally resul n he same suaon! Emhaszng he eual role for and n Hamlonan Formalsm!
4 Canoncal Transformaon: Tye Tye : F F (,, ) P P (old) (new), where F s a funcon of and P + me F F(,, ) (One can hnk of as he Legendre ransform of n exchangng he varables and P.) Subsung no our defnng euaon for canoncal ransformaon, E. (): df F F F H P K P P P d P Agan, wrng he euaon n dfferenal form: F F F d dp K H d 0 P
5 Canoncal Transformaon: Tye, P Snce all he are ndeenden, her coeffcens mus vansh ndeendenly. Ths gves he followng se of euaons: F P,, F F, P, K H P F P For a gven secfc exresson for, e.g.,, F, P, P F P,, P P F, P, P K H Thus, he deny ransformaon s also a canoncal ransformaon!
6 Canoncal Transformaon: Tye Le consder a slghly more general examle for ye : wh F, P, f,,, P g,,, n n where f and g are funcon of s only + me Gong hrough he same rocedure, we wll ge: F F, P, P F f P g F P f,, n f K H P g Noce ha he euaon s he general on ransformaon n he confguraon sace. In order for hs o be canoncal, he P and H ransformaons mus be handled carefully (no necessary smle funcons).
7 Canoncal Transformaon: Summary The remanng wo basc yes are Legendre ransformaon of he remanng wo varables: F F (,, ) 3 F F (,, ) & 4 P P P (Resuls are summarzed n Table 9. on. 373 n Goldsen.) Canoncal Transformaons form a grou wh he followng roeres:. The deny ransformaon s canoncal (ye examle). If a ransformaon s canoncal, so s s nverse 3. Two successve canoncal ransformaons ( roduc ) s canoncal 4. The roduc oeraon s assocave
8 Canoncal Transformaon: 4 Tyes Tye : F F(,, ) Tye : F F (,, ) P P df H(,,) P K(, P,) old, new, d F,, P,, F P,, P,, de var nd var F F K H P (,, ) P (,, ) F F K H P Tye 3: F F (,, ) 3 Tye 4: F F (,, ) 4 P P F 3 3,, P,, F 4 4, P,, P, F F3 K H F F4 K H P
9 Canoncal Transformaon (more) If we are gven a canoncal ransformaon P (,, ) P (,, ) (*) How can we fnd he arorae generang funcon F? - Le say, we wsh o fnd a generang funcon of he s ye,.e., F F(,, ) (Noe: generang funcon of he oher yes can be oban hrough an arorae Legendre ransformaon.) - Snce our chosen generang funcon ( s ye) deends on,, and exlcly, we wll rewre our and P n erms of and usng E. (*): (,, ) P P (,, )
30 Canoncal Transformaon (more) Now, from he ar of euaons for he Generang Funcon Dervaves (Table 9.), we form he followng dff es, F(,, ) (,, ) P can hen be obaned by drecly negrang he above euaons and combnng he resulng exressons. F,, F,, P(,, ) Noe: Snce df s an exac dfferenal wr and, hen P FP,, F FF,, (We wll gve he full ls of relaons laer.)
3 Canoncal Transformaon (more) Examle (G8.): We are gven he followng canoncal ransformaon for a sysem wh dof: (, ) cos sn P P(, ) sn cos (HW: showng hs rans. s canoncal) ( and P s roaed n hase sace from and by an angle ) We seek a generang funcon of he s knd: F (, ) Rewre he ransformaon n erms of and (nde. vars of F ): F (, ) co sn F P P (, ) sn sn co sn cos cos sn
3 Canoncal Transformaon (more) Frs, noce ha he cross second dervaves for F are eual as reured for a canoncal ransformaon: F co sn sn F co sn sn Now, negrang he wo aral dfferenal euaons: F co h sn Comarng hese wo exresson, one ossble soluon for s, F, co sn F co sn F g
33 Canoncal Transformaon (more) Now, le say we wsh o seek a generang funcon of he nd ye: F ( P, ) As we have dscuss revously, hs reures us o use he followng Legendre ransform of F n E. (): F F (,, ) P P Subsung hs erm n E. (), resuls n relacng he erm by n our condon for a canoncal ransformaon (G9.): HP K df d P P Conseuenally, hs gves us he wo aral dervaves relaons for F : F P,, F P,, P
34 Canoncal Transformaon (more) To solve for F ( P,, ) n our examle, agan, we rewre our gven canoncal ransformaon n and P exlcly. (, ) cos sn P P(, ) sn cos F P (, ) P an cos Inegrang and combnng gve, F (, P) P P cos cos P an cos sn sn cos P F, P P an cos
35 Canoncal Transformaon (more) 0 Noce ha when, sn 0 (, ) cos sn P P(, ) sn cos so ha our coordnae ransformaon s us he deny ransformaon: and P, P CANNOT be wren exlcly n erms of and! so ha our assumon for usng he ye generang funcon (wh and as nd var) canno be fulflled. Conseuenly, blow u and canno be used o derve he canoncal ransformaon: F, F, co as 0 sn Bu, usng a Tye generang funcon wll work.
36 Canoncal Transformaon (more) Smlarly, we can see ha when, cos 0 our coordnae ransformaon s a coordnae swch, P, CANNOT be wren exlcly n erms of and P! so ha he assumon for usng he ye generang funcon (wh and P as nd var) canno be fulflled. F Conseuenly, blow u and canno be used o derve he canoncal ransformaon:, P (, ) cos sn P P(, ) sn cos P F, P P an as 0 cos Bu, usng a Tye generang funcon wll work n hs case.
37 Canoncal Transformaon (more) - A suable generang funcon doesn have o conform o only one of he four yes for all he degrees of freedom n a gven roblem! - There can also be more han one soluon for a gven CT - Frs, we need o choose a suable se of ndeenden varables for he generang funcon. For a generang funcon o be useful, should deends on half of he old and half of he new varables As we have done n he revous examle, he rocedure n solvng for F nvolves negrang he aral dervave relaons resuled from conssence consderaons usng he man condon for a canoncal ransformaon,.e., df H(,,) P KP (,,) ( G9.) d
38 Canoncal Transformaon (more) For hese aral dervave relaons o be solvable, one mus be able o feed-n n ndeenden coordnae relaons (from he gven CT) n erms of a chosen se of ½ new + ½ old varables. - In general, one can use ANY one of he four yes of generang funcons for he canoncal ransformaon as long as he RHS of he ransformaon can be wren n erms of he assocaed ars of hase sace coordnaes: (,, ), (, P, ), (,, ), or (, P, ). T T T3 T 4 - On he oher hand, f he ransformaon s such ha he RHS canno wren n erm of a arcular ar: (,, ), (, P, ), (,, ), or (, P, ), hen ha assocaed ye of generang funcons canno be used.
39 Canoncal Transformaon: an examle wh wo dofs - To see n racce how hs mgh work Le say, we have he followng ransformaon nvolvng dofs:,,,, P,, P ( a) P ( b) ( a) P ( b) - As we wll see, hs wll nvolve a mxure of wo dfferen basc yes.
40 Canoncal Transformaon: an examle wh wo dofs ( a) P ( b) - Frs, le see f we can use he smles ye (ye ) for boh dofs,.e., F wll deend only on he - s: F (,,,, ) ( a) P ( b), Noce ha E (a) s a relaon lnkng only, hey CANNOT boh be ndeenden varables Tye (only) WON T work!,,, - As an alernave, we can ry o use he se as our ndeenden varables. Ths wll gve an F whch s a mxure of Tye 3 and.,, P, (In Goldsen (. 377), anoher alernave was usng resuled n a dfferen generang funcon whch s a mxure of Tye and.)
4 Canoncal Transformaon: an examle wh wo dofs From our CT, we can wre down he followng relaons: Deenden varables,, P, P Indeenden varables,,, P P (*) Now, wh hs se of ½ new + ½ old ndeenden varable chosen, we need o derve he se of aral dervave condons by subsung no E. 9. (or look hem u from he Table). F(,,,, )
4 Canoncal Transformaon: an examle wh wo dofs The exlc ndeenden varables (hose aear n he dfferenals) n E. 9. are he - s. To do he converson:,,,,,, we wll use he followng Legendre ransformaon: (E. 9. s exlc nd vars) (our referred nd vars) F F'(,,,, ) Subsung hs no E. 9., we have: H P P K df d F' F' F' F' P P K F '
43 Canoncal Transformaon: an examle wh wo dofs F ' F ' F ' F ' F ' H P P K Comarng erms, we have he followng condons: F ' F ' P P F ' F ' K H F ' As adversed, hs s a mxure of Tye 3 and of he basc CT.
44 Canoncal Transformaon: an examle wh wo dofs Subsung our coordnaes ransformaon [E. (*)] no he aral dervave relaons, we have : F ' F ' F ' F ' P P F' f (,, ) F' g(,, ) F' h(,, ) F' k(,, ),, P, F' (Noe: Choosng nsead, Goldsen has. Boh of hese are vald generang funcons.) F'' P P P
45 Canoncal Transformaon: Revew df H(,,) P K(, P,) old, new, d P (,, ) P (,, ) Tye : F F(,, ) Tye : F F (,, ) P P F,, P,, F P,,, P, F F K H F F K H P Tye 3: F F (,, ) 3 Tye 4: F F (,, ) 4 P P F 3 3,, P,, F 4 4, P,, P, F F3 K H F F4 K H P
46 Canoncal Transformaon: Revew - Generang funcon s useful as a brdge o lnk half of he orgnal se of coordnaes (eher or ) o anoher half of he new se (eher or P). - In general, one can use ANY one of he four yes of generang funcons for he canoncal ransformaon as long as he ransformaon can be wren n erms of he assocaed ars of hase sace coordnaes: (,, ), (, P, ), (,, ), or (, P, ). - On he oher hand, f he ransformaon s such ha canno be wren n erm of a arcular ar: (,, ), (, P, ), (,, ), or (, P, ), hen ha assocaed ye of generang funcons canno be used. - he rocedure n solvng for F nvolves negrang he resulng aral dervave relaons from he CT condon