From the Hamilton s Variational Principle to the Hamilton Jacobi Equation

Transcription

1 A. La osa Lecure Noes U-hyscs 4/5 ECE 598 I N T O D U C T I O N T O U A N T U M M E C A N I C ro he alon s Varaonal rncle o he alon Jacob Equaon ef: alean an Croer Theorecal Mechancs Wley 97. Ths s one of he bes boo I have ever encounere. I hghly recoen.. Golsen Classcal Mechancs Ason Wesley 4.. The Lagrange forulaon an he alon s varaonal rncle 4.A ecfcaon of he sae of oon 4.B Te evoluon of a classcal sae: alon s varaonal rncle Defnon of he classcal acon The varaonal rncle leas o he Newon s Law The Lagrange equaon of oon obane fro he varaonal rncle Eale: The Lagrangan for a arcle n an elecroagnec fel 4.C Consans of oon Cyclc coornaes an he conservaon of he generalze oenu Lagrangan neenen of e an he conservaon of he alonan Case of a oenal neenen of he veloces: alonan s he echancal energy 4. The alon forulaon of echancs 4.A Legenre ransforaon 4.B The alon Equaons of Moon ece for solvng robles n echancs roeres of he alonan 4.C Defnon of he osson brace : nng consan of oon before calculang he oon self Loong for funcons whose osson brace wh he alonan vanshes Cyclc coornaes 4.D The ofe alon s rncle: Dervaon of he alon s equaons fro a varaonal rncle 4.3 The osson brace 4.3A alonan equaons n ers of he osson braces 4.3B unaenal braces 4.3C The osson brace heore: reservng he escron of he classcal oon n ers of a alonan a Eale of oon escrbe by no alonan

2 b Change of coornaes o aan a alonan escron 4.4 Canoncal ransforaons 4.4A Canono ransforaons.e. no que canoncal reservaon of he canoncal equaons wh resec o a arcular alonan 4.4B Canoncal ransforaons Defnon Canoncal ransforaon heore Canoncal ransforaon an he nvarance of he osson brace 4.4C esrce canoncal ransforaon 4.5 ow o generae resrce canoncal ransforaons 4.5A Generang funcon of ransforaons 4.5B Classfcaon of resrce canoncal ransforaons 4.5C Te evoluon of a echancs sae vewe as seres of canoncal ransforaons The generaor of he eny ransforaon Infnesal ransforaons The alonan as a generang funcon of canoncal ransforaons Te evoluon of a echancal sae vewe as a canoncal ransforaon 4.6 Unversaly of he Lagrangan 4.6A Invaran of he Lagrangan equaon wh resec o he confguraon sace coornaes 4.6B The Lagrangan equaon as an nvaran oeraor 4.7 The alon Jacob equaon The alon rncal funcon urher hyscal sgnfcance of he alon rncal funcon

3 ro he alon s Varaonal rncle o he alon Jacob Equaon 4. The Lagrange forulaon an he alon s varaonal rncle 4.A Classcal secfcaon of he sae of oon The saal confguraon of a syse coose by N on asses s coleely escrbe by 3N Caresan coornaes y z y z N y N z N. If he syse s subjece o consrans hen he 3N Caresan coornaes are no neenen varables. If n s he leas nuber of varables necessary o secfy he os general oon of he syse hen he syse s sa o have n egrees of freeo. The confguraon of a syse wh n egrees of freeo s fully secfe by n generalze oson-coornaes q q q n The objecve n classcal echancs s o fn he rajecores q = q = 3 n or sly q = q where q sans collecvely for he se q q q n. 4.B Te evoluon of a classcal sae: alon s varaonal rncle One of he os elegan ways of eressng he conon ha eernes he arcular ah q ha a classcal syse wll acually follow ou of all oher ossble ahs s he alon s rncle of leas acon whch s escrbe below. The classcal acon One frs eresses he Lagrangan L of he syse n ers of he generalze oson an he generalze veloces coornaes q an q = n. L q q Tycally L=T-V where T s he nec energy an V s he oenal energy of he syse. or eale for a arcle of ass ovng n a oenal V he Lagrangan s L = / - V 3 Then for a coule of fe en ons a an b he classcal acon s efne as

4 b q a L q q Classcal acon 4 uncon or se of funcons n-enson case Nuber -enson case or se of nubers n-enson case Noce vares eenng on he arbrarly selece ah q ha jons he enng on a an b a he corresonng es an. ee fgure below. We ehasze ha n 4 q sans for a funcon or se of funcons n he n-enson case Insncly we wll also call a ah q s a nuber or se of nubers n he n-enson case I s he value of he funcon q when he arguen s. L s evaluae a a funcon.e. q s evaluae a a nuber or se of nubers n he n-enson case alon s varaonal rncle for conservave syses Ou of all he ossble ahs ha go fro a o b he syse aes only one. On wha bass such a ah s chosen? Answer: The ah followe by he syse s he one ha aes he funconal an eree. e. a au or a nu. q q = The varaonal rncle 5 Tha s ou of all ossble ahs by whch he syse coul ravel fro an nal oson a e o a fnal oson a e wll acually ravel along he ah for whch he negral s an ereu wheher a au or a nu. The fgure below shows he case for he one-ensonal oon.

5 b C a C gure. or a fe ons a an b aong all he ossble ahs wh he sae en ons he ah C aes he acon an ereu. Eale: The varaonal rncle leas o he Newon s law Conser a arcle ovng uner he nfluence of a conservave force be V gravaonal force srng force whose assocae oenal s V.e. = -. nce he nec energy s gven by T= / where eans / hen L = / - V an b a b L = V a On he lef se s evaluae on a ah. 6 On he rgh se he values an have o be use nse he negral bu for slcy we have jus wren an resecvely Dfferen ahs gve n general fferen values for for fe a an b. Le s assue C s he ah ha aes he value of ereu. One way o oban a ore elc for of C s o robe eresson 6 wh a faly of ral ahs ha are nfnesally neghbors o ha arcular ah. Le s ry for eale = C + h 7 uncon whch eans calar araeer = C + h 7 where h s an arbrary funcon subjece o he conon

6 h = h = 8 an s an arbrary scalar araeer In eresson 7 when choosng sall values for b fferen han he ah C. becoes a ah jus a Noe: In he leraure he ah fference - C s soees calle. ere we are ong o use h a nuber an h a funcon nsea jus o ehasze ha - C s a fference beween wo ahs an no he fference of jus wo nubers. Evaluang eresson 6 a he ah gves = b { a C + h V C + h } 9 The conon ha C s an ereu becoes b a C h gure. or an arbrary funcon h a araerc faly of ral ahs = C + h s use o robe he acon gven by eresson 6. ro 6 b a { C + h h V h } where V sans for he ervave of he oenal V = b a { C h V h } + b a h The las er wll be cancelle when evaluae a =.

7 b a { C h V C h } The frs er on he rgh se of he equaly above can be negrae by ars b a C h = h C - b a C h Accorng o 8 h vanshes a an ; herefore he frs ers on he rgh se vanshes b a C h = - b a C h Eresson becoes b a { C + V C } h nce hs las eresson has o be equal o zero for any funcon h us haen ha C + V C = ; or Tha s V C = C =. b a L = V aes an eree value when he ah. b a 3 V sasfes he Newon s Law = =

8 In he ne secon a ey eresson ha we wll use hrough he ervaon s he followng: Conser an arbrary funcon u v where u u u u an v v v v 3 3 or arbrary ncreens an he value of u Δ v δ can be aroae by u Δ v δ u v u u u3 + 3 v v v 3 + oees he followng noaon s use u u u u3 Accorngly u Δ v δ u v + Δ δ u + v 3 + The Lagrange equaons of oon obane fro he alon s varaonal rncle In a ore general case he syse ay be coose by n arcles. Usng = n n he acon s efne as b L 4 a where a an b are wo fe on n he confguraon sace.

9 Usng a faly of ral funcons of he for = + h 5 where h = h = Eresson 5 s a coac noaon of = n n h = h h h n h h h n We loo for an eree value of he funcon = L 6 ro 5 an 6 one obans Noce he er L h L h } { L oes no aear n he las eresson. Inegrang by ars he secon er L h } + { L h - L { h } nce h = h = one obans { L h } - L h } { = { L L - } h In hs eresson L L an are evaluae a + h.

10 { L L - } h In hs eresson L L an are evaluae a. The varaonal rncle requres ha { L L - for arbrary ral funcons h. Ths coul be sasfe only f } h = 7 L L for = n. The Lagrange Equaons 8 These are secon orer equaons. The oon s coleely secfe f he nal values of he n coornaes an he n veloces are secfe. Tha s he an for a colee se of n neenen varables for escrbng he oon. ear: alon s varaonal rncle nvolves hyscal quanes T an V whch can be efne whou reference o a arcular se of generalze coornaes. The se of Lagrange equaons s herefore nvaran wh resec o he choce of coornaes.!

11 Eale: The Lagrangan for a arcle n an elecroagnec fel Ths s a case where he force een on he velocy q E v B 9 nce he Mawell s equaon saes ha B hen B can be eresse n ers of a vecor oenal A A B A because he vergence of he roaonal s encally equal o zero. On he oher han he hr Mawell s equaon E B/ can hen be wren A as E or E A/. The quany wh vanshng curl can be wren as he graen of soe scalar funcon ha s E A/ or where A E Wh hese alernave eressons for E an B he equaon of oon q E v B can be eresse as q A q + q A I can be eonsrae see hoewor assgnen ha A A A 3 - Accorngly he vecor eresson can be eresse n er of he coonens q A A + A = 3 + The secon an he fourh ers on he rgh se consue a eac fferenal of A A A 4 TAK: We have o fgure ou he Lagrangan L such ha he Lagrangan equaons 8 lea o 4.

12 Noce f here were no agnec fel L woul be q / L. The agnec fel nrouces a er eenng on he velocy. ence le s ry f q L 5 where 3 f q L f L f f f L where sans for 3 sans for 3 L L les f f f f q 6 whch shoul be coare wh 4 q + A q q A Noce he rgh se of 6 conans a er n whch s no resen n 4. Therefore we requre f = Or equvalenly f = for = 3 7 A arcular soluon of 7 s

13 elacng 8 n 6 8 f G G G 3G3 q G G G The las wo ers on he rgh se consue a eac fferenal of G G q G 9 whch shoul be coare wh 4 q + q A A q A coarson beween 4 an 9 ncaes ha 8 gves G qa. elacng hs soluon n f q A 3 The Lagrangan n 5 s hen gven by L q q A 3 Lagrangan for a arcle n an elecroagnec fel escrbe by a scalar oenal an vecor oenal A. E B A A 4.C Consans of oon Cyclc coornaes an he conservaon of he generalze oenu There s anoher way o enfy consan of oon. or eale f he Lagrangan of he syse oes no conan a gven coornae he corresonng Lagrangan L L equaon becoes L

14 L I eans ha he generalze oenu s consan Defnon: If a coornae oes no aear n he Lagrangan he varable s sa o be cyclc. = consan for a cyclc coornae 3 CAE: Lagrangan neenen of e an conservaon of he alonan L Conser a Lagrangan ha oes no een elcly on e.e. =. The oal e ervave of L L 33 s L L + L ro he Lagrange equaon L L L L + L L L he revous equaon can be wren as L L - Ths les L L = consan 34 when L oes no een on elcly. We have jus foun a consan of oon. The quany on he lef of eresson 34 s calle he alonan of he syse.

15 L L 35 A ore rgorous efnon of he alonan funcon wll be gven n he followng secons. Eresson 34 ncaes ha when L oes no een elcly on e he alonan quany s a consan of oon. Case: oenal neenen of he veloces or he case n whch V = V L = T V L = = V 36 L = = T L = T L L = T T V = T V = Energy 37 Thus when he oenal oes no een on he he velocy an L oes no een on he e elcly s he energy of he syse an a he sae e a consan of oon. 4. The alonan forulaon of classcal echancs An alernave o he Lagrangan escron of echancs oulne above s he alonan forulaon. Insea of ealng wh a se of n fferenal equaons of n - orer gven n 8 one s resore o solve n fferenal equaons of s orer as we wll see below. owever one ay en u wh a slar nensy of ffculy when solvng he corresonng equaons. The avanages of he alonan forulaon le

16 no necessarly n s use as a calculaon ool bu raher n he eeer nsgh affors no he foral srucure of echancs. Is ore absrac forulaon s of neres because of her essenal role n consrucng he ore oern heores n hyscs. In hs course s use as a on of earure for elaborang a quanu heory. In hs secon we show ha he new forulaon s leene hrough: a change of varables o be secfe laer n n an nsea of L he use anoher funcon = n n = ow o oban such a ransforaon? ow o choose? o be secfelaer n n To ge an ea of wha ransforaon s convenen. e. wha arcular cobnaon beween he an varables s suable for our urose le s falarze wh he followng uch sler Legenre ransforaon 4.A Legenre ransforaons Conser a hyscal quany s escrbe by a wo-varable funcon L=L y 3 A fferenal change of L s gven by L L L = + y y = u + v y 33 I ay haen ha a escron coul ore convenen n ers of an Accorngly we woul le hen o erfor a change of varables y v 34 where v = L y y We wll assue ha he relaonsh v = L y y L. y allow us o u y n ers of an v. One way o oban a quany B o relace L whose fferenal-change coul be eresse recly n ers of an v nsea of an y s by efnng B=B v as follows B v L y y v 35

17 L y = L y y y Is corresonng fferenal s gven by B = L y v v y usng 33 = u + v y y v v y B = u y v 36 As we sa above we assue ha he efnon v = L y / y allows o u y n ers of an v. Also snce u= L/ he above assuon ensures u can be u n ers of an v as well. Thus B ens u beng eresse n ers of he fferenals neenen varables whch s wha we wane. The las eresson also gves B B u = an y = v In suary an v he new 37 y v v = L/ y L y B v B v L y y v = L y y L/ y 38 L/ L = L/ + y y = u + v y B = u y v Wh hs bacgroun we wll see below ha he ransforaon o rans fro he Lagrangan forulaon o he alonan forulaon s of he Legenre s ye. Jus enfy he varable wh he y varable we have use n hs secon. 4.B The alon Equaons of Moon The equaon of oon of a syse s escrbe by a Lagrange funcon L L leang o a se of fferenal equaons of secon orer L L = n

18 Conser he followng ransforaon of coornaes 39 L where L On one han les = - L 4 has o be eresse n ers of - L The alonan funcon 4 On he oher han he eresson for - L gven n 4 les = L L L = L Thus he frs an fourh su-ers cancel ou L 4 ro 4 an 4 one obans

19 - L - L an = n alon canoncal equaons n frs-orer equaons 43 L Noce f L oes no een on elcly neher oes. 44 ece for solvng robles n echancs e u he Lagrangan L. L Oban he canoncal oena Oban - L n ers of an. Eale. Two ensonal oon of a arcle n a cenral oenal V r =Vr. L r r = T - V = r r - Vr. Usng he efnon 39 L r r = r an L = r Usng he efnon 4 r r - r r L r r r r r r Vr. L The eressons r r an L are nvere o eress r an n ers of r

20 r r r r r r = Vr. r r r r Usng 43 one obans he equaon of oon for each of he 4 neenen varables r r. r r = r V r = r r = r r = roeres of he alonan The sgnfcance of he alonan was alreay observe n econ 4.C n whch L for he case when he alonan consues a consan of oon. Now ha we have a ore foral efnon of he alonan gven n eresson 4 we can re-sae he followng anner: 45 roof: - The frs an secon su-er cancel ou whch roves he saeen. Tha s a alonan ha oes no een elcly on e consues a consan of oon 59 s a consan of oon f. 46 L oes no een elcly on.

21 L Ths follows fro he wo general resuls 44 L nce les an hence. an 45 or a Lagrangan L=T-V where he oenal V s neenen of an he nec energy T s hoogeneous quarac 47 T n hen s he oal energy an L hen has he for L V One obans L = L an = T = L L = T L= T T V = T + V. ay be a consan of he oon bu no he energy f T s no hoogeneous quarac.. L ay be he energy bu no consan f.. ay be neher he energy nor a consan of oon. In he alonan aroach A echancal syse s coleely secfe a any e by gvng all he an coornaes. Tha s he sae of a syse s secfe by a on n hase sace. The as s o fn how hs on oves n e.e. he oon n hase sace The nal conon ell us where n hase sace he syse sars bu s he alonan whch ells us hough he canoncal equaons 43 how rocees fro here.

22 = o hase sace Alernavely he alonan eernes all he ossble oons he syse can erfor n hase sace he nal conons cng ou he arcular oon whch s he soluon o a arcular roble. 4.C Defnon of he osson brace nng consans of oon before calculang he oon self I s ossble o use he alonan o eerne recly how a gven ynacal funcon vares along he soluon-oon even before calculang he oon self. or eale eernng wheher such a quany s a consan of oon or no. To ha effec le s conser a ynacal funcon = an calculae s oal fferenal change wh e - 48 The frs er on he rgh se s calle he osson brace beween an o be escrbe n ore eale n he secons below. 49 osson brace beween an he alonan. In ers of he osson brace he e eenence of a hyscal quany s eresse as

23 48 a Loong for hyscal quanes whose osson brace wh he alonan vanshes A quany ha oes no een elcly on e wll be a consan of oon f he osson brace beween an he alonan vanshes. Ceranly one woul have o have a lo of nuon o fgure ou such a funcon. We wll see laer however ha here es syseac ehos o fn jus ha. ere we jus wan o show he ossbly of fnng consan of oon whou solvng he equaons. To see how hs wors le s ae he falar eale of he wo ensonal syerc sle haronc oscllaor. or slcy ae = an =. Then L 49 L for = 5 L - L - where we have o relace he n ers of he gven n 5 5 Whou fnng he elc soluon le s show ha eresson 48 ell us ha he angular oenu q q angular oenu 5 Is a consan of he oon. roof. rs le s calculae

24 whch gves Accorngly eresson 48 gves = Tha s whou elcly calculang he soluon we now ha for hs roble he angular oenu s a consan of oon. 53 b Cyclc coornaes Accorng o he efnon gven n econ 4.C a cyclc coornae s one ha oes L L no aear n he Lagrangan. The Lagrangan equaon hen les ha L he generalze oenu = s a consan of oon. Bu he alonan equaon les also ha n hs case ; ha s a coornae ha s cyclc.e. absen n he Lagrangan 54 wll also be absen fro he alonan Ths concluson can also be obane fro he efnon of he alonan - L. Noce ffers fro L by whch oes no nvolve he coornaes elcly. alonan wh full cyclc coornaes The soluon of he alon s equaon s rval for he arcular case n whch he alonan oes no een elcly on e s a consan of oon an all coornaes are cyclc. = n oes no aear n he alonan Uner hose conons all he conjugae oena wll be consan.

25 The alonan hen ay be wren n he for... n an he equaon of oon for he coornaes wll be whose soluons are consan I s rue ha rarely occurs n racce ha all he coornaes are cyclc for hus ang avanage of he easy way o fn he soluons. owever a gven roble can be escrbe by fferen ses of coornaes. I becoes oran hen o fn a syseac roceure for ransforng fro one se of varables o anoher se of varables where he soluon s ore convenenly racable. Tha wll be he subjec of usng canoncal ransforaons o be escrbe n secon D The ofe alon s rncle: Dervaon of he alon s equaons fro a varaonal rncle. alon s canoncal equaon can be obane fro a varaonal rncle slar o he way he Lagrange equaons were obane n econ 4.B above. owever he varaons wll be over ahs n he hase-sace whch has n ensons wce he n ensons of he confguraon-sace. Ineresngly enough he funcon nse he negral uon whch he varaonal rncle wll be ale s agan he Lagrangan L bu now consere as a funcon of. L - 55 As a frs se le s aly he varaonal rncle o L 56 Inse he negral we have jus wren for slcy bu shoul be use nsea resecvely. In alyng he varaonal rncle we realze s very slar o he case when we ale n he confguraon sace. Ths e we jus have ore neenen varables. The resul s

26 L L L L an for = n. 57 Alyng 57 o he Lagrangan gven n 55 L - L L L On he oher han L L L L L - les - les 58a 58b Tha s we have oban n 58 he canoncal alonan equaons The osson brace Eresson 48 evaluaes how a gven ynac quany = vares as a funcon of e whle an evolves accorng o he alon equaons. The frs er of he rgh han n 48 urns ou o be an oran eresson n self; s calle he osson brace of an. In general he osson brace of he ynacal varable = wh he ynacal varable = s efne as osson brace of he ynac quanes an 55 Noe: oees wll be convenen o eress he osson brace as 56 o ehasze ha he ervaves n he brace are aen wh resec o he varables. or ay haen ha an nverble ransforaon of coornaes an

27 ay have aen lace. Therefore he ynacal quanes wll have a eenence on an an he osson brace of he an can be aen wh resec o he new varables 57 In ers of he osson brace eresson 48 can be eresse as A The alonan equaons n ers of he osson braces In he arcular case ha = eresson 58 gves Bu noce ha Therefore = = = = In he arcular case ha = eresson 58 gves Bu noce ha Therefore = = = Thus alon canoncal equaons n frs-orer equaons 59 = n consues an alernave way o eress he alon canoncal equaons.

28 4.3B unaenal braces = for any... n = for any... n = for = for = Tha s 6.. ear. Noce he resul 6 aears o be rval. I s. In fac hs s a roery of he osson brace self regarless of he esence of a alonan. I urns ou however ha f he brace of he varables were calculae wh resec o oher arbrary varables he value of woul be n general fferen han. 6 or an arbrary ransforaon Bu for a arcular ye of ransforaon of coornaes calle canoncal ransforaons whch are nrouce n connecon o alonan escron of oon o be escrbe n eal n he followng secons he osson brace has he rearable roery of reanng consan ha s regarless of he new canoncal varables use o escrbe he oon. ence he usefulness of he osson brace; hels o enfy such oran canoncal ransforaons.

29 ere we erve he chan rule ha relaes he osson braces evaluae wh resec o fferen coornaes. If = an = where an = + Thus

30 or = an = where an 6 Noce f a arcular ransforaon of coornaes sasfes = = an 63 = hen 6 gves. Tha s Val only for hose arcular rans- 64 foraon ha sasfy 63 Thus we have foun ha hose ransforaon of coornaes sasfyng 63 are very secal: he brace of wo arbrary ynacal quanes an rean nvaran neenen of wheher we use or o evaluae he brace. 4.3C The osson brace heore: reservng he escron of he classcal oon n ers of a alonan a Eale of a oon for whch here s no alonan o escrbe Gven a alonan we say ha he e evoluon of he classcal syse s generae by accorng o he canoncal equaons 43 an 59. There are cases however n whch no alonan generaes a arcular oon. Conser for eale he followng oon: 65 -

31 Tha here s no whose assocae canoncal equaon ens u n 6 can be seen by frs requrng - The frs equaon gves 4 an he secon equaon gves The fac ha les ha such a funcon oes no es. 66 b Change of coornaes o aan a alonan escron I s neresng o noe n he eale above ha uner a roer change of coornaes a alonan funcon K can be foun such ha K K an -. The followng ransforaon wll o he rc. 67 The equaons of oon for an are accorng o 65 = = / + accorng o 65

32 = / - + = / - / = / / Bu In shor he oon escrbe n he coornaes as - for whch here s no a alonan s alernavely escrbe n he new coornaes by he followng equaon of oon 68 These equaons o have a alonan. Inee K seng gves K = / + f Therefore seng K gves K = / + g K = / + / 69 c The osson brace heore The above rear ncaes ha when a alonan canno be foun ay be ha we are usng he wrong coornaes. Wrong n he sense ha oes no es a alonan K o escrbe he oon n ers of he canoncal equaons 43 an 59. The followng osson brace heore coes hany hen as a guance o recognze suaons n whch he oon n he coornaes beng use o escrbe can be generae by a alonan.

33 Le be he e eveloen of a syse on hase sace. Ths eveloen s generae by a alonan f an only f every ar of ynac varables sasfes he relaon 7 An oulne of he roof s gven n he Aen- a he en of hs chaer. Ths heore s use o verfy wheher or no a gven oon s a alonan oon. If we new = an = we woul aly 7 wh he choce of =q an =. If 7 were no fulflle hen here wll be no alonan o escrbe such a oon. 4.4 Canoncal Transforaons In any occasons ay be convenen o ae a ransforaon of coornaes. The ovaon ay no be necessarly o slfy he aheacal buren n solvng he alonan equaons n he new coornaes; nsea ore ofen s o enfy ore clearly hose hyscal quanes ha rean consans hroughou he oon even whou solvng he equaons of oon. In oher occasons he ransforaon of coornaes allows a beer nerreaon of classcal echancs forals as a on of earure for elaborang a quanu heory. The laer s ore ernen o hs course. The eale gven n he revous secon llusrae however ha f we wan o ee a alonan forals o escrbe he oon we have o be careful when ang a roer change of coornaes. Oherwse we ay en u wh no alonan assocae o ha oon when escrbe n hose new coornaes. In hs secon we suy hose yes of ransforaons of coornaes ha allow eeng he escron of he classcal oon whn a alonan forals. e. s escron accorng o he canoncal equaons 43 an 59. They are calle canoncal ransforaons. We wll fn ha he syseac roceure for generang such ransforaons nvolves he arcaon of four yes of so calle generang funcons. Each generang funcon rouces a corresonng ye of canoncal ransforaon. In he course of consrucng a conssen forals for obanng canoncal ransforaons a forunae urn of evens occurs. I urns ou ha fnng he roer generang funcon o oban a canoncal ransforaon of secfc characerscs s equvalen o fnng he soluon o he canoncal alonan equaons! Ths wll be shown n econ 4.5C below. urher a arcular canoncal ransforaon wll lea o a new alonan ha s encally equal o zero; he equaon ha such a generang funcon us sasfy s he celebrae alon Jacob equaon hs wll be aresse n econ 4.7. In he subsequen chaers when elaborang a quanu echancs forals we wll requre ha he quanu echancs equaon shoul have he alon Jacob equaon as a l. The laer jusfy our curren effor for aanng a

34 goo unersanng of he alon Jacob equaon. We sar by aressng he conce of canoncal ransforaons frs. Transforaon of coornaes Conser ha a classcal oon s escrbe by 7 for whch a alonan ess;.e. he changes of an are governe by a alonan : an 7 Conser a ransforaon of coornaes 73 ; As an change he varables an also change accorngly. owever nohng ensures ha he e evoluon of he laer varables reserves he alonan forals gven n eresson 43. Tha s oesn always ess a funcon K such ha K K an. Those arcular ransforaons ha allow reservng he alonan for of he equaon of oon are calle canoncal ransforaon. In hs secon we aress a eho o generae such ransforaons an nqure how o selec a canoncal ransforaon ha reners a alonan K ha s a consan funcon for n such a case he equaon of oon becoes obvously very sle. = o Transforaon = o hase sace gure 3. hase sace

35 ; 4.4A Canono ransforaons.e. no que canoncal reservaon of he canoncal equaons wh resec o a arcular alonan Conser a alonan =. An nverble ransforaon of varables such ha he e evoluon of he new varables an 74 reserve he for of he alonan equaons.e. here ess a funcon K such ha K K an s calle canono.e. no que canoncal wh resec o.. Noe: I urns ou a ransforaon ha s canono wh resec o a gven alonan nee no be so wh resec o anoher. 4.4B Canoncal ransforaons Defnon A ransforaon ha s canono wh resec o all alonans 75 s calle canoncal. Canoncal ransforaon heore 3 Le be a se of general coornaes on hase sace. The osson braces of any wo ynac varables an wll be secfe as Conser an nverble ransforaon ;

36 The followng hree saeens are equvalen: a The ransforaon ; s canoncal. 76 b There ess a nonzero consan z such ha any ynacal varables an sasfy z Tha s he osson brace s raccally neenen of he coornaes use o calculae. The ransforaon ; s canono wh resec o all quarac alonans of he for n = C + c c' + n n + where = ' = ' = 77 ' " 78 An oulne of he roof s gven n he Aen-. Canoncal ransforaon an he nvarance of he osson brace In shor he heore above esablshes ha A ransforaon s canoncal f an only f 79 I reserves he osson brace o whn a consan facor z. 4.4C esrce canoncal ransforaons A canoncal ransforaon ; s calle 8 resrce-canoncal ransforaon f n eresson 78 z=. Tha s. Noe: In he leraure he resrce canoncal ransforaons are ofen sly calle canoncal ransforaons. 4.5 ow o generae resrce canoncal ransforaons 4.5A Generang funcons of ransforaons

37 In he reanng of he noes whenever referrng o canoncal ransforaons we wll assue ha hey are resrce z=; ha s. A way o generae resrce canoncal ransforaons wll resul along he way of aeng o classfy he. Towar hs en le s analyze frs he followng eresson - 8 where he varables correson o he ransforaon 73 ; 8 I urns ou when 8 s wren as a funcon of reveals uch abou he ransforaon. I s foun ha when he ransforaon 8 s resrce canoncal eresson 8 becoes an eac oal ervave of a scalar funcon. The laer s hen use o generae he ransforaon The sraegy s o show ha here ess a funcon ha allows eress 8 n he followng alernave for - = else To hs en le s ean he lef se n ers of he ol coornaes - = = The ne se s hen o enfy er by er eressons 83 an 84. Aen-3 oulnes he eonsraon ha here ess such a scalar funcon such ha

38 an K 85 urher relacng 85 n 84 Or - = + - K - - K = - - K = 86 Wha s rearable s ha he lef se s an eac fferenal. The funcon s calle he generang funcon of he ransforaon for as we wll see once s gven he ransforaon equaons B Classfcaon of resrce canoncal ransforaons Eresson 86 suggess ha n orer o effec he ransforaon beween he wo ses of canoncal varables us be a funcon of boh he ol an he new varables. s a funcon of 4n varables lus he e. Bu only n of hese are neenen because he wo ses an are connece by he n ransforaon equaons 8. Accorngly here are four oenal fors o eress whch wll een on he crcusances cae by he secfc roble: = = = 3 = 4

39 Case : Assue he funcon s gven. In hs case 86 saes - - K = = nce he an he are neenen hen he coeffcens of equal 87 an shoul be = n 88a These are use o solve for he n varables as a funcon of = n 88b nce he have been eerne n 88a hs eresson gves he n varables as a funcon of whch leaves 87 wh K 88c Case : Assue he funcon s gven. In hs case 86 saes - K = - = 89 On he lef se we ean he er - - K =. or convenence we wre as

40 = 89 where we wll relace = Inverng he orer of he suaon = 9 elacng 9 n K = = 9 nce he an he are neenen n 9 hen he coeffcens of an shoul be equal = 9 K - = I s no sraghforwar o vsualze ha fro he frs equaon how he can be solve as a funcon of snce he are nvolve here. ence we erfor an era se. rs noce ha he frs equaon n 9 can be wren as 93a Also noce n he secon equaon of 9

41 ence he secon equaon n 9 can be eresse as = = 93b In er of he funcon ' eressons 9a an 9b gve ' = ' 94 K - = ' A he en here wll be no neres n he funcon. We alreay foun a ha f eresse n ers of as neenen varables wll generae a canoncal ransforaon. ence le s jus renae he as. In suary = K - = solve for Once he are nown hs gves Inver he o oban

42 K = + ' K = + ' + 4.5C Evoluon of he echancal sae vewe as seres of canoncal ransforaons The generaor of he eny ransforaon Conser he generang funcon 95 Le s fn ou he canoncal ransforaon generaes. Usng = We fn he new coornaes are he sae ol coornaes. Tha s he funcon n generaes he eny ransforaon. I Infnesal ransforaons Conser he generang funcon + G 97 where s an nfnesal nuber an G an arbrary funcon o be secfe laer. Due o he sall value of an snce generaes he eny ransforaon we eec he new coornaes wll ffer fro he ol ones also by nfnesal values; ha s Le s fgure ou he values of Alyng 94 o 97 gves = = 98 an.

43 G 99a = G base on 99a one obans G G 99b In shor we have obane: The funcon + G generaes he ransforaon G G The alonan as a generang funcon of canoncal ransforaons Noce n f G s he alonan an s chosen o be he ncreenal e fferenal one obans G = Usng he alonan equaons 43 = Bu s he ncreen of ue o he oon = Tha s we have obane: G = Usng he alonan equaons 43 = =

44 The funcon + generaes he ransforaon where an are resecvely he changes n an ue o he oon governe by he alona. 7 Noce n ha bascally he alonan can be vewe as he generaor of he ransforaon. I ransfors he value of he coornaes a he e o he value of hose coornaes a he e +. Ths resul s very neresng. I ells us ha he evoluon of he sae of oon can be vewe as a seres of canoncal ransforaons generae by he funcon as he resul alyng successvely one afer anoher fferenal e. v Te evoluon of a echancal sae vewe as a canoncal ransforaon ybolcally he resul n can be eresse hs way: Defnng ' + - ' ". 3 Evoluon of he classcal sae over e. The alonan n consues he generaor of he sae s evoluon wh e. Eresson 3 also hns an aroach a leas conceually of how o aan a soluon of he equaon of oon. In effec snce he successve alcaons of canoncal ransforaons s a canoncal ransforaon fnng he soluon of he alon equaons can be vew as he as of fnng a canoncal ransforaon ha aes he nal conons ol coornaes o he values of he sae a he e he new coornaes. Canoncal ransforaon 4 ol coornaes new coornaes

45 4.6 Unversaly of he Lagrangan 4.6A Invaran of he Lagrangan equaon wh resec o he coornaes use n he confguraon sace I was sae n he secons 4.B above ha he Lagrangan equaons L L 5 have he arcular neresng roery of beng neenen of he arcular coornaes use n he confguraon sace. egarless of he coornaes beng use he Lagrangan equaons loo he sae. ro each new coornaes we woul be able o oban a corresonng alonan by followng he roceure of econ 4.B. uch a saeen ay aear o be a os wh he conces oulne n econ 4.4 where we ha o be careful n no ang he wrong ransforaon of coornaes oherwse we woul en u wh a no alonan escrbng he syse. The ransforaon ha o be canoncal. The followng eale hels clarfy he ssue. Conser he generang funcon f 6 where he f are arbrary funcons. Usng 94 = f 7 = f Tha s he new coornaes een only on he ol coornaes an e bu o no nvolve he ol oena. uch a ransforaon s hen an eale of he class of on ransforaon one n he confguraon sace whch le o he conseraon of he nvaran of he Lagrangan equaon wh resec o a ransforaon of coornaes. In hs cone such a on ransforaon 7 are canoncal an herefore reserve he alonan escron of he oon. We have herefore a vew of he nvarance of he Lagrangan fro a canoncal ransforaon of coornaes ersecve. 4.6B The Lagrangan equaon as an nvaran oeraor In econ 4.D he alcaon of he ofe alon s varaonal rncle o he generalze Lagrangan L le o L L L L an for = n 6

46 or slfcaon uroses n hs secon le s use ζ o we eress he Lagrangan as L ζ ζ 7 An he equaon 6 ae he coac for L ζ ζ for = n 8 or new coornaes obane hrough a canoncal ransforaon whch n our new noaon wll be eresse as ζ η 9 The alcaon of he varaonal rncle o he Lagrangan wll lea o L' η η L' η η On he oher han eresson 86 ell us ha - - K = for = n or n he new noaon L ζ ζ - L' η η Now coe he e o hghlgh furher he orance of eresson. I urns ou ha relacng n 8 one obans L' η η 3 Tha L ζ ζ rearable. an L' η η are he sae s

47 I was obane no obane because L ζ ζ an L' η η are equaons for he sae se of oon n he hase-sace. I was obane only because L an L ffer by a funcon I sugges ha s a n of oeraor neenen of he arcular coornaes beng use. or f he Lagrangan s eresse as L ζ ζ or L' η η he alcaon of he oeraor escron of he sae se of oon n he hase-sace. reners he 4.7 The alon Jacob equaon The alon rncal funcon We have been envsonng ways o oban sle ways o solve he alonan equaons. In one case we allue o he convenence of fnng a syseac way o ransfor fro one se of coornaes o anoher se of coornaes n such a way ha he new alonan ens u havng all he coornaes = n beng cyclc. Anoher aroach was oulne n econ 4.5D wh a canoncal ransforaon relang he ol coornaes aen as he sae varables a a gven e o he new coornaes aen as he sae varables a a e. We wll follow hs laer aroach for beng ore convenen snce ales even when he alonan eens on e.e. s no a consan of oon. The ouloo wll have a slgh varaon. We wll be loong for a canoncal ransforaon such ha ransfor he coornaes o a new se of coornaes he laer are consan values of he nal conons. In our cuso ernology 4 New coornaes consan n e K

48 One can auoacally ensure ha he new varables are consan n e by requrng ha he new alonan K be encally zero! for hen he equaons of oon are K K K s relae o he ol alonan an he generang funcon by whch wll be zero f K 5 6 I s convenen o choose as he neenen coornaes; ha s whch gves he = 7 = 8 Equaon 6 aes he for... n Gven hs consues an equaon for. I s cusoary o enoe he soluon of hs equaon by whch s calle he alon s rncal funcon n alon-jacob Equaon wh = consan alon s rncal funcon When solvng 9 he n consans of negraon can be aen as he s.

49 Eresson evaluae a gves a relaonsh beween he consan value of an he nal conons. Then usng eresson 8 = one has a relaonsh o fn he s. In arcular he s can be obane by evaluang hs eresson a. Inverng = one obans whch gves he he coornaes as a funcon of e an he nal conons. In shor when solvng he alon-jacob equaon we are a he sae e obanng a soluon of he echancal roble urher hyscal sgnfcance of he alon rncal funcon Le s evaluae he oal ervave of. because he s are consan. Usng eresson 8 gves Usng he alon Jacob equaon o The er on he rgh s he Lagrangan L ; hus L L s hen an nefne e negral of he Lagrangan L ' ' ' '

50 Aen- osson brace heore Le be he e eveloen of a syse on hase sace. Ths eveloen s generae by soe alonan f an only f

51 every ar of ynac varables Oulne of he roof: sasfes he relaon lyng s sraghforwar. n: The esence of a alonan les ha any varable changes accorng o. Use he las eresson wh =. Tha les he esence of a alonan governng he varaons of an s ore elaborae. The varable for eale wll have he for = = n larly = = n Below we eonsrae ha les = We wan o rove an can be obane fro a funcon such ha / an /. We nee o eonsrae he esence of. Noe: I ay be suffcen o eonsrae ha = or wll ensure no conracng he fac ha f ese woul have o sasfy nce regarless of he esence of a alonan or no sar wh he followng resul:. Eresson les Usng = an = one obans we

52 or + = = + = In oher wors oes no conrac he ques for fnng a funcon ha sasfes / an /. Aen- Canoncal ransforaon heore Le be a se of general coornaes on hase sace. The osson braces of any wo ynac varables an wll be secfe as Conser an nverble ransforaon ; The followng hree saeens are equvalen: a The ransforaon ; s canoncal. b There ess a nonzero consan z such ha any ynacal varables an sasfy z Tha s he osson brace s raccally neenen of he coornaes use o calculae. c The ransforaon ; s canono wh resec o all quarac alonans of he for n = C + c c'

53 + n n " ' where = ' = ' = roof: Tha b les a s a b sraghforwar. All nees o be one s o show. or accorng o he osson brace heore shown above he laer les ha he evoluon of an s generae by a alonan. ro b: z z Assung ha he evoluon of s governe by a alonan osson brace heore les } { z z Usng b agan Tha c les b s ore elaborae. Elong he fac ha ; s canono wh resec o any quarac alonan we wll fgure ou frs he value of =? =? an =? whch hen be use n he resul for obane n 36. Les sar conserng he case ha ; s canono wh resec o =C consan. consan les cons an cons 3

54 nce ; s canono wh resec o he alonan =cons eans accorng o he osson brace heore ha here es a alonan K= K accorng o whch a quanes le an woul evolve n e as Bu accorng o 4 all he an whch gves Le s u funcon of ; ; an 4 an as a f 5 g s or a alonan =cons f g s he e varaon of f g an s are gven by f f f = f g g g = g ; an

55 s s = g s nce accorng o 4 an one obans f g s o neher nor f g ;. s ; an.. 6 een elcly on he e varable.. Noe: Noce ha alhough we have use he alonan =consan o oban he resul 6 neher of he values een on he arcular alonan use. They een only on he ransforaon coornaes. Conser now he alonan = c Ths les n c' c' an c 7 nce ; s canono wh resec o hs alonan eans accorng o he osson brace heore ha here es a alonan K = K ' accorng o whch a quanes le an woul evolve n e as an or

56 Usng 7 ' c ; c ; an c Thus f g s ence n ers of he alonan = f f f g g g s s s Usng 6 f ' n c c f ' n n f c f c ' n n f c f c

57 Ths shoul be rue for all arbrary values of c an c whch les f an f for =. n larly 8 g an g for =. n. ro 6 an 8 he quanes an 9 o no een on he varables.. Tha s hey are consan values.. Now we nee o rove ha all hose consan values are of he sae agnue z. To ha effec conser he alonan = n n " ' " " ' " ' ' ' nce ; s canono wh resec o hs alonan eans = =

58 = All hese quanes are equal o zero accorng o 9. Usng les " ' " ' " ' " ' = ' + ' - G + G = les G = G =I les G s syerc = les = =I les n syerc larly usng les ' " ' ' " ' = G G = = les = = = les =; =I les = = = les G =G ; =I les G=G Usng les

59 ' ' ' ' = G + - G + 3 = les G = G ; =I les G s syerc = les ' = - ; '=I les = - = ro an 3 = or f = = or s = 4 G syerc g = g We can furher eerne he g values for. or eale: nce he ar s arbrary le s conser one n whch only 3 = 3 are he only eleens. G g... g... g g g g 3... g g g g g.. Eresson 3 requres G = G whch les g = g 33 an g 3 = g 3n = for n>3 We arrve o he concluson hen ha 3 g 3 g 3 3 g 3 g g = cons 5 The consan value eens on he arcular ransforaon use an s referre o as z n he e of he heore. Thus = or f = or s G syerc g = = 6 = z ro he chan rule eresson 4 n he an e of hs chaer

60 ... z z z whch gves z The osson brace s nvaran uner a canoncal ransforaon u o a consan value z. Aen-3 ow o generae canoncal ransforaons A way o generae resrce canoncal ransforaons wll resul along he way of aeng o classfy he. Towar hs en les analyze he followng eresson - where he varables correson o he ransforaon ;. I urns ou when s wren as a funcon of reveals uch abou he ransforaon. Bascally he sraegy conss n showng ha when he ransforaon s resrce canoncal ha s here es a = an a K=K

61 eresson urns ou o be an eac oal ervave of a scalar funcon =. - = eanng he secon er = - an enanng an enanng - The objecve s o show ha here ess a funcon ha aes equal o les les whch gves } { he laer s he Lagrange brace Whou roof we sae:

62 The ransforaon ; s canoncal f an only f 3 } { an } { } { Therefore one obans Ths ncaes ha here es a funcon such ha 4 larly les les whch gves } { = Ths ncaes ha here es a funcon such ha 5 Ψ les Ψ an Ψ les les an Ths gves

63 Ψ an Ψ 6 On one han can be eerne by he alonan K=K K On he oher han s a quany ha can also be race by he alonan = Thus K K larly K K urher K K Uon subracon K K K K

64 K K Now sung on he varable see eresson 6 K K K K K } { } { K Ψ 7 Bu accorng o 4 K Ψ K Ψ K Ψ 8 larly we eec o oban K Ψ 9 Ths les ha K Ψ s a funcon of alone. owever 4 an 5 eerne u o an ave funcon f; he laer can be choose such ha

65 K Ψ ro whch efnes an K ro whch efnes an 4 ro whch efnes an 5. Golsen Classcal Mechancs Ason-Wesley ublshng 959. alean an Croer Theorecal Mechancs Wley 97. age 8. 3 ef. age88.