8. HAMILTONIAN MECHANICS

Transcription

1 8. HAMILTONIAN MECHANICS In ode o poceed fom he classcal fomulaon of Maxwell's elecodynamcs o he quanum mechancal descpon a new mahemacal language wll be needed. In he pevous secons he elecomagnec feld was descbed usng paal dffeenal equaons Maxwell's equaons fo he feld componens and he veco and scala poenals. These equaons povded he bass fo he developmen of he equaons of moon of chaged pacles embedded n he elecomagnec feld. Howeve hese equaons of moon wee smplfed descpons of he acual moons of lage numbes of chages n a conducng maeal. In hs pevous fomulaon, he elecomagnec feld was an absac mahemacal eny. Ths appoach was a consequence of he classcal naue elecomagnesm snce he feld s eaed as an eheeal eny ha seves as he medum o cay elecomagnec waves, he enegy and momenum. Specal mehods have been developed by Lagange n ode o deal wh he lage possbly nfne numbe of pacles. By fomulang Newon s nd Law n ems of he knec and poenal enegy as funcons of he coodnae sysem n whch he pacles ae movng. Lagange succeeded n genealzng he use of he coodnaes. Ths appoach allows he equaons of moon o be solaed fom a specfc coodnae sysem, whch n ems allows he vaaonal pncpal o be appled o a vaey of poblems ncludng he descpon of he elecomagnec feld and quanum mechancal fomulaon. The fomulaon of he elecomagnec feld can be esaed n ems of Hamlon's heoy of mechancs usng he elecomagnec feld's veco poenal as a sang pon [Hel81]. Ths mehod povdes a classcally conssen anson o he quanum mechancal descpon of he effec of he elecomagnec feld on chaged pacles. In ode o poceed wh hs fomulaon seveal new conceps mus be pesened. The quanum naue of mae wll be befly descbed followed by he descpon of Hamlonan mechancs. The fomulaon of he equaons of moon was fs used n classcal mechancs [Gold55], bu now seves as he noducoy mehod o quanzng he elecomagnec feld NEWTON S EQUATIONS IN LAGRANGIAN FORM Specal mehods have been developed by Lagange n ode o deal wh he lage and possbly nfne numbe of pacles o be descbed by he equaons of moon. By fomulang Newon' nd Law n ems of Copygh 000,

2 he knec and poenal enegy as funcons of he coodnae sysem n whch he pacles ae movng. Lagange succeed n genealzng he use of coodnaes. Ths appoach allows he equaons of moon o be solaed fom a specfc coodnae sysem, whch n un allows he vaaonal pncpal o be appled o a vaey of poblems ncludng he descpon of he elecomagnec feld and s quanum mechancal fomulaon. Isaac Newon fomulaed he laws of moon usng a calculus of hs own nvenon. Usng he Caesan coodnae sysem, Newon s equaons fo he h pacle wh mass m ae: dx m = X d dy m = Y 1,,, = n (8.1) d m dz d = Z whee, X, Y and Z ae he hee componens of he foce acng on he h pacle. The ansfomaon of he equaons fom Newonan fom o he Lagangan fom wll be make use of boh he knec and poenal enegy defnon n he Caesan coodnae sysem. The knec enegy, T, s defned as: 1 m T = x + y + z + + x + y + z m n 1 = m ( + + n x y z ). = 1 n ( ) ( n n n), (8.) If only consevave sysems of pacles ae consdeed, hen he poenal enegy, V, can be defned as a funcon of he coodnaes xyz,,,, xyz,, of all he pacles. In hs appoach he foce n n n expeenced by each pacle s equal o he paal devave of he poenal enegy, such ha, 8 Copygh 000, 001

3 V X = x V Y = = 1,,, n. (8.3) y V Z = z These equaons can now be used o esae Newon s equaon of moon. By emovng he efeences o he ndvdual coodnaes, he noaon fo he equaons of moon can be smplfed. Assumng ha he foce appled o he pacle can be found fom a poenal, whch s a V,, accodng o he elaonshp funcon of boh poson and me, ( ) (, ) = (, ) F V. Subsung hs expesson no Newon s equaon of moon gves, (, ) V m + = 0. (8.4) F Snce he momenum of he pacle s = ṙ p m, Newon s equaon can be ewen as, ( ) V, p + = 0. (8.5) mx F Snce he mass of he pacle, m, s a consan, he momenum can be ewen n ems of he pacle s knec enegy, ( m ) d d m d p= ( m ) = = d d d m p (8.6) By defnng he knec enegy as, T = m ṙ, and usng hs expesson o elmnae he momenum p fom Newon s Law gves, d T V + = 0. (8.7) d p F The knec enegy s now a funcon of ṙ bu no. The poenal enegy V as a funcon of bu no ṙ. Ths allows he Lagangan o be defned agan as usng he elaons L =T and L = V as, Copygh 000,

4 (, ) = ( ) ( ) L T V. (8.8) Ths now allows Newon s Law o be saed as he Eule lagange equaon (n Caesan coodnaes), (, ) (, ) d L L = 0. (8.9) d m So fa he changes fom he descpon of Newon s Law as he smple equaons of moon, o he Lagangan descpon has no smplfed anyhng. In he nex secons, he Lagangan descpon of he moon of a pacle wll be used o emove he dependence on he Caesan coodnae sysem. In addon he Hamlonan descpon of he pacles moon wll be developed. Ths descpon wll used as he bass fo he quanum mechancal descpon of he elecomagnec feld neacng wh chaged mae. F 8.. VARIATIONAL DESCRIPTION OF THE EQUATIONS OF MOTION The equaons of moon of an objec movng n a Caesan coodnae sysem was fs descbed by Isaac Newon. In Newon s mechancs he moon of a pacle s unquely deemned by he vecoal foce acng on he pacle a evey nsance of me [Lanc70], [Byo70], [Byo69], [Chan95]. In Newon s mechancs he acon of a foce s descbed by he momenum poduced by ha foce. Thee ae ohe descpons of he acon of a foce. One such descpon was povde by Gofed Wlhelm Lebnz ( ), who was a conempoay of Newon s. Lebnz fomulaon ncluded a quany knows as vs vva (Lan fo lvng foce) whch n moden ems s call he knec enegy [Asm66]. Lebnz eplaced Newon s momenum by he knec enegy and eplaced Newon s foce by he wok of he foce. Ths wok of he foce was lae eplaced by he wok funcon. Lebnz s now ceded wh foundng a second banch of mechancs analycal mechancs, whch s based on he manenance of he equlbum beween he knec enegy and he wok funcon. In moden ems he foce funcon s eplaced by he poenal enegy. Ths appoach lad he foundaon fo he Pncpal of Leas Acon. I s convenen o dvde he developmen of classcal mechancs no hee peods, he fs based on Newon's Phlosphae Nauals Pncpa Mahemaca publshed n 1687 [Cajo6], he second based on Lagange's Mécanque Analyque (Analycal Mechancs) publshed n 1788 [Lag88] and he hd based on Hamlon's Geneal Mehod of Dynamcs publshed n 1834 and 1835 [Ham34] and well as Cal Gusav Jacob Jacob's ( Copygh 000, 001

5 1851) Volesunge übe Dynamcs publshed by Clebsh n These woks esablshed mechancs as a mahemacal scence complee wh heoecal explanaon of he behavo of objecs and lke he pevous descpons of Foue's mahemacal physcs woks, fomed a paadgm fo he mehods used by Maxwell and he descpon of elecomagnec phenomenon. On New Yea s Day 1697, Johann Benoull ( ) of he Unvesy of Basal posed he queson o he shapes mahemacans n he whole wold gven wo pons A and B n a vecal plane, fnd he pah A M B whch he movable pacle M wll avese n he shoes me, assumng he acceleaon on M s due solely o gavy. Usng Benoull s descpon [Su86], he cuve ACEDB shown n Fgue 8.1, has a pah of leas me fom A o B. Leng C and D be wo pons on he cuve, Benoull sad CED mus have he same pah of leas me. Ths s he essenal pon of Benoull s agumen and he powe of hs developmen n moden physcs. Any cuve whch has a mnmum popey globally (n he lage) mus also have hs popey locally (n he small). If wee no he leas me pah han hee would be some ohe pah CFD whch would be fase. If ha wee he case, he new pah ACFDB would be fase han he pah ACEDB, whch would be conay o he ognal hypohess. The esul s Benoull s conbuon o moden physcs.. he pah quckes oveall mus be he quckes n beween any nemedae pons, and he popey whch holds globally mos also hold locally. A C F E D B Copygh 000,

6 Fgue 8.1 Benoull s cuve whch descbes he leas me pah beween o pons A and B. The mahemacal poblem was posed o he wold s mahemacans as a challenge. Alhough he soluon was aleady known o Benoull, a smple soluon was also known o Newon, who dd no espond o Benoull s challenge. Ths poblem s known as he backsochome poblem bachsos = shoes and chonos = me and maks he begnnng of he geneal nees n he calculus of vaaons [Byo69], [Chan95], [Su86], [Red69]. [1] Usng Benoull s appoach he local pncpal allows he negal equaons of moon o be ansfomed no dffeenal equaons of moon. The esul s a gealy smplfed mehod of analyzng he moon of he pacle M along he pah of leas me. In any change whch occus n naue, he sum of he poduc of each body mulpled by he space aveses and by s speed (efeed o as he acon ) s always he leas possble. Pee Lous de Maupeus [Maup46], [Doug90]. The developmen of analycal mechancs s assocaed wh Leonhad Eule ( ), Joseph Lous Lagange ( ), Smeon Posson ( ) and Wllam Rowan Hamlon ( ). I s essenally a efomulaon of Newon's mechancs whch allows many poblems o be solved moe smply. 1 When Benoull fs ssued he challenge hee wee no esponses. He fowaded he poblem o Chales Monagu ( ), who was he pesden of he Royal Socey. Isaac Newon esponded o he queson wh an anonymous soluon n a lee daed Jan 30, The esuls wee publshed n Phlosophcal Tansacons, fo Januay 1696/7. Alhough Benoull s gven ced fo he soluon o hs poblem, Newon s soluon was ecognzed by Benoull n a lee o Basange de Beauval alhough s auho, n excessve modesy, does no eveal hs name, we can be cean beyond any doub ha he auho s he celebaed M. Newon [Chan95] When Benoull solved he bachsochone poblem he boased of havng dscoveed a wondeful soluon, bu dd no publsh mmedaely. Insead he poceeded o challenge ohe mahemacans, especally hs elde bohe, Jacob ( ). Benoull caed on a be feud n whch he publcly chaacezed hs bohe as ncompeen. He fnally publshed hs soluon n 1697 whch descbed he moon of a bob avelng on a cyclod pah. Befoe Benoull publshed hs wok Huygens had dscoveed ha a mass pon oscllang whou fcon unde he nfluence of gavy on a vecal cyclod has a peod ndependen of amplude. Ths cyclod was called a auochone wh Benoull's dscovey, hs cuve was enamed he bachsochone [Cou56]. 8 6 Copygh 000, 001

7 Newonan mechancs was founded on he concep of pon masses, ha s objecs wh no dmensonal fom. Newon's equaons of moon ae saed n ems of he Caesan coodnaes of he pacle n moon. Whle he poblems of dynamcs can heoecally be solved by such means, n sysems conanng lage numbes of pacles, he negaon of he equaons of moon s geneally oo complex. Specal mehods wee developed o deal wh hs complexy. Lagange s appoach makes use of an negal equaon conanng he poenal and knec eneges. The knec enegy (T) depends on he objec s velocy v = dx d, whle he poenal enegy (V), depends only on he objec s poson x. The fom of Lagange s soluon s he dffeence beween he knec and poenal eneges. Lagange fomulaed he soluon o he equaons of moon by means of genealzed coodnaes,.e. any se of vaables suffcen n numbe o defne unambguously he confguaon of he sysem. The genealzed coodnaes n he Lagange and Hamlon descpons of moon ulzng he expessons fo knec and poenal enegy as funcons of hese coodnaes. In he classcal descpon of moon, wo measuable quanes of a pacle n moon ae s spaal poson and momenum. If hese values ae known fo any pon n space and me, he pacle s moon o pah can be calculaed fom Newon s second law of moon and knowledge of he exenal foce law acng on he pacle. If he pacle s moon s obseved ove a small poon of s pah s momenum s nealy consan. The poduc of he pacle s momenum and small dsance s called he ncease n he pacle s acon. Ths acon s a scala quany ha he pacle caes wh and nceases as he pacle moves along s pah. [] 8.3. CALCULUS OF VARIATIONS The pncpal of saonay acon appeaed n Heo of Alexanda s (6 A. D.) Caapca (Opcs) whch descbed he eflecon of lgh fom a plane mo as he shoes pah aken. Pee de Fema ( ) Ths descpon of acon dffes fom he ognal concep developed by P. Maupeus who poposed ha bachsochome poblem could be bee solved by no consdeng he ans me of he movable pacle, bu ahe by a quany called acon. Maupeus ncoecly defned hs acon as he poduc of he dsance he pacle avels and s speed [Moz89] Copygh 000,

8 efomulaed hs concep as he pncpal of Leas Tme n Fema saed ha a lgh ay equed he leas me even f devaed fom he shoes physcal pah,... naue opeaes by he smples and mos expedous way and means. Fema s pncpal was capable of poducng he coec law of eflecon and lead o he law elang he angle of ncdence and eflecon a an neface o he ao of he efacve ndces of he meda. The elaonshp was confmed expemenally by Wllebod Snell van Royen ( ) n 161 and s known as Snell s law. The calculus of vaaons and he pncple of leas acon combne o fom a poweful mehod of nvesgang poblems n dynamcs. Pee Lous Moeaude Maupeus ( ), he auho of he Pncple of Leas Acon n 1774, declaed o be a meaphyscal pncple on whch all canons of moon ae based. [3] The Newonan equaons of moon can be wen n a fom whch makes he anson o quanum mechancs appea naual [Byo69]. The concep and pncple of leas acon wee genealzed by Hamlon o nclude he popagaon of lgh as well as he moon of pacles. By placng a escon on he defnon of acon Newonan mechancs can be ansfomed no quanum mechancs. Newonan mechancs assumes ha a pacles moon can be followed n nfne deal and nfne pecson. If hs wee possble han he moon of a 3 Egheenh cenuy phlosophe scenss leaned o compue he pahs aken by planes and objecs usng Newon's equaons of moon. A Fench geomee and phlosophea, Pee Lous Moeau de Maupeus [Maup46] along wh Joseph Lous Lagange showed ha he pahs aken by hese objecs ae always he mos economcal when he knec and poenal enegy ae compued as a sngle quany. In he way he movng objec mnmzes acon a quany based on he objecs velocy, mass and he space hough whch avels. No mae wha foces wee appled o he objec, somehow choose he cheapes of all possble pahs. Unlke he oal enegy of an objec s knec and poenal enegy whch ae always conseved, he quany of acon s consanly changng. No mae wha value he acon may assume dung he objecs flgh, a he desnaon he acon wll a mnmum of all he possble acons ha could have occued. In hs vew of mechancs, he objec seems o choose s pah, wh he knowledge of all possble pahs a he begnnng of he moon. Maupeus woe... I s no n he lle deals... ha we mus look fo he supeme Beng, bu n phenomena whose unvesaly suffes no excepon and whose smplcy lay hem que open o ou sgh. [Gle9], [Feyn64], [You68]. 8 8 Copygh 000, 001

9 pacle could be descbed by he pacles poson and momenum a a sngle pon n space and me. Ths pocess would be obsevable f all physcal enes wee nfnely dvsble. Howeve f naue s somehow lmed n s dvsbly han he acon dung a pocess can change only by a fne amoun ħ, han he pecse deemnaon of a pacle s moon can neve be deemned. In ode o deemne he pacles moon pecsely, he momenum and poson mus be known a he same pon n space and me. Snce he acon s he poduc of momenum and a measued spaal neval ha mus be aken as nfnesmal, he acon becomes nfnesmal and hus smalle han some lm ħ. The esul s ha he momenum becomes nfne, losng all knowledge of he pacles acon. The esul s ha he pacle s acon becomes quanzed so ha s poson and momenum can no be smulaneously known. The full mpac of hs esul wll be developed n lae secons ORDINARY MAXIMUM AND MINIMUM THEORY The calculus of vaaons has been an mpoan banch of mahemacal physcs fo nealy hee cenues. The ask of fndng pons a whch a funcons possesses a maxmum o mnmum s common n he analyss physcal poblems. In he calculus of vaaons, funconal foms ae found n whch negals assume maxmum o mnmum values. These foms may conan seveal vaables and descbe muldmensonal pocesses. Befoe consdeng maxma and mnma of an negal funcon, he heoy of he calculus of funcons of a sngle vaable wll be examned. Le fx () be a connuous funcon of a sngle vaable, x, havng a maxmum o mnmum value a x= a. The fo a suffcenly smallε, hee s a maxmum a, and a mnmum a, fa ( +ε ) fa () < 0, (8.10) fa ( +ε ) fa () > 0. (8.11) Takng he maxmum case and assumng fa ( +ε) can be expanded n posve negal powes of ε, by Taylo's heoem, gves, +ε =ε + ε 3 fa ( ) fa () fa ()½ fa () + O ( ε ). (8.1) Copygh 000,

10 The Landau symbol, O, has he meanng: O ( ε 3 ) possesses he popey 3 3 ha as ε 0, he quany 1 ε O ( ε) s bounded. Fom Eq. (8.10) and Eq. (8.1) a a maxmum o a mnmum he sgn of fa ( +ε ) fa () s ndependen of he sgn of ε, and so fom Eq. (8.1) fa ( ) = 0. Fom Eq. (8.10) and Eq. (8.1) follows ha a a maxmum ( fa ) s negave and fom Eq. (8.11) and Eq. (8.1) ha a a mnmum ( fa ) s posve. Alenavely a a maxmum fa ( ) s a deceasng funcon of a and a a mnmum s an nceasng funcon of a. I s possble ha fa ( ) = 0 and ha fa () s nehe a maxmum o mnmum of fx (). Such a condon occus when fa ( ) = 0 and () fa = 0, and ( fa ) 0. I s hen cusomay o say ha fa () s a saonay value of fx (). In geneal all oos of fx ( ) = 0 ae sad o gve se o saonay values of fx (). Wh hs bef backgound he Lagangan fomalsm wll developed n he nex secon Lagangan Fomalsm and he Calculus of Vaaons The Lagange fomalsm wll be developed hough a smple example he moon of a pacle wh mass m n a hamonc oscllao poenal gven by Vx () = kx /. Accodng o Newon's second law of moon, he acceleaon of he pacle s deemned by, mx = kx, (8.13) whch has he well known soluon, x = x cos( ω +φ), (8.14) 0 whee ω= km, s he angula fequency, he consans x 0 and φ ae deemned by he nal condons. Consde wo mes 1 when he pacle s a poson x 1 and when he pacle s a poson x. The pah he pacle follows beween mes 1 and can be descbed by he quany, 1 dx S = L x, d, (8.15) d 8 10 Copygh 000, 001

11 whee he dffeence beween he knec enegy T and he poenal enegy V, L = T Vs called Lagangan. The quany S, whch n he pas was called Hamlon s Pncpal Funcon, bu s now called he acon funcon. Dmensonally, he acon s an enegy mes a me and has smla dmensons as Planck's consan. The acon, S, s a funconal of x, ha s s a funcon of he funcon x, () whch descbes he pah sasfyng he wo consans ha x () a me = 1 assumes he value x 1, whle x () a me = assumes he value x. Apa fom hese consans he pah may be abay. The acon S s hen a funcon of he dffeen pahs sasfyng he wo consans. In ode o fomulae he exemum on S, a famly of funcons s consdeed, gven by, x (,) α = x (,0) +αη(), (8.16) whee he funcon x (,0) s he one coespondng o he exemum. The funcon η( ) s abay, excep ha s sasfes he consans η () =η ( ) = 0. 1 The acon S s hen a funcon S () α of he paamee α, ( ) S() α = dl x (,), α x (,), α. (8.17) 1 Ths expesson allows fo he possbly ha he exemum may depend explcly on me wh he foce consan of he hamonc oscllao beng a funcon of me, k = k. () The exemum condon s gven by, S = 0, α 0 By dffeenang Eq. (8.18) wh espec o he paamee α gves, 1 α= S L d L = η( ) α d x d x (8.18) (8.19) Copygh 000,

12 Inegang by pas [4] n ode o eplace η by η, esuls n, S L d L = η α (), d x d x 1 (8.0) snce, L L d L η = η η d () d, x x d x d L = 0 η () d. d x 1 (8.1) The exemum condon Eq. (8.1) hen becomes, d L L = 0, d x x snce η( ) s abay excep fo he condon η () =η ( ) = 0. 1 (8.) Eq. (8.) s named he Lagange equaon. The devave of Lagange's equaon saed fom a consdeaon of he nsananeous sae of he sysem and small vual dsplacemens abou he nsananeous sae,.e. fom a dffeenal pncple such as D'Alembe's pncple. [5] I s also possble o oban Lagange's equaons fom a 4 The mehod of negaon by pas employs he deny f ( dg d) = d( fg) d g( df d ), wee f and g boh ae funcons of. When boh sdes of hs equaon ae negaed wh espec o ove he neval fom 1 o he esul s { ( )} = [ ] {( ) } f dg d d fg df d g d I s supsng ha hee can be seveal fomulaons of he pncpals of mechancs. Once s undesood ha mechancs s a descpon of moon, hen dffeen mehods of descbng hs moon can seve dffeen puposes. Alhough Newon s mehod of descbng he moon of pacles has long been he mos useful smple appoach ohe fomulaons have been ceaed whch aemp o smplfy he soluons of vaous ypes of poblems. These alenave fomulaons dffe consdeably n he conceps of mass and foce. Some ae esaemens of Newon s laws, whle ohes noduce new conceps. D Alembes s pncpal s a esaemen of Newon s Laws whch seeks o educe dynamcs o sacs usng Newon s concep of mass and foce. D Alembe fomulaed hs pncpal n 1743 n he wok Taê de Dynamque, whch was evsed n 1758 as A geneal pncpal fo fndng he moons of seveal bodes whch eac on each ohe n any fashon. In 8 1 Copygh 000, 001

13 pncple whch consdes he ene moon of he sysem beween mes 1 and and small vual vaaons of he ene moon fom he acual moon GENERALIZED COORDINATES I s no always convenen o use Caesan coodnaes when solvng poblems n Newonan mechancs. Alenave coodnae sysems esul n smple soluons. The analyss of he moon of a pendulum s an example. The Lagangan fomulaon of he equaons of moon s well sued fo hese non Eucldean o consaned dynamcal vaables. The genealzed coodnaes pesened n hs secon ae no alenaves o Eucldean coodnae sysems, bu ae descpons of he confguaon of he mechancal sysem wh z degees of feedom. An example of such a D Alembe s fomulaon he concep of vual dsplacemens s used o descbe he moons of pacles n he pesence of exenal foces. Ths pncpal can be saed n a geneal fom as:... f hee ae n pacles 1,, 3,..., n aced on by foces F, F, F,, 1 3 especvely, and f hese ae gven abay (vual) dsplacemens d, d, d,, whee s he poson veco of he pacle, he condon 1 3 of equlbum unde he acon of he foces s F d F d F d F d [Lnd56] = n n The second ype of fomulaon employees he concep of enegy, Hamlon s pncpal beng he one ulzed hee. In 1894 Hench Hez publshed Pncpals of Mechancs n whch he e esablshed he pncpals of mechancs wh logcal a conssency no found n he usual Newonan pesenaons of he day. Such lae 19h cenuy woks usually conaned meaphyscal unceanes and vagueness. Hez aemped o educe dynamcs o knemacs, avodng conceps lke foce, mass and enegy. The fundamenal pncpal of Hezan mechancs s: Evey fee sysem emans ehe s a sae of es o n unfom moons along a saghes pah [Lnd56]. Snce mos sysem encouneed n paccal suaons ae non fee Hez assumes ha evey pa of a non fee sysem s pa of a fee sysem. Evey moon of a fee sysem o s non fee pas obeys he fundamenal pncpal called naual moon, and Hezan mechancs s only concened wh naual moon. Thee wee seous poblems wh he descpon of moon, snce n ode o make he pncpal wok was necessay o nvoke he exsence of ohe pacles, whch may no be mmedaely dscenble. The wods concealed became assocaed wh Hez s pncpals. Even hough Hez s concep dd no lead o a pacle mehod of compung he moons of pacles dd lay he foundaon fo Hamlon s pncpal, n whch he concealed aspecs of he moon becomes he enegy of he sysem whch s mnmzed dung he pacles moon. Copygh 000,

14 sysem s n pacles each wh mass m and coodnaes z, whee = 1,,, n. By choosng any ndependen funcons of he ognal 3n dynamcal coodnaes z, = ( ) = ( ) = ( ) j ( ) q q q qz, whee j = 1,,,3n, he new dynamcal vaables can be defned. These new vaables ae he genealzed coodnaes, q = q ( ) and he genealzed veloces, q = dq d [Doug90]. ( ) I s saghfowad o genealze he Lagange fomalsm o sysems wh moe degees of feedom han ae found n classcal mechancs. Ths may be done by consdeng a sysem descbed by he se of genealzed coodnaes q, whee assumes he values 1,,,s. As befoe, he Lagangan s he dffeence beween he knec and poenal eneges s gven by L = T V. The Lagange equaons ae deved by equng he acon, ( 1,,,, ; 1,,,, ; ) 1 S = L qq q qq q d, (8.3) o have an exemum. The equaons of moon o Lagange's equaons, hen become, d L L = 0, (8.4) d q q whee he me devave of he genealzed coodnae s now gven by, dq q = [Wh37]. [6] Ths fomulaon of Lagange's equaons s mos d poweful fo heoecal puposes and wll be used n he developmen of he elecomagnec feld Hamlonan. The mos sgnfcan popey of Eq. (8.4) s s nvaance wh espec o abay coodnae ansfomaons. Poof of hs mpoan abue s gven n [You68], pp The Lagangan of a sysem s no unque. The oal me devave of an abay funcon can be added o he Lagangan L o gve a new Lagangan L, such ha, 6 Fo a pacle movng n a me ndependen poenal, he Lagangan does no depend explcly on me. If he sysem s solaed n such a manne so ha me ansfomaon nvaance s peseved, he Lagange equaons can be shown whou an explc me dependence and wen as, L( q, q ) Copygh 000, 001

15 ( qq,, ) ( qq,, ) fq (, ) L = L +, (8.5) whch has he same popees as he nal Lagangan L wh espec o he pncple of leas acon. The acon funcon S elave o he Lagangan L s gven as, 1 ( (), ) ( (), ) S = L d = S+ fq fq. (8.6) 1 1 Snce he nal and fnal posons ae fxed S and S have he same exemum and dffe only by a consan HAMILTONIAN FORMALISM If he only use of he Lagangan acon pncple s o egeneae he equaons of moon would be consdeed neesng bu edundan. Howeve, he Lagangan acon pncple povdes a descpon of he dynamcs of a sysem whch conans moe nfomaon han suppled by Newon's equaons of moon. Fs he acon funcon, S, s a global saemen abou he sysem, fom whch a local dffeenal equaon can be deved by mposng he exemum condon. The acon s global n he sense ha eceves conbuons fom he ene ajecoy of he pacle n moon. As such he acon ecods he hsoy of he pacle's moon. Second, alhough he acon s exemzed by he se of classcal ajecoes ha ae he soluon o Eq. (8.4), he acon can be evaluaed fo any ajecoy. Thd, he acon appoach allows he defnon of he canoncal coodnaes of poson and momenum o be genealzed by descbng he enegy of he sysem hough he Hamlonan. The soluon of a dynamcal poblem by Lagange's mehod eques he negaon of n second ode dffeenal equaons n he n unknowns q 1,, q n. An alenave sysem poposed by Hamlon consss of n fs ode dffeenal equaons n n unknowns, and has he advanage ha s smple and concse n s fomulaon [Ham35] [7]. In 7 Hamlon was an asonome and mahemacan n Dubln, Ieland. As a chld podgy he was able o anslae fom Lan and Geek a age 5 and had maseed 13 languages by age 13. He suded a Tny College, Cambdge and was apponed pofesso of asonomy a age. Hamlon s woks nclude mechancs and opcs as well as he dscovey of quanenons, whch genealze complex numbes o a non communave algeba. Alhough Hamlon ded 35 yeas befoe Planck publshed hs heoy of quanum mechancs, Hamlon has been mmoalzed hough hs assocaon wh he enegy opeao n Schödnge s wave equaon. Copygh 000,

16 Hamlon's equaons, he canoncal equaons consue he bass fo he quanum mechancal fomulaon of elecodynamcs. An ognal concep noduced by Hamlon s he genealzed momenum, whch s defned as, δs L p () =. (8.7) δq () q () Defnng he genealzed momena as p = mx allows he noducon of he Hamlonan by he ansfomaon, The dffeenal d H s, d ( pq) = pq ( pq, ) ( qqpq (,) ) H, L,. (8.8) snce, dh d = + = L pdq qdp dl pdq + qdp =, (8.9) L ṗ =. (8.30) q I follows fom Eq. (8.9) ha he equaons of moon fo he sysem may be wen as, H q H q p, q s L = ps s q q p q s L L q s = ps, s q s s q qs q q q s d L L qs = ps, s qs d q s q q s q p = p. p, p [8] (8.31) 8 In hs devaon of he Lagange equaons he ndex subscp o s s used o ndcae he h o sh coodnae, n ode o dsngush beween each genealzed coodnae n a muldmensonal coodnae space. The suffx o he backeed devave ndcaes ha he q's ae kep consan whle he ndex s summed ove he s's Copygh 000, 001

17 H p H p q q s L = q + ps, s p p = q., q L q s L q s = q + ps, s q p s q p q s q s q (8.3) whch consue Hamlon's equaons. The change of vaables fom (,) qq o ( qp,) esuls fom he ansfomaon n Eq. (8.31) whch s known as a Legende ansfomaon. [9] Resang he equaons of moon n ems of he coodnaes and he momena esuls n he Hamlonan of he sysem, H( pq,,) = (,,) pq L qq, (8.33) whch descbes he dynamcs of he sysem n ems of he sum of he knec enegy and he poenal enegy. Hamlon's pncple eques ha he pah aken by any physcal sysem beween wo saes a specfed mes and wh fxed values of he vaables mus be such ha he value of he funcon δ 1 ( T Vd ) mus be an exemum. In hs fom Hamlon's o pncple s suffcen o geneae boh he equaons of moon of he sysem and he bounday condons fo any connuous feld wh localzed foms of enegy. [10] Fo he acual soluon of poblems, he equaons of Lagange ae moe convenen han hose of Hamlon, snce he fs sep n negang 9 The change n bass fom ( x, x, ) o ( x, p, ) s accomplshed hough he Legende ansfomaon. Consde a funcon of he vaables f ( x, y) so ha a dffeenal of f has he fomdf = udx + vdy, whee u = f x and v = f y. To change he bass of he descpon fom x, y o he ndependen vaables u, y, so ha dffeenal quanes ae expessed n ems of he dffeenals du and dy. Le g be a funcon of u and y defned by he equaon g = f ux. A dffeenal of g s hen gven as dg = df udx xdu whch has he desed fom. The quanes x and v ae now funcons of he vaables u and y by he elaons x ( g u) and v. =, v ( g y) =, whch ae he convese of he above elaons fo u 10 Fo physcal applcaons, L( x, x, ) mus be chosen so ha he Eule Lagange equaons epesen he coec equaons of moon. Copygh 000,

18 Hamlon's equaons s o educe he numbe by half, an opeaon whch leads back o he ognal Lagange equaons. The domnae poson of he equaons of Lagange n he hsoy of dynamcs can bes be ced n Hamlon's own wods. The heoecal developmen of he laws of moon of bodes s of such nees and mpoance, ha has engaged he aenon of all he mos emnen mahemacans, snce he nvenon of dynamcs as a mahemacal scence by Galleo... Among he successos of hose llusous men, Lagange has pehaps done moe han any ohe analys, o gve exen and hamony o such deducve eseaches, by showng he moons of sysems of bodes may be deved fom one adcal fomula; he beauy of he mehod so sung he dgny of he esuls, as o make hs gea wok a knd of scenfc poem. [Ham34] Canoncal Coodnaes and Posson Backes The Hamlonan and he Lagangan ae elaed as, H L =. λ pq, λ pq, (8.34) The Hamlonan equaons of moon ae also known as he canoncal equaons, esulng n, q = H = = ; p H ; H L. p q (8.35) The Hamlonan fomulaon povdes an elegan descpon of mechancs n whch he poson and momenum of each pacle s eaed as hough hey wee ndependen quanes. The coodnaes q and he momenum p ae acually allowed o be moe geneal han jus Caesan coodnaes. In pacula Hamlon's equaons fom he bass of he quanum fomulaon of Maxwell's equaons, snce he feld poenal becomes an negal pa of he canoncal momenum and s eaed as f wee a genealzed coodnae n he Lagange equaons. [11] 11 Equaons Eq. (8.31) and Eq. (8.3) ae also vald fo a sysem of N pacles wh coodnaes ( x 1, x,, x 3N ). The foces beween he pacles can be epesened by he poenal enegy V ( x1, x,, x3 N ). The Newonan equaons of moon n Eq. (8.33) may be 8 18 Copygh 000, 001

19 Hamlon s equaons can be esaed usng he Posson backe noaon. Fo a sysem of s genealzed coodnaes and s genealzed upq, momena, he Posson backe can be defned fo any wo funcons ( ) and vpq (, ) s an ansymmec opeaon gven as, { uv, } s u v v u = q p q p = 1 (8.36) Ths fom can be ewen as, S u u u = q + p = 1 q p (8.37) fo he vaaon of any physcal quany u. Usng Hamlon s equaons, u = H, u. (8.38) Subsung he genealzed coodnaes of poson and momenum gves, and q q, H, (8.39) = { } p p, H. (8.40) = { } Seveal denes esul fom he Posson backe noaon ha wll be useful n he fomulaon of quanum mechancs, Fs, {, } =δ, qp. (8.41) j The Posson backes nvolvng only p o q vansh as, consdeed as Eule equaons coespondng o he equemen ha he funcon S should be an exemum. Ths alenave concep s mpoan because enables he equaons of moon o be expessed n a fom ha s nvaan wh espec o he coodnaes. The exemum equemen (Hamlon's Pncple) conans only physcal quanes such as knec and poenal enegy, whch ae ndependen of he coodnae sysem. Fo any abay coodnae sysem, he momena p = L x do no n geneal have dmensons of ue momenum, wh he coodnae x j beng a dmensonless angula quany. The poduc of any momenum p j wh s assocaed coodnae x j always has he dmensons of acon (enegy canoncally conjugae. j j me). The momenum p j and he coodnae x j ae sad o be Copygh 000,

20 { } { } pp, = qq, = 0. (8.4) Quanes whose Posson backes ae zeo, commue, and hose whose Posson backes ae equal o 1 ae canoncally conjugae. Fom Eq. (8.39), can be seen ha any quany ha commues wh he Hamlonan does no vay wh me. Usng Eq. (8.36), f he Posson backe of a funcon u wh a consan c gves, and, and, { uc, } = 0, (8.43) {, } = {, } Usng he ules of dffeenaon, uv vu. (8.44) { +, } = {, } + {, } u vw uw vw, (8.45) {, } = {, } + {, } uvw uvw vuw. (8.46) Usng he Posson backes and Hamlon's equaons of moon, and du u = + d { u, } pq, H (8.47) {, } =δ, qq (8.48) j pq, j I s he Posson backe fomalsm ha wll seve as he bass of he quanum mechancal commuao algeba developed n he subsequen secons STANDARD LAGRANGIAN OF CLASSICAL ELECTRODYNAMICS To hs pon n he monogaph he adaon feld densy has been eaed n fom descbed by Eq. (4.33). As such he deals behnd hs equaon have no been developed. Befoe poceedng wh he opeao appoach o he quanum feld equaons, he Hamlonan fom of he feld equaons wll be addessed. The feld enegy descbed n Eq. (4.33) can be deved hough he expanson of Maxwell's equaons usng Hamlon's equaons of moon 8 0 Copygh 000, 001

21 wh equaons ( VI ) and ( VII ) epesenng he elecc and magnec enegy denses of he feld pope. These eneges ae consdeed o esde n he feld and o be localzed by ( VI ) and ( VII ) n evey volume elemen, such ha a volume dv conans feld enegy n he amoun 1 ( E + B ) dv. V The developmen of he Hamlonan fom of he feld enegy, sang fom Maxwell's equaons s one appoach. [1] Anohe appoach s o consuc he feld equaons and he assocaed Hamlonan, by seachng fo he Lagangan ha esuls n he pope feld equaons. Tha s he appoach aken hee and by [You68]. [13] The Lagange and Hamlon equaons of moon developed n he pevous secon wll be used o deve he Hamlonan fo he adaon feld, whch n un wll be used o deve he opeao fomulaon of he elecomagnec adaon feld. Ths appoach wll be woked ou n deal sang wh he equaons of moon fo a chaged paal n an elecomagnec feld and concludng wh he Hamlonan fo he same feld. In hs case he foces on he pacle ae no deved fom he feld poenal, bu ahe ase fom he velocy of he pacle as avels hough he feld. The acceleaon (change n momenum) of he chaged pacle s gven by he Loenz equaon, { E ( B) } F p = e + v (8.49) whee v of he velocy of he pacle, e s he pacle's chage and E and B ae he elecc and magnec felds. The E and B felds ae deved fom he veco poenals n he usual manne, The acceleaon equaon, Eq. (8.49), becomes, A E = φ, (8.50) B = A. 1 Fo a moe goous developmen of he adaon feld enegy and he assocaed consevaon laws deved fom Maxwell's equaons ahe hen he Lagangan, see 11 of [Eyge7]. 13 Ths appoach depends on developng he dealed Lagangan fo he adaon feld equaons n boh vacuum and souce foms. Ths secon and he efeence [You68] povdes he lowes deals of he Lagangan appoach o quanum feld poblems. As such gves nsgh o he cuen eseach acves n pacle physcs and quanum feld heoy. Copygh 000,

22 A { q A } p = e φ + ( ), A = e φ + { ( q A)( q ) A}. (8.51) In ode o expess he equaons n he Lagangan fom, Eq. (8.8), he followng Lagangan funcon s used, The genealzed momena, usng Eq. (8.3), as, L = T eφ+ e Aq (8.5) p L = q, = p + ea. (8.53) The fom of Eq. (8.53) s smla o he Gauge ansfomaons gven n Eq. (4.7) and Eq. (4.8) and developed fuhe. Fnally he Hamlonan s sll equal o he oal enegy, usng Eq. (8.5), H = pq. L (8.54) Movng o he Lagangan fo a chaged body ahe han a sngula pon, he Lagangan s gven by, L = T + ρ A v φ dv. (8.55) whee v s he velocy of he chage a any pon, ρs he chage densy and dv s an elemen of volume. To oban Eq. (8.55) he chaged body s consdeed as a sysem of muually aacng pacles. The v's and ρ ae eaed as funcons of he genealzed coodnaes used o defne he sysem. Accodng o Eq. (8.55), he genealzed momena wll be deemned by, vs p = p + ρ AsdV, q and he Hamlonan, usng he noaon of Eq. (8.33) s gven as, s (8.56) 8 Copygh 000, 001

23 v s H = H qp, ρ AsdV + ρφ dv. (8.57) s q Tme Independen Lagangan The developmen of he acual elecomagnec feld equaons depends on a fundamenal dffeence beween he pevous equaons. Up o hs pon he feld equaons conaned one ndependen vaable,, and seveal dependen vaables, q. In he elecomagnec feld equaons boh he q 's and ae ndependen vaables, and he quanes specfyng he feld ae he dependen vaables. Ths suaon can be descbed by consdeng a feld defned by he quanes fq (,). A Lagangan, L, can be found whch s a funcon of he f f 's, he 's and he f 's, so chosen ha Lagange's dffeenal q equaons, f f f + + = 0 s q1 q1 q1 s f f f + + = 0 q q s q s f f f + + s q q q s = 0 (8.58) fo he negal L dqd o be saonay esulng n he equaons of he feld. Rewng Eq. (8.58) by consdeng f 's o be funcons of he genealzed coodnaes evaluaed a a specfc pon n space, esuls n he odnay Lagange equaons fo L and Eq. (8.58) fo L, L L L + = f 0, (8.59) f s qs f q Copygh 000,

24 ae now equvalen, ha s Lagange's equaons ae vald fo boh pon chages and dsbued chages movng n he elecomagnec feld. The Lagangan L, whch s a funcon of he dynamcal vaables can be ewen as, whee L s now called he Lagangan densy. [14] = 3 L L dq, (8.60) Lagangan Densy In ode o develop he undelyng mahemacs necessay fo he quanum feld descpon of elecomagnesm, he sandad Lagangan wll be exended. Wha s needed s a fomalsm ha descbes he obseved phenomena of he adaon neacon wh chaged mae. Ths descpon mus defne he oal Lagangan L and manan nenal conssency, when he numbe of degees of feedom becomes nfne. The Lagangan fomalsm fo sysems of pon pacles and he devaon of he Hamlonan povdes an easy anson o quanum mechancs. The sysems pesened so fa conss of a fne numbe of vaables. Alhough hee ae many physcal sysems wh a fne numbe of degees of feedom, he elecomagnec feld s no one of hem. Thee ae ohe cases such as gases o lquds all of whch have one o moe 14 In he developmen of elecodynamcs, he Lagangan densy s a funcon of he dynamcal vaables A ( ) and da ( ) d whee descbes all he ndvdual pons n he dscee space and descbes all he possble coodnae values. The Lagangan densy funcon of he coodnaes A ( ) and he veloces da ( ) d A, whose pesence shows ha he moon of he coodnae A ( ) and he spaal devaves, denoed by j o he moon of a neghbong pon n he same manne he dscee vaable q depends on q 1 and q +1 [Cohe89]. s coupled These spaal devaves ae no new ndependen vaables bu ae lnea combnaons of genealzed coodnaes. In he sudy of elecomagnec heoy he Lagangan densy akes on he fom ( A, da d, A ) L. j The Lagangan densy ha s used n elecodynamcs conans genealzed coodnae devaves. Such a sucue allows Maxwell's equaons o be descbe he moons of felds coupled fom pon o pon n space. The absence of hese spaal devaves n he Lagangan densy would lead o a heoy whee he elecomagnec feld evolves ndependenly a each pon n space. Snce Maxwell's equaons nvolve he spaal devave of he feld, he Lagangan densy also depends on spaal devaves. 8 4 Copygh 000, 001

25 vaables whch ae funcons of connuous vaables. Thee ae vaous mehods of ansfomng a dscee sysem o a connuous one. One mehod s o consde a connuous lnea elasc sucue as he lm of a sysem of pon pacles and hen o genealze he esuls. A second mehod s o consuc a genealzed vaaonal pncple and he hd mehod s o employ he Foue ansfom o consuc a genealzed se of vaables n Foue space. Thee ae seveal advanages o he hd appoach. Fs he connuous sysem whch was a funcon of he connuous vaable x s ansfomed o a dscee sysem of vaables wh an ndex of k, as long as he sysem s enclosed n a fne volume. By combnng he fs and hd appoach usng he Foue spaal descpon of a lnea elasc medum a anson o he quanum feld descpon can be made. The sang pon sang pon fo hs new Lagangan wll be he same as he anson fom classcal o quanum mechancs Classcal Hamlonan dynamcs. A genealzed appoach φ epesen he dsplacemen amplude can be fomulaed by leng ( ) of he feld a a pon. Ths esuls n he feld havng an nfne numbe of degees of feedom whch mus be specfed a each pon whee whee. Fo a connuous space he summaon n Eq. (8.59) becomes nfne L and mus be eplaced by he negal n Eq. (8.60). In hs way ( ) depends on he amplude of he feld a o nea he pon. Ths φ self and mus conan he me amplude mgh be a funcon of ( ) devaves of φ, jus as he Lagangan of a pacle conans he knec enegy, whch s a funcon of velocy. The Lagangan densy L mus φ, ohewse hee would be no also depend on he spaal devaves of ( ) connecon beween he feld ampludes a neghbong pons n space. The sysem s moe easly quanzed as a dscee fomulaon, snce he Foue coeffcens can be decly noduced as ceaon and annhlaon opeaos. Thee ae some dffcules wh hs appoach, bu hey wll be deal wh n he secon on gauge heoy. Because of he dffculy a smplfed mechancal example wll be used n whch a longudnal wave n one dmenson s used o llusae he dea. Sang wh he one dmenson longudnal wave descbed by he wave equaon, φ φ ρ µ = x 0, (8.61) Copygh 000,

26 whee φ ( x, ) s he dsplacemen of pon x a me. The densy of he elasc medum s ρ and he esong foce of he meda s µ. The lengh of he one dmensonal meda s L whch eques φ x vanshes a he boundaes. Gven hese condons, he dsplacemen funcon can be expanded n a Foue sees as, ( ) ( ) φ x, = φ snkx, (8.6) k whee k has he peodc values nπ Ln, = 0,1,, The peodc bounday condon φ x =0 can now be eplaced by, (, ) (, ) φ x+ L =φ x. (8.63) Anohe smplfcaon s o expand he Foue sees s an exponenal such ha, 1 φ = φ L ( x, ) ( ) k k e kx, (8.64) whee k s now he wave numbe whch can be posve as well as negave. Snce he Foue expanson gven n Eq. (8.64) nvolves complex numbes and φ ( x, ) s a eal quany n ems of φ k and φ k ae elaed o each ohe hough hs complex conjugae φ k =φ k esulng n k ndependen vaables. h The k nsance of φ can be obaned fom he connuous funcon hough Foue ansfom, 1 kx φ = φ( ) k L xedx. (8.65) By leng he exen of he medum end o nfny he followng lms, L and usng and, k L π dk, (8.66) φ k π φ L ( k ). (8.67) The Foue ansfom pas can be gven as, 8 6 Copygh 000, 001

27 φ ( x, ) = 1 φ( ) π kx kedx,, (8.68) φ ( k, ) = 1 φ( ) π kx xe, dx. (8.69) The Foue ansfom of he wave equaon now becomes, φk ρ + k φ k = 0 (8.70) whch s now he equaons of moon of he sysem, bu conanng an nfne numbe of degees of feedom. These equaons can be deved fom he Lagangan, ( φ 1 φ ) = ρ φφ 1, µ φφ k L. (8.71) k k k k k k k k Usng he elaons gven n Eq. (81) and Eq. (80) hese ansons fom he dscee fomulaon usng φ k o he connuous fomulaon usng φ( x ) can be made. Consdeng he fs em of he Lagangan, 1 1 k k k ( k) ( ) ρ φφ = ρ φ φ kdk, 1 = ρ π φ φ kdk, 1 = ρ φ φ ( x) ( ) kx ( xdx ) e ( ) = T = φ φ 1 ( xdx ) ( x) ( x) xdx, whch s he knec enegy densy. The second em,. (8.7) Copygh 000,

28 1 1 k k k ( ) ( ) ( ) µ k φφ = µ k φ k φ kdk, kx ( xke ) ( ) 1 = µ π φ φ kdxdk, 1 kx = µ π φ( x) e φ( kdxdk ), x 1 kx = µ π φ( x) dx φ ( ) e kdk, x 1 = µ φ φ x φ 1 = µ dx, x = V xdx. ( x) dx ( x), (8.73) whch s he poenal enegy densy. 8 8 Copygh 000, 001

29 Fusa f pe plua, quod fe poes pe paucoa OR Essena non sun mulplcanda paee necessaem (I s van o do wh moe wha one can do wh less) OR (Enes ae no be mulple beyond necessy) [15] Occam s Razo 15 Abued o Wllam of Occam, o Ockham, o pobably Oakham n Suey ( ), Oxfod schola n he Ode of Fancscan Fas. Occam s azo s wdely used n scenfc analyss wh an nepeaon akn o: One should always choose he smple of wo ohewse compeng descpons of physcal phenomena. [Doug90]. Copygh 000,