CHAPTER 3 AGRANGIAN MECHANICS 3 Inoucon The usual way of usng newonan mechancs o solve a poblem n ynamcs s fs of all o aw a lage, clea agam of he sysem, usng a ule an a compass Then mak n he foces on he vaous pas of he sysem wh e aows an he acceleaons of he vaous pas wh geen aows Then apply he euaon F ma n wo ffeen econs f s a wo-mensonal poblem o n hee econs f s a heemensonal poblem, o τ I f oues ae nvolve Moe coecly, f a mass o a momen of nea s no consan, he euaons ae F p an τ In any case, we ave a one o moe euaons of moon, whch ae ffeenal euaons whch we negae wh espec o space o me o fn he ese soluon Mos of us wll have one many, many poblems of ha so Somemes s no all ha easy o fn he euaons of moon as escbe above Thee s an alenave appoach known as lagangan mechancs whch enables us o fn he euaons of moon when he newonan meho s povng ffcul In lagangan mechancs we sa, as usual, by awng a lage, clea agam of he sysem, usng a ule an a compass Bu, ahe han awng he foces an acceleaons wh e an geen aows, we aw he velocy vecos (nclung angula veloces) wh blue aows, an, fom hese we we own he knec enegy of he sysem If he foces ae consevave foces (gavy, spngs an seche sngs), we we own also he poenal enegy Tha one, he nex sep s o we own he lagangan euaons of moon fo each coonae These euaons nvolve he knec an poenal eneges, an ae a lle b moe nvolve han F ma, hough hey o ave a he same esuls I shall eve he lagangan euaons of moon, an whle I am ong so, you wll hnk ha he gong s vey heavy, an you wll be scouage A he en of he evaon you wll see ha he lagangan euaons of moon ae nee ahe moe nvolve han F ma, an you wll begn o espa bu o no o so! In a vey sho me afe ha you wll be able o solve ffcul poblems n mechancs ha you woul no be able o sa usng he famla newonan mehos, an he spee a whch you o so wll be lme solely by he spee a whch you can we Inee, you scacely have o sop an hnk You know sagh away wha you have o o Daw he agam Mak he velocy vecos We own expessons fo he knec an poenal eneges, an apply he lagangan euaons I s auomac, fas, an enoyable Incenally, when agange fs publshe hs gea wok a méchanue analyue (he moen Fench spellng woul be mécanue), he pone ou wh some pe n hs noucon ha hee wee no awngs o agams n he book because all of mechancs coul be one analycally e wh algeba an calculus No all of us, howeve, ae as gfe as agange, an we canno om he fs an vey mpoan sep
of awng a lage an clea agam wh ule an compass an makng all he velocy vecos 3 Genealze Coonaes an Genealze Foces In wo-mensons he posons of a pon can be specfe ehe by s ecangula coonaes (x, y) o by s pola coonaes Thee ae ohe possbles such as confocal concal coonaes ha mgh be less famla In hee mensons hee ae he opons of ecangula coonaes (x, y, z), o cylncal coonaes (ρ,, z) o sphecal coonaes (,, ) o agan hee may be ohes ha may be of use fo specalze puposes (nclne coonaes n cysallogaphy, fo example, come o mn) The sae of a molecule mgh be escbe by a numbe of paamees, such as he bon lenghs an he angles beween he bons, an hese may be vayng peocally wh me as he molecule vbaes an wss, an hese bons lenghs an bon angles consue a se of coonaes whch escbe he molecule We ae no gong o hnk abou any pacula so of coonae sysem o se of coonaes Rahe, we ae gong o hnk abou genealze coonaes, whch may be lenghs o angles o vaous combnaons of hem We shall call hese coonaes (,, 3,) If we ae hnkng of a sngle pacle n hee-mensonal space, hee wll be hee of hem, whch coul be ecangula, o cylncal, o sphecal If hee wee N pacles, we woul nee 3N coonaes o escbe he sysem unless hee wee some consans on he sysem Wh each genealze coonae s assocae a genealze foce P, whch s efne as follows If he wok eue o ncease he coonae by δ s P δ, hen P s he genealze foce assocae wh he coonae I wll be noe ha a genealze foce nee no always be mensonally euvalen o a foce Fo example, f a genealze coonae s an angle, he coesponng genealze foce wll be a oue One of he hngs ha we shall be wanng o o s o enfy he genealze foce assocae wh a gven genealze coonae 33 Holonomc consans The complee escpon of a sysem of N unconsane pacles eues 3N coonaes You can hnk of he sae of he sysem a any me as beng epesene by a sngle pon n 3N-mensonal space If he sysem consss of molecules n a gas, o a cluse of sas, o a swam of bees, he coonaes wll be connually changng, an he pon ha escbes he sysem wll be movng, pehaps compleely unconsane, n s 3N-mensonal space
3 Howeve, n many sysems, he pacles may no be fee o wane anywhee a wll; hey may be subec o vaous consans A consan ha can be escbe by an euaon elang he coonaes (an pehaps also he me) s calle a holonomc consan, an he euaon ha escbes he consan s a holonomc euaon If a sysem of N pacles s subec o k holonomc consans, he pon n 3N-mensonal space ha escbes he sysem a any me s no fee o move anywhee n 3Nmensonal space, bu s consane o move ove a suface of menson 3N k In effec only 3N k coonaes ae neee o escbe he sysem, gven ha he coonaes ae connece by k holonomc euaons Incenally, I looke up he wo holonomc n The Oxfo Englsh Dconay an sa ha he wo was fom he Geek őλος, meanng whole o ene an νóµ-ος, meanng law I also sa apple o a consane sysem n whch he euaons efnng he consans ae negable o aleay fee of ffeenals, so ha each euaon effecvely euces he numbe of coonaes by one; also apple o he consans hemselves As an example, conse a ba of we soap slheng aoun n a hemsphecal basn of aus a You can escbe s poson n he basn by means of he usual wo sphecal angles (, ); he moon s ohewse consane by s emanng n conac wh he basn; ha s o say s subec o he holonomc consan a Thus nsea of neeng hee coonaes o escbe he poson of a oally unconsane pacle, we nee only wo coonaes O agan, conse he ouble penulum shown n fgue XIII, an suppose ha he penulum s consane o swng only n he plane of he pape o of he sceen of you compue mono x FIGURE XIII l y l (x, y ) (x, y ) Two unconsane pacles woul eue sx coonaes o specfy he posons bu hs sysem s subec o fou holonomc consans The holonomc euaons z 0
4 an z 0 consan he pacles o be movng n a plane, an, f he sngs ae kep au, we have he aonal holonomc consans x + y l an ( x x ) + ( y y) l Thus only wo coonaes ae neee o escbe he sysem, an hey coul convenenly be he angles ha he wo sngs make wh he vecal 34 The agangan Euaons of Moon Ths secon mgh be ough bu on be pu off by I pomse ha, afe we have go ove hs secon, hngs wll be easy Bu n hs secon I on lke all hese summaons an subscps any moe han you o Suppose ha we have a sysem of N pacles, an ha he foce on he h pacle ( o N) s F If he h pacle unegoes a splacemen δ, he oal wok one on he sysem s F δ The poson veco of a pacle can be wen as a funcon of s genealze coonaes; an a change n can be expesse n em of he changes n he genealze coonaes Thus he oal wok one on he sysem s F, δ 34 whch can be wen F δ 34 Bu by efnon of he genealze foce, he wok one on he sysem s also P δ 343 Thus he genealze foce P assocae wh genealze coonae s gven by P F 344 Now F m, so ha P m 345
5 Also + 346 Subsue fo fom euaon 346 no euaon345 o oban m P 347 Tha s, m P 348 Ths vey las sep mgh no be obvous, so le me llusae by an example Conse he elaon beween ecangula an sphecal coonaes cos sn x Then sn sn cos cos an cos sn x x Also sn sn cos cos cos sn + x Theefoe, sn sn cos cos x an so we see ha x x We connue The knec enegy T s m m T 349
6 Theefoe T m 340 T an m 34 On subsung hese n euaon 348 we oban P T T 34 Ths s one fom of agange s euaon of moon, an ofen helps us o answe he ueson pose n he pas senence of secon 3 namely o eemne he genealze foce assocae wh a gven genealze coonae If he vaous foces n a pacula poblem ae consevave (gavy, spngs an seche sngs, nclung valence bons n a molecule) hen he genealze foce can be obane by he negave of he gaen of a poenal enegy funcon e V P In ha case, agange s euaon akes he fom T T V 343 In my expeence hs s he mos useful an mos ofen encounee veson of agange s euaon The uany T V s known as he lagangan fo he sysem, an agange s euaon can hen be wen 0 344 I s my unesanng ha hs las veson s of eep heoecal sgnfcance, alhough I on ecall my eve havng use n solvng any acual poblem Euaon 343 s my favoe, pove he foces ae consevave (Fo consevave foces, see Chape 9, hough n pacce means gavy, spngs an sngs)
7 In each of euaons 34, 3 an 4 one of he s has a o ove You can see whch one s by hnkng abou he mensons of he vaous ems Do has menson T So, we have now eve agange s euaon of moon I was a ha suggle, an n he en we obane hee vesons of an euaon whch a pesen look ue useless Bu fom hs pon, hngs become ease an we aply see how o use he euaons an fn ha hey ae nee vey useful 35 Acceleaon Componens In secon 34 of chape 3 of he Celesal Mechancs book, I eve he aal an ansvese componens of velocy an acceleaon n wo-mensonal coonaes The aal an ansvese velocy componens ae faly obvous an scacely nee evaon; hey ae us ρ an ρ Fo he acceleaon componens I epouce hee an exac fom ha chape: The aal an ansvese componens of acceleaon ae heefoe ( ρ ρ ) an ( ρ + ρ ) especvely I also eve he aal, meonal an azmuhal componens of velocy an acceleaon n hee-mensonal sphecal coonaes Agan he velocy componens ae ahe obvous; hey ae, an sn, whle fo he acceleaon componens I epouce hee he elevan exac fom ha chape On gaheng ogehe he coeffcens of ˆ, ˆ, ˆ, we fn ha he componens of acceleaon ae: Raal: Meonal: Azmuhal: sn + sn cos sn + cos + sn You mgh lke o look back a hese evaons now Howeve, I am now gong o eve hem by a ffeen meho, usng agange s euaon of moon You can ece fo youself whch you pefe We ll sa n wo mensons e R an S be he aal an ansvese componens of a foce acng on a pacle ( Raal means n he econ of nceasng ρ; ansvese means n he econ of nceasng ) If he aal coonae wee o ncease by δρ,
8 he wok one by he foce woul be us R δρ Thus he genealze foce assocae wh he coonae ρ s us P ρ R If he azmuhal angle wee o ncease by δ, he wok one by he foce woul be Sρ δ Thus he genealze foce assocae wh he coonae s P Sρ Now we on have o hnk abou how o sa; n agangan mechancs, he fs lne s always T, an I hope you ll agee ha T m( ρ + ρ ) 35 If you now apply euaon 34 n un o he coonaes ρ an, you oban P m( ρ ρ ) an P mρ( ρ + ρ ), 35a,b ρ an so R m( ρ ρ ) an S m( ρ + ρ ) 353a,b Theefoe he aal an ansvese componens of he acceleaon ae ( ρ ρ ) an ( ρ + ρ ) especvely We can o exacly he same hng o fn he acceleaon componens n heemensonal sphecal coonaes e R, S an F be he aal, meonal an azmuhal (e n econ of nceasng, an ) componens of a foce on a pacle If nceases by δ, he wok one s R δ If nceases by δ, he wok one s S δ If nceases by δ, he wok one s F sn δ Theefoe R, P S an P F sn P Sa: T m( + + sn ) 354 If you now apply euaon 34 n un o he coonaes, an, you oban P m( sn ), 355 P m( + sn cos ) 356 an P m( sn + sn cos + sn ) 357 Theefoe R m( sn ), 358 S m( + sn cos ) 359
9 an F m( sn + cos + sn ) 350 Thus he acceleaon componens ae Raal: Meonal: Azmuhal: sn + sn cos sn + cos + sn Be sue o check he mensons Snce o has menson T, an hese expessons mus have he mensons of acceleaon, hee mus be an an wo os n each em 36 Slheng Soap n Concal Basn We magne a slppey (no fcon) ba of soap slheng aoun n a concal basn An solae ba of soap n negalacc space woul eue hee coonaes o specfy s poson a any me, bu, f s subec o he holonomc consan ha s o be n conac a all mes wh a concal basn, s poson a any me can be specfe wh us wo coonaes I shall, fs of all, analyse he poblem wh a newonan appoach, an hen, fo compason, I shall analyse usng lagangan mehos Ehe way, we sa wh a lage agam In he newonan appoach we mak n he foces n e an he acceleaons n geen See fgue XIII The sem vecal angle of he cone s α R sn α mg FIGURE XIII α
0 The wo coonaes ha we nee ae, he sance fom he veex, an he azmuhal angle, whch I ll ask you o magne, measue aoun he vecal axs fom some abay ogn The wo foces ae he wegh mg an he nomal eacon R of he basn on he soap The acceleaons ae an he cenpeal acceleaon as he soap moves a angula spee n a ccle of aus sn α s sn α We can we he newonan euaon of moon s vaous econs: Hozonal: R cosα m( sn α sn α) e R m an α ( ) 36 Vecal: R sn α mg m cosα 36 Pepencula o suface: R mg sn α m sn α cosα 363 Paallel o suface: g cosα sn α 364 Only wo of hese ae nepenen, an we can choose o use whcheve wo we wan o a ou convenence Thee ae, howeve, hee uanes ha we may wsh o eemne, namely he wo coonaes an, an he nomal eacon R Thus we nee anohe euaon We noe ha, snce hee ae no azmuhal foces, he angula momenum pe un mass, whch s sn α, s conseve, an heefoe s consan an eual o s nal value, whch I ll call l Ω Tha s, we sa off a a sance l fom he veex wh an nal angula spee Ω Thus we have as ou h nepenen euaon l Ω 365 Ths las euaon shows ha as 0 One possble ype of moon s ccula moon a consan hegh (pu 0 ) Fom euaons 36 an s easly foun ha he conon fo hs s ha g 366 sn α an α
In ohe wos, f he pacle s poece nally hozonally ( 0 ) a l an Ω, wll escbe a hozonal ccle (fo eve) f Ω g sn an l α α / Ω C, say 367 If he nal spee s less han hs, he pacle wll escbe an ellpcal ob wh a mnmum < l; f he nal spee s geae han hs, he pacle wll escbe an ellpcal ob wh a maxmum > l Now le s o he same poblem n a lagangan fomulaon Ths me we aw he same agam, bu we mak n he velocy componens n blue See fgue XIII3 We ae ealng wh consevave foces, so we ae gong o use euaon 343, he mos useful fom of agange s euaon sn α α FIGURE XIII3 We nee no spen me woneng wha o o nex The fs an secon hngs we always have o o ae o fn he knec enegy T an he poenal enegy V, n oe ha we can use euaon 343 T m( + sn α ) 368
an V mg cos α + consan 369 Now go o euaon 343, wh, an wok ou all he evaves, an you shoul ge, when you apply he lagangan euaon o he coonae : sn α g cosα 360 T V Now o he same hng wh he coonae You see mmeaely ha an T T ae boh zeo Theefoe s zeo an heefoe s consan Tha s, m sn α s consan an so s consan an eual o s nal value l Ω Thus he secon lagangan euaon s l Ω 36 Snce he lagangan s nepenen of, s calle, n hs connecon, an gnoable coonae an he momenum assocae wh, namely m, s consan Now s ue ha we ave a boh of hese euaons also by he newonan meho, an you may no feel we have gane much Bu hs s a smple, noucoy example, an we shall soon appecae he powe of he lagangan meho, Havng go hese wo euaons, whehe by newonan o lagangan mehos, le s exploe hem fuhe Fo example, le s elmnae beween hem an hence ge a sngle euaon n : 4 l Ω sn α g cosα 36 3 v We know enough by now (see Chape 6) o we as v, whee v, an f we 4 le he consans l Ω sn α an g cos α eual A an B especvely, euaon 36 becomes v A v B 363 3 (I may us be useful o noe ha he mensons of A an B ae 4 T an T especvely Ths wll enable us o keep ack of mensonal analyss as we go)
3 If we sa he soap movng hozonally (v 0) when l, hs negaes, wh hese nal conons, o v A + B( l ) 364 l Agan, so ha we can see wha we ae ong, le an euaon364 becomes A + Bl C (noe ha [C] T ), l A v C B 365 Ths gves v ( ) as a funcon of The pacle eaches s maxmum o mnmum hegh when v 0; ha s whee 3 B C + A 0 366 We aleay know one oo of hs cubc euaon, namely l, because we ou nal conon was ha v 0 a l, an, wh A, B an C all posve, fom he heoy of euaons, hee ae no moe han wo posve eal oos Example Wh nal conons l an Ω ΩC, we oban A 4l 3 g cosα, B g cosα an C 6gl cosα Wh ' / l, he euaon becomes 3 ' 3' + 0 367 The wo posve eal oos ae ' (whch we aleay know) an + 3 (The negave eal oo s 3) Execse Show ha f he nal conon s 045l he low pon s eache a ΩC Execse Show ha f he nal conon s cωc he low o hgh pon s eache a he posve oo of he euaon ' c ' c' 0 Show ha hs agees wh he wo pevous examples Wha s he oo f c? Wha f c s vey small say 00? Wha f s zeo?
4 37 Slheng Soap n Hemsphecal Basn Suppose ha he basn s of aus a an he soap s subec o he holonomc consan a - e ha emans n conac wh he basn a all mes Noe also ha hs s us he same consan of a penulum fee o swng n hee-mensonal space excep ha s subec o he holonomc consan ha he sng be au a all mes Thus any conclusons ha we each abou ou soap wll also be val fo a penulum We ll sa wh he newonan appoach, an I ll aw n e he wo foces on he soap, namely s wegh an he nomal eacon of he basn on he soap Fgue XIII4 R mg FIGURE XIII4 We ll make use of he expessons fo he aal, meonal an azmuhal acceleaons fom secon 35 an we ll we own he euaons of moon n hese econs: Raal: mg cos R m( sn ), 37 Meonal: mg sn m( + sn cos ), 37 Azmuhal: 0 m( sn + cos + sn ) 373 We also have he consan ha a an hence ha 0, afe whch hese euaons become mg cos R ma( + sn ), 374
5 g sn a( sn cos ), 375 0 sn + cos 376 These, hen, ae he newonan euaons of moon If you sll pefe he newonan meho o he lagangan meho, an you wsh o negae hese an fn expessons, an R sepaaely, by all means go ahea an o so bu I m now gong o y he lagangan appoach Alhough agange hmself woul no have awn a agam, we shall no om ha sep bu nsea of makng n he foces, we ll mak n he velocy componens, an hen we ll mmeaely we own expessons fo he knec an poenal eneges Inee he fs lne of a lagangan calculaon s always T a a sn FIGURE XIII5 T ma ( + sn ) 377 V mga cos + consan 378 Now apply euaon 343 n un he coonaes an : a asn cos g sn 379
6 : As fo he concal basn, we see ha T V an ae boh zeo ( s an "gnoable T coonae") an heefoe s consan an eual o s nal value If he nal values of an ae Ω an α especvely, hen Ths s meely sang ha angula momenum s conseve sn sn α Ω 370 We can easly elmnae fom euaons 349 an 0 o oban g sn k co csc, a 37 4 whee k sn α Ω 37 We as n he usual way an negae o oban he fs space negal: g (cos cosα) k(csc csc α) 373 a The uppe an lowe bouns fo occu when 0 Example Suppose ha he nal value of s α 45 o an ha we sa by pushng he soap hozonally ( 0 ) a an nal angula spee Ω 3 a s, so ha k 5 a s Suppose ha he aus of he basn s a 96 m an ha g 98 m s You can hen solve euaon fo 0 One soluon, of couse, s α We coul fn he ohe soluon by Newon-Raphson eaon, o by pung csc /( cos ) an solvng as a cubc euaon n cos Alenavely, y hs: ak e g / a l, ( Ωsn α) l cosα m, n, cos x, cosα c, g so ha csc /( x ) an csc α /( c ) The euaon hen becomes (wok hough he algeba on us ake my wo fo ) ( x c)[( c ) x nx + c( m + c)] 0
7 One soluon s obvously α In ou example he uaac pa f he euaon becomes 05x 05x + 0340900974 0 The only eal soluon fo s 50 o 53' 38 Moe Examples x M m The uppe pulley s fxe n poson Boh pulleys oae feely whou fcon abou he axles Boh pulleys ae lgh n he sense ha he oaonal neas ae small an he oaon conbues neglgbly o he knec enegy of he sysem The ms of he pulleys ae ough, an he opes o no slp on he pulleys The gavaonal acceleaon s g The mass M acceleaes upwas a a ae x wh espec o he uppe, fxe, pulley, an he smalle pulley acceleaes ownwas a he same ae The mass m acceleaes upwas a a ae y wh espec o he small pulley, an conseuenly s acceleaon n laboaoy space s x y The acceleaon of he mass m s heefoe x + y n laboaoy space The obec s o fn x an y n ems of g x y m x + y FIGURE XIII6 The knec enegy s T Mx + m ( x y ) + m ( x + y ) 38 The poenal enegy s V g Mx m ( x y) m ( x + y)] consan 38 [ + Apply agange s euaon (343) n un o he coonaes x an y:
8 x: M x + m x y) + m ( x + y ) g( M m ) 383 ( m y: m x y) + m ( x + y ) g( m ) 384 ( m These wo euaons can be solve a one s lesue fo x an y A hoop of mass M an aus a olls whou slppng on a hozonal plane A bea of mass m sles smoohly aoun he m of he hoop Descbe he moon a a a FIGURE XIII7 I have make n he seveal velocy vecos The hoop s ollng a angula spee Conseuenly he lnea spee of he cene of mass of he hoop s a, an he bea also shaes hs velocy In aon, he bea s slng elave o he hoop a an angula spee an conseuenly has a componen o s velocy of a angenal o he hoop We ae now eay o sa The knec enegy of he hoop s he sum of s anslaonal an oaonal knec eneges: The knec enegy of he bea s M ( a ) + ( Ma ) Ma
9 ma ( + cos ) Theefoe T Ma + ma ( + cos) 385 The poenal enegy s V consan mga cos 386 The lagangan euaon n becomes The lagangan euaon n becomes a ( cos) + g sn 0 387 (M + m ) m( cos sn ) 388 (Some algeba was neee o ave a hese If you nee help, le me know) These, hen, ae wo ffeenal euaons n he wo vaables The lagangan pa of he analyss s ove; we now have o see f we can o anyhng wh hese euaons I s easy o elmnae an hence ge a sngle ffeenal euaon n : (M + msn ) a + masn cos + (M + m) g sn 0 389 If you ae goo a ffeenal euaons, you mgh be able o o somehng wh hs, an ge as a funcon of he me In he meanme, I hnk I can ge he fs space negal (see Chape 6) e as a funcon of Thus, he oal enegy s consan: Ma + ma ( + cos) mga cos E 380 Hee I am measung he poenal enegy fom he cene of he ccle Also, f we assume ha he nal conon s ha a me 0 he knec enegy was zeo an α, hen E mga cosα Euaon 388 can easly be negae once wh espec o me, snce cos sn ( cos), as woul have been appaen ung he evaon of euaon 388 Wh he conon ha he knec enegy was nally zeo, negaon of euaon388 gves
0 ( M + m) m cos 38 Now we can easly elmnae beween euaons 380 an, o oban a sngle euaon elang an : b ( + csn ) cos 0, 38 Mma whee, m, mga b c secα 383a,b,c (M + m) E M E a sn a FIGURE XIII8 As n example, we have a hoop of aus a an mass M, an a ng of mass m whch can sle feely an whou fcon aoun he hoop Ths me, howeve, he hoop s no ollng along he able, bu s spnnng abou a vecal axs a an angula spee The ng has a velocy componen a because s slng aoun he hoop, an a componen a sn because he hoop s spnnng The esulan spee s he ohogonal sum of
hese The knec enegy of he sysem s he sum of he anslaonal knec enegy of he ng an he oaonal knec enegy of he hoop: T ma ( + sn ) + ( Ma ) 384 If we efe poenal enegy o he cene of he hoop: V mga cos 385 The lagangan euaons wh espec o he wo vaables ae : a ( sn cos ) g sn 0 386 : m sn + M consan 387 The consan s eual o whaeve he nal value of he lef han se was Eg, maybe he nal values of an wee α an ω Ths fnshes he lagangan pa of he analyss The es s up o you Fo example, woul be easy o elmnae beween hese wo euaons o oban a ffeenal euaon beween an he me If you hen we as / n he usual way, I hnk wouln be oo ffcul o oban he fs space negal an hence ge as a funcon of I haven e, bu I m sue ll wok v l ( l + ) FIGURE XIII9
Fgue XIII0 shows a penulum The mass a he en s m I s a he en no of he usual nflexble sng, bu of an elasc spng obeyng Hooke s law, of foce consan k The spng s suffcenly sff a gh angles o s lengh ha emans sagh ung he moon, an all he moon s esce o a plane The unseche naual lengh of he spng s l, an, as shown, s exenson s The spng self s lgh n he sense ha oes no conbue he he knec o poenal eneges (You can gve he spng a fne mass f you wan o make he poblem moe ffcul) The knec an poenal eneges ae ( + ( l + ) T 388 ) m an V consan mg( l + )cos + k 389 Apply agange s euaon n un o an o an see whee leas you v Anohe example suable fo lagangan mehos s gven as poblem numbe n Appenx A of hese noes agangan mehos ae paculaly applcable o vbang sysems, an examples of hese wll be scusse n a lae chape Snce hese chapes ae beng wen n moe o less anom oe as he sp moves me, ahe han n logcal oe, pesen ncaons ae ha s lkely o be chape 7 - afe he unlkely seuence of elavy an hyosacs 39 Hamlon s Vaaonal Pncple Hamlon s vaaonal pncple n ynamcs s slghly emnscen of he pncple of vual wok n sacs, scusse n secon 94 of Chape 9 When usng he pncple of vual wok n sacs we magne sang fom an eulbum poson, an hen nceasng one of he coonaes nfnesmally We calculae he vual wok one an se o zeo I am slghly emne of hs when scussng Hamlon s pncple n ynamcs Imagne some mechancal sysem some conapon nclung n s consucon vaous wheels, one os, spngs, elasc sngs, penulums, nclne planes, hemsphecal bowls, an laes leanng agans smooh vecal walls an smooh hozonal floos I may eue N genealze coonaes o escbe s confguaon a any me Is confguaon coul be escbe by he poson of a pon n N- mensonal space O pehaps s subec o k holonomc consans n whch case he pon ha escbes s confguaon n N-mensonal space s no fee o move
3 anywhee n ha space, bu s consane o slhe aoun on a suface of menson N k The sysem s no sac, bu s evolvng I s changng fom some nal sae a me o some fnal sae a me The genealze coonaes ha escbe ae changng wh me an he pon n N-space s slheng oun on s suface of menson N k One can magne ha a any nsan of me one can calculae s knec enegy T an s poenal enegy V, an hence s lagangan T V You can mulply a some momen by a small me neval δ an hen a up all of hese poucs beween an o fom he negal Ths uany of menson M T an SI un J s s somemes calle he acon Thee ae many ffeen ways n whch we can magne he sysem o evolve fom s nal sae o s fnal sae an hee ae many ffeen oues ha we can magne mgh be aken by ou pon n N-space as s moves fom s nal poson o s fnal poson, as long as moves ove s suface of menson N k Bu, alhough we can magne many such oues, he manne n whch he sysem wll acually evolve, an he oue ha he pon wll acually ake s eemne by Hamlon s pncple; an he oue, accong o hs pncple, s such ha he negal s a mnmum, o a maxmum, o an nflecon pon, when compae wh ohe magnable oues Sae ohewse, le us suppose ha we calculae ove he acual oue aken an hen calculae he vaaon n f he sysem wee o move ove a slghly ffeen aacen pah Then (an hee s he analogy wh he pncple of vual wok n a sacs poblem) hs vaaon δ fom wha woul have been ove he acual oue s zeo An hs s Hamlon s vaaonal pncple The nex uesons wll suely be: Can I use hs pncple fo solvng poblems n mechancs? Can I pove hs bal asseon? e me y o use he pncple o solve wo smple an famla poblems, an hen move on o a moe geneal poblem
4 The fs poblem wll be hs Imagne ha we have a pacle han can move n one menson (e one coonae fo example s hegh y above a able suffces o escbe s poson), an ha when s coonae s y s poenal enegy s Is knec enegy s, of couse, V mgy 39 T my 39 We ae gong o use he vaaonal pncple o fn he euaon of moon e we ae gong o fn an expesson fo s acceleaon I magne a he momen you have no ea wha s acceleaon coul possbly be bu on woy, fo we know ha he lagangan s my mgy, 393 an we ll make sho of wh Hamlon s vaaonal pncple an soon fn he acceleaon Accong o hs pncple, y mus vay wh n such a manne ha δ m ( y gy) 0 394 e us vay y by δ y an y by δy, an see how he negal vaes The negal s hen m ( y δy g δy), 395 whch I ll call I I y Now y, an f y vaes by δy, he esulng vaaon n y wll be δ y δy, o δy δy Theefoe I m y δy 396 (If unconvnce of hs, conse e cos e sn e sn ) By negaon by pas: I δ [ my y] δ m y y 390
5 The fs em s zeo because he vaaon s zeo a he begnnng an en pons In he secon em, y y, an heefoe I m y δy 39 â δ m ( y + g) δy, 39 an, fo hs o be zeo, we mus have y g 393 Ths s he euaon of moon ha we sough You woul neve have guesse hs, woul you? Now le s o anohe one-mensonal poblem Only one coonae, x, escbes he pacle s poson, an, when s coonae s x we ll suppose ha s poenal enegy s x V mω, an s knec enegy s, of couse, T mx The euaon of moon, o he way n whch he accelaon vaes wh poson, mus be such as o sasfy δ ( x ω x ) 0 394 m If we vay x by δ x an x by δx, he vaaon n he negal wll be m δ ( x δx ω x δx) I I, say 395 By pecsely he same agumen as befoe, he fs negal s foun o be m x δx Theefoe δ m x δx mω x δx, 396 an, fo hs o be zeo, we mus have x ω x 397 These wo examples mus have gven he mpesson ha we ae ong somehng vey ffcul n oe o eve somehng ha s mmeaely obvous bu he examples wee us nene o show he econ of a moe geneal agumen we ae abou o make
6 Ths me, we ll conse a vey geneal sysem, n whch we we he lagangan as a funcon of he (seveal) genealze coonaes an he me aes of change - e ), ( - whou specfyng any pacula fom of he funcon an we ll cay ou he same so of agumen o eve a vey geneal euaon of moon We have δ + δ δ δ 0 398 As befoe,, δ δ so ha δ δ δ δ δ 399 â 0 δ δ 390 Thus we ave a he geneal euaon of moon 0 39 Thus we have eve agange s euaon of moon fom Hamlon s vaaonal pncple, an hs s nee he way s ofen eve Howeve, n he chape, I eve agange s euaon ue nepenenly, an hence I woul ega hs evaon no so much as a poof of agange s euaon bu as a vncaon of he coecness of Hamlon s vaaonal pncple
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