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1 LOGISMOS METABOLWN & EFARMOGES STH MAJHMATIKH MONTELOPOIHSH PTUQIAKH ERGASIA DIONUSHS JEODOSHS-PALIMERHS A.M. : 311/ EPIBLEPWN: NIKOLOPOULOS QRHSTOS A PANEPISTHMIO AIGAIOU TMHMA MAJHMATIKWN SAMOS 2008

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3 PROLOGOS O Logismìc Metbol n eðni o kl doc twn mjhmtik n pou nfèreti sth melèth twn sunrthsik n. SqetÐzeti me kl douc twn Mjhmtik n, ìpwc h GewmetrÐ ki oi Diforikèc Exis seic, ll ki me th Fusik, idiðter me th Mhqnik. Lìgw twn efrmog n tou se probl mt beltistopoðhshc ki elègqou, o Logismìc Metbol n s mer, brðskei efrmogèc ki se llouc tomeðc ìpwc h oikonomð ki h biologð. H ptuqik ut ergsð dièpeti pì thn poyh ìti t Mjhmtik eðni ergleðo gi thn nk luyh thc l jeic ki mèso gi th diereônhsh twn finomènwn thc fôshc ki thc koinwnðc pou mc perib llei. 'Etsi, to fhrhmèno exhgeðti di tou sugkekrimènou. H prgmtopoðhsh thc ergsðc ut c eðni potèlesm dirkoôc ki usthr c doulei c, kj c prosp jhs, pr t n ktl bw ki èpeit n pod sw me nlutikì trìpo, t mjhmtik probl mt ki tic efrmogèc touc sth Fusik. Autì bèbi, de j mporoôse n epiteuqjeð qwrðc th bo jei tou kjhght k. Q. Nikolìpoulou, o opoðoc me empisteôjhke, me sumboôleuse ki me kjod ghse me upomon. Gi utì, de j tn rketì mìno èn ' euqrist '. 3

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5 Perieqìmen 1 EISAGWGH STO LOGISMO METABOLWN Sunrthsoeid To prìblhm tou Brqustoqrìnou ANAGKAIES SUNJHKES GIA AKROTATA GrmmikoÐ q roi me nìrm H Pr th Metbol HExÐswsh Euler-Lgrnge K poiec Eidikèc Peript seic To prìblhm tou Brqustoqrìnou (sunèqei) GENIKEUSEIS Pr gwgoi uyhlìterhc t xhc Pollèc sunrt seic Probl mt pollpl n oloklhrwm twn Fusikèc Sunorikèc Sunj kec QAMILTONIANH JEWRIA HArq tou Hmilton Exis seic Hmilton To AntÐstrofo Prìblhm ISOPERIMETRIKA PROBLHMATA Probl mt MeEn Periorismì Probl mt MePollploÔc PeriorismoÔc IKANES SUNJHKES GIA AKROTATA H DeÔterh Metbol H sunj kh tou Legendre H sunj kh tou Jcobi Iknèc sunj kec gi krìtto

6 6 PERIEQ OMENA 7 PROSDESH PETRELAIOFORWN (MOORING) Eisgwgik DÔo Diforetikèc ProseggÐseic Trìpoi Prìsdeshc To Montèlo

7 Kef lio 1 EISAGWGH STO LOGISMO METABOLWN... sugkekrimèn, epeid to sq m tou sômpntoc eðni to teleiìtero, ki kìmh perissìtero, epeid eðni sqedismèno pì ton sofìtero dhmiourgì tðpote den mporeð n sumbeð ston kìsmo qwrðc n l mpei k pou sto b joc k poioc nìmoc megðstou elqðstou. Leonrd Euler 1.1 Sunrthsoeid 'En pì t pio shmntik probl mt tou ApeirostikoÔ LogismoÔ eðni h megistopoðhsh elqistopoðhsh mic prgmtik c sun rthshc mic metblht c. AnzhtoÔme, dhld, t shmeð x 0 gi t opoð f (x 0 )=0ki se k je tètoio shmeðo elègqoume to prìshmo thc deôterhc prg gou f (x 0 ). An f (x 0 ) < 0 to x 0 eðni topikì mègisto thc f ki n f (x 0 ) > 0 to x 0 eðni topikì el qisto thc f. O Logismìc Metbol n eðni ènc kl doc twn mjhmtik n pou sqoleðti me genikeôseic tou prp nw probl mtoc tou ApeirostikoÔ LogismoÔ. AntÐ gi thn eôresh sunjhk n, upì tic opoðec di forec sunrt seic èqoun krìtt, o Logismìc Metbol n sqoleðti me thn elqistopoðhsh megistopoðhsh genikìterwn posot twn pou onom zonti sunrthsoeid upì thn morf orismènou oloklhr mtoc. 'En sunrthsoeidèc J epð tou A, ìpou to A eðni èn 7

8 8 KEF ALAIO 1. EISAGWGH STO LOGISMO METABOLWN kl c orismèno sônolo sunrt sewn, eðni mi peikìnish pou ntistoiqðzei se k je sun rthsh y A ènn prgmtikì rijmì pou sumbolðzeti me J(y). KÔrio prìblhm tou LogismoÔ Metbol n eðni h eôresh twn sunrt sewn enìc sunìlou A oi opoðec elqistopoioôn to J. To kl c orismèno sônolo A lègeti sônolo twn podekt n sunrt sewn ki mporeð n eðni to sônolo twn suneq n sunrt sewn pou orðzonti se èn di sthm [, b], to sônolo twn suneq c diforðsimwn sunrt sewn sto [, b] pou plhroôn th sunj kh f() =0 opoiod pote llo sônolo, rkeð n eðni kl c orismèno. To prìblhm elqistopoðhshc megistopoðhshc enìc sunrthsoeidoôc J epð tou A lègeti metbolikì prìblhm. 1.2 To prìblhm tou Brqustoqrìnou H istorð tou LogismoÔ Metbol n ousistik xekðnhse me èn prìblhm pou èjese o Johnn Bernoulli to 1696 wc prìklhsh gi th mjhmtik koinìtht thc epoq c ki idiðter proc ton derfì tou Jcob Bernoulli. To prìblhm eðqe wc ex c : 'En s m m zc m f neti n olisj sei, me rqik tqôtht mhdèn, kt m koc mic kmpôlhc y = y(x) pì to shmeðo (x 0,y 0 ) sto (x 1,y 1 ) me x 0 < x 1 ki y 0 > y 1. Poi kmpôlh dðnei ton tqôtero qrìno kjìdou n to s m kineðti mìno upì thn ep rei thc brôthtc ki qwrðc trib. To prìblhm prosèlkuse polloôc mjhmtikoôc ìpwc touc L Hopitl, Leibniz, Newton ki rgìter touc Euler ki Lgrnge. Sto tèloc tou 1696 o Johnn Bernoulli ìpwc ki o Leibniz eðqn eðdh d sei tic lôseic touc gi to sugkekrimèno prìblhm to opoðo ki onìmsn fi To prìblhm tou Brqustoqrìnou fl. O Newton prìlo pou èmje gi to sugkekrimèno prìblhm me 6 m nec kjustèrhsh, to èluse to Ðdio pìgeum ki èsteile n num th lôsh tou ston Johnn Bernoulli. Tèloc o Jcob Bernoulli prostèjhke sth lðst twn mjhmtik n pou èlusn epituq c to prìblhm. Oi lôseic pou èdinn oi mjhmtikoð ekeðnhc thc epoq c tn kurðwc gewmetrik c fôsewc. H lôsh ìmwc pou èdwsn oi Euler ki Lgrnge od ghse se genikèc mejìdouc pou podeðqthkn qr simec gi thn epðlush mic poikilðc tètoiwn problhm twn. H kmpôlh pou dðnei th lôsh eðni èn tìxo enìc kukloeidoôc, thc kmpôlhc pou perigr feti pì thn kðnhsh enìc shmeðou epð thc perifèreic kuliìmenou troqoô.

9 1.2. TO PR OBLHMA TOU BRAQUSTOQR ONOU 9 y y 1 y 2 x 1 x 2 x Sq m 1.1: H kmpôlh tou brqustoqrìnou 'Estw y = y(x) h kmpôlh pou sundèei t (x 0,y 0 ) ki (x 1,y 1 ). Skopìc mc eðni n upologðsoume ton qrìno kjìdou T gi thn y = y(x). H tqôtht kt m koc thc kmpôlhc eðni u = ds dt dt = ds u = dx2 + dy 2 u = 1+( dy dx )2 u dx = 1+y (x) 2 ìpou s to m koc tìxou epð thc kmpôlhc. Opìte o sunolikìc qrìnoc kjìdou dðneti pì thn: x1 1+y (x) T = 2 dx u x 0 Efìson èqoume kðnhsh qwrðc trib, èqoume sunthrhtikì pedðo dun mewn.'ar h enèrgei dithreðti kt th di rkei thc kðnhshc, dhld V + T (th qronik stigm t =0)=V + T (th qronik stigm t>0), ìpou V h dunmik enèrgei ki T hkinhtik enèrgei. Gi t =0èqoume T =0 foô to s m xekin pì thn hremð. 'Ar LÔnontc wc proc u, èqoume 0+mgy 0 = 1 2 mu2 + mgy u = 2g(y 0 y(x)) u dx,

10 10 KEF ALAIO 1. EISAGWGH STO LOGISMO METABOLWN Sunep c, o qrìnoc pou piteðti gi thn k jodo thc sfðrc eðni : T (y) = x1 x 0 1+y (x) 2 2g(y0 y(x)) dx, ìpou èqoume gr yei T (y) gi n deðxoume ìti to T exrt ti pì thn kmpôlh y pou n kei sto sônolo twn suneq c diforðsimwn sunrt sewn A pou orðzonti sto di sthm x 0 <x<x 1 ki plhroôn tic sunorikèc sunj kec y(x 0 )=y 0 ki y(x 1 )=y 1. To ntðstoiqo metbolikì prìblhm sunðstti sthn elqistopoðhsh tou T (y), ìpou y A. Lìgw twn sugkekrimènwn sunorik n sunjhk n to prìblhm lègeti prìblhm sugkekrimènwn (stjer n) krwn. Th lôsh tou probl mtoc tou brqustoqrìnou j th doôme sthn enìtht (2.4.1).

11 Kef lio 2 ANAGKAIES SUNJHKES GIA AKROTATA 2.1 GrmmikoÐ q roi me nìrm 'Opwc eðdme, mi ngkð sunj kh gi n èqei mi sun rthsh f topikì el - qisto se k poio shmeðo x 0 eðni h f (x 0 )=0. T krìtt sunrthsoeid n mporoôn n prosdioristoôn me trìpo n logo m' utìn gi sunrt seic poll n metblht n. Skopìc mc gi rq, eðni n orðsoume thn pr gwgo enìc sunrthsoeidoôc J se k poio stoiqeðo y 0 A. Gi n to ktfèroume utì upojètoume ìti to sônol o twn podekt n sunrt sewn A eðni uposônolo enìc grmmikoô q rou me nìrm, ètsi ste n mporoôme n sugkrðnoume thn tim J(y 0 ) me thn tim J(y) gi y flkont fl sto y 0 (dhld, h y y 0 n eðni mikr ). Orismìc : Dinusmtikìc ( Grmmikìc) q roc eðni ènmhkenìsônolo V efodismèno me thn pr xh thc prìsjeshc fi+fl ki tou bjmwtoô pollplsismoô. Gi opoid pote stoiqeð f,g,h V ki, b R oi prp nw pr xeic èqoun tic kìloujec idiìthtec : 1. f + g V 2. f + g = g + f 3. f +(g + h) =(f + g)+h 4. Up rqei mondikì stoiqeðo 0 V tètoio ste f +0=f, f V 5. f V, ( f) V tètoio ste f +( f) =0 11

12 12 KEF ALAIO 2. ANAGKAIES SUNJHKES GIA AKROTATA 6. f V 7. (f + g) =f + g 8. ( + b)f = f + bf 9. (b)f = (bf) f = f Pr deigm To sônolo C[, b], ìpou t stoiqeð tou eðni suneqeðc sunrt seic pou orðzonti sto di sthm [, b]. An f, g C[, b] h prìsjesh orðzeti wc (f + g)(x) =f(x)+g(x) ki o pollplsismìc wc ìpou R (f)(x) =f(x) Pr deigm To sônolo C 2 [, b], ìpou tstoiqeð tou eðni suneqeðc sunrt seic me suneq deôterh pr gwgo eðni epðshc grmmikìc q roc. Orismìc 'Estw V ènc grmmikìc q roc, tìte mi nìrm epð tou V eðni mi peikìnish pou se k je y V ntistoiqðzei ènn mh rnhtikì prgmtikì rijmì pou sumbolðzeti me y, h opoð lègeti nìrm tou y, ki plhroð tic kìloujec sunj kec : 1. y =0n kimìno n y =0 2. y = y gi k je R,y V 3. y 1 + y 2 y 1 + y 2 gi k je y 1,y 2 V

13 2.2. H PR WTH METABOL H 13 Pr deigm O q roc C[, b] twn suneq n sunrt sewn epð tou [, b] eðni grmmikìc q roc me nìrm, n orðsoume th nìrm th nìrm y M =mx x b y(x) y 1 = y(x) dx. Pr deigm Ston grmmikì q ro C 1 [, b] h nìrm pou orðzeti wc y w =mx y(x) +mx x b x b y (x) onom zeti sjen c nìrm ki pðzei shmntikì rìlo sto logismì metbol n. H Ðdi nìrm orðzeti epðshc ston C n [, b], me n H Pr th Metbol 'Estw J : A R èn sunrthsoeidèc epð tou A, ìpou A V ki V grmmikìc q roc me nìrm. Tìte to J lème ìti èqei topikì el qisto sto ŷ A n up rqei ε>0 tètoio ste J(ŷ) J(y) gi k je y A me y ŷ <ε. AntÐstoiq lème ìti to J èqei topikì mègisto sto ŷ A n to ŷ eðni topikì el qisto gi to J. Skopìc mc eðni n orðsoume thn pr gwgo enìc sunrthsoeidoôc. Sn form j qrhsimopoi soume ton orismì thc prg gou mic sun rthshc f : R R upì th morf f(x 0 + x) f(x 0 )=f (x 0 ) x + o( x)

14 14 KEF ALAIO 2. ANAGKAIES SUNJHKES GIA AKROTATA ìpou o( x) 1 sumbolðzei ìrouc uyhlìterhc t xhc wc proc x me o( x) lim x 0 x =0, dhld sugklðnoun sto mhdèn tqôter pì to x. PrthroÔme ìti to diforikì thc f sto x 0, pou orðzeti wc df (x 0, x) =f (x 0 ) x, eðni o grmmikìc ìroc wc proc th metbol x thc olik c metbol c f f(x 0 + x) f(x 0 ). 'Estw J : A R, ìpou A uposônolo enìc grmmikoô q rou me nìrm V ki y 0 A. Gi n metb loume to y 0, jewroôme èn stoiqeðo h V tètoio ste to y 0 + εh n n kei sto A gi k je rket mikrì prgmtikì rijmì ε. H metbol εh lègeti metbol thc sun rthshc y 0 ki sumbolðzeti suqn me δy 0. OrÐzoume tìte thn olik metbol tou J pou ofeðleti sth metbol εh tou y 0 wc: J = J(y 0 + εh) J(y 0 ). Stìqoc mc eðni o upologismìc tou grmmikoô mèrouc ut c thc metbol c. OrÐzoume th sun rthsh: J (ε) J(y 0 + εh). H J eðni sun rthsh mic prgmtik c metblht c ε, pou orðzeti me ton upologismì tou sunrthsoeidoôc J epð thc monoprmetrik c oikogèneic sunrt sewn y 0 + εh. AnptÔssoume thn J (ε) qrhsimopoi ntc to je rhm Tylor, upojètontc bèbi ìti h J (ε) eðni eprk c diforðsimh. 'Eqoume: J (ε) =J (0) + J (0)ε + J (0) ε ! J(y 0 + εh) J(y 0 )=J (0)ε + J (0) ε ! Opìte ktl goume ston epìmeno orismì. 1 Εστω f(ε) και g(ε) συναρτήσεις ορισμένες σε μια περιοχή του ε =0(η οποία μπορεί να μην περιέχει το 0). Θα γράφουμε και θα γράφουμε f(ε) =o(g(ε)) καθώς ε 0 αν lim ε 0 f(ε) g(ε) f(ε) =O(g(ε)) καθώς ε 0 =0, αν υπάρχει σταθερά M > 0 τέτοια ώστε f(ε) M g(ε) για κάθε ε σε μια περιοχή του μηδενός (η οποία δεν περιέχει κατ ανάγκη το μηδέν).

15 2.2. H PR WTH METABOL H 15 Orismìc 'Estw J : A R sunrthsoeidèc epð tou A, ìpou A V ki V grmmikìc q roc me nìrm. 'Estw y 0 A ki h V tètoi ste y 0 +εh A gi k je rket mikrì ε. Sthn perðptwsh ut, h pr th metbol ( metbol Gâteux) tou J sto y 0 kt thn kteôjunsh h orðzeti wc δj(y 0,h) J (0) d dε J(y 0 + εh) ε=0 (2.1) me thn proôpìjesh ìti up rqei h pr gwgoc. Mi kteôjunsh h gi thn opoð up rqei h pr gwgoc ìpwc orðzeti sth (2.1), onom zeti podekt metbol sto y 0. Prt rhsh PrthroÔme ìti n gr youme thn (2.1) se morf orðou, dhld wc J(y 0 + εh) J(y 0 ) δj(y 0 ) = lim ε 0 ε up rqei mi nlogð me thn kteujunìmenh pr gwgo mic prgmtik c sun rthshc f(x) :R R sto X 0 =(x 0,y 0 ) sthn kteôjunsh n pou orðzeti wc f(x 0 + ε n) f(x 0 ) D n f(x 0 ) lim. ε 0 ε SunoyÐzontc, ìtn jèloume n upologðsoume thn pr th metbol δj(y 0,h), upologðzoume pr t to J sto y 0 + εh, ìpouto h eðni tètoio ste y 0 + εh A ki èpeit upologðzoume thn pr gwgo wc proc ε, thn opoð pðrnoume gi ε =0. Je rhm 'Estw J : A R sunrthsoeidèc, A V, ìpou V grmmikìc q roc me nìrm. AngkÐ sunj kh gi n èqei to J topikì krìtto sto y 0 wc proc th nìrm eðni gi ìlec tic podektèc metbolèc h. δj(y 0,h)=0 (2.2) Apìdeixh 'Estw J : A R ki y 0 A shmeðo ìpou to J lmb nei topikì el qisto. OrÐzoume thn sun rthsh mic prgmtik c metblht c ε wc J (ε) =J(y 0 +εh) h opoð èqei topikì el qisto sto ε =0,meh V tètoio ste to y 0 + εh A gi mikrèc timèc tou ε. 'Opwc gnwrðzoume pì ton Apeirostikì Logismì j prèpei n èqoume J (0) = 0 δj(y 0,h)=0ki utì sumbðnei nexrt twc pì thn epilog tou h.

16 16 KEF ALAIO 2. ANAGKAIES SUNJHKES GIA AKROTATA Pr deigm 'Estw to sunrthsoeidèc J(y) = 1 0 (1 + y (x) 2 )dx ì- pou y C 1 [0, 1] ki y(0) = 0,y(1) = 1. 'Estw y 0 (x) =x ki h(x) =x(1 x). Hoikogènei kmpul n dðneti pì thn y 0 +εh = x+εx(1 x). UpologÐzontc to J epð ut c thc oikogèneic pðrnoume J (ε) = J(y 0 + εh) = = = 2+ ε2 3 [1 + (y 0 (x)+εh (x)) 2 ]dx [1 + (1 + ε(1 2x)) 2 ]dx 'Ar J (ε) = 2 3 ε ki J (0) = 0. Opìte èqoume ìti δj(y 0,h)=0, dhld ìti to J eðni st simo sto y 0 = x wc proc thn kteôjunsh h = x(1 x).

17 2.3. H EX ISWSH EULER-LAGRANGE H ExÐswsh Euler-Lgrnge 'Opwc nfèrme to Jemeli dec Prìblhm tou LogismoÔ Metbol n eðni h eôresh mic sun rthshc y, enìc kl c orismènou sunìlou sunrt sewn A (pq C 2 [, b]), h opoð elqistopoieð to sunrthsoeidèc J(y) = L(x, y, y )dx me y() =y 0 ki y(b) =y 1, ìpou h L eðni mi sun rthsh dôo forèc suneq c diforðsimh wc proc t trðorðsmt thc ki onom zeti Lgkrnzin. AnzhtoÔme loipìn mi ngkð sunj kh gi krìtto. Bsizìmenoi sthn prohgoômenh pr grfo j upologðsoume thn pr th metbol. Ac upojèsoume ìti to J prousi zei topikì el qisto sto y. 'Opwc ki st prohgoômen ènc trìpoc gi n pr goume èn podektì sônolo sunrt sewn ŷ eðni n prosjèsoume se mi podekt sun rthsh èn mikrì pollpl sio mic sun rthshc h(x) C 2 [, b] me h() =h(b) =0. Tìte h ŷ = y + εh eðni podekt sun rthsh ki J(ŷ) = = L(x, ŷ, ŷ )dx UpologÐzoume t r thn pr th metbol : d dε J(ŷ) ε=0 dl = dε = = 0 L(x, y + εh, y + εh )dx ε=0 dx [ dŷ ŷ dε + dŷ ŷ dε PrgwgÐzontc thn ŷ = y + εh wc proc ε èqoume : dŷ dε = h 'Ar h pr th metbol gi ε =0gÐneti: δj(y, h) = d dε J(y + εh) ε=0 =, dŷ dε = h. ] dx ε=0 [ y h + ] y h dx =0 Oloklhr nontc kt pr gontec ton deôtero ìro thc prp nw sqèshc è- qoume : y h dx = h b y h d ( ) dx dx y [ ( )] d = hdx dx y

18 18 KEF ALAIO 2. ANAGKAIES SUNJHKES GIA AKROTATA y x 1 x 0 x 2 b x Sq m 2.1: H grfik pr stsh thc h(x). Epomènwc èqoume ìti δj(y, h) = [ y d dx ( y )] hdx =0. (2.3) L mm An mi sun rthsh f(x) eðni suneq c sto [, b] ki iknopoieð th sqèsh f(x)h(x)dx =0 gi k je dôo forèc suneq c prgwgðsimh sun rthsh h me h() =h(b) =0 tìte f(x) =0 x [, b]. Apìdeixh 'Estw ìti x 0 (, b) tètoio ste f(x 0 ) > 0. Epeid h f eðni suneq c sto [, b], x 1,x 2 tètoi ste <x 1 <x 0 <x 2 <bki f(x) > 0 x (x 1,x 2 ). JewroÔme th sun rthsh h(x) tètoi ste h(x) = { (x x1 ) 3 (x 2 x) 3, n x 1 x x 2 0, diforetik, thc opoðc h grfik pr stsh fðneti sto Sq m (2.1). J deðxoume t r ìti h h eðni dôo forèc suneq c prgwgðsimh. 'Eqoume : lim x x + 1 h(x) h(x 1 ) x x 1 = lim x x + 1 (x x 1 ) 3 (x 2 x) 3 0 x x 1 = lim x x + 1 (x x 1 ) 2 (x 2 x) 3 = 0

19 2.3. H EX ISWSH EULER-LAGRANGE 19 ki lim x x 1 h(x) h(x 1 ) x x 1 = lim x x x x 1 =0 Opìte h =0. EpÐshc èqoume : ki lim x x + 1 h (x) h (x 1 ) x x 1 = lim x x + 1 lim x x 1 3(x x 1 ) 2 (x 2 x) 2 (x 2 + x 1 2x) 0 x x 1 = lim 3(x x 1 )(x 2 x) 2 (x 2 + x 1 2x) =0 x x + 1 h (x) h (x 1 ) x x 1 = lim x x x x 1 =0 Opìte h (x 1 )=0. Me prìmoio trìpo mporoôme n deðxoume ìti h (x 2 )=0 H deôterh pr gwgoc thc h eðni : 6(x x 1 )(x 2 x){(x x 1 ) 2 + h (x) = (x 2 x) 2 3(x x 1 )(x 2 x)}, n x 1 x x 2 0, diforetik. 'Ar ki lim h (x) =h (x 1 )=0 x x 1 lim h (x) =h (x 2 )=0 x x 2 Epomènwc h h C 2 [, b]. Sthn perðptwsh ut, èqoume f(x)h(x)dx = x2 x 1 f(x)(x x 1 ) 3 (x 2 x) 3 dx > 0, foô h f eðni jetik sto (x 1,x 2 ). 'Omwc t r ktl xme se topo foô gnwrðzoume pì thn upìjesh tou L mmtoc ìti f(x)h(x)dx = 0. 'Ar f(x) =0 x [, b]. Qrhsimopoi ntc t r to prp nw L mm sthn (2.3) ktl goume sto epìmeno je rhm :

20 20 KEF ALAIO 2. ANAGKAIES SUNJHKES GIA AKROTATA Je rhm 'Estw J(y) sunrthsoeidèc thc morf c J(y) = L(x, y, y )dx me y C 2 [, b] ki y() =y 0, y(b) =y 1. An hsun rthsh y eðni shmeðo ìpou to sunrthsoeidèc prousi zei topikì krìtto tìte y (x, y, y ) d ( ) dx y (x, y, y ) =0, x [, b]. (2.4) H exðswsh (2.4) lègeti exðswsh Euler-Lgrnge ki eðni mi sun jhc diforik exðswsh deôterhc t xhc pou prist mi ngkð sunj kh gi krìtto. An to y eðni topikì krìtto tìte δj(y, h) =0 h ki lème ìti to J eðni st simo sto y ìti to y eðni st simh sun rthsh gi to J. PÐrnontc thn pr gwgo sto x ston deôtero ìro thc (2.4) èqoume : y 2 L y x (x, y, y ) 2 L y y (x, y, y )y 2 L y y (x, y, y )y =0 (2.5) Opìte t r eðni profnèc ìti h exðswsh (2.4) eðni mi deôterhc t xhc diforik exðswsh. Pr deigm N brejoôn t krìtt tou sunrthsoeidoôc J(y) = 1 0 (y 2 +3y +2x)dx, y(0) = 0, y(1) = 1. H Lgkrnzin eðni L = y 2 +3y +2x. 'Eqoume y =3 ki =2y y ki h exðswsh Euler-Lgrnge gðneti 3 d dx (2y )=0 y = 3 2, h opoð eðni mi grmmik diforik exðswsh 2hc t xhc. Oloklhr nontc dôo forèc brðskoume : y = 3 4 x2 + C 1 x + C 2.

21 2.3. H EX ISWSH EULER-LAGRANGE 21 Qrhsimopoi ntc t r tic sunj kec y(0) = 0, y(1) = 1 brðskoume ìti C 1 = 1 4 ki C 2 =0. Opìte to pijnì krìtto tou J pou iknopoieð tic prp nw sunj kec eðni h sun rthsh y = 3 4 x x. Pr deigm (Gewdisikèc grmmèc sto epðpedo) 'Estw dôo shmeð P :(, y 0 ) ki Q :(b, y 1 ) sto R 2. N brejeð h kmpôlh pou en nei t shmeð P, Q ki èqei to el qisto m koc. To metbolikì prìblhm se ut thn perðptwsh eðni h elqistopoðhsh tou sunrthsoeidoôc pou dðnei to m koc tìxou, dhld tou J(y) = 1+y (x) 2 dx, epð tou sunìlou twn suneq c diforðsimwn sunrt sewn se èn di sthm x b me sunorikèc sunj kec y() =y 0,y(b) =y 1. Gi n èqei to J topikì el qisto sto y, ngkð sunj kh eðni n plhroð to y thn exðswsh Euler-Lgrnge y d ( ) =0. dx y H Lgkrnzin eðni 1+y (x) 2. Epomènwc èqoume y =0, y = y 1+y 2 ki ( d dx y 1+y 2 'Ar h exðswsh Euler-Lgrnge gðneti Oloklhr nontc dôo forèc pðrnoume ) = y (x) (1 + y 2 ) 3 2 y (x) =0.. y(x) =C 1 x + C 2 (2.6)

22 22 KEF ALAIO 2. ANAGKAIES SUNJHKES GIA AKROTATA me C 1,C 2 stjerèc pou proèkuyn pì tic dôo oloklhr seic. Qrhsimopoi ntc tic sunorikèc sunj kec sthn (2.6) pðrnoume y 0 = C 1 + C 2 y 1 = C 1 b + C 2 LÔnontc t r to prp nw sôsthm wc proc C 1 ki C 2 pðrnoume Sunep c h y eðni thc morf c C 1 = y 1 y 0 b και C 2 = y 0b + y 1. b y(x) = ( y1 y 0 b Ton ìro C 2 mporoôme n ton gr youme wc 'Etsi h y gðneti C 2 = y 0b + y 1 b y(x) = ) x + y 0b + y 1. b = y 0b + y 1 + y 0 y 0 b = (y 1 y 0 )+y 0 ( b) b = (y 1 y 0 ) b = (y 1 y 0 ) b + y 0 + y 0. ( ) y1 y 0 (x )+y 0, b h opoð eðni h exðswsh thc eujeðc pou dièrqeti pì t shmeð P ki Q. 2.4 K poiec Eidikèc Peript seic H exðswsh Euler-Lgrnge eðni mi sun jhc diforik exðswsh deôterhc t - xhc ki h lôsh thc perièqei en gènei dôo ujðretec stjerèc. Pollèc forèc tètoiou eðdouc exis seic eðni dôskolo n lujoôn kìm ki n plopoihjoôn. Up rqoun ìmwc k poiec peript seic ìpou eðni duntìn n plopoi soume mèswc ut thn exðswsh ki utì sumbðnei ìtn h Lgkrnzin L den exrt ti eujèwc pì mð pì tic metblhtèc x, y y.

23 2.4. K APOIES EIDIK ES PERIPT WSEIS 23 PerÐptwsh I H Lgkrnzin den exrt ti pì th metblht y perðptwsh to sunrthsoeidèc j eðni thc morf c Se ut thn J(y) = L(x, y )dx. Tìte h exðswsh Euler-Lgrnge gðneti ( ) d =0 = C, dx y y ìpou C stjer pou proèkuye pì olokl rwsh. Pr deigm 'Estw to sunrthsoeidèc J(y) = e x 1+y 2 dx. H exðswsh Euler-Lgrnge utoô tou sunrthsoeidoôc eðni 'Eqoume y 1, dhld C 1+y ex. 2 LÔnontc thn (2.7) wc proc y pðrnoume y = ex y 1+y 2 = C (2.7) y = C e 2x C 2. Oloklhr nontc t r thn prp nw exðswsh pðrnoume ( ) e y(x) =sec 1 x +C 1, C ìpou C 1 stjer pou proèkuye pì thn olokl rwsh. PerÐptwsh II H Lgkrnzin den exrt ti pì th metblht y perðptwsh h exðswsh Euler-Lgrnge èqei th morf pou eðni mi lgebrik exðswsh. y =0, Se ut thn

24 24 KEF ALAIO 2. ANAGKAIES SUNJHKES GIA AKROTATA Pr deigm 'Estw to sunrthsoeidèc ((e x +1)y 2 y)dx. H Lgkrnzin eðni L = (e x +1)y 2 y, opìte s' ut thn perðptwsh h exðswsh Euler-Lgrnge èqei th morf y =2y(ex +1) 1=0 'Etsi ktl xme se mi lgebrik exðswsh pou èqei lôsh wc proc y y = 1 2(e x +1). PerÐptwsh III H Lgkrnzin den exrt ti eujèwc pì th metblht x 'Estw t r ìti sunrthsoeidèc eðni thc morf c PrthroÔme ìti : [ d L y ] dx y J(y) = [ dx L(y, y )dx. = dy y dx + ( dy dy 2 y y dx + L y y y = y y + [ ( y y y y + y 2 L y y y + ( ) = y y 2 L y 2 y 2 L y y y = 0 dy dx + )] 2 L dy y y dx )] 2 L y y y H pr stsh sthn teleutð prènjesh sumpðptei me thn exðswsh (2.5) plopoihmènh gi thn perðptwsh ìpou to L den exrt ti pì to x. Opìte se ut thn perðptwsh h exðswsh Euler-Lgrnge j eðni thc morf c L(y, y ) y y (y, y )=C. (2.8)

25 2.4. K APOIES EIDIK ES PERIPT WSEIS 25 Oi exis seic pou mc proèkuyn stic peript seic I,III eðni pr t oloklhr mt thc exðswshc Euler-Lgrnge. 'En pr to olokl rwm mic diforik c exðswshc deôterhc t xhc F (x, y, y,y )=0eÐni mi sun rthsh thc morf c g(x, y, y ), pou perilmb nei pr gwgo qmhlìterhc t xhc, h opoð eðni stjer ìtn h y eðni lôsh thc rqik c exðswshc F (x, y, y,y )=0. Sunep c h g(x, y, y )=C, ìpou C stjer, sunist mi olokl rwsh thc exðswshc deôterhc t xhc. Pr deigm Ac upojèsoume ìti èqoume th Lgkrnzin L = 1 2 [(y ) 2 ω 2 y 2 ] H exðswsh Euler-Lgrnge eðni To pr to olokl rwm eðni y + ω 2 y =0 (y ) 2 + ω 2 y 2 = C. (2.9) J exet soume t r n pr gmtito pr to olokl rwm eðni stjerì. H genik lôsh thc exðswshc Euler-Lgrnge eðni ìpou A, B stjerèc. EpÐshc èqoume y(x) =A cos (ωx)+b sin (ωx), y (x) = ωa sin (ωx)+bω cos (ωx). An ntiktst soume t r thn genik lôsh sthn (2.9) pðrnoume: ( ωa sin (ωx)+bω cos (ωx)) 2 + ω 2 (A cos (ωx)+b sin (ωx)) 2 = ω 2 (A 2 + B 2 ). 'Ar to pr to olokl rwm eðni stjerì. Pr deigm (H Arq tou Fermt) Upojètoume èn optikì mèso me metblhtì deðkth di jlshc n = n(x, y, z). Apì thn kumtik jewrð tou fwtìc èqoume ìti h tqôtht tou fwtìc sto optikì mèso eðni Ðsh me c/n, ìpou c h tqôtht tou fwtìc sto kenì. Tìte o qrìnoc pou piteðti pì mi ktðn fwtìc n txidèyei metxô dôo shmeðwn A(r 1 ) ki B(r 2 ) eðni T = r2 r 1 n ds (2.10) c

26 26 KEF ALAIO 2. ANAGKAIES SUNJHKES GIA AKROTATA H rq tou Fermt (1650) thc gewmetrik c optik c lèei ìti Mi ktðn fwtìc pou sundèei dôo shmeð enìc mèsou, me deðkth di jlshc n = n(r), koloujeð p ntote to drìmo tou elqðstou qrìnou fwtein c didrom c. Dhld h prgmtik didrom eðni ut pou elqistopoieð to olokl rwm (2.10). E n upojèsoume ìti h ktðn tou fwtìc brðsketi sto xy epðpedo ki ìti o deðkthc di jlshc eðni sun rthsh mìno tou y, h rq tou Fermt gr feti T (y) = x2 x 1 n(y) 1+y 2 dx = x2 x 1 L(y, y )dx, ìpou ds = (dx) 2 +(dy) 2 = 1+y 2 dx. Epeid h Lgkrnzin eðni nex rthth tou x h exðswsh Euler-Lgrnge eðni thc morf c L y y = c n(y) 1+y 2 = c, ìpou c stjer.

27 2.4. K APOIES EIDIK ES PERIPT WSEIS To prìblhm tou Brqustoqrìnou (sunèqei) Epistrèfoume t r sto prìblhm tou brqustoqrìnou me skopì n elqistopoi soume to sunrthsoeidèc T (y) = x1 x 0 1+y 2 2g(y0 y) dx pou upìkeiti stic sunorikèc sunj kec y(x 0 )=y 0,y(x 1 )=y 1 me y 0 >y 1, ìpou T o qrìnoc pou piteðti gi thn k jodo thc sfðrc kt m koc thc kmpôlhc y = y(x). H Lgkrnzin tou sunrthsoeidoôc eðni nex rthth tou x ki sunep c odhgoômste sthn fi PerÐptwsh III fl. Epomènwc h exðswsh Euler-Lgrnge j eðni thc morf c L(y, y ) y (y, y )=C. y 'Eqoume L = 1+y 2 y0 y Sunep c, h exðswsh eðni, 1 2 = y (1 + y 2 ) y 2g(y0 y). 1+y 2 y0 y (y ) 2 (1 + y 2 ) 1 2 y0 y = C, ìpou o ìroc 2g mporeð n porrofhjeð sth stjer C foô den j mc ephre sei to krìtto tou T. H prp nw exðswsh gðneti : 1+y 2 (y ) 2 (1 + y 2 ) 1 2 = C y0 y. Uy nontc t r sto tetr gwno ki ektel ntc k poiec pr xeic ktl goume sthn ( ) 2 dy = 1 (y 0 y)c 2 dx C 2 (y 0 y) dy dx = 1 (y 0 y)c 2. C 2 (y 0 y) Kt m koc thc kmpôlhc y(x) èqoume ìti y(x) <y 0, gi k je x [x 0,x 1 ] ki ètsi dy < 0, ìpwc fðneti ki sto Sq m (1.1) (sel. 9). QwrÐzontc t r dx metblhtèc ki lmb nontc up ' ìyin ìti dy/dx < 0 pðrnoume y0 y dx = dy, (2.11) A (y0 y) ìpou A = 1 C 2. Gi n lôsoume thn prp nw exðswsh mporoôme n qrhsimopoi soume thn

28 28 KEF ALAIO 2. ANAGKAIES SUNJHKES GIA AKROTATA Sq m 2.2: Anestrmmèno kukloeidèc - l Ôsh tou brqustoqrìnou trigwnometrik ntikt stsh y 0 y = A sin 2 φ 2 'Etsi èqoume me dy = A sin φ 2 cos φ 2 dφ. dx = A sin 2 φ 2 dx = A (1 cos φ)dφ (2.12) 2 foô gnwrðzoume ìti sin 2 φ 2 = 1 cos φ 2. Oloklhr nontc thn (2.12) pðrnoume Oi exis seic x = A (φ sin φ)+b. 2 y 0 y = A sin 2 φ 2 x = A (φ sin φ)+b 2 eðni oi prmetrikèc exis seic enìc kukloeidoôc ìpwc fðneti sto Sq m (2.2).

29 Kef lio 3 GENIKEUSEIS Mèqri t r eðdme sunrthsoeid thc morf c L(x, y, y )dx, ìpou h Lgkrnzin exrt ti pì tic metblhtèc x, y ki y. Up rqoun ìmwc poll probl mt st opoð h Lgkrnzin exrt ti pì uyhlìterhc t xhc prg gouc pì perissìterec sunrt seic. H mèjodoc pou qrhsimopoi sme ste n brðskoume t krìtt sunrthsoeid n mporeð n genikeujeð gi genikèc peript seic ìpwc oi prp nw qwrðc ousistikèc llgèc. 3.1 Pr gwgoi uyhlìterhc t xhc Ac jewr soume to sunrthsoeidèc thc morf c J(y) = L(x, y, y,y )dx, y A, ìpou A to sônolo C 4 [, b] sunrt sewn pou iknopoioôn tic sunorikèc sunj kec y() =A 1, y () =A 2, y(b) =B 1, y (b) =B 2. Upojètoume ìti h L èqei suneqeðc merikèc prg gouc trðthc t xhc wc proc kjèn pì t tèsser orðsmt thc. 'Estw ìti to J prousi zei sto y topikì krìtto. Pr goume to podektì sônolo sunrt sewn ŷ = y+εh me h tètoi ste h() =h () =h(b) =h (b) =0 29

30 30 KEF ALAIO 3. GENIKEUSEIS ki ŷ A. Jèloume n upologðsoume ki p li thn pr th metbol. UpologÐzoume th difor J = J(ŷ) J(y) me th bo jei tou jewr mtoc Tylor J = ε ( h y ) + h + h dx + O(ε 2 ). y y Opìte h pr th metbol gi utì to sunrthsoeidèc eðni δj(y, h) = d dε ε=0 J(y + εh) = d L(x, y + εh, y + εh,y + εh )dx dε ε=0 ( = h ) + h + h dx y y y (3.1) 'Opwc ki prohgoumènwc mporoôme n oloklhr soume kt pr gontec ki n pleðyoume tic prg gouc thc h. 'Eqoume h b b ( dx = h y y h d dx y = h d ( ) b + dx y ( ) d 2 = dx, dx 2 y ìpou qrhsimopoi sme tic sunorikèc sunj kec h d2 dx 2 h() =h () =h(b) =h (b) =0. ) dx ( y ) dx EpÐshc h dx = h y y = b h d ( dx y ( ) dx. h d dx y ) dx Sunep c h(3.1)gðneti : δj(y, h) = { h y d ( dx y )+ d2 dx 2 ( y )} dx.

31 3.1. PAR AGWGOI UYHL OTERHS T AXHS 31 AfoÔ to J prousi zei sto y topikì krìtto, tìte pì to Je rhm (2.2.1) j prèpei δj(y, h) =0 Dhld j èqoume { h y d dx ( y )+ d2 dx 2 ( y )} dx =0. Gi thn olokl rwsh thc didiksðc j qrhsimopoi soume mi genðkeush tou L mmtoc (2.3.1). L mm 'Estw f suneq c sun rthsh sto [, b]. An isqôei f(x)h(x)dx =0 gi k je h C n [, b] pou iknopoieð tic sunj kec h() = h () =... = h (n) () =h(b) =h (b) =...= h (n) (b) =0, tìte f(x) 0, x [, b]. Apìdeixh 'Estw ìti f(x 0 ) > 0 gi k poio x 0 [, b]. Epeid h f eðni suneq c j isqôei ìti f(x) > 0 se k poio di sthm [x 1,x 2 ] [, b] to opoðo perièqei to x 0. JewroÔme th sun rthsh h(x) = { (x x1 ) n+1 (x 2 x) n+1, n x 1 x x 2 0, diforetik, h opoð iknopoieð tic sunj kec tou L mmtoc. Tìte ìmwc èqoume f(x)h(x)dx = f(x)(x x 1 ) n+1 (x 2 x) n+1 dx > 0 foô to olokl rwm eðni jetikì sto di sthm [x 1,x 2 ]. Autì eðni ntðjeto b sthn upìjesh ìti f(x)h(x)dx =0. 'Ar f(x) 0 x [, b]. Qrhsimopoi ntc t r to L mm (3.1.1) gi n = 4 sumperðnoume ìti mi ngkð sunj kh gi n eðni to y topikì krìtto eðni h y d ( ( ) )+ d2 =0, x b. dx y dx 2 y H prp nw exðswsh eðni h exðswsh Euler-Lgrnge tou probl mtoc ki eðni mi sun jhc diforik exðswsh tètrthc t xhc wc proc y. H genik lôsh thc j perilmb nei tèsseric stjerèc pou mporoôn n upologistoôn pì tic sunorikèc sunj kec y() =A 1, y () =A 2, y(b) =B 1, y (b) =B 2.

32 32 KEF ALAIO 3. GENIKEUSEIS Pr deigm 'Estw to sunrthsoeidèc J(y) = 1 0 ((y ) 2 2py)dx, ìpou p stjer ki y(0) = y (0) = 0, y(1) = y (1) = 1. H exðswsh Euler- Lgrnge gi utì to sunrthsoeidèc eðni : me genik lôsh y (4) (x) =p, y(x) = 1 4! px4 + C 1 x 3 + C 2 x 2 + C 3 x + C 4. Qrhsimopoi ntc tic rqikèc sunj kec pðrnoume y(0) = 0 C 4 =0 y (0) = 0 C 3 =0 y(1) = 1 p 4! + C 1 + C 2 =1 y (1) = 1 p 3! +3C 1 +2C 2 =1 LÔnontc to prp nw sôsthm èqoume ìti C 1 = 1 p, 12 to krìtto dðneti pì thn C 2 =2+ p 24. 'Ar y(x) = 1 24 px4 (1 + p 12 )x3 +(2+ p 24 )x2. H mèjodoc pou perigr yme prp nw mporeð n genikeujeð gi sunrthsoeid thc morf c J(y) = L(x, y, y,...,y (n) )dx, y A, ìpou A to sônol otwn sunrt sewn C n [, b] pou plhroôn tic sunorikèc sunj kec y() =A 1,y () =A 2,...,y (n 1) () =A n y(b) =B 1,y (b) =B 2,...,y (n 1) (b) =B n. JewroÔme thn ŷ = y(x)+εh(x) ìpou h(x) C n [, b] me h() =h () =...= h (n 1) () =0

33 3.2. POLL ES SUNART HSEIS 33 h(b) =h (b) =...= h (n 1) (b) =0. UpologÐzoume th metbol tou sunrthsoeidoôc J = J(y + εh) J(y) me th bo jei tou jewr mtoc Tylor ki pðrnoume : J = ε ( y h + y h ) h(n) dx + O(ε 2 ). y (n) Sunep c h pr th metbol tou sunrthsoeidoôc J(y) eðni ( δj(y, h) = y h + y h ) h(n) dx. y (n) AngkÐ sunj kh gi n èqoume topikì krìtto eðni h δj(y, h) =0. 'Ar ( y h + y h ) h(n) dx =0. y (n) Oloklhr nontc kt pr gontec ki qrhsimopoi ntc tic sunorikèc sunj kec èqoume [ y d ( dx y )+ d2 dx 2 ( ( )...+( 1) n dn y dx n y (n) )] hdx =0. Efrmìzontc t r to L mm (3.1.1) sumperðnoume ìti h exðswsh Euler eðni h y d ( ( ( ) )+ d2 )...+( 1) n dn =0 dx y dx 2 y dx n y (n) ki eðni mi sun jhc diforik exðswsh t xhc 2n. 3.2 Pollèc sunrt seic Mi diforetik genðkeush eðni n jewr soume ìti to sunrthsoeidèc J e- xrt ti pì perissìterec pì mi sunrt seic. Gi n plopoi soume to prìblhm c jewr soume ìti to sunrthsoeidèc exrt ti pì dôo sunrt seic y 1,y 2 C 2 [, b]. To sunrthsoeidèc tìte j eðni thc morf c me sunorikèc sunj kec J(y 1,y 2 )= L(x, y 1,y 2,y 1,y 2 )dx y 1 () =A 1,y 1 (b) =B 1,y 2 () =A 2,y 2 (b) =B 2.

34 34 KEF ALAIO 3. GENIKEUSEIS Upojètoume ìti oi y 1 ki y 2 elqistopoioôn to J. Metb loume k je mi pì utèc tic sunrt seic nex rtht pì thn llh epilègontc h 1,h 2 C 2 [, b] pou n iknopoioôn tic h 1 () =h 2 () =h 1 (b) =h 2 (b) =0 ètsi ste n p roume to podektì zeôgoc sunrt sewn ŷ 1 ki ŷ 2 me ŷ 1 = y 1 +εh 1 ki ŷ 2 = y 2 +εh 2. Skopìc mc ki ut th for eðni n upologðsoume thn pr th metbol. Me qr sh tou jewr mtoc Tylor upologðzoume th metbol tou sunrthsoeidoôc J = J(y 1 + εh 1,y 2 + εh 2 ) J(y 1,y 2 ) ( = ε h 1 + h 2 + h 1 + h 2 y 1 y 2 y 1 y 1 y 2 Sunep c h pr th metbol eðni ( δj = h 1 + h 2 + h 1 + h 2 y 1 y 2 ) dx + O(ε 2 ). y 2 ) dx Oloklhr nontc kt pr gontec touc dôo teleutðouc ìrouc ki qrhsimopoi ntc to Je rhm (2.2.1) pðrnoume : ( ( {( y 1 d dx y 1 )) h 1 + ( y 2 d dx y 2 ))h 2 } dx =0, ìpou qrhsimopoi sme tic sunj kec h 1 () =h 2 () =h 1 (b) =h 2 (b) =0. An epilèxoume h 2 =0gi x [, b] h prp nw exðswsh gðneti ( ( y 1 d dx y 1 )) h 1 dx =0 ki efrmìzontc to L mm (2.3.1) pðrnoume d ( ) =0. (3.2) y 1 dx y 1 An t r epilèxoume h 1 =0gi x [, b] tìte ( d ( )) h 2 dx =0 y 2 dx ki efrmìzontc to Je rhm (2.2.1) pðrnoume d ( ) =0. (3.3) y 2 dx y 2 y 2

35 3.2. POLL ES SUNART HSEIS 35 AfoÔ oi y 1,y 2 ntistoiqoôn se topikì el qisto tou J tìte j prèpei n iknopoioôn to sôsthm twn dôo sun jwn diforik n exis sewn (3.2) ki (3.3), oi opoðec eðni oi exis seic Euler-Lgrnge tou sunrthsoeidoôc J(y 1,y 2 )= L(x, y 1,y 2,y 1,y 2 )dx. Pr deigm Ac jewr soume to sunrthsoeidèc J(y 1,y 2 )= 1 0 ( y 2 1 +( y 2 1) 2 + y1 2 + y 1y 2 )dt. Oi exis seic Euler-Lgrnge tou sunrthsoeidoôc eðni ÿ 1 y y 2 = 0 (1) ÿ y 1 = 0 (2) J lôsoume t r utì to sôsthm me th mèjodo thc ploif c. Pollplsi zoume thn (1) epð 2 ki thn prgwgðzoume dôo forèc. Opìte n thn prosjèsoume sth (2) ktl goume sth grmmik diforik exðswsh tètrthc t xhc To qrkthristikì polu numo ut c eðni y (4) 1 y (2) y 1 =0. (3.4) p(λ) = λ 4 λ =0, to opoðo èqei rðzec tic λ 1,2 = 1 ± µ 1,2 = ±i 'Ar ìpwc gnwrðzoume pì th jewrð twn sun jwn diforik n exis sewn h genik lôsh thc (3.4) eðni gi µ = y 1 (t) =c 1 e λ 1t + c 2 e λ 2t + c 3 cos (µt)+c 4 sin (µt) Apì thn genik lôsh thc (3.4) ki thn exðswsh (1) brðskoume thn genik lôsh thc y 2 (t) h opoð eðni [ y 2 (t) =2 e λ1t c 1 ( 1+λ 2 1)+e λ2t c 2 ( 1+λ 2 2)

36 36 KEF ALAIO 3. GENIKEUSEIS ] (1 + µ 2 )c 3 cos (µx) (1 + µ 2 )c 4 sin (µx). Ac genikeôsoume t r t prp nw gi sunrthsoeid pou perièqoun y i, i =1,...,n sunrt seic. To sunrthsoeidèc j èqei th morf me J(y 1,...,y n )= y i () =A i,y i (b) =B i L(x, y 1,...,y n,y 1,...,y n )dx, (i =1,...,n) Metb lloume k je y i kt h i me h i () =h i (b) =0. Tìte h metbol tou sunrthsoeidoôc me qr sh tou jewr mtoc Tylor gðneti J = J(y 1 + εh 1,...,y n + εh n ) J(y 1,...,y n ) n ( = ε h i + ) h y i y i i dx + O(ε 2 ) i=1 Opìte h pr th metbol eðni : δj = n ( h i + ) h y i y i i dx. i=1 Dilègoume didoqik k poio h i 0ki jètoume ìl t upìloip Ðs me mhdèn. Tìte ( h i + ) h y i y i i dx =0 (i =1...,n). Oloklhr nontc kt pr gontec ton teleutðo ìro ki qrhsimopoi ntc to L mm (2.3.1) ktl goume sto sôsthm : y i d dx ( y i ) =0, (i =1,...,n).

37 3.3. PROBL HMATA POLLAPL WN OLOKLHRWM ATWN Probl mt pollpl n oloklhrwm - twn J orðsoume t r sunrthsoeid pou orðzonti pì pollpl oloklhr mt. Esti zoume thn prosoq mc sthn pio pl perðptwsh ìpou h oloklhrwtè sun rthsh eðni dôo metblht n. 'Estw Ω èn kleistì qwrðo sto R 2. JewroÔme to sônolo twn suneq n sunrt sewn u = u(x, y) pou orðzonti sto Ω ki èqoun suneqeðc prg gouc deôterhc t xhc ki to sumbolðzoume me C 2 (Ω). H u(x, y) prist mi leð epif nei p nw pì to qwrðo Ω. To metbolikì prìblhm se ut thn perðptwsh sunðstti sthn eôresh mic sun rthshc u C 2 (Ω) tètoi ste n elqistopoieð to sunrthsoeidèc J(u) = L(x, y, u(x, y),u x (x, y),u y (x, y))dxdy upì tic sunj kec Ω u(x, y) =f(x, y), (x, y) Ω, ìpou f mi dedomènh sun rthsh pou orðzeti sto sônoro tou Ω. J kolouj soume t Ðdi b mt ìpwc ki prohgoumènwc. 'Estw ìti h u(x, y) eðni topikì krìtto tou J. JewroÔme thn oikogènei twn podekt n sunrt sewn û(x, y) =u(x, y) +εh(x, y) ìpou h C(Ω) ki h(x, y) =0gi (x, y) Ω. Me qr sh tou jewr mtoc Tylor èqoume J = J(û) J(u) { = ε h u + h x + h y u x u y Ω Epomènwc h pr th metbol eðni { δj(u, h) = h u + h x + h y u x u y Ω Ω } dxdy + O(ε 2 ) } dxdy PrthroÔme ìti ( h x + ) h y dxdy = Ω u x u y [ ( h ) + ( h )] [ ( ) dxdy h + ( )] dxdy Ω x u x y u y Ω x u x y u y To pr to pì t dôo prp nw oloklhr mt mporoôme n to metsqhmtðsoume se epikmpôlio me th bo jei tou Jewr mtoc Green pou lèei ìti ( Q x P ) ( ) dxdy = Pdx+ Qdy, y Ω

38 38 KEF ALAIO 3. GENIKEUSEIS z y 2 x 1 x 0 y 1 y 0 y x x 2 Sq m 3.1: H grfik pr stsh thc h(x, y) gi k je sun rthsh Q, P : Ω R me P, Q, P y,q x suneqeðc. Efìson oi L, h eðni suneqeðc sunrt seic mesuneqeðc prg gouc èqoume : [ ( h ) + ( h )] ( dxdy = h dx ) dy =0, Ω x u x y u y Ω u y u x foô h(x, y) =0gi (x, y) Ω. Sunep c h pr th metbol gðneti [ δj(u, h) = h(x, y) u ( ) ( )] dxdy x u x y u y Ω An h u(x, y) eðni topikì krìtto tìte pì to Je rhm (2.2.1) èqoume ìti δj(u, h) =0. H kìloujh genðkeush tou L mmtoc (2.3.1) j mc bohj sei sthn eôresh mic ngkðc sunj khc gi krìtto. L mm An g C(Ω) ki g(x, y)h(x, y)dxdy =0 Ω gi k je h(x, y) sto C 2 (Ω) me h(x, y) =0sto sônoro tou Ω tìte g(x, y) =0 gi (x, y) Ω. Apìdeixh 'Estw ìti g(x 0,y 0 ) > 0 se k poio shmeðo (x 0,y 0 ) sto Ω. Epeid h g eðni suneq c, j èqoume g(x, y) > 0 gi k je x, y se èn qwrðo D =

39 3.3. PROBL HMATA POLLAPL WN OLOKLHRWM ATWN 39 {(x, y) Ω x 1 x x 2, y 1 y y 2 } pou perièqei to (x 0,y 0 ). JewroÔme th sun rthsh h(x, y) = { (x x1 ) 3 (x 2 x) 3 (y y 1 ) 3 (y 2 y) 3, n (x, y) D 0, diforetik, me grfik pr stsh ìpwc fðneti sto Sq m (3.1) Tìte èqoume g(x, y)h(x, y) > 0 Ω foô h g(x, y) eðni jetik sto D. Autì ìmwc eðni topo lìgw thc upìjeshc tou L mmtoc. 'Ar g(x, y) =0gi (x, y) Ω. Me th qr sh tou prp nw L mmtoc sth sunj kh δj(u, h) =0pÐrnoume u ( ) ( ) =0 (3.5) x u x y u y pou eðni h exðswsh Euler-Lgrnge tou probl mtoc. H (3.5) eðni mi merik diforik exðswsh deôterhc t xhc gi th u = u(x, y) ki oi lôseic thc lègonti krìtt krìttec epif neiec. Pr deigm N brejeð h exðswsh Euler-Lgrnge tou sunrthsoeidoôc [ 1 J(u) = Ω 2 u2 x + 1 ] 2 u2 y + p(x, y)u dxdy, ìpou p(x, y) mi dedomènh sun rthsh epð tou Ω. 'Eqoume u = p(x, y), = u x, u x 'Ar h exðswsh Euler-Lgrnge eðni : pou eðni h exðswsh P oisson. u y = u y p(x, y) x u x y u y =0 u xx + u yy = p(x, y)

40 40 KEF ALAIO 3. GENIKEUSEIS Pr deigm 'Estw Ω odðskoc pou orðzeti pì x 2 + y 2 < 1 ki èstw to sunrthsoeidèc J(u) = (u 2 x + u2 y )dxdy. N brejeð h exðswsh Euler-Lgrnge tou sunrthsoeidoôc. 'Eqoume Ω u =0, =2u x, u x 'Ar h exðswsh Euler Lgrnge eðni pou eðni gnwst wc exðswsh Lplce. x u x + y u y =0 u xx + u yy =0 u y =2u y. O Bèlgoc fusikìc Joseph Plteu ( ), pì to 1830 wc to 1869 prgmtopoðhse pl joc peirm twn me ntikeðmeno thn epifneik t sh ki t triqoeid finìmen, peir mt pou eðqn ter sti epirro ekeðnh thn epoq ki epnl fjhkn pì xiwshmeðwtouc fusikoôc tou dek tou en tou i n, ìpwc o Michel Frdy ( ). 'En pì t peir mt tou tn to ex c: An bujðsoume èn sôrm se spoôni di lum glukerðnhc, sun jwc nsôroume èn leptì str m spounioô pou sunoreôei me to sôrm. B sh twn prthr sewn tou, o Plteu èjese to mjhmtikì prìblhm: Gi dedomèno sônoro (sôrm), p c mporeð n deðxei kneðc thn Ôprxh mic tètoic epif neic (membr nh spounioô) ki pìsec tètoiec epif neiec eðni duntìn n up rxoun? H fusik rq pou krðbeti pðsw pì to er thm, eðni ìti h fôsh teðnei n elqistopoi sei to embdìn. Dhld, h epif nei pou sqhmtðzeti j prèpei n eðni h epif nei el qistou embdoô n mes se ekeðnec pou èqoun gi sônoro th dedomènh kmpôlh. J exet soume t r to prìblhm tou Plteu 1. 1 Το πρόβλημα αυτό το έθεσε πρώτα ο Joseph-Louis Lgrnge το 1760, αλλά πήρε το όνομα του Plteu λόγω των πειραμάτων του με τις μεμβράνες σαπουνιών.

41 3.3. PROBL HMATA POLLAPL WN OLOKLHRWM ATWN 41 Pr deigm Dedomènhc mic kleist c kmpôlhc Γ stoq ronbrejeð h epif nei u = u(x, y) me sônoro Γ thc opoðc to embdìn eðni el qisto. Stìqoc mc, gi rq, eðni n broôme to sunrthsoeidèc pou mc dðnei to embdìn mic epif neic ki èpeit n broôme th sun rthsh u = u(x, y) h opoð to elqistopoieð. Apì ton Apeirostikì Logismì èqoume ton ex c orismì Orismìc An mi leð epif nei S, pou orðzeti p nw pì to qwrðo D, dðneti pì thn exðswsh r(u, υ) =x(u, υ)i + y(u, υ)j + z(u, υ)k, me (u, υ) D, tìte to embdìn thcepif neic S eðni A(S) = r u r υ da, ìpou r u = x u i + y u j + z u k ki r υ = x υ i + y υ j + z υ k. D Sugkekrimèn gi thn epif nei u = u(x, y) èqoume tic prmetrikèc exis seic x = x, y = y ki z = u(x, y). 'Etsi ( ) u u x = i + k x ki Opìte u y = j + ( ) u k. y i j k u x u y = 1 0 u x 0 1 u y = u x i u y j + k. Epomènwc to sunrthsoeidèc pou mc dðnei to embdìn epif neic eðni J(y) = Ω 1+ ( ) 2 ( ) 2 u u + dxdy, (3.6) x y ìpou Ω eðni to qwrðo pou perikleðeti pì thn probol thc Γ epð tou epipèdou x, y ki Ω to sônoro tou Ω.

42 42 KEF ALAIO 3. GENIKEUSEIS Jèloume n broôme mi sun rthsh u = u(x, y) h opoð n elqistopoieð to sunrthsoeidèc (3.6) upì th sunorik sunj kh u(x, y) =f(x, y), (x, y) Ω, ìpou f sun rthsh pou orðzeti sto Ω. H Lgkrnzin tou sunrthsoeidoôc eðni L = 1+p 2 + q 2, ìpou p = u x 'Eqoume ki q = u y. u = 0 p q = = p 1+p2 + q 2 q 1+p2 + q 2 'Etsi h exðswsh Euler-Lgrnge gðneti u x x 1+u 2 x + u 2 y y u y 1+u 2 x + u 2 y UpologÐzontc tic merikèc prg gouc pðrnoume =0. (1 + u 2 y )u xx u y u x u xy +(1+u 2 x )u yy =0 (3.7) H(3.7)eÐni h zhtoômenh exðswsh el qisthc epif neic ki eðni mi mh grmmik merik diforik exðswsh deôterhc t xhc. Gi polô perissìter pì 70 qrìni, mjhmtikoð ìpwc oi Riemnn, W- eierstrss, Schwrz, Drboux ki Lebesgue prosp jhsn n pnt soun sthn prìklhsh tou Plteu. To 1931 to prìblhm lôjhke oristik pì ton Jesse Dougls ston opoðo d jhke to pr to sthn istorð brbeðo Fields Medl (1936) gi tic prosp jeiec tou. 'Omwc poll probl mt se sqèsh me tic membr nec spounioô eðni np ntht, ki ut h perioq èreunc prmènei zwntn stic mèrec mc. H genðkeush se m-pl oloklhr mt eðni mesh. 'Estw u = u(x 1,...,x m ) mi sun rthsh m metblht n. To sunrthsoeidèc j eðni thc morf c J(u) =... Ω m L ( x 1,...,x m,u, u x 1,..., u x m ) dx 1...dx m

43 3.3. PROBL HMATA POLLAPL WN OLOKLHRWM ATWN 43 ìpou Ω m kleistì qwrðo ston m-di stto eukleðdeio q ro. Se ut thn perðptwshhexðswsh Euler Lgrnge eðni : u x 1 u x1 x 2... u x2 x m u xm =0.

44 44 KEF ALAIO 3. GENIKEUSEIS 3.4 Fusikèc Sunorikèc Sunj kec Mèqri t r sqolhj kme me thn eôresh krìttwn shmeðwn sunrthsoeid n thc morf c J(y) = L(x, y, y )dx upì tic sunorikèc sunj kec y() =A,y(b) =B. EÐdme epðshc ìti n h Lgkrnzin perilmb nei prg gouc uyhlìterhc t xhc tìte qreizìmste perissìterec sunorikèc sunj kec gi ton prosdiorismì thc st simhc sun rthshc. 'Omwc pollèc forèc sunnt me probl mt st opoð den mc dðneti o kt llhloc rijmìc sunjhk n. Gi pr deigm c jewr soume to kìloujo prìblhm : 'Estw ìti h tqôtht tou reômtoc enìc potmoô me pr llhlec ìqjec pou pèqoun b mon dec m kouc eðni u(x, y) =u(x)j ìpou j eðni to mondiðo di - nusm sth dieôjunsh y. Upojètoume ìti mi pì tic ìqjec eðni o xonc y ki ìti to (0, 0) eðni to shmeðo ekkðnhshc. To er thm pou tðjeti eðni to ex c Poi didrom j prèpei n kolouj sei mi b rk ste n ft sei sthn pènnti ìqjh ston suntomìtero duntì qrìno? Upojètoume ìti h tqôtht thc b rkc se kðnhto nerì eðni c, ìpou c 2 >u 2. y u y y x 0,0 x b x Sq m 3.2: H didrom y = y(x) kt m koc enìc potmoô me tqôtht u = u(x)j. Prt rhsh To prìblhm utì difèrei pì ìs èqoume dei mèqri t r sto ìti to shmeðo fixhc epð thc eujeðc x = b den eðni kjorismèno.

45 3.4. FUSIK ES SUNORIAK ES SUNJ HKES 45 Tètoiou eðdouc probl mt onom zonti probl mt eleôjerou krou ki ìtn mð pì tic sunj kec ki ìlec den eðni prosdiorismènec lègonti fusikèc sunorikèc sunj kec. Gi n sugkekrimenopoi soume tic ènnoiec jewroôme to sunrthsoeidèc ìpou y C 2 [, b] ki J(y) = L(x, y, y )dx y() =A, y(b) :µη πρoσδιoρισµένo. 'Estw ŷ = y(x)+εh(x) to sônolotwn podekt n sunrt sewn, me h C 2 [, b] ki h() =0. Epeid oi podektèc sunrt seic den eðni prosdiorismènec sto dexiì kro den piteðti kmð sunj kh gi thn h sto x = b. Qrhsimopoi ntc touc Ðdiouc sullogismoôc ìpwc ki st prohgoômen ktl goume ìti h pr th metbol eðni ( h ) + h dx =0. y y Oloklhr nontc kt mèrh to deôtero ìro tou oloklhr mtoc èqoume h b ( y + h y d ( )) dx =0. dx y Epeid h() =0èqoume ( y d ( )) hdx + dx y y h(b) =0 (3.8) x=b gi k je h sto C 2 [, b]. Efìson h (3.8) isqôei gi k je h sto C 2 [, b], j prèpei ni isqôei ki gi tic h pou iknopoioôn thn h(b) =0. 'Ar ( y d dx y ) hdx =0. Qrhsimopoi ntc to L mm (2.3.1) èqoume ìti mi ngkð sunj kh gi n eðni h y krìtto eðni ìti j prèpei n iknopoieð thn exðswsh Euler-Lgrnge y d ( ) =0. dx y Antikjist ntc t r thn exðswsh Euler-Lgrnge sthn (3.8) pðrnoume y h(b) =0, x=b

46 46 KEF ALAIO 3. GENIKEUSEIS h opoð j isqôei gi opoid pote h(b), opìte j isqôei y =0 (3.9) x=b pou eðni mi sunj kh epð tou krìttou y sto x = b. HexÐswsh (3.9) onom - zeti fusik sunorik sunj kh. Tèloc gi ton prosdiorismì enìc krìttou, enìc probl mtoc eleôjerou krou mc rkoôn h exðswsh Euler-Lgrnge, h dedomènh sunorik sunj kh ki h fusik sunorik sunj kh. Me n logouc sullogismoôc èpeti ìti n h tim sto risterì kro den eðni prosdiorismènh, tìte h fusik sunorik sunj kh sto x = gi krìtto eðni y =0. x= MporoÔme n genikeôsoume to prp nw prìblhm sthn perðptwsh ìpou knèn pì t dôo kr den eðni prosdiorismèno wc ex c : JewroÔme to sônol otwn podekt n sunrt sewn ŷ = y + εh. Me qr sh tou jewr mtoc Tylor h metbol tou sunrthsoeidoôc eðni ( J = ε y h + ) y h dx + O(ε 2 ). 'Ar h pr th metbol eðni δj = ( y h + ) y h dx Oloklhr nontc kt mèrh èqoume ( δj = y d ( )) h(x)dx + dx y y h(b) x=b y h(). (3.10) x= JewroÔme ìti isqôei h() =h(b) =0. Tìte mi ngkð sunj kh gi n èqoume krìtto sto y eðni δj =0. Me th qr sh tou L mmtoc (2.3.1) ktl goume ki p li sthn exðswsh Euler-Lgrnge y d ( ) =0. dx y Me ntikt stsh thc exðswshc Euler-Lgrnge sthn (3.10) èqoume ìti y h(b) x=b y h() =0. x=

47 3.4. FUSIK ES SUNORIAK ES SUNJ HKES 47 AfoÔ h prp nw exðswsh prèpei n isqôei gi opoid pote h j èqoume ìti y =0 & x= y =0. x=b Opìte gi n lôsoume èn tètoio prìblhm, ìpou kmð pì tic sunorikèc sunj kec den eðni prosdiorismènh, prèpei pr t n broôme thn exðswsh Euler- Lgrnge ki èpeit n qrhsimopoi soume tic dôo fusikèc sunorikèc sunj kec. Pr deigm N brejeð h diforik exðswsh ki oi sunorikèc sunj kec gi to krìtto tou metbolikoô probl mtoc J(y) = 1 0 (p(x)y 2 q(x)y 2 )dx, y(0) = 0,y(1) : µη πρoσδιoρισµένo, ìpou p ki q eðni jetikèc omlèc sunrt seic orismènec sto [0, 1]. 'Eqoume ki y = 2q(x)y y =2p(x)y. Epomènwc h exðswsh Euler-Lgrnge eðni : q(x)y + d dx (p(x)y )=0, 0 <x<1. H fusik sunorik sunj kh sto x =1dÐneti pì thn y =0 2p(1)y (1) = 0. x=1 Epeid ìmwc p>0, oi sunorikèc sunj kec gðnonti y(0) = 0 y (1) = 0

48 48 KEF ALAIO 3. GENIKEUSEIS Pr deigm (To prìblhm tou Zermelo) N brejeð h fusik sunorik sunj kh sto x = b gi to prìblhm tou di plou kt pl toc enìc potmoô. 'Estw c h tqôtht thc b rkc, φ h gwnð pou sqhmtðzei me ton xon x ki u(x) h tqôtht tou reômtoc tou potmoô. H kðnhsh thc b rkc mporeð n nlujeð stouc xonec x,y. Ston xon x kineðti me tqôtht en ston xon y me dy dt dx dt Sundu zontc tic (3.11) ki (3.12) pðrnoume = c cos φ, (3.11) = c sin φ + u. (3.12) y = dy/dt dx/dt = c sin φ + u c cos φ. (3.13) O qrìnoc, gi n disqðsei h b rk ton potmì, dðneti pì t = = = T dt dt dx dx 1 dx, (3.14) c cos φ me c cos (φ) 0lli c o qrìnoc j tn peiroc. Apì thn (3.13) pðrnoume c cos φy = u + c sin φ c cos φy = u ± c 1 cos 2 φ c cos φy u = ±c 1 cos 2 φ (c cos φy u) 2 = c 2 (1 cos 2 φ) y 2 c 2 cos 2 φ 2uy c cos φ + u 2 c 2 + c 2 cos 2 φ =0 (y 2 +1)(ccos φ) 2 2uy c cos φ (c 2 u 2 )=0 Diir ntc t r me (c cos φ) 2, to opoðo eðni di foro tou mhdenìc, pðrnoume ( ) 2 ( ) 1 1 (c 2 u 2 ) +2uy (y 2 +1)=0. c cos φ c cos φ

49 3.4. FUSIK ES SUNORIAK ES SUNJ HKES 49 Jètoume p = 1 c cos φ. Tìte (c 2 u 2 )p 2 +2uy p (y 2 +1)=0. LÔnontc wc proc p thn prp nw exðswsh pðrnoume p = 2uy ± (2uy ) 2 +4(c 2 u 2 )(y 2 +1) 2(c 2 u 2 ) = uy ± c 2 (y 2 +1) u 2. c 2 u 2 Gi n per sei h b rk ton potmì j prèpei Epomènwc dx dt > 0, dhld c cos φ>0. 1 c cos φ = c2 (y 2 +1) u 2 uy c 2 u 2 (3.15) Me ntikt stsh thc (3.15) sth (3.14) pðrnoume to sunrthsoeidèc c2 (1 + y J(y) = 2 ) u 2 uy dx, c 2 u 2 (3.16) 0 to opoðo mc dðnei to qrìno pou qrei zeti h b rk gi n dipleôsei kt pl toc ton potmì kolouj ntc mi dedomènh didrom y = y(x). J broôme t r th sunorik sunj kh sto x = b. 'Eqoume y = 1 c 2 u 2 H sunorik sunj kh sto x = b gðneti ( c 2 y ). c2 (1 + y 2 ) u u 2 c 2 y (b) c2 (1 + y (b) 2 ) u(b) 2 u(b) =0 c2 y (b) =u(b) c 2 (1 + y (b) 2 ) u(b) 2. Uy nontc eic th deutèr ki k nontc pr xeic ktl goume sthn y (b) = u(b) c 'Etsi h klðsh upì thn opoð h b rk proseggðzei thn ìqjh sto x = b eðni o lìgoc thc tqôthtc tou neroô sthn ìqjh proc thn tqôtht thc b rkc se kðnhto nerì. ProsdiorÐzontc t r thn exðswsh Euler-Lgrnge j mporèsoume n broôme poi sun rthsh j mc elqistopoi sei to sunrthsoeidèc (3.16). 'Eqoume = 0 y ( ) 1 c 2 y = y (c 2 u 2 ) c2 (1 + y 2 ) u u 2.

50 50 KEF ALAIO 3. GENIKEUSEIS u y u c x 0,0 x b Sq m 3.3: LÔsh tou probl mtoc tou Zermelo 'Ar h exðswsh Euler-Lgrnge eðni c 2 y (c 2 u 2 + c 2 y 2 ) 3 2 =0, me sunorikèc sunj kec y(0) = 0 ki y (b) = u(b) c. LÔnontc ut th diforik exðswsh pðrnoume y(x) = u(b) c x, dhld h poreð pou j kolouj sei h b rk gi n ft sei sthn pènnti ìqjh (x = b) ston suntomìtero duntì qrìno eðni h eujeð me klðsh u(b) c ìpwc fðneti sto Sq m (3.3).

51 Kef lio 4 QAMILTONIANH JEWRIA S utì to kef lio j doôme ìti h Klsik Mhqnik mporeð n jemeliwjeð se mi nè rq, thn rq tou Hmilton, ntð twn xiwm twn tou NeÔtwn. J broôme èn nèo sôsthm diforik n exis sewn oi opoðec perigr foun thn kðnhsh tou sust mtoc. Mi mèjodoc gi thn exgwg ut n twn exis sewn eðni pì mi metbolik rq ki bsðzeti sthn idè ìti èn sôsthm j prèpei n exelðsseti qronik p nw sth didrom el qisthc ntðdrshc. Tètoiou eðdouc rqèc èqoun meg lh istorð ki qronologoônti pì touc ArqÐouc 'Ellhnec. O 'Hrwn o AlexndreÔc ditôpwse mi rq elqðstou pou for th didrom twn nkl menwn ktðnwn tou fwtìc. Ton dèkto èbdomo i n ditup jhke h Arq tou Fermt,ìti oi ktðnec tou fwtìc kinoônti epð thc didrom c elqðstou qrìnou. To 1746, h rq tou G llou mjhmtikoô Mupertuis nèfere ìti èn sôsthm prèpei n exelðsseti pì mi kt stsh se mi llh kt tètoion trìpo ste h dr sh n eðni el qisth. Uposthriktèc tètoiwn rq n up rxn ki oi Guss ki Lgrnge. T pr t qrìni tou dèktou èntou i n o Hmilton ditôpwse mi periektik, komy rq pì thn opoð prokôptoun oi exis seic Euler-Lgrnge. 'En shmntikì pleonèkthm thc rq c tou Hmilton eðni ìti me th bo jei thc mporoôme n epekteðnoume tic mejìdouc thc Klsik c Mhqnik c ki se llouc tomeðc thc Fusik c. 4.1 H Arq tou Hmilton 'Estw y 1,...,y n èn sônolo genikeumènwn suntetgmènwn enìc dedomènou dunmikoô sust mtoc, pou kjorðzoun pl rwc thn kt stsh tou sust mtoc se opoid pote qronik stigm. JewroÔme, dhld, ìti oi y 1,...,y n eðni sunrt seic tou qrìnou ki mporeð n eðni m kh, gwnðec otid pote llo. Oi pr gwgoi wc proc to qrìno ẏ 1,...,ẏ n onom zonti genikeumènec tqôthtec. H kinhtik enèrgei T sthn pio genik perðptwsh eðni mi tetrgwnik morf 51

52 52 KEF ALAIO 4. QAMILTONIANH JEWRIA wc proc tic ẏ i, dhld T = n n ij (y 1,...,y n )ẏ i ẏ j, i=1 j=1 ìpou ij gnwstèc sunrt seic twn genikeumènwn suntetgmènwn y 1,...,y n. H dunmik enèrgei V mporeð n eðni sun rthsh twn t, y i ki ẏ i, dhld thc morf c V = V (t, y 1,...,y n, ẏ 1,...,ẏ n ). OrÐzoume th Lgkrnzin wc L = T V. H Arq tou Hmilton gi tètoi sust mt ditup neti wc ex c : JewroÔme èn mhqnikì sôsthm pou perigr feti pì genikeumènec suntetgmènec y 1,...,y n me Lgkrnzin L = T V. Tìte hkðnhsh tou sust mtoc pì thn qronik stigm t 0 mèqri th qronik stigm t 1 eðni tètoi ste to sunrthsoeidèc J(y 1,...,y n )= t1 t 0 L(t, y 1,...,y n, ẏ 1,...,ẏ n )dt (4.1) n eðni st simo. 'Eqoume upojèsei ìti h L eðni dôo forèc suneq c diforðsimh sun rthsh ki y i C 2 [t 0,t 1 ]. Mi llh ditôpwsh thc Arq c tou Hmilton eðni h ex c : 'Otn èn sôsthm kineðti pì mi jèsh S 0 :(y 1 (t 0 ),...,y n (t 0 )) sth jèsh S 1 : (y 1 (t 1 ),...,y n (t 1 )), pou orðzonti pì tic genikeumènec suntetgmènec y i th qronik stigm t 0 ki t 1 ntðstoiq, tìte metxô ìlwn twn dunt n troqi n pou sundèoun t dôo prp nw shmeð tou q rou nprst sewn, h prgmtik kðnhsh j l bei q r kt m koc ekeðnhc thc didrom c pou krottopoieð to olokl rwm (4.1). H Arq tou Hmilton suqn ditup neti ki wc δ Ldt =0, enno ntc krib c thn prp nw ditôpwsh. EpÐshc, to sunrthsoeidèc Ldt lègeti suqn olokl rwm dr shc. Efìson to sunrthsoeidèc (4.1) pðrnei st simh tim gi tic y i = y i (t) i = 1,...,n èpeti pì to Je rhm (2.3.1) ìti h y i (t) prèpei n iknopoieð tic exis seic Euler-Lgrnge d ( ) =0 i =1,...,n. y i dt ẏ i