Anaz thsh eustaj n troqi n se triplˆ sust mata swmˆtwn

Tmhma Fusikhc Aristoteleio Panepisthmio Jessalonikhc Ptuqiakh Ergasia Anaz thsh eustaj n troqi n se triplˆ sust mata swmˆtwn Ajanˆsioc MourtetzÐkoglou A.E.M.:13119 epiblèpwn kajhght c G. Bougiatz c 8 IoulÐou 2016

Perieqìmena 1 Eisagwg sto prìblhma twn tri n swmˆtwn 5 1.1 Genik IstorÐa............................... 5 1.2 To Prìblhma twn Tri n Swmˆtwn.................... 6 1.2.1 Orismìc tou Probl matoc.................... 6 1.2.2 Oloklhr mata thc kðnhshc.................... 6 1.2.3 KÐnhsh wc proc to kèntro mˆzac................. 7 1.3 LÔseic - Troqièc.............................. 8 1.3.1 Eidikèc periodikèc lôseic tou probl matoc............ 8 1.3.2 Qaotikèc lôseic.......................... 10 1.4 To prìblhma twn tri n swmˆtwn sthn Ourˆnia Mhqanik....... 11 2 To peristrefìmeno montèlo 13 2.1 KinoÔmeno sôsthma anaforˆc....................... 15 2.2 Metasqhmatismìc GXY G 1 xy.................... 17 2.2.1 Jèsh-taqÔthta G 1 wc proc to GXY.............. 17 2.2.2 Parˆllhlh metatìpish (suntetagmènec sto G 1 xy)....... 17 2.2.3 Peristrof............................ 17 2.3 Genikeumènec suntetagmènec....................... 19 2.3.1 Apì to adraneiakì sto peristrefìmeno sôsthma........ 20 2.4 H sunˆrthsh tou Lagkrˆnz kai oi exis seic tou sust matoc...... 21 2.4.1 Kinhtik enèrgeia......................... 21 2.4.2 Dunamikì............................. 22 2.4.3 Oloklhr mata Enèrgeiac kai Stroform c............ 23 2.5 Diaforikèc exis seic thc kðnhshc.................... 24 2.6 EÔresh periodik n troqi n........................ 24 2.7 Eustˆjeia periodik n troqi n...................... 28 2.7.1 O monìdromoc pðnakac...................... 29 2.7.2 Eustˆjeia grammik n Qamiltonian n susthmˆtwn........ 31 3

4 PERIEQŸOMENA 3 Anˆlush apotelesmˆtwn 33 3.1 Troqièc.................................. 35 3.1.1 Oikogèneia Circ05........................ 36 3.1.2 Oikogèneia Circ1......................... 42 3.1.3 Oikogèneia Circ2......................... 49 3.1.4 Oikogèneia Circ5......................... 54 3.1.5 Troqiˆ 9.............................. 58 3.2 Enèrgeia, Stroform, PerÐodoc..................... 59 3.2.1 Enèrgeia.............................. 60 3.2.2 Stroform............................. 62 3.2.3 PerÐodoc.............................. 64 3.3 'Oria sunèqishc wc proc tic mˆzec kai eustˆjeia............ 66 3.3.1 Oikogèneia Circ05........................ 66 3.3.2 Oikogèneia Circ1......................... 67 3.3.3 Oikogèneia Circ2......................... 68 3.3.4 Oikogèneia Circ5......................... 69 4 Sumperˆsmata 71

Kefˆlaio 1 EISAGWGH STO PROBLHMA TWN TRIWN SWMATWN 1.1 Genik IstorÐa To prìblhma twn 3 swmˆtwn asqoleðtai me ton prosdiorismì thc kðnhshc 3 swmˆtwn ìtan oi jèseic, taqôthtec kai mˆzec touc eðnai gnwstèc. Oi dôo jewrðec pou qrhsimopoioôntai gia na epilujeð to prìblhma eðnai o nìmoc thc pagkìsmiac èlxhc kai h klasik mhqanik tou NeÔtwna. O Ðdioc o NeÔtwnac sto biblðo tou Principia(1687) asqol jhke kai me to prìblhma twn 2 kai twn 3 swmˆtwn. Se antðjesh me to prìblhma twn 3 swmˆtwn, to prìblhma twn 2 swmˆtwn to melèthse sqedìn pl rwc. Pˆnw se autˆ ta apotelèsmata sth sunèqeia o J.Bernoulli apèdeixe ìti oi kin seic twn 2 swmˆtwn mporoôn na eðnai mìno elleiptikèc, uperbolikèc kai parabolikèc. Tèloc, o L.Euler apèdeixe ìti ìlo to prìblhma mporeð na anaqjeð sthn eôresh thc kðnhshc enìc s matoc se pedðo kentrik n dunˆmewn me mˆza thn anhgmènh mˆza µ tou sust matoc µ = m 1m 2 m 1 +m 2 O pr toc pou asqol jhke me to prìblhma twn 3 swmˆtwn anafèrame dh ìti tan o NeÔtwnac, o opoðoc asqol jhke me to selhniakì prìblhma, dhlad èna sôsthma 'Hlioc - Gh - Sel nh kai prospˆjhse na brei an h troqiˆ enìc tètoiou sust matoc eðnai eustaj c. To prìblhma ìmwc autì pou prospajoôse na lôsei o NeÔtwnac an ke sthn kathgorða tou periorismènou probl matoc 3 swmˆtwn, ìpwc onomˆsthke apì ton Euler ìpou mìno ta 2 s mata èqoun mˆza en to trðto kineðtai upì thn barutik touc epðdrash. PolloÐ epist monec asqol jhkan me to periorismèno prìblhma, qwrðc ìmwc na mporèsoun na prosfèroun mða telik lôsh, ìpwc o NeÔtwnac gia to prìblhma twn 2 swmˆtwn, an kai br kan kˆpoiec memonwmènec lôseic. To tèloc aut c thc èreunac to èdwse o H.Poincare ìtan apèdeixe ìti to prìblhma twn 3 swmˆtwn den èqei analutik lôsh. Metˆ thn apìdeixh aut tou Poincare ˆllaxe ˆrdhn o trìpoc prosèggishc tou 5

6 KEFŸALAIO 1. EISAGWGŸH STO PRŸOBLHMA TWN TRIŸWN SWMŸATWN probl matoc, kaj c plèon o mìnoc trìpoc na paraqjoôn apotelèsmata tan mèsw thc arijmhtik c olokl rwshc, mða mèjodoc pou eðnai arketˆ epðponh gia na gðnei qwrðc th qr sh kˆpoiac upologistik c mhqan c, dhlad 'me to qèri'. 'Etsi, h megˆlh ˆnjish sto prìblhma autì ep lje me thn dhmiourgða twn hlektronik n upologist n kai thn dunatìthta pou eðqan na kˆnoun touc apaitoômenouc upologismoôc se mhdaminì qrìno mprostˆ ston ˆnjrwpo, kai to fainìmeno autì suneqðzetai ìso belti nontai oi upologistèc allˆ kai oi mèjodoi arijmhtik c olokl rwshc. 'Opwc anafèrame to genikì prìblhma twn 3 swmˆtwn anafèretai se 3 s mata ta opoða allhlepidroôn metaxô touc me barutikèc dunˆmeic. Prokeimènou na melet - soume autì to prìblhma ja prèpei na katagrˆyoume tic exis seic pou perigrˆfoun to sôsthma kai na epilèxoume to katallhlìtero sôsthma suntetagmènwn ste na melet soume tic troqièc twn swmˆtwn. 1.2 To Prìblhma twn Tri n Swmˆtwn 1.2.1 Orismìc tou Probl matoc 'Estw trða s mata me mˆzec m 1, m 2, m 3, ta opoða kinoôntai ston 3-diˆstato EukleÐdio q ro, upì thn epðdrash twn barutik n dunˆmewn metaxô touc. OrÐzoume adraneiakì sôsthma anaforˆ OXY Z kai ta dianôsmata jèshc twn tri n swmˆtwn R i (i = 1, 2, 3) wc proc autì. Dedomènwn arqik n sunjhk n, to prìblhma ègkeitai sthn eôresh twn troqi n twn tri n swmˆtwn. H sunolik dônamh pou askeðtai sto èna s ma apì ta ˆlla dôo isoôtai me F i = G 3 j i kai oi exis seic kðnhshc dðnontai apì tic sqèseic R i = G 3 j i 1.2.2 Oloklhr mata thc kðnhshc To olokl rwma thc enèrgeiac èqei th morf m i m j R i R j 3 ( R i R j ) (1.1) m j R i R j 3 ( R i R j ) (1.2) E = 1 2 3 m i V i 2 G i=1 3 j>i m i m j R ij (1.3) ìpou R ij h apìstash twn dôo swmˆtwn kai V i = Ri. An dialèxoume tic kartesianèc suntetagmènec wc tic genikeumènec(x i, Y i, Z i ) kai orðsoume kai tic antðstoiqec ormèc

1.2. TO PRŸOBLHMA TWN TRIŸWN SWMŸATWN 7 (P x,i, P y,i, P z,i ) tìte h sunˆrthsh Hamilton tou sust matoc tautðzetai me thn olik enèrgeia, H = E. H olik orm P kai h olik stroform L tou sust matoc dðnontai apì tic sqèseic 3 P = m i Ri (1.4) kai L = i=1 3 R i P i = i=1 1.2.3 KÐnhsh wc proc to kèntro mˆzac 3 m i ( R i V i ) (1.5) To kèntro mˆzac tou sust matoc orðzetai to shmeðo K, to diˆnusma jèshc tou opoðou (wc proc to O)prèpei na ikanopoieð th sqèsh i=1 R K = 3 i=1 m i R i 3 i=1 m i (1.6) apì thn opoða prokôptei ìti V K = 3 i=1 RK = m i R i 3 i=1 m i = 3 i=1 P i 3 i=1 m i (1.7) kai 3 i=1 R K = m i R i 3 i=1 m i = 3 i=1 F i 3 i=1 m i (1.8) 'Omwc apì tic exis seic (1.2) kai (1.8) prokôptei ìti 3 i=1 m i R i = 3 i=1 F i = 0 kai exaitðac autoô R K = 0 V K = (P/ 3 m i ) = c 1 R K = c 1 t + c 2 (1.9) i=1 ìpou c i stajerˆ dianôsmata. ProkÔptei logikˆ ìti to kèntro mˆzac tou sust matoc ekteleð eujôgrammh kai omal kðnhsh se sqèsh me to adraneiakì sôsthma anaforˆc kai h orm tou(upojètontac ìti h mˆza ìlou tou sust matoc sugkentr netai sto K) isoôtai me thn sunolik orm tou sust matoc. Qrhsimopoi ntac to parapˆnw sumpèrasma mporoôme na orðsoume èna nèo adraneiakì sôsthma anaforˆc me arq to K, Kxyz, me tic exis seic kai ta pr ta oloklhr mata thc kðnhshc wc proc to kèntro mˆzac na dðnontai apì tic sqèseic r K,i = G 3 j>i m j R i R j 3 ( R i R j ) (1.10)

8 KEFŸALAIO 1. EISAGWGŸH STO PRŸOBLHMA TWN TRIŸWN SWMŸATWN E K = 1 2 3 m i V K,i 2 G i=1 3 j>1 m i m j R ij (1.11) P K = 3 m i rk,i (1.12) i=1 L K = 3 r K,i P K,i = i=1 3 m i ( r K,i V K,i ) (1.13) i=1 Oi lôseic tou sust matoc diaforik n exis sewn onomˆzontai barukentrikèc troqièc. ParathroÔme ìti: Apì thn exðswsh tou kèntro mˆzac (diat rhsh thc olik c orm c) prokôptei ìti h kðnhsh enìc apì ta trða s mata mporeð na prosdioristeð pl rwc apì thn kðnhsh twn ˆllwn dôo. 'Ara, an broôme ta c i mei noume thn tˆxh tou sust matoc twn D.E., ftˆnontac se èna isodônamo prìblhma èxi bajm n eleujerðac. H diat rhsh twn E kai L prosdiorðzei akìma tèsseric stajerèc. 'Ara prokôptoun dèka stajerèc thc kðnhshc. ApodeiknÔetai ìti den upˆrqoun ˆlla oloklhr mata thc kðnhshc gia to genikì prìblhma twn tri n swmˆtwn(ja qreiazìmastan sunolikˆ 3 3 2 = 18 stajerèc). Epomènwc, katal goume sto sumpèrasma ìti den upˆrqei genik lôsh tou sust matoc, parˆ mìno eidikèc lôseic. Aut h mh oloklhrwsimìthta tou sust matoc sunepˆgetai thn Ôparxh qaotik n kin sewn. 1.3 LÔseic - Troqièc 1.3.1 Eidikèc periodikèc lôseic tou probl matoc Ekmetaleuìmenoi gewmetrikèc summetrðec mporoôme na entopðsoume periodikèc lôseic sto adraneiakì sôsthma oi opoðec onomˆzontaiqorografðes (choreographies ). Up- ˆrqoun treic oikogèneiec oi opoðec suqnˆ anafèrontai wc klasikˆ paradeðgmata, eðte lìgw aplìthtac kai majhmatik c eukolðac (Euler, Lagrange) eðte lìgw monadikìthtac thc troqiˆc: 1. H lôsh tou Euler, ìpou ta trða s mata pˆnta eðnai suneujeiakˆ, me to èna apì ta trða na brðsketai sto kèntro. Tètoiec lôseic eðnai pˆnta astajeðc kai den anamènetai na brejoôn sthn pragmatikìthta.

1.3. LŸUSEIS - TROQIŸES 9 2 0.48 1 0 1 2 2 1 0 1 2 2. H lôsh tou Lagrange, ìpou ta trða s mata kinoôntai se mða èlleiyh kai sqhmatðzoun metaxô touc èna isìpleuro trðgwno. Ta sust mata autˆ den eðnai pˆnta eustaj. 2 0.75 1 0 1 2 2 1 0 1 2 3. H lôsh tou oqtarioô (figure of 8) ìpou ta trða s mata kinoôntai se èna sq ma pou moiˆzei me èna orizìntio oqtˆri( to sômbolo tou apeðrou). Se autèc tic lôseic èqeic dojeð kai to ìnoma qorografðec, lìgw thc omoiìthtac me kin seic omˆdac qoreut n mpalètou. H lôsh aut protˆjhke jewrhtikˆ apì ton Richard Moeckel to 1988 kai brèjhke kai peiramatikˆ to 1993 apì ton Christopher Moore.

10 KEFŸALAIO 1. EISAGWGŸH STO PRŸOBLHMA TWN TRIŸWN SWMŸATWN 2 0.75 1 0 1 2 2 1 0 1 2 1.3.2 Qaotikèc lôseic Oi lôseic pou anafèrame parapˆnw eðnai spˆniec kai den antiproswpeôoun thn pleioyhfða twn troqi n pou prokôptoun. Oi perissìterec arqikèc sunj kec mac odhgoôn se qaotikèc troqièc, dhlad troqièc pou parousiˆzoun polô megˆlh euaisjhsða stic arqikèc sunj kec. O pr toc pou suneidhtopoðhse pragmatikˆ aut n thn idiìthta tou probl matoc tan o Poincare, se antðjesh me tic prohgoômenec prospˆjeiec, stic opoðec pðsteuan ìti aplˆ apètuqan na broun analutik lôsh. Aut h qaotik sumperiforˆ tou sust matoc, pou èkane polô dôskolh thn eôresh lôsewn mìno me to qèri, eðnai pou to katèsthse wc èna ˆneu endiafèrontoc prìblhma (sth genik tou morf ) mèqri na exeliqjoôn oi upologistèc kai na mporoôn na gðnoun arketˆ akribeðc arijmhtikèc proseggðseic (dekaetðec 60-70).

1.4. TO PRŸOBLHMA TWN TRIŸWN SWMŸATWN STHN OURŸANIA MHQANIKŸH 11 2 0.99 1 0 1 2 2 1 0 1 2 1.4 To prìblhma twn tri n swmˆtwn sthn Ourˆnia Mhqanik O lìgoc pou xekðnhse h asqolða thc episthmonik c koinìthtac me to prìblhma twn tri n swmˆtwn tan gia na mporèsoun majhmatikˆ na exhg soun tic kin seic twn ourˆniwn swmˆtwn. AfoÔ lôjhke analutikˆ to prìblhma twn dôo swmˆtwn apì ton NeÔtwna, h logik sunèqeia tan na pˆne se èna pio perðploko sôsthma me ap tero skopì na ermhneôsoun to hliakì mac sôsthma. Autìc o skopìc paramènei akìma kôrioc sthn anaz thsh, upologistikˆ plèon, periodik n lôsewn, miac kai h dunamik enìc hliakoô sust matoc eðnai arketˆ idiaðterh, lìgw thc megˆlhc diaforˆc maz n anˆmesa sta s mata (p.q. 'Hlioc, plan thc, dorufìroc). 'Etsi èqoume thn dhmiourgða kai melèth katˆllhla periorismènwn montèlwn. An kai sthn ergasða asqoloômaste me to genikì prìblhma twn tri n swmˆtwn ìpou ìlec oi mˆzec eðnai thc Ðdiac tˆxhc megèjouc, en gènei sthn ourˆnia mhqanik epikentr nontai sto ierarqikì prìblhma twn tri n swmˆtwn, sto opoðo trða astèria dhmiourgoôn mða duˆda kai mða monˆda, me apotèlesma na moiˆzei me to prìblhma twn dôo swmˆtwn, tou opoðou thn analutik lôsh thn xèroume. Autì sumbaðnei diìti to diplì astèri, an èqei arket apìstash me to trðto astèri, sumperifèretai dunamikˆ

12 KEFŸALAIO 1. EISAGWGŸH STO PRŸOBLHMA TWN TRIŸWN SWMŸATWN wc èna astèri pou brðsketai sto kèntro mˆzac twn dôo. Me bˆsh aut n thn prosèggish, h èreuna ston tomèa thc ourˆniac mhqanik c gðnetai gia na brejoôn oi diˆforec allhlepidrˆseic metaxô tou zeugarioô kai tou monoô, kai ti shmasða èqei kat' epèktash èna diplì astèri sthn exèlixh enìc astrikoô sust matoc. 'Eqoun parathrhjeð peript seic pou to trðto astèri èqei diafôgei, èqei antikatast sei èna apì ta dôo me apotèlesma na dhmiourghjeð èna kainoôrio diplì astèri kai na diafôgei to astèri pou prohgoumènwc an ke sto diplì. Aut h èreuna èqei megˆlh shmasða ste na mporèsoun na brejoôn oi tˆseic tou plhjusmoô twn dipl n kai twn mon n asteri n, kai na exhghjeð h epðdrash aut n twn plhjusm n sthn exèlixh enìc sm nouc asteri n. Stìqoc thc ergasðac aut c eðnai h eôresh peratwmènwn eustaj n troqi n stic opoðec mporoôn na kinoôntai ta trða s mata to èna gôrw apì to ˆllo. H mejodologða pou efarmìzoume eðnai h sunèqish eustaj n kuklik n periodik n troqi n wc proc th mˆza. Xekin ntac apì eustajeðc kuklikèc troqièc tou planhtikoô probl matoc, dhlad se sust mata me dôo mikrèc mˆzec, arqðzoume na auxˆnoume th mˆza twn dôo mikr n swmˆtwn (touc arqikoôc plan tec) mèqri autèc na gðnoun sugkrðsimec me th mˆza tou trðtou s matoc (to arqikì astèri). Tètoiec troqièc eðnai gnwstèc apì th melèth tou planhtikoô probl matoc twn tri n swmˆtwn.

Kefˆlaio 2 To peristrefìmeno sôsthma suntetagmènwn kai periodikèc troqièc H pr th perigraf tou probl matoc gðnetai me to adraneiakì sôsthma suntetagmènwn, sto opoðo to kèntro mˆzac twn 3 swmˆtwn eðnai kai to kèntro tou sust matoc. Sthn ergasða aut jewroôme ìti ta s mata kinoôntai sto Ðdio epðpedo. ExaitÐac aut c thc exˆrthshc metaxô twn swmˆtwn mporoôme na upologðsoume th jèsh tou trðtou s matoc an xèroume tic jèseic twn ˆllwn dôo. 13

14 KEFŸALAIO 2. TO PERISTREFŸOMENO MONTŸELO y P 2 R 2 R G G R 1 P 1 O x R 0 P 0 Sq ma 2.1: OQU:adraneiakì sôsthma, s mata P i me mˆzec m i ìpou i = 0, 1, 2 Gia to kèntro Mˆzac G isqôoun oi parakˆtw exis seic gia th jèsh kai thn taqôthtˆ tou 1 R G = (m 0 + m 1 + m 2 ) (m 0R 0 + m 1R1 + m 2R2 ) (2.1) U G = 1 (m 0 + m 1 + m 2 ) (m 0 U 0 + m 1 U1 + m 2 U2 ) (2.2) ìpou h taqôthta isoôtai me thn parˆgwgo thc jèshc U Ṙ Gia tuqoôsec suntetagmènec X i, Y i jètoume to G san arq tou adraneiakoô sust -

2.1. KINOŸUMENO SŸUSTHMA ANAFORŸAS 15 matoc me thn metatrop X i X i X G Y i Y i Y G Ẋ i Ẋi U G,X Ẏ i Ẏi U G,Y (2.3) 2.1 KinoÔmeno sôsthma anaforˆc 'Estw to adraneiakì sôsthma GXY ìpou G to kèntro mˆzac P 0, P 1, P 2. JewroÔme t ra èna sôsthma to opoðo akoloujeð thn peristrof tou s matoc 1 gôrw apì to s ma 0. Sugkekrimmèna h dieôjunsh tou ˆxona twn x sto sôsthma autì orðzetai pˆnta apì thn eujeða twn dôo aut n swmˆtwn kai me forˆ apì to 0 proc to 1. To kèntro tou sust matoc G 1 eðnai to kèntro mˆzac twn swmˆtwn 0 kai 1 kai o ˆxonac twn y orðzetai kˆjeta ston x kai me forˆ ste to sôsthma na eðnai dexiìstrofo. To sôsthma autì eðnai to G 1 xy ìpwc faðnetai sto sq ma 2.1.

16 KEFŸALAIO 2. TO PERISTREFŸOMENO MONTŸELO y Y P 2 P 1 x G θ X G 1 P 0 Sq ma 2.2: To peristrefìmeno sôsthma suntetagmènwn Sto peristrefìmeno sôsthma oi jèseic twn swmˆtwn 0 kai 1 orðzontai mìno apì thn sunist sa touc x 0 kai x 1, antðstoiqa, afoô ja eðnai pˆnta y 0,1 = 0. Dedomènou ìti to h arq G 1 eðnai to kèntro mˆzac, mporoôme na jewr soume anexˆrthth metablht mìno th jèsh x 1. To trðto s ma, tou opoðou tic suntetagmènec (x 2, y 2 ) prospajoôme na broôme, den èqei periorismoôc sthn kðnhs tou pˆnw sto peristrefìmeno epðpedo xy. H allag aut twn suntetagmènwn apì to adraneiakì sto peristrefìmeno gðnetai ìpwc perigrˆfetai parakˆtw.

2.2. METASQHMATISMŸOS GXY G 1 XY 17 2.2 Metasqhmatismìc GXY G 1 xy 2.2.1 Jèsh-taqÔthta G 1 wc proc to GXY H jèsh kai h taqôthta tou G 1 brðsketai apì tic parakˆtw sqèseic X G1 = 1 m 0 +m 1 (m 0 X 0 + m 1 X 1 ) Ẋ G1 = 1 m 0 +m 1 (m 0 Ẋ 0 + m 2 Ẋ 1 ) Y G1 = 1 m 0 +m 1 (m 0 Y 0 + m 1 Y 1 ) Ẏ G1 = 1 m 0 +m 1 (m 0 Ẏ 0 + m 2 Ẏ 1 ) (2.4) 2.2.2 Parˆllhlh metatìpish (suntetagmènec sto G 1 xy) 'Olec oi suntetagmènec prèpei na oristoôn wc proc to kainoôrio kèntro, to opoðo eðnai to G 1, ˆra gia kˆje tuqaðo s ma ja prèpei na isqôoun oi parakˆtw tèssereic sqèseic (dôo gia jèsh, dôo gia taqôthta) x = X X G1 y = Y Y G1 ẋ = Ẋ ẊG 1 ẏ = Ẏ ẎG 1 (2.5) 2.2.3 Peristrof To nèo mac sôsthma eðnai upì gwnða θ se sqèsh me to adraneiakì. Gia na mporèsoume na broôme tic nèec mac suntetagmènec ja prèpei na qrhsimopoi soume ton pðnaka peristrof c ( ) cosθ sinθ R = (2.6) sinθ cosθ ìpou x = R x gia jèsh kai ẋ = R x gia taqôthta Kˆnontac ton pollaplasiasmì brðskoume tic suntetagmènec jèseic x = x cosθ + y sinθ y = x sinθ + y cosθ (2.7) ParagwgÐzontac thn jèsh tou s matoc eôkola katal goume sthn taqôthta pou brðsketai apì tic dôo autèc sqèseic: ẋ = ẋ cosθ + ẏ sinθ + y θ ẏ = ẋ sinθ + ẏ cosθ x θ (2.8) To teleutaðo b ma pou apomènei eðnai na broôme ta θ, θ apì to adraneiakì sôsthma, me to θ na eðnai h gwnða pou sqhmatðzei h eujeða pou sundèei ta P 0 kai P 1, ˆra brðsketai

18 KEFŸALAIO 2. TO PERISTREFŸOMENO MONTŸELO me ton ex c trìpo: ìpou θ = arctan( Y 1 Y 0 ) X 1 X 0 (2.9) θ = 1 r ( X 2 01 Ẏ01 Y 01 Ẋ01) (2.10) r 2 = X 2 01 + Y 2 01 Ẏ01 = Ẏ1 Ẏ0 Ẋ01 = Ẋ1 Ẋ0

2.3. GENIKEUMŸENES SUNTETAGMŸENES 19 2.3 Genikeumènec suntetagmènec y P 2 P 1 (1-μ) r θ μr G 1 r x P 0 Sq ma 2.3: Genikeumènec suntetagmènec sto peristrefìmeno sôsthma Oi genikeumènec suntetagmènec kai taqôthtec, loipìn, pou perigrˆfoun to sôsthma eðnai oi: 1. Jèsh P 2 (x 2, y 2 ) 2. Sqetik apìstash P 0 P 1, dhlad h r = P 0 P 1 3. GwnÐa θ tou peristrefìmenou se sqèsh me to adraneiakì 4. Genikeumènec taqôthtec: ẋ 2, ẏ 2, ṙ, θ

20 KEFŸALAIO 2. TO PERISTREFŸOMENO MONTŸELO kai ja orðsoume thn stajerˆ µ = m 0 m 0 +m 1, h opoða eðnai h anhgmènh mˆza twn dôo eswterik n swmˆtwn. H sqetik apìstash r twn swmˆtwn 0 kai 1 brðsketai apì ta x 0, x 1 r = (x 1 x 0 ) 2 = x 1 x 0 (2.11) x 0 = (1 µ)r x 1 = µr kai isqôei ìti y 0 = y 1 = 0 to opoðo sunepˆgetai ìti (2.12) ṙ = ẋ0 1 µ (2.13) ṙ = ẋ1 µ 2.3.1 Apì to adraneiakì sto peristrefìmeno sôsthma Genikeumènec suntetagmènec r, θ, x 2, y 2 ṙ, θ, ẋ 2, ẏ 2 Mèsw twn (2.12) gnwrðzoume ta x i, y i i = 0, 1 Mèsw twn (2.13) gnwrðzoume ta ẋ i, ẏ i 'Ara prèpei apì ta x i, y i, θ, θ na katal xoume sta X i, Y i Metasqhmatismìc jèsewn sta X i, Y i tou G i X Y xèroume ìti gia tic jèseic isqôei x = R x x = R 1 x ìpou ( ) R cosθ sinθ 1 = sinθ cosθ Kˆnontac tic prˆxeic katal goume sta ex c: x 0 = x 0 cosθ y 0 sinθ y 0 = x 0 sinθ + y 0 cosθ x 1 = x 1 cosθ y 1 sinθ y 1 = x 1 sinθ + y 1 cosθ Epeid ìmwc èqoume orðsei ta y 0 = y 1 = 0 ftˆnoume eôkola sta ex c: x 0 = x 0 cosθ = (1 µ)rcosθ y 0 = x 0 sinθ = (1 µ)rsinθ x 1 = x 1 cosθ = µrcosθ y 1 = x 1 sinθ = µrsinθ (2.14) (2.15) (2.16)

2.4. H SUNŸARTHSH TOU LAGKRŸANZ KAI OI EXISŸWSEIS TOU SUSTŸHMATOS 21 AntÐstoiqa gia to P 2 isqôei: x 2 = x 2 cosθ y 2 sinθ y 2 = x 2 sinθ + y 2 cosθ (2.17) Metasqhmatismìc taqut twn sto G 1 x y Gia na broôme tic taqôthtec ja prèpei na paragwgðsoume tic jèseic tic opoðec upologðsame. 'Ara qrhsimopoi ntac tic (2.16)kai (2.17) kai paragwgðzontac gia y i 0 (genikˆ lìgw tou ìti ja prokôyoun parˆgwgoi ẏ i pou eðnai mh-mhdenikèc) ja katal x- oume se sqèseic gia tic taqôthtec: (ẋ ) ) ( i (ẋi = R ẏ ) θy 1 + i i ẏ 2 θx (2.18) i Gia tic taqôthtec mporeð na qrhsimopoihjeð h (2.16) ste na katal xoume gia autèc stic sqèseic: ẋ 0 = (1 µ)(ṙcosθ rsinθ θ) ẏ 0 = (1 µ)(ṙsinθ rcosθ θ) ẋ 1 = µ(ṙcosθ rsinθ θ) (2.19) ẏ 1 = (1 µ)(ṙsinθ rcosθ θ) Prosdiorismìc tou G wc proc to G 1 x y GnwrÐzontac ìla ta x i kai y i mporoôme aplˆ na broôme tic suntetagmènec tou G 1 R G = (m (m 0 +m 1 +m 2 ) 0R 0 + m 1R 1 + m 2R 2 ), R i = (x i, y i) 1 R G = (m (m 0 +m 1 +m 2 ) 0 R 0 + m 1 R 1 + m 2 R 2 ), R i = (ẋ i, ẏ i) Epomènwc brðskoume ìti (2.20) X = X X G (2.21) ìpou tou X eðnai to diˆnusma jèshc/taqôthtac tou s matoc kai XG = (R G, RG ) 2.4 H sunˆrthsh tou Lagkrˆnz kai oi exis seic tou sust matoc 2.4.1 Kinhtik enèrgeia TeleutaÐo b ma sto na orðsoume majhmatikˆ to montèlo mac eðnai na upologðsoume thn sunˆrthsh Lagrange ekfrasmènh stic genikeumènec suntetagmènec tou peristrefìmenou sust matoc.

22 KEFŸALAIO 2. TO PERISTREFŸOMENO MONTŸELO 'Eqoume tic genikeumènec suntetagmènec r, ṙ, θ, θ, x, y ìpou x x 2, y y 2. Apì autèc sômfwna me thn klasik mhqanik brðskoume thn kinhtik enèrgeia na eðnai Ðsh me: T = 1 2 m 0 R 0 2 + 1 2 m 1 R 1 2 + 1 2 m 2 R 2 2 (2.22) 'Opou èqoume ìti R 0 = r 0 G 1 G = r 0 m 2 m r 2 R 1 = r 1 G 1 G = r 1 m 2 m r 2 R 2 = m 0 + m 1 m r 2 kai eôkola katal goume sthn ex c èkfrash gia thn kinhtik enèrgeia T = 1 2 (m 0 + m 1 )( m 2 m r 2 2 + m 1 m 0 r 2 1 ) (2.23) alli c, exaitðac ìti r1 = m 0 m 1 r0, mporeð na grafteð wc T = 1 2 (m 0 + m 1 )( m 2 m r 2 2 + m 0 m 1 r 2 0 ) (2.24) 'Eqoume ìti ṙ 2 0 = ẋ 2 0 + ẏ 2 0 kai mèsw thc (2.16) paðrnoume ìti ṙ 1 0 = (1 µ) 2 (ṙ 2 + r 2 θ2 ) (2.25) AntistoÐqwc apì thn ṙ 2 2 = ẋ 2 2 + ẏ 2 2 kai mèsw thc (2.18) brðskoume ìti ṙ 2 2 = ẋ 2 + ẏ 2 + θ 2 (x 2 + y 2 ) + 2 θ(xẏ ẋy) (2.26) Gia thn telik morf thc kinhtik c enèrgeiac qreiˆzetai aplˆ na antikatast soume tic (2.25) kai (2.26) sthn (2.24) kai ja katal xoume ston tôpo T = 1 2 (m 0+m 1 )[ m 0 m 1 (1 µ) 2 (ṙ 2 +r 2 θ2 )+ m 2 m (ẋ2 +ẏ 2 + θ 2 (x 2 +y 2 )+2 θ(xẏ ẋy))] (2.27) 2.4.2 Dunamikì H sunolik dunamik enèrgeia V tou sust matoc eðnai to ˆjroisma twn energei n metaxô twn tri n swmˆtwn. 'Ara isqôei ìti V = m 0m 1 r 01 m 0m 2 r 02 m 1m 2 r 12 (2.28)

2.4. H SUNŸARTHSH TOU LAGKRŸANZ KAI OI EXISŸWSEIS TOU SUSTŸHMATOS 23 ìpou ta r 01, r 02, r 12 brðskontai eôkola apì tic r 2 01 = r 0 2 + r 1 2 = r 2 r 2 02 = (x 0 x 2 ) 2 + (y 0 y 2 ) 2 = [(1 µ)r x] 2 + y 2 r 2 12 = (x 1 x 2 ) 2 + (y 1 y 2 ) 2 = [ µr x] 2 + y 2 = (µr + x) 2 + y 2 (2.29) Mèsw twn (2.28) kai (2.29) katal goume sto sumpèrasma ìti to dunamikì eðnai sunˆrthsh twn r, x, y, dhlad V = V (r, x, y) Qrhsimopoi ntac autˆ ta sumperˆsmata ftˆnoume sth lagkrantzian tou sust - matoc pou isoôtai me L = T V 2.4.3 Oloklhr mata Enèrgeiac kai Stroform c ParathroÔme ìti h sunˆrthsh Lagrange den exartˆtai apì thn genikeumènh suntetagmènh θ kai ˆra h antðstoiqh genikeumènh orm (stroform ) p θ = L θ eðnai olokl rwma thc kðnhshc. Sugkekrimmèna brðskoume ìpou p θ = M 0 r 2 θ + M1 ( θ(x 2 + y 2 ) + xẏ ẋy) (2.30) M 0 = m 0 m 1 (m 0 + m 1 )(1 µ) 2 = m 0m 1 m 0 + m 1 eðnai h anhgmènh mˆza tou sust matoc P 0 P 1 kai M 1 = m 2(m 0 + m 1 ) m 0 + m 1 + m 2 = (m 0 + m 1 )m 2 m 0 + m 1 + m 2 h anhgmènh mˆza tou sust matoc (P 0 P 1 )P 2. EpÐshc, afoô o qrìnoc t eðnai agno simh suntetagmènh, ja upˆrqei to olokl rwma tou Jacobi J = p r ṙ + p θ dotθ + p x ẋ + p y ẏ L, ìpou p r = L/ ṙ, p x = L/ ẋ kai p y = L/ ẏ oi genikeumènec ormèc. Epeid o metasqhmatismìc apì to adraneiakì sto peristrefìmeno sôsthma den perièqei to qrìno, to olokl rwma tou Jacobi sumpðptei me thn enèrgeia tou sust matoc E = 1 2 M 0(ṙ 2 + r 2 θ2 ) + 1 2 M 1(ẋ 2 + ẏ 2 + 2 θ(xẏ ẋy) + θ 2 (x 2 + y 2 )) + V (2.31)

24 KEFŸALAIO 2. TO PERISTREFŸOMENO MONTŸELO 2.5 Diaforikèc exis seic thc kðnhshc GnwrÐzontac plèon thn L mporoôme eôkola na broôme tic exis seic kðnhseic tou sust - matoc qrhsimopoi ntac thn exðswsh Euler-Lagrange : d dt ( L q ) L q = 0, kai katal goume stic treic diaforikèc exis seic 2hc tˆxhc gia tic genikeumènec suntetagmènec: M 1 ẍ = m 0m 2 ((1 µ)r x) r 2 02 + m 1m 2 (µr + x) r 2 12 + M 1 (x θ 2 + 2 θẏ + y θ) (2.32) M 1 ÿ = m 0m 2 y r 2 02 + m 1m 2 y r 2 12 + M 1 (y θ 2 2 θẋ x θ) (2.33) M 0 r = m 0m 1 + m 0m 2 (1 µ)((1 µ)r x) + m 1m 2 µ(µr + x) + M r 2 r02 2 r12 2 0 r θ 2 (2.34) 'Opou r 2 02 = ((1 µ)r x) 2 + y 2 kai r 2 12 = (µr + x) 2 + y 2 kai ta M 0, M 1 oi anhgmènec mˆzec pou anafèrame prohgoumènwc. MporoÔme na parathr soume ìti oi diaforikèc exis seic paramènoun analloðwtec kˆtw apì thn summetrða Σ : (x, y, t) (x, y, t) 2.6 EÔresh periodik n troqi n 'Ena ergaleðo twn majhmatik n sto opoðo prèpei na anaferjoôme, diìti me th bo jeia tou brðskoume periodikèc troqièc, eðnai oi tomèc Poincare. Oi tomèc Poincare eðnai ousiastikˆ stigmiìtupa thc kðnhshc ston q ro twn fˆsewn, anˆ qronikì diˆsthma ìso h perðodoc thc troqiˆc pou anazhtoôme. Kˆje forˆ pou h troqiˆ pernˆei apì to epðpedo to opoðo zht same, tìte af nei èna stðgma. Mia kanonik periodik troqiˆ af nei èna kai monadikì stðgma, ènan peperasmèno akèraio tètoio ste o sunolikìc qrìnoc thc kðnhshc pou melet same na eðnai pollaplˆsioc thc periìdou thc troqiˆc. Oi troqièc pou af noun ˆpeira stðgmata ta opoða ìmwc katanèmontai pˆnw se epifˆneiec me kanonikì (taktikì) trìpo onomˆzontai hmiperiodikèc troqièc. Oi qaotikèc troqièc af noun ˆpeira stigmata ta opoða den parousiˆzoun kamða kanonikìthta allˆ mˆllon katanèmontai tuqaða pˆnw sth tom. JewroÔme to adiatˆrakto (ìpou ta dôo elafrìtera s mata den allhlepidroôn lìgw mhdenik n maz n) sôsthma sto opoðo m 1 = m 2 = 0, h dunamik enèrgeia

2.6. EŸURESH PERIODIKŸWN TROQIŸWN 25 isoôtai me UK 2 = F (r) r m kai ta s mata ekteloôn kuklikèc troqièc me aktðnec R 1 kai R 2, taqôthtec m0 υ i =, i = 1, 2 R i kai perðodo T i = 2π m0 R 3/2 i. JewroÔme ìti arqikˆ ta s mata brðskontai ston ˆxona GQ (tou adraneiakoô sust matoc) o opoðoc sumpðptei me ton ˆxona G 1 x tou peristrefìmenou sust matoc. Tìte sto peristrefìmeno sôsthma ja èqoume tic arqikèc sunj kec x 10 = R 1, ẋ 10 = 0, x 20 = R 2, y 20 = ẋ 20 = 0, ẏ 20 = υ 2 υ 1. (2.35) y. 2 x 1 x 2 x ApodeiknÔetai ìti metˆ apì qrìno T = 2π T 1 T 2 1 ta s mata epanèrqontai stic Ðdiec arqikèc sunj kec gia to peristrefìmeno sôsthma. 'Ara oi parapˆnw arqikèc sunj kec antistoiqoôn se periodik troqiˆ periìdou T gia to adiatˆrakto sôsthma sto peristrefìmeno sôsthma anaforˆc. Jèloume t ra na broôme kˆpoia periodik troqiˆ allˆzontac tic mˆzec, ˆra den eðmaste pia sto adiatˆrakto sôsthma. Ja jèsoume m 1 = m 1 kai m 2 = m 2 kai ja yˆxoume na broôme tic allagèc pou prèpei na kˆnoume stic arqikèc mac sunj kec ste me autèc tic kainoôriec mˆzec na broôme mia periodik troqiˆ polô kontˆ sthn antðstoiqh troqiˆ tou adiatˆraktou sust matoc. 'Eqontac plèon ìti m 1, m 2 0 jewroôme tic lôseic gia tic arqikèc sunj kec tou

26 KEFŸALAIO 2. TO PERISTREFŸOMENO MONTŸELO adiatˆraktou probl matoc : x 1 = x 1 (t; x 10, x 20, y 20, ẋ 10, ẋ 20, ẏ 20, m 1, m 2 ) x 2 = x 2 (t; x 10, x 20, y 20, ẋ 10, ẋ 20, ẏ 20, m 1, m 2 ) y 2 = y 2 (t; x 10, x 20, y 20, ẋ 10, ẋ 20, ẏ 20, m 1, m 2 ) ẋ 2 = ẋ 2 (t; x 10, x 20, y 20, ẋ 10, ẋ 20, ẏ 20, m 1, m 2 ) ẏ 2 = ẏ 2 (t; x 10, x 20, y 20, ẋ 10, ẋ 20, ẏ 20, m 1, m 2 ) (2.36) OrÐzoume ton qrìno thc apeikìnishc Poincaré t ètsi ste na isqôei y 2 (t ) = 0. EpÐshc ja jewr soume summetrikèc periodikèc troqièc oi opoðec eðnai summetrikèc wc proc ton ˆxona x, dhlad eðnai analloðwtec kˆtw apì th summetrða Σ. Sthn perðptwsh aut h troqiˆ tèmnei kˆjeta ton ˆxona twn x dôo forèc kai ˆra mporoôme na jewr soume arqikèc sunj kec ètsi ste na isqôei ìti: ẋ 10 (t ) = 0 ẋ 20 (t ) = 0 (2.37) 'Eqoun apomeðnei plèon treic arqikèc sunj kec na broôme ste na upologðsoume th nèa periodik troqiˆ, oi x 10, x 20, ẏ 20. 'Omwc mporôme na jewr soume to x 10 stajerì, qwrðc na upˆrqei prìblhma me th genikìthta tou probl matoc, kai ja metabˆlloume mìno tic ˆllec dôo metablhtèc ste na ikanopoi soume tic periodikèc sunj kec ẋ 1 (t ; x 10, x 20 + x 2, y 20 = 0, ẋ 10 = 0, ẋ 20 = 0, ẏ 20 + ẏ 2, m 1, m 2 ) = 0 ẋ 2 (t ; x 10, x 20 + x 2, y 20 = 0, ẋ 10 = 0, ẋ 20 = 0, ẏ 20 + ẏ 2, m 1, m 2 ) = 0 (2.38) JewroÔme ìti h metabol sthn tim twn maz n eðnai arketˆ mikr, ˆra kai oi diorj seic x 2, ẏ 2 ja eðnai mikrèc. AnaptÔssw thn (2.38) se seirˆ Taylor gôrw apì thn arqik lôsh kai katal gw sth sqèsh: ẋ 1 (t ; x 10, x 20, y 20 = 0, ẋ 10 = 0, ẋ 20 = 0, ẏ 20 ; m i 0) + ( ẋ 1 x 20 ) x 20 + ( ẋ 1 ẏ 20 ) ẏ 2 = 0 ẋ 2 (t ; x 10, x 20, y 20 = 0, ẋ 10 = 0, ẋ 20 = 0, ẏ 20 ; m i 0) + ( ẋ 2 x 20 ) x 20 + ( ẋ 2 ẏ 20 ) ẏ 2 = 0 (2.39) Xèroume ìti ta ẋ i (t ; x 10, x 20, y 20 = 0, ẋ 10 = 0, ẋ 20 = 0, ẏ 20 ; m i = 0) eðnai Ðsa me 0. Efìson isqôei ìti m i 0 allˆ mikrèc tìte katal goume ìti: ẋ 1 (t ; x 10, x 20, y 20 = 0, ẋ 10 = 0, ẋ 20 = 0, ẏ 20, m i 0) = ẋ 1 ẋ 2 (t ; x 10, x 20, y 20 = 0, ẋ 10 = 0, ẋ 20 = 0, ẏ 20, m i 0) = ẋ 2 (2.40) 'Ara to anˆptugma Taylor ja grafteð wc ex c: ẋ 1 + ( ẋ 1 x 20 ) x 20 + ( ẋ 1 y 20 ) ẏ 20 = 0 ẋ 2 + ( ẋ 2 x 20 ) x 20 + ( ẋ 2 y 20 ) ẏ 20 = 0 (2.41)

2.6. EŸURESH PERIODIKŸWN TROQIŸWN 27 kai lônontac wc proc tic diorj seic x 20, ẏ 20 ja ftˆsw sthn: ( ẋ 1 x 20 ) x 20 + ( ẋ 1 y 20 ) ẏ 20 = ẋ 1 ( ẋ 2 x 20 ) x 20 + ( ẋ (2.42) 2 y 20 ) ẏ 20 = ẋ 2 ) ( ẋ1 ẋ 1 x 20 ẏ 20 ẋ 2 ẋ 2 x 20 ẏ 20 t=t }{{} ( x20 ẏ 20 M ) = ( x20 ẏ 20 ( ) 1 ẋ1 ẋ 1 x 20 ẏ 20 ẋ 2 ẋ 2 x 20 ẏ 20 ) ( ) ẋ1 = ẋ 2 t=t ( ) ẋ1 ẋ 2 (2.43) (2.44) Me autìn ton trìpo brðskw tic diorj seic se 1h tˆxh stic arqikèc sunj kec, tètoiec ste na èqw periodik troqiˆ: x 10 = x 10, x 20 = x 20 + x 20, y 20 = 0 ẋ 10 = 0, ẋ 20 = 0, ẏ 20 = ẏ 20 + ẏ 20 (2.45) Epeid oi diorj seic eðnai 1hc tˆxhc, ˆra ìqi apìluta akribeðc, epanalambˆnoume th diadikasða mèqri ta ẋ 1, ẋ 2 na eðnai tìso kontˆ sto mhdèn ìso apaiteð h akrðbeia pou orðsame. Gia na upologðsoume ton pðnaka M ja prèpei na upologðsoume tic parag gouc. Autì mporoôme na to epitôqoume arijmhtikˆ. P.q. 'Estw ìti jèlw na upologðsw tic ẋ 1 x 20 t=t ẋ 2 x 20 t=t kai Oloklhr nw tic exis seic twn ẋ 1, ẋ 2 me arqikèc sunj kec x 10, x 20, y 20 = ẋ 10 = ẋ 20 = 0, ẏ 20 kai gia qrìno t. BrÐskw ston t pìso eðnai ta : ẋ 1 (t ; x 10, x 20, y 20 = 0, ẋ 10 = 0, ẋ 20 = 0, ẏ 20, m i 0) ẋ 2 (t ; x 10, x 20, y 20 = 0, ẋ 10 = 0, ẋ 20 = 0, ẏ 20, m i 0) (2.46) Sth sunèqeia oloklhr nw tic sqèseic gia arqikèc sunj kec kai brðskw: x 10, x 20 + x 20, y 20 = ẋ 10 = ẋ 20 = 0, ẏ 20 ẋ 1(t ; x 10, x 20 + x 20, y 20 = 0, ẋ 10 = 0, ẋ 20 = 0, ẏ 20, m i 0) ẋ 2(t ; x 10, x 20 + x 20, y 20 = 0, ẋ 10 = 0, ẋ 20 = 0, ẏ 20, m i 0) (2.47) ìpou autìc o t eðnai diaforetikìc apì ton prohgoômeno kai eðnai kˆje forˆ o qrìnoc thc tom c Poincare.

28 KEFŸALAIO 2. TO PERISTREFŸOMENO MONTŸELO 'Ara qrhsimopoi ntac ton orismì thc parag gou brðskw ẋ 1 x 20 t=t ẋ 1 ẋ 1 x 20 (2.48) kai ẋ 2 x 20 t=t ẋ 2 ẋ 2 x 20 (2.49) Gia na upologðsw ta ẋ 1 ẏ 10, ẋ 1 ẏ 20 akolouj thn Ðdia diadikasða aut th forˆ allˆzontac to ẏ 20. Oloklhr nontac autˆ ta b mata brðskw mða periodik troqiˆ gia m 1 = m 1, m 2 = m 2, kai suneqðzw mèqri ìpoia tim mˆzac jèlw na epitôqw antikajist ntac kˆje forˆ thn prohgoômenh mˆza me m i = m i + m i. 2.7 Eustˆjeia periodik n troqi n Gia na upologðsoume thn eustˆjeia thc troqiˆc ja prèpei na thn grammikopoi soume. To pr to b ma gia na to kˆnoume autì eðnai na grˆyoume to sôsthma exis sewn kðnhshc pou èqoume wc èna sôsthma èxi diaforik n exis sewn pr thc tˆxhc, antð gia tri n deôterhc tˆxhc. 'Ara ja ftˆsoume se èna sôsthma thc morf c Ẋ i = F i (X 1, X 2, X 3, X 4, X 5, X 6 ), i = 1, 2, 3, 4, 5, 6 (2.50) ìpou X 1 = x 1, X 2 = x 2, X 3 = y 2, X 4 = ẋ 1, X 5 = ẋ 2, X 6 = ẏ 2. Autì isqôei gia kˆje sôsthma exis sewn to opoðo exasfalðzei thn analutikìthta thc lôshc. 'Estw ìti h X i (t), i = 1, 2, 3, 4, 5, 6 (2.51) eðnai lôsh tou sust matoc, h opoða antistoiqeð stic arqikèc sunj kec X i0 = X i (0) gia t = 0(apì th stigm pou to sôsthmˆ mac eðnai autìnomo, mporoôme na dialèxoume wc arq tou qrìnou to 0 qwrðc na qajeð h genikìthta). 'Estw ìti kai X i(t) ˆllh lôsh, pou antistoiqeð se nèec arqikèc sunj kec X i0 = X i0 +ξ i0, ìpou to ξ i0 jewreðtai mikrì. MporoÔme na grˆyoume th nèa lôsh sth morf X i(t) = X i (t) + ξ i (t), i = 1, 2, 3, 4, 5, 6 (2.52) 'Opou ξ i (t) eðnai sthn pragmatikìthta oi diataraqèc thc arqik c lôshc. MporoÔme na broôme èna nèo sôsthma diaforik n exis sewn apì to opoðo proèrqontai oi sunart - seic ξ i (t), proseggðzontac grammikˆ tic arqikèc diataraqèc ξ i0. An antikatast soume tic nèec lôseic sto arqikì mac sôsthma, kai anaptôxoume tic sunart seic F i se seirˆ

2.7. EUSTŸAJEIA PERIODIKŸWN TROQIŸWN 29 Taylor kai krat soume mìno touc ìrouc pr thc tˆxhc thc ξ i, ja pˆroume, dedomènou ìti oi X i (t) ikanopoioôn to sôsthmˆ mou, to grammikì sôsthma ξ i = 6 P ij (t)ξ j, i, j = 1, 2, 3, 4, 5, 6 (2.53) j=1 ìpou ( ) Fi P ij (t) =, i, j = 1, 2, 3, 4, 5, 6 (2.54) X j 0 kai to mhdèn sto deðkth shmaðnei ìti oi merikèc parˆgwgoi èqoun upologisteð gia thn arqik mac lôsh. Autì shmaðnei ìti tic P ij (t) eðnai gnwstèc sunart seic sto qrìno. To sôsthma twn ξ i eðnai èna grammikì sôsthma me suntelestèc pou genikˆ eðnai sunart seic tou qrìnou. Autì to sôsthma antistoiqeð sthn arqik lôsh, diìti oi suntelestèc P ij (t) eðnai upologismènoi gia autèc tic sunart seic, kai mac dðnoun tic diataraqèc ξ i (t) se grammik prosèggish. To sôsthma twn ξ i onomˆzetai sôsthma exis sewn metabol n, pou antistoiqeð sth lôsh X i (t). Apì th genik jewrða grammik n susthmˆtwn diaforik n exis sewn gnwrðzoume ìti h genik lôsh tou (2.53) mporeð na ekfrasteð wc o grammikìc sunduasmìc èxi grammik c anexˆrthtwn lôsewn. Ac upojèsoume t ra èxi lôseic thc (2.53) pou antistoiqoôn se arqikèc sunj kec (1, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1) antðstoiqa, se qrìno t = 0 kai onomˆsoume (t) to jemeli dh pðnaka twn lôsewn tou opoðou oi st lec eðnai proanaferjeðsec èxi lôseic. Autì shmaðnei ìti (0) = I, ìpou I o monadiaðoc 6x6 pðnakac. Tìte, oi lôseic ξ i tou sust matoc, h opoða antistoiqeð se arqikèc sunj kec ξ i0, grˆfetai se morf pðnaka wc ξ = (t)ξ 0 (2.55) ìpou ξ = ξ 1 ξ 2 ξ 3 ξ 4 ξ 5 ξ 6, ξ 0 = ξ 10 ξ 20 ξ 30 ξ 40 ξ 50 ξ 60 (2.56) FaÐnetai ìti an gnwrðzoume to (t) tìte mporoôme na broôme th genik lôsh thc (2.53), h opoða antistoiqeð stic arqikèc sunj kec ξ 0. O pðnakac (t) onomˆzetai matrizant tou sust matoc. 2.7.1 O monìdromoc pðnakac Sthn prohgoômenh anˆlush den kˆname kamða upìjesh gia th lôsh X i (t), ˆra ìlec oi idiìthtec stic opoðec katal xame isqôoun se ìlec tic peript seic. T ra, ja peri-

30 KEFŸALAIO 2. TO PERISTREFŸOMENO MONTŸELO orðsoume touc eautoôc mac stic idiìthtec twn exis sewn metabol n pou antistoiqoôn se periodik troqiˆ X i (t) me perðodo T. X i (t + T ) = X i (t), i = 1, 2, 3, 4, 5, 6 (2.57) Oi exis seic metabol n se aut thn perðptwsh eðnai èna grammikì sôsthma diaforik n exis sewn me periodikoôc suntelestèc, me perðodo T, ìpwc faðnetai apì thn (2.54). MporoÔme t ra na grˆyoume tic exis seic metabol n wc pðnaka sth morf ξ = p(t)ξ (2.58) ìpou p(t) ènac 6x6 periodikìc pðnakac, me perðodo T, p(t + T ) = p(t),tou opoðou ta stoiqeða eðnai oi sunart seic P ij (t), ìpwc orðsthkan sthn (2.54). 'Estw o (t) o matrizant tou pðnaka twn ξ. O pðnakac (T ), pou ton lambˆnoume apì ton (t) gia t = T, onomˆzetai monìdromoc pðnakac, o opoðoc antistoiqeð sthn periodik lôsh X i (t), kai eðnai shmantikìc sthn melèth thc grammik c eustˆjeiac thc periodik c troqiˆc. H melèth thc eustˆjeiac gðnetai me th bo jeia twn idiotim n tou pðnaka (T ). ApodeiknÔetai ìti oi idiotimèc tou monìdromou pðnaka taxinomoôntai katˆ antðstrofa zeôgh, dhlad gia kˆje λ i Λ upˆrqei kai to λ 1 i Λ. Epiplèon parathroôme ìti oi lôseic pou apoteloôn tic st lec tou jemeli dh pðnaka lôsewn (t) jewroôntai pragmatikèc kai sunep c o monìdromoc pðnakac eðnai pragmatikìc. Autì shmaðnei ìti h qarakthristik exðswsh (T ) λi = 0 (2.59) èqei pragmatikoôc suntelestèc kai sunep c oi idiotimèc taxinomoôntai kai katˆ suzug migadikˆ zeôgh. Me bˆsh ta parapˆnw mporoôn na apodeiqjoôn oi parakˆtw idiìthtec gia tic idiotimèc tou monìdromou pðnaka enìc grammikoô QamiltonianoÔ sust matoc: 1. Eˆn upˆrqei mða rðza pragmatik kai Ðsh me th monˆda, λ 1 = 1, tìte upˆrqei kai h antðstrof thc, λ 2 = 1, dhlad mia monadiaða idiotim eðnai pˆntote dipl ( pollapl ). To Ðdio isqôei kai gia idiotim Ðsh me -1. 2. Eˆn upˆrqei sto sônolo Λ twn idiotim n mia pragmatik rðza megalôterh thc monˆdac, λ 1 > 1, upˆrqei epðshc kai mia rðza mikrìterh thc monˆdac, λ 2 = 1 λ 1 < 1, kai epðshc λ 2 > 0. OmoÐwc gia λ 1 < 1, upˆrqei kai h 1 < λ 2 < 0. 3. Eˆn upˆrqei mia idiotim migadik me mètro Ðso me th monˆda, λ 1 = e iφ, tìte upˆrqei kai mia idiotim λ 2 = e iφ, ste λ 1 λ 2 = 1. ParathroÔme ìti oi dôo idiotimèc autèc eðnai epð tou monadiaðou kôklou sto migadikì epðpedo kai suzugeðc migadikèc.

2.7. EUSTŸAJEIA PERIODIKŸWN TROQIŸWN 31 4. Eˆn upˆrqei mia idiotim migadik me mètro diˆforo thc monˆdac, λ 1 = Re iφ, ìpou p.q. R > 1, tìte upˆrqoun kai oi ex c treic idiotimèc:λ 2 = R 1 e iφ, λ 3 = Re iφ, λ 4 = R 1 e iφ, diìti mìno se aut n thn perðptwsh oi idiotimèc diatˆssontai se antðstrofa kai migadikˆ suzug zeôgh(λ 1 λ 4 = 1, λ 2 λ 3 = 1, λ 1 = λ 3, λ 2 = λ 4 ). 5. O monìdromoc pðnakac èqei pˆnta dôo idiotimèc Ðsec me th monˆda. 'Ara efìson ja broôme èxi idiotimès(6x6 eðnai o pðnakac), oi ˆllec tèsseric eðnai pou prèpei na broôme stic peript seic 1,2,3,4 Stic peript seic 3 kai 4 eðnai dunatìn oi idiotimèc na eðnai pollaplèc, thc Ðdiac ìmwc pollaplìthtac 2.7.2 Eustˆjeia grammik n Qamiltonian n susthmˆtwn Lambˆnontac upìyin th morf thc genik c lôshc enìc grammikoô QamiltwnianoÔ sust matoc, katal goume sto sumpèrasma ìti h lôsh ξ(t), gia tuqoôsec arqikèc sunj kec, den eðnai peratwmènh ìtan upˆrqei èstw kai mða idiotim me mètro diaforetikì thc monˆdac. MporoÔme ìmwc sthn perðptwsh aut na epilèxoume ètsi tic arqikèc sunj kec ste h lôsh pou ja prokôyei na eðnai peratwmènh. Autì mporeð na gðnei an sto diˆnusma twn arqik n sunjhk n ξ(0) den upˆrqei sunist sa katˆ th dieôjunsh tou idiodianôsmatoc pou antistoiqeð sthn idiotim me mètro megalôtero thc monˆdac. H mìno perðptwsh pou èqoume peratwmènh lôsh eðnai ekeðnh ìpou ìlec oi idiotimèc tou monìdromou pðnaka èqoun mètro Ðso me th monˆda.

32 KEFŸALAIO 2. TO PERISTREFŸOMENO MONTŸELO

Kefˆlaio 3 Anˆlush apotelesmˆtwn Sto tm ma autì thc ergasðac ja analôsoume ta apotelèsmata pou proèkuyan apì th melèth mac. Oi arijmhtikèc oloklhr seic twn exis sewn èginan me thn mèjodo Bulirsch Stoer kai me akrðbeia 14 shmantik n yhfðwn. XekinoÔme apì upologismènec eustajeðc kuklikèc troqièc tou planhtikoô probl - matoc twn tri n swmˆtwn (ìpou oi dôo mˆzec eðnai polô mikrèc kai h mˆza tou trðtou s matoc sqedìn Ðsh me th monˆda) kai ja kˆnoume sunèqish wc proc tic mˆzec gia kˆpoiec apì autèc tic troqièc. H sunèqish wc proc tic mˆzec eðnai diparametrik, dhlad h sunèqish mporeð na gðnetai allˆzontac thn m 1 thn m 2 anexˆrthta. Autì bèbaia apaiteð pˆra polloôc upologismoôc ste na kalôyoume ìlh th didiˆstath pollaplìthta sthn opoða brðskontai oi periodikèc troqièc. Sth melèth mac akoloujoôme monoparametrik sunèqish krat ntac ton lìgo maz n m 2 /m 1 stajerì. QrhsimopoioÔme oikogèneiec kuklik n troqi n se sugkekrimènec analogðec maz n 1 : 2,1 : 1,2 : 1,5 : 1, ìpou pr ta anafèroume thn mˆza tou exwterikoô s matoc, Σ 2, kai metˆ tou eswterikoô s matoc, Σ 1. Apì ed kai sto ex c oi tèsseric autèc troqièc ja onomˆzontai antðstoiqa, Circ05, Circ1, Circ2, Circ5. Sto parakˆtw sq ma deðq noume to p c allˆzei to x 1 se sqèsh me to n 1 n 2, ìpou n 1 n 2 o lìgoc thc mèshc kðnhshc twn dôo swmˆtwn, n = 2π T. 33

34 KEFŸALAIO 3. ANŸALUSH APOTELESMŸATWN Sq ma 3.1: Metabol tou x 1 kata m koc twn oikogenei n kuklik n troqi n (dhlad kata thn metabol tou lìgou suqnot twn n 1 /n 2 gia lìgouc maz n 1/2 (pˆnw aristerˆ), 1/1 (pˆnw dexiˆ), 2/1 (kˆtw aristerˆ) kai 5/1 (kˆtw dexiˆ) SÔmfwna me thn ergasða twn Bozis, Hadjidemetriou(1976) apodeðqjhke ìti mporeð mða periodik troqiˆ tou probl matoc twn tri n swmˆtwn na suneqisteð wc proc tic mˆzec twn swmˆtwn. Krat ntac ton lìgo twn dôo maz n stajerì kai brðskontac thn exˆrthsh twn tri n metablht n wc proc th mˆza pou allˆzoume kai thn perðodo, x 10 = x 10 (m 2, T ),x 20 = x 20 (m 2, T ),ẏ 20 = ẏ 20 (m 2, T ), katal goume sto sumpèrasma ìti oi exis seic autèc se parametrik morf orðzoun mða disdiˆstath epifˆneia ston

3.1. TROQIŸES 35 trisdiˆstato q ro x 10 x 20 ẏ 20, thn opoða apokaloôme qarakthristik epifˆneia. Kˆje shmeðo thc epifˆneiac orðzei pl rwc mða troqiˆ. Anˆloga me to pìsec metablhtèc allˆzoume kratˆme stajerèc mporoôme na dhmiourg soume monoparametrikèc diparametrikèc oikogèneic troqi n pou an koun se aut n thn epifˆneia. Efìson emeðc allˆzoume mða mˆza kai thn perðodo, en oi ˆllec dôo mˆzec prokôptoun apì to s- tajerì lìgo kai thn kanonikopoðhsh sth monˆda, ja brðc koume mono-parametrikèc oikogèneiec. H diadikasða pou ektelèsame emeðc gia na ftˆsoume sta apotelèsmatˆ mac tan h ex c: Gia kˆje mða oikogèneia apì tic tèsseric arqikèc, ( Circ05, Circ1, Circ2, Circ5) epilègoume kˆpoiec apì tic troqièc, tètoiec ste na èqoume mða pl rh ˆpoyh thc sumperiforˆc thc sunèqishc anˆloga me ta arqikˆ n1 n 2, kai se autèc trèxame to prìgrammˆ mac ste na kˆnoume th sunèqish wc proc tic mˆzec. O parakˆtw pðnakac parousiˆzei tic timèc twn n 1 n 2 pou p rame gia kˆje oikogèneia. 3.1 Troqièc Parakˆtw parousiˆzoume kˆpoia antiproswpeutikˆ paradeðgmata troqi n ta opoða p rame apì touc upologismoôc me th metabol twn maz n. Katˆ th metabol aut h gewmetrða twn troqi n allˆzei me èna suneq trìpo kai parapl siec timèc maz n dðnoun parapl sia apotelèsmata. Parousiˆzoume parakˆtw xeqwristˆ gia kˆje kuklik oikogèneia tic troqièc, me tic arqikèc sunj kec kˆje troqiˆc na anagrˆfontai kˆtw apì thn eikìna.

36 KEFŸALAIO 3. ANŸALUSH APOTELESMŸATWN 3.1.1 Oikogèneia Circ05 Troqiˆ 1 Eustaj c T 2 T 1 = 5.8023 Eustaj c

3.1. TROQIŸES 37 Eustaj c Eustaj c

38 KEFŸALAIO 3. ANŸALUSH APOTELESMŸATWN

3.1. TROQIŸES 39 Troqiˆ 8 Eustaj c T 2 T 1 = 3.2004 Eustaj c

40 KEFŸALAIO 3. ANŸALUSH APOTELESMŸATWN Eustaj c Eustaj c

3.1. TROQIŸES 41 Troqiˆ 15 Eustaj c T 2 T 1 = 2.15

3.1. TROQIŸES 47 Troqiˆ 7 Eustaj c gia polô qamhlèc mˆzec h oikogèneia aut T 2 T 1 = 3.0280 Astaj c

48 KEFŸALAIO 3. ANŸALUSH APOTELESMŸATWN Troqiˆ 13 Eustaj c T 2 T 1 = 2.12

3.1. TROQIŸES 49 3.1.3 Oikogèneia Circ2 Troqiˆ 1 Eustaj c T 2 T 1 = 5.4008 Eustaj c

3.1. TROQIŸES 51

52 KEFŸALAIO 3. ANŸALUSH APOTELESMŸATWN Troqiˆ 8 Astaj c epeid brðsketai sqedìn ston suntonismì 3:1 T 2 T 1 = 3.0108 Astaj c

3.1. TROQIŸES 57 Troqiˆ 7 Eustaj c gia polô mikrèc mˆzec allˆ gr gora gðnetai astaj c lìgw tou suntonismoô 3:1 T 2 T 1 = 3.0673 Astaj c

58 KEFŸALAIO 3. ANŸALUSH APOTELESMŸATWN 3.1.5 Troqiˆ 9 Eustaj c gia polô mikrèc mˆzec h opoða gðnetai astaj c lìgw tou suntonismoô 3:1 allˆ eðnai h monadik oikogèneia pou melet same pou epistrèfei sthn eustˆjeia gia mia perioq megalôterwn maz n se autìn ton suntonismì (sto sq ma pou deðqnoume eðnai eustaj c) T 2 T 1 = 2.9839

3.2. ENŸERGEIA, STROFORMŸH, PERŸIODOS 59 Troqiˆ 13 Eustaj c T 2 T 1 = 2.1502 3.2 Enèrgeia, Stroform, PerÐodoc Epìmeno b ma sthn ˆnalush eðnai na melet soume to p c allˆzei h enèrgeia, h stroform kai h perðodoc se sqèsh me tic mˆzec, kaj c autèc auxˆnontai katˆ m koc thc sunèqishc. Dialèxame tic pr tec troqièc kˆje oikogèneiac (autèc me ton megalôtero lìgo periìdwn metaxô twn dôo planht n, ˆra kai pio eustajeðc), epeid suneqðzontai gia uyhlìterec mˆzec kai ˆra mporoôme kalôtera na parathr soume thn sumperiforˆ twn parapˆnw megej n.

60 KEFŸALAIO 3. ANŸALUSH APOTELESMŸATWN 3.2.1 Enèrgeia Oikogèneia Circ05 Oikogèneia Circ1

3.2. ENŸERGEIA, STROFORMŸH, PERŸIODOS 61 Oikogèneia Circ2 Oikogèneia Circ5

62 KEFŸALAIO 3. ANŸALUSH APOTELESMŸATWN 3.2.2 Stroform Oikogèneia Circ05 Oikogèneia Circ1

64 KEFŸALAIO 3. ANŸALUSH APOTELESMŸATWN 3.2.3 PerÐodoc Oikogèneia Circ05 Oikogèneia Circ1

66 KEFŸALAIO 3. ANŸALUSH APOTELESMŸATWN 3.3 'Oria sunèqishc wc proc tic mˆzec kai eustˆjeia H teleutaða susqètish pou ja melet soume eðnai to poiec eðnai oi mègistec mˆzec (m 1, m 2 ), stic opoðec ftˆnei h eustˆjeia, se sqèsh me ton lìgo aktðnwn twn troqi n twn dôo planht n sthn arqik kuklik troqiˆ. 3.3.1 Oikogèneia Circ05 Se aut n thn oikogèneia parathroôme ìti gia uyhloôc lìgouc periìdwn (pˆnw apì 3:1), oi mˆzec twn dôo arqik n planht n ftˆnoun mèqri megˆlec timèc, xepern ntac se mègejoc to arqikì astèri. To prìgrammˆ mac den mporeð na suneqðsei thn sunèqish pèra apì ekeðnec tic mˆzec. Autìc eðnai kai o lìgoc pou blèpoume mða sqedìn eujeða gramm lðgo pˆnw apì to m 1 = 0.6. O lìgoc pou den mporeð na gðnei h sunèqish eðnai mˆllon to gegonìc ìti h pollaplìthta twn periodik n troqi n parousiˆzei ptuq (anadðplwsh wc proc th mˆza). Sth sunèqeia parathroôme mða katakìrufh pt sh ston suntonismì 3:1, ston opoðo eðnai gnwstì ìti emfanðzetai astˆjeia sto sôsthma. Apì ekeð kai metˆ den brðskoume megˆlec mˆzec sta exwterikˆ s mata, an kai emfanðzontai kˆpoiec upologðsimec mˆzec prin ton suntonismì 2:1. Ta dôo epiplèon shmeða pou emfanðzontai sto lìgo aktðnwn 1, 8 tou ˆxona Q upˆrqoun diìti

3.3. ŸORIA SUNŸEQISHS WS PROS TIS MŸAZES KAI EUSTŸAJEIA 67 h oikogèneia mac pèrase se astajeðc troqièc arketˆ qamhlˆ allˆ epan lje gia ekeðnh thn perioq maz n sthn eustˆj eia. 3.3.2 Oikogèneia Circ1 Se aut n thn oikogèneia parathroôme, ìpwc kai sthn prohgoômenh, ìti stouc megˆlouc suntonismoôc ftˆnoun oi mˆzec se polô uyhlèc timèc, allˆ aut th forˆ stamatˆme thn sunèqish diìti oi troqièc gðnontai astajeðc. EpÐshc, an h astˆjeia eðnai isqur to prìgramma adunateð na upologðsei tic epìmenec troqièc giatð oi diaforikèc proseggðseic apoklðnoun. 'Opwc akrib c kai sthn parapˆnw oikogèneia, emfanðzetai mða katakìrufh pt sh ston suntonismì 3:1, kai h Ðdia sumperiforˆ anˆmesa stouc suntonismoôc 3:1 kai 2:1, me mða mikr aôxhsh thc mègisthc mˆzac qwrðc ìmwc aut th forˆ na upˆrqei epistrof thc oikogèneiac sthn eustˆjeia ìpwc sunèbh sthn Circ05.

68 KEFŸALAIO 3. ANŸALUSH APOTELESMŸATWN 3.3.3 Oikogèneia Circ2 Se antðjesh me tic dôo prohgoômenec oikogèneiec, t ra blèpoume mða omal pt sh twn mègistwn maz n apì touc megˆlouc lìgouc aktðnwn ( periìdwn) wc kai thn gnwst astˆjeia tou 3:1. H oikogèneia aut ftˆnei se qamhlìterec mègistec mˆzec apì tic ˆllec dôo. Anˆmesa stouc suntonismoôc 3:1 kai 2:1 katˆ ta gnwstˆ emfanðzei polô qamhlèc mègistec mˆzec, kai ìpwc eðdame kai sthn pr th oikogèneia, emfanðzontai shmeða sta opoða h oikogèneia epistrèfei sthn eustˆjeia gia mia perioq twn maz n, h opoða perioq ìmwc eðnai arketˆ megˆlh kai ftˆnei se timèc shmantikèc giatð eðnai sthn Ðdia tˆxh megèjouc me th mˆza tou asterioô, perðpou èna èkto kai èna trðto antðstoiqa oi m 1, m 2.

3.3. ŸORIA SUNŸEQISHS WS PROS TIS MŸAZES KAI EUSTŸAJEIA 69 3.3.4 Oikogèneia Circ5 H teleutaða mac oikogèneia moiˆzei arketˆ me th Circ2, diìti kai aut parousiˆzei mða stajer pt sh twn mègistwn maz n wc ton suntonismì 3:1 (gia thn akrðbeia sqedìn grammik ). EmfanÐzontai oi gnwstèc perioqèc mikr n mègistwn maz n anˆmesa stouc suntonismoôc 3:1 kai 2:1, allˆ se antðjesh me tic ˆllec emfanðzei megalôterec perioqèc stic opoðec epistrèfei h oikogèneia se eustˆjeia, kai mˆlista eðnai h mình h opoða èqei eustajeðc troqièc se arketˆ megˆlec mˆzec sqedìn pˆnw ston suntonismì 3:1.

Kefˆlaio 4 Sumperˆsmata Sthn ergasða aut anazht same eustajeðc troqièc gia triplˆ sust mata swmˆtwn upì thn allhlepðdrash barutik n sunˆmewn. Qrhsimopoi same to montèlo tou genikoô probl matoc twn tri n swmˆtwn to opoðo ìmwc to perigrˆfoume mèsa apì èna peristrefìmeno sôsthma susntetagmènwn. Se aut th perigraf to dunamikì mac sôsthma eðnai tri n bajm n eleujerðac. 'Otan ta dôo s mata èqoun polô mikrèc mˆzec se sqèsh me to trðto, tìte perigrˆfoume èna planhtikì sôsthma. Sto sôsthma autì, to opoðo èqei melethjeð diexodikˆ apì ton I.D.QatzhdhmhtrÐou, mporoôn na entopistoôn periodikèc troqièc oi opoðec sqhmatðzoun oikogèneiec sto q ro twn fˆsewn tou sust matoc. Oi troqièc autèc mporoôn na suneqistoôn metabˆllontac tic mˆzec twn swmˆtwn (Bozis and Hadjidemetriou, 1976). Sthrizìmenoi se autˆ ta apotelèsmata, jewr same tic oikogèneiec kuklik n periodik n troqi n tou planhtikoô probl matoc tic opoðec tic epekteðname wc proc tic mˆzec twn arqikˆ mikr n swmˆtwn (planht n) diathrìntac ton lìgo touc stajerì. Ta sumperˆsmata sta opoða katal xame apì touc upologismoôc mac sunoyðzontai sta ex c: 1. Oi oikogèneiec kuklik n periodik n troqi n tou planhtikoô probl matoc m- poroôn genikˆ na suneqistoôn èwc kai polô megˆlec mˆzec, ìpou polô megˆlec ennooôme sugkrðsimec thc Ðdiac tˆxhc megèjouc. H sunèqish stamatˆei eðte kontˆ se sugkroôseic twn swmˆtwn se isqurˆ astajeðc troqièc se ptuqèc thc pollaplìthtac pˆnw sthn opoða ˆn koun oi periodikèc troqièc ( Bozis and Hadjidemetriou, 1976). 2. H sunèqish wc proc tic mˆzec mac upologðzei eustajeðc troqièc gia megˆlec mˆzec stic oikogèneiec pou xekðnhsan apì lìgw periìdwn T 2 T 1 megalôtero tou 3, qwrðc ìmwc na apokleðontai eidikèc peript seic se qamhlìterouc lìgouc. 3. H suntriptik pleioyhfða twn eustaj n troqi n pou brðskontai sto prìblhma twn tri n swmˆtwn eðnai thc morf c tou ierarqikoô sust matoc, ìpou ta dôo 71

72 KEFŸALAIO 4. SUMPERŸASMATA apì ta trða s mata kinoôntai se èna isqurì barutikì zeugˆri kai sqedìn wc zeugˆri (prosomoiˆzontac èna s ma) allhlepidroôn me to trðto. 4. To shmantikìtero apì ta eur matˆ mac eðnai eustajeðc troqièc ìpou den parousiˆzetai to ierarqikì sôsthma allˆ emfanðzetai mia idiaðterh gewmetrik morf troqi n stic opoðec h barutik allhlepðdrash eðnai isqur anˆmesa kai sta trða s mata, eðte enallˆssetai diadoqikˆ h ierarqik dom me diaforetikˆ s mata. 5. Tèloc, ja prèpei na epishmˆnoume ìti ìlec tic troqièc pou upologðsame, ja mporoôsame me touc katˆllhlouc metasqhmatismoôc monˆdwn na tic fèroume stic diastˆseic pragmatik n swmˆtwn (astèrwn) kai na orðzoun èna pragmatikì (eustajèc) triplì sôsthma astèrwn.

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