Gkoliac Iwannhc H PROSEGGISH HILL STHN MELETH THS...

ARISTOTELEIO PANEPISTHMIO JESSALONIKHSSQOLH JETIKWN EPISTHMWN

TMHMA FUSIKHS

Gkoliac Iwannhc

H PROSEGGISH HILL STHN MELETH THS KINHSHS

TWN FUSIKWN DORUFORWN

Diplwmatikh Ergasia

Epiblepwn Kajhghthc : Gewrgioc Bougiatzhc

Jessalonikh, Dekembrioc 2009

PerÐlhyh

Sthn paroÔsa diplwmatik ergasÐa meletoÔme thn kÐnhsh twn fusik¸ndorufìrwn qrhsimopoi¸ntac thn prosèggish Hill. Oi exis¸seic kÐnhshc toukuklikoÔ probl matoc Hill proèrqontai apì ekeÐnec tou antÐstoiqou kuklikoÔprobl matoc twn tri¸n swmtwn en efarmìsoume ton metasqhmatismì Hillkai proume thn oriak diadikasÐa µ → 0. To nèo sÔsthma exis¸sewn eÐnaipolÔ pio aplì apì majhmatik c apìyewc, wstìso mporeÐ na qrhsimopoihjeÐsthn perigraf poll¸n problhmtwn thc ourniac mhqanik c. H melèth tousust matoc gÐnetai me th qr sh twn tom¸n Poincare kaj¸c kai me thn eÔreshtwn periodik¸n tou troqi¸n. Sth sunèqeia meletoÔme to elleiptikì prìblhmaHill. Oi exis¸seic kÐnhshc proèrqontai apì autèc tou elleiptikoÔ periori-smènou probl matoc twn tri¸n swmtwn me thn Ðdia diadikasÐa pou qrhsimo-poi same kai gia tic exis¸seic tou kuklikoÔ probl matoc. UpologÐzoume miaseir apì nèec oikogèneiec periodik¸n troqi¸n tou elleiptikoÔ probl matockai meletoÔme thn eustjeia touc. Sta plaÐsia tou elleiptikoÔ probl matocmeletoÔme kat pìso oi eustajeÐc periodikèc troqièc tou kuklikoÔ probl ma-toc diathroÔn thn eustjei touc ìtan arqÐzei kai auxnetai h ekkentrìthtathc troqic twn dÔo prwteuìntwn swmtwn. Gia th melèth aut qrhsimopoioÔ-me to deÐkth qaotikìthtac Fast Lyapunov Indicator (FLI) kai kataskeuzoumeqrtec eustjeiac diafìrwn perioq¸n tou q¸rou twn fsewn. Tèloc meletoÔ-me ta diagrmmata Henon gia diforec timèc ekkentrìthtac kai efarmìzoumeta apotelèsmat mac sthn eustjeia twn makrin¸n dorufìrwn twn meglwnplanht¸n tou hliakoÔ mac sust matoc.

Abstract

In the present diploma thesis we are dealing with the motion of natu-ral satellites in the framework of the Hill’s approximation. The equationsof motion of the circular Hill’s problem are derived from these of the circu-lar restricted three body problem, if we apply the Hill transformation andlet µ → 0. The new equations of motion are simpler from the mathema-tical point of view, however they can describe various problems of celestialmechanics. A systematic exploration of the system is carried out by usingPoincare Surfaces of Section and also by computing the periodic orbits of thesystem. Additionally a systematic exploration of the elliptic Hill’s problem isalso carried out. The equations of motion can be derived from the equationsof the elliptic restricted three body problem with the same procedure usedin the circular case. A series of new families of periodic orbits of the elli-ptic problem is calculated and we examine their stability. Using the ellipticHill’s problem equations we are testing if the stable periodic orbits of thecircular problem remain stable when the eccentricity of the two primaries isdifferent from zero. In order to do that the Fast Lyapunov Indicator (FLI) isbeing used for creating stability maps in different regions of the phase space.Finally, we produce Henon diagrams for different values of the eccentricityand we apply our methods to the study of the distant satellites of the giantplanets in our solar system.

3

Perieqìmena

1 Eisagwg 6

I To Kuklikì Prìblhma Hill 10

2 Exis¸seic KÐnhshc 112.1 Prìblhma tri¸n swmtwn . . . . . . . . . . . . . . . . . . . . . 112.2 Periorismèno prìblhma tri¸n swmtwn . . . . . . . . . . . . . 12

2.2.1 Orismìc tou probl matoc . . . . . . . . . . . . . . . . . 122.2.2 Peristrefìmeno sÔsthma suntetagmènwn . . . . . . . . 142.2.3 Exis¸seic kÐnhshc sto peristrefìmeno sÔsthma . . . . . 152.2.4 Jacobi,ZVC kai shmeÐa isorropÐac . . . . . . . . . . . . 18

2.3 Prosèggish Hill . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Metafor sust matoc sunt/nwn . . . . . . . . . . . . . 202.3.2 Metasqhmatismìc klÐmakac . . . . . . . . . . . . . . . . 212.3.3 Exis¸seic kuklikoÔ Hill . . . . . . . . . . . . . . . . . 222.3.4 Jacobi,ZVC kai shmeÐa isorropÐac . . . . . . . . . . . . 24

2.4 Qamiltonian tou kuklikoÔ Hill . . . . . . . . . . . . . . . . . . 26

3 Epifneiec Tom c Poincare 283.1 MejodologÐa . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2 Epifneiec tom c tou probl matoc Hill . . . . . . . . . . . . . 30

3.2.1 Epifneiec tom c gia CH ≥ 4.5 . . . . . . . . . . . . . . 303.2.2 Epifneiec tom c gia 4.5 ≥ CH > 4.326 . . . . . . . . . 313.2.3 Epifneia tom c gia CH ≤ 4.326749 . . . . . . . . . . . 32

4 Periodikèc Troqièc 354.1 MejodologÐa . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4

4.1.1 Periodikèc troqièc . . . . . . . . . . . . . . . . . . . . . 354.1.2 Summetrikèc periodikèc troqièc . . . . . . . . . . . . . . 364.1.3 Exis¸seic metabol¸n . . . . . . . . . . . . . . . . . . . 364.1.4 Eustjeia periodik¸n troqi¸n . . . . . . . . . . . . . . 384.1.5 EÔresh periodik¸n troqi¸n . . . . . . . . . . . . . . . . 39

4.2 DÐktuo summetrik¸n periodik¸n troqi¸n . . . . . . . . . . . . . 414.2.1 Oikogèneiec a kai c . . . . . . . . . . . . . . . . . . . . 414.2.2 Oikogèneia f . . . . . . . . . . . . . . . . . . . . . . . 424.2.3 Oikogèneia g . . . . . . . . . . . . . . . . . . . . . . . . 434.2.4 Oikogèneia g′ . . . . . . . . . . . . . . . . . . . . . . . 434.2.5 Oikogèneiec g3 kai Hg . . . . . . . . . . . . . . . . . . 44

II To Elleiptikì Prìblhma Hill 46

5 Exis¸seic KÐnhshc 475.1 Genikì prìblhma twn tri¸n swmtwn . . . . . . . . . . . . . . . 475.2 Periorismèno elleiptikì prìblhma twn 3 swmtwn . . . . . . . 485.3 To Elleiptikì Prìblhma Hill . . . . . . . . . . . . . . . . . . . 50

5.3.1 Exis¸seic KÐnhshc . . . . . . . . . . . . . . . . . . . . 53

6 Periodikèc Troqièc 556.1 MejodologÐa . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.2 Eustjeia Periodik¸n Troqi¸n . . . . . . . . . . . . . . . . . 566.3 Nèec Oikogèneiec Periodik¸n Troqi¸n . . . . . . . . . . . . . . 59

6.3.1 Oikogèneia f . . . . . . . . . . . . . . . . . . . . . . . 606.3.2 Oikogèneia g . . . . . . . . . . . . . . . . . . . . . . . . 676.3.3 Oikogèneia g′ . . . . . . . . . . . . . . . . . . . . . . . 696.3.4 Oikogèneia Hg . . . . . . . . . . . . . . . . . . . . . . 716.3.5 Sumpersmata . . . . . . . . . . . . . . . . . . . . . . . 73

7 Qrtec Eustjeiac 747.1 O deÐkthc qaotikìthtac F.L.I . . . . . . . . . . . . . . . . . . . 747.2 Qrtec eustjeiac kat m koc twn oikogenei¸n f, g kai g′ . . . 757.3 Diagrmmata Henon . . . . . . . . . . . . . . . . . . . . . . . . 78

7.3.1 Digramma Henon gia ekkentrìthta mhdèn . . . . . . . . 787.3.2 Diagrmmata Henon gia touc meglouc plan tec . . . . 807.3.3 Sumpersmata . . . . . . . . . . . . . . . . . . . . . . . 86

5

Keflaio 1

Eisagwg

H allhlepÐdrash dÔo swmtwn upì thn epÐdrash amoibaÐac elktik c dÔnamhceÐnai èna klasikì prìblhma thc dunamik c pou qronologeÐtai apì thn epoq touIsak NeÔtwna (1687). O NeÔtwnac katfere na epilÔsei to prìblhma gia dÔosfairik antikeÐmena ta opoÐa kinoÔntai ktw apì thn amoibaÐa barutik toucèlxh. To 1710 o Johann Bernoulli apèdeixe pwc h kÐnhsh tou enìc s¸matocse sqèsh me to llo mporeÐ na perigrafeÐ me mia kwnik tom : èlleiyh,parabol uperbol . O Euler to 1744 epexergsthke tic leptomèreiec touprobl matoc twn dÔo swmtwn, deÐqnontac ìti to prìblhma twn dÔo swmtwnmporeÐ na anaqjeÐ sto prìblhma kÐnhshc enìc s¸matoc me mza Ðsh me thnanhgmènh mza tou sust matoc.

H profan c epèktash tou probl matoc twn dÔo swmtwn eÐnai to prìblh-ma twn tri¸n swmtwn. H kÐnhsh tri¸n swmtwn lìgw thc amoibaÐac metaxÔtouc barutik c èlxhc melet jhke gia pr¸th for apì ton Euler to 1776 kailÐgo argìtera apì ton Lagrange to 1788. To prìblhma mporeÐ na tejeÐ apl :JewroÔme trÐa s¸mata ta opoÐa kinoÔntai lìgw thc amoibaÐac metaxÔ touc ba-rutik c èlxhc kai sth sunèqeia , dedomènwn twn arqik¸n sunjhk¸n, jèloumena prosdiorÐsoume thn epakìloujh kÐnhs touc. To prìblhma twn tri¸n sw-mtwn suqn anafèretai wc to poio dishmo apì ìla ta probl mata dunamik c.'Enac apì touc lìgouc pou èqei gÐnei tìso dishmo eÐnai h paraplanhtik touaplìthta. Wstìso o H.Poincare èdeixe ìti to prìblhma den mporeÐ na lujeÐanalutik. Sunep¸c to mìno pou mporoÔme na knoume eÐnai na melet soumearijmhtik tic idiìthtec twn lÔsewn twn exis¸sewn kÐnhshc.

To prìblhma twn tri¸n swmtwn aplopoieÐtai en to elafrìtero apì tas¸mata jewrhjeÐ ìti den ephrezei thn kÐnhsh twn dÔo llwn swmtwn (lè-gontai kai prwteÔonta s¸mata). To periorismèno (kuklikì) prìblhma twn

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tri¸n swmtwn pragmateÔetai thn kÐnhsh enìc apeiroelqista mikroÔ swmati-dÐou upì thn epÐdrash thc barutik c èlxhc twn dÔo prwteuìntwn swmtwn taopoÐa kinoÔntai se kuklik troqi to èna se sqèsh me to llo. Oi exis¸seictou periorismènou probl matoc twn tri¸n swmtwn katal goun se katllhlhgia melèth morf se èna peristrefìmeno sÔsthma suntetagmènwn me kèntroto barÔkentro twn dÔo prwteuìntwn.

Sq ma 1.1: O Leonard Euler(1707-1783) arister kai o Henry Poincare(1854-1912) dexi, dÔo spoudaÐoi epist monec pou asqol jhkan me to prìblhma twntri¸n swmtwn

Ac upojèsoume t¸ra to periorismèno prìblhma twn tri¸n swmtwn giata s¸mata 'Hlioc - Plan thc - Dorufìroc ìpou o plan thc kai o dorufìrocèqoun polÔ mikrìterh mza apì ton 'Hlio. En gènei h amoibaÐa èlxh metaxÔ touplan th kai tou dorufìrou mporeÐ na agnohjeÐ kai to prìblhma katal gei sthnupèrjesh dÔo problhmtwn twn dÔo swmtwn. Wstìso en h apìstash metaxÔplan th kai dorufìrou gÐnei arket mikr , tìte h amoibaÐa èlxh touc gÐnetaithc Ðdia txhc megèjouc me thn èlxh tou liou kai den mporeÐ na agnohjeÐ.

O pr¸toc pou efrmose thn parapnw idèa sto sÔsthma HlÐou - Ghc -Sel nhc tan o G. W. Hill sta tèlh tou 19o ai¸na. Sthn ergasÐa tou aut xekin¸ntac apì to periorismèno kuklikì prìblhma twn tri¸n swmtwn, me-tèfere to sÔsthma suntetagmènwn apì to barÔkentro HlÐou-Ghc sth Gh kaiefrmose metasqhmatismì klÐmakac kat µ1/31. Sthn sunèqeia je¸rhse thn

1µ = mp

mp+msόπου mp, ms η μάζα του πλανήτη και του ηλίου αντίστοιχα

7

oriak diadikasÐa µ → 0 kai katèlhxe se èna sÔsthma exis¸sewn to opoÐomelet thn kÐnhsh thc Sel nhc gÔrw apì th Gh, diataragmèno apì ton 'Hliopou brÐsketai se polÔ meglh apìstash. H prosèggish aut ègine gnwst wcprosèggish Hill.

H prosèggish Hill eÐnai idiaÐtera qr simh sthn ournia mhqanik kaj¸c miaseir apì probl mata mporoÔn na perigrafoÔn apì tic exis¸seic tou kuklikoÔprobl matoc Hill. Tètoiou eÐdouc probl mata eÐnai:

• H kÐnhsh fusik¸n dorufìrwn gÔrw apì plan th o opoÐoc brÐsketai semeglh apìstash apì ton 'Hlio.

• H allhlepÐdrash metaxÔ swmatidÐwn se planhtikoÔc daktulÐouc kai do-rufìrwn pou brÐskontai se kontinèc troqièc.

• H proswrin sÔllhyh enìc kom th apì èna plan th.

• H katanom swmatidÐwn gÔrw apì th Gh.

Epiprìsjeta, apì majhmatik c apìyewc to prìblhma Hill apoteleÐ thn a-ploÔsterh, mh-oloklhr¸simh perÐptwsh tou probl matoc twn N-swmtwn. Oiexis¸seic kÐnhshc eÐnai aploÔsterec akìmh kai apì autèc tou kuklikoÔ perio-rismènou probl matoc twn tri¸n swmtwn kai den perièqoun kamÐa parmetro.Parola aut oi lÔseic parousizoun poluplokìthta, pou eÐnai qarakthristi-kì ìlwn twn mh-oloklhr¸simwn susthmtwn.

Arketèc dekaetÐec met ton orismì tou probl matoc apì ton Hill, me thnèleush twn hlektronik¸n upologist¸n kai thn almat¸dh prìodo touc rqisanna gÐnontai oi pr¸tec arijmhtikèc melètec tou genikoÔ probl matoc twn tri¸nswmtwn kai twn upoproblhmtwn tou (periorismèno,elleiptikì,Hill k.a). Methn prosèggish Hill pr¸toc asqol jhke o M.Henon arqik se jewrhtikìepÐpedo kai me dÔo diadoqikèc ergasÐec (1969-1970) , upolìgise arijmhtiktic periodikèc troqièc tou probl matoc kai sth sunèqeia qrhsimopoi¸ntac tictomèc Poincare katfere na diamorf¸sei mia genik eikìna gia to sÔsthma.

To elleiptikì prìblhma Hill prokÔptei apì to antÐstoiqo elleiptikì pe-riorismèno prìblhma me thn Ðdia diadikasÐa pou qrhsimopoioÔme gia to kuklikìHill. Shmantik doulei pnw sto jèma autì eÐqe gÐnei apì ton kajhght tou tm matìc mac S.Iqtiroglou, o opoÐoc afoÔ apèdeixe thn sunèqish twnperiodik¸n troqi¸n tou kuklikoÔ probl matoc sto elleiptikì sth sunèqeiaupolìgise kai arijmhtik tic oikogèneiec autèc. KamÐa apì tic oikogèneiec pouupologÐsthkan den perieÐqe eustajeÐc periodikèc troqièc.

8

H paroÔsa ptuqiak ergasÐa qwrÐzetai se dÔo mèrh. Sto pr¸to mèrocmeletoÔme to kuklikì prìblhma Hill. UpologÐzoume tic periodikèc troqièc touprobl matoc kai meletme thn eustjei touc. MeletoÔme ,epÐshc, to q¸rotwn fsewn qrhsimopoi¸ntac th mèjodo twn tom¸n Poincare. Sto deÔteromèroc thc ergasÐac meletoÔme thn epÐdrash thc ekkentrìthtac thc troqic twndÔo prwteuìntwn swmtwn sthn eustjeia thc troqic tou trÐtou s¸matoc.Autì to petuqaÐnoume qrhsimopoi¸ntac tic eustajeÐc periodikèc troqièc toukuklikoÔ probl matoc tic opoÐec diatarssoume sta plaÐsia tou elleiptikoÔprìblhmatoc Hill. Kataskeuzoume qrtec eustjeiac kat m koc twn oikoge-nei¸n tou kuklikoÔ probl matoc kai diagrmmata Henon gia na ereun soumeto q¸ro twn fsewn. Tèloc upologÐzoume mia seir apì nèec oikogèneiectroqi¸n tou elleiptikoÔ probl matoc kai meletoÔme thn eustjeia touc.

9

Mèroc I

To Kuklikì Prìblhma Hill

10

Keflaio 2

Exis¸seic KÐnhshc

Sto keflaio autì ja doÔme pwc prokÔptoun oi exis¸seic tou kuklikoÔ pro-bl matoc Hill xekin¸ntac apì tic exis¸seic tou probl matoc twn tri¸n sw-mtwn. Jewr¸ntac pwc to trÐto s¸ma èqei amelhtèa mza se sqèsh me talla dÔo pernme sto periorismèno prìblhma twn tri¸n swmtwn. Sth sunè-qeia efarmìzoume ton metasqhmatismì Hill kai ja katal goume sto kuklikìprìblhma Hill. Sthn teleutaÐa pargrafo parousizoume ton qamiltonianìformalismì tou probl matoc, gia lìgouc plhrìthtac all kai kalÔterhc an-tÐlhyhc tou probl matoc.

2.1 To prìblhma twn tri¸n swmtwn

JewroÔme trÐa s¸mata pou allhlepidroÔn metaxÔ touc mìno mèsw barutik¸ndunmewn. H dÔnamh pou askeÐtai se kajèna apì ta trÐa s¸mata sÔmfwna meton nìmo thc pagkìsmia èlxhc eÐnai

F i = G ·3∑i,j

mi ·mj

r3ij

· rij

analutik :

F 1 = Gm1 ·m2

r312

· r12 +Gm1 ·m3

r313

· r13

F 2 = Gm2 ·m1

r321

· r21 +Gm2 ·m3

r323

· r23 (2.1)

F 3 = Gm3 ·m1

r331

· r31 +Gm3 ·m2

r332

· r32

11

Sq ma 2.1: Digramma twn dunmewn kai twn apostsewn sto prìblhma twntri¸n swmtwn.

Apì to deÔtero nìmo tou NeÔtwna gnwrÐzoume ìti F i = mi · Ri opìte jaeÐnai :

F 1 = m1 · R1, F 2 = m2 · R2, F 3 = m3 · R3 (2.2)

Apì tic sqèseic (2.1) kai (2.2) kai afoÔ diairèsoume thn katllhlh kje formza paÐrnoume tic exis¸seic kÐnhshc tou probl matoc twn tri¸n swmtwn:

R1 = Gm2

r312

· r12 +Gm3

r313

· r13

R2 = Gm1

r321

· r21 +Gm3

r323

· r23 (2.3)

R3 = Gm1

r331

· r31 +Gm2

r332

· r32

'H se sÔntomh morf

Ri = G3∑i,j

mj

r3ij

· rij

2.2 To periorismèno prìblhma twn tri¸n

swmtwn

2.2.1 Orismìc tou probl matoc

Sto prìblhma twn tri¸n swmtwn an upojèsoume pwc to èna apì ta trÐas¸mata èqei agno simh mza se sqèsh me tic llec dÔo (ac upojèsoume to

12

m3) tìte mporoÔme na jewr soume ìti den ephrezei thn kÐnhsh twn llwndÔo prwteuìntwn swmtwn. Oi sqèseic (2.3) grfontai:

R1 = Gm2

r312

· r12

R2 = Gm1

r321

· r21

R3 = Gm1

r331

· r31 +Gm2

r332

· r32 (2.4)

Ta dÔo megla s¸mata kinoÔntai mìno lìgw tic metaxÔ touc barutik cèlxhc se troqièc pou problèpontai apì to prìblhma twn dÔo swmtwn. GiaaploÔsteush tou probl matoc mporoÔme na jewr soume arqikèc sunj kec giata dÔo s¸mata ¸ste na kinoÔntai se kuklik troqi gÔrw apì to kèntro mzactouc (barÔkentro). H exÐswsh kÐnhshc gia to 3o s¸ma se dianusmatik morf eÐnai h exÐswsh (2.4)

R3 = Gm1

r331

· r31 +Gm2

r332

· r32

JewroÔme t¸ra èna sÔsthma suntetagmènwn (x,h,z). Oi jèseic twn prwteuìn-twn1 swmtwn eÐnai (ξ1, η1, ζ1) kai (ξ2, η2, ζ2). Ta dÔo prwteÔonta s¸mataèqoun stajer apìstash, Ðdia gwniak taqÔthta kai koinì kèntro mzac. 'E-stw t¸ra G(m1 +m2) = 1 kai m1 > m2 tìte an jèsoume µ = m2

m1+m2ja eÐnai

µ1 = Gm1 = 1 − µ kai µ2 = Gm2 = µ . Sto adraneiakì sÔsthma sunte-tagmènwn (ξ, η, ζ) pou jewr same oi diaforikèc exis¸seic kÐnhshc tou trÐtous¸matoc (2.4) paÐrnoun thn paraktw morf :

ξ = (1− µ)ξ1 − ξr3

1

+ µξ2 − ξr3

2

η = (1− µ)η1 − ηr3

1

+ µη2 − ηr3

2

(2.5)

ζ = (1− µ)ζ1 − ζr3

1

+ µζ2 − ζr3

2

An upojèsoume ìti h kÐnhsh tou trÐtou swmatidÐou gÐnetai sto epÐpedo(ξ, η) pou orÐzoun ta lla dÔo s¸mata tìte mporoÔme na agnooÔme thn teleu-taÐa exÐswsh kai gia thn kÐnhsh tou trÐtou swmatidÐou èqoume to paraktw

1Πρωτεύοντα ονομάζονται τα σώματα των οποίων η μάζα δεν είναι αμελητέα. Στην προ-

κειμένη περίπτωση είναι τα m1 και m2.

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sÔsthma exis¸sewn:

ξ = (1− µ)ξ1 − ξr3

1

+ µξ2 − ξr3

2

(2.6)

η = (1− µ)η1 − ηr3

1

+ µη2 − ηr3

2

(2.7)

ìpou t¸ra r21 = (ξ1−ξ)2 +(η1−η)2 kai r2

2 = (ξ2−ξ)2 +(η2−η)2. To sÔsthmatwn parapnw exis¸sewn apoteleÐ èna mh autìnomo dunamikì sÔsthma dÔobajm¸n eleujerÐac.

2.2.2 Peristrefìmeno sÔsthma suntetagmènwn

Gia na persoume apì to adraneiakì sÔsthma suntetagmènwn (ξ, η) se ènaperistrefìmeno sÔsthma (x, y) efarmìzoume ton paraktw metasqhmatismìstrof c:

ξ = xcosωt− ysinωt (2.8)

η = xsinωt+ ycosωt (2.9)

O parapnw metasqhmatismìc mporeÐ na grafeÐ kai me morf pÐnaka ¸c ex c:(ξη

)=

(cosωt −sinωtsinωt cosωt

)(xy

)

Sq ma 2.2: H sqèsh metaxÔ twn kartesian¸n (x,h) kai twn suntetagmènwnsto peristrefìmeno sÔsthma (x,y).

14

ParagwgÐzontac tic sqèseic (2.8),(2.9) wc proc to qrìno èqoume:

ξ = xcosωt− xωsinωt− ysinωt− yωcosωt⇒

ξ = (x− yω)cosωt− (y + ωy)sinωt (2.10)

η = xsinωt+ xωcosωt+ ycosωt− yωcosωt⇒

η = (x− yω)sinωt+ (y + xω)cosωt (2.11)

Oi sqèseic (2.10),(2.11) mporoÔn na grafoÔn kai me th morf pÐnaka:(ξη

)=


)(x− yωy + xω

)Me mia deÔterh parag¸gish twn sqèsewn (2.8),(2.9) wc proc to qrìno jaèqoume:

ξ = xcosωt− xωsinωt− xsinωt− xω2cosωt−

−ysinωt− yωcosωt− yωcosωt+ yω2sinωt⇒

ξ = (x− 2yω − xω2)cosωt− (y + 2xω − yω2)sinωt (2.12)

η = xsinωt+ xωcosωt+ xωcosωt− xω2sinωt+

+ycosωt− yωsinωt− yωsinωt− yω2cosωt⇒

η = (x− 2yω − xω2)sinωt+ (y + 2xω − yω2)cosωt (2.13)

Oi sqèseic (2.12) kai (2.13) mporoÔn kai autèc na grafoÔn me th morf pÐnaka:(ξη

)=


)(x− 2yω − xω2

y + 2xω − yω2

)

2.2.3 Oi exis¸seic kÐnhshc sto peristrefìmeno

sÔsthma suntetagmènwn

An epilèxoume o xonac x tou peristrefìmenou sust matoc na eÐnai h eujeÐapou en¸nei ta dÔo prwteÔonta s¸mata tìte oi suntetagmènec touc ja eÐnai thcmorf c (x1, 0) kai (x2, 0) sunep¸c apì tic sqèseic (2.8),(2.9) èqoume:

ξ1 = x1cosωt ξ2 = x2cosωt (2.14)

η1 = x1sinωt η2 = x2sinωt (2.15)

15

Antikajist¸ntac stic exis¸seic (2.6),(2.7) tic sqèseic (2.8),(2.9),(2.12),(2.13)kaj¸c kai tic (2.14),(2.15) èqoume gia tic exis¸seic kÐnhshc sto peristrefì-meno sÔsthma suntetagmènwn:

(x− 2yω − xω2)cosωt− (y + 2xω − yω2)sinωt =

= (1− µ)x1cosωt− xcosωt+ ysinωt

r31

+ µx2cosωt− xcosωt+ ysinωt

r31

⇒

(x− 2yω − xω2)cosωt− (y + 2xω − yω2)sinωt =

=

[(1− µ)

x1 − xr3

1

+ µx2 − xr3

2

]cosωt+

(1− µr3

1

+µ

r32

)ysinωt (2.16)

(x− 2yω − xω2)sinωt− (y + 2xω − yω2)cosωt =

= (1− µ)x1sinωt− xsinωt+ ycosωt

r31

+ µx2sinωt− xsinωt+ ycosωt

r31

⇒

(x− 2yω − xω2)sinωt+ (y + 2xω − yω2)cosωt =

=

[(1− µ)

x1 − xr3

1

+ µx2 − xr3

2

]sinωt+

(1− µr3

1

+µ

r32

)ycosωt (2.17)

An t¸ra stic parapnw sqèseic orÐsoume tic paraktw posìthtec:

a = x− 2yω − ω2x b = y + 2xω − ω2y

c = (1− µ)x1 − xr3

1

+ µx2 − xr3

2

d =

(1− µr3

1

+µ

r32

)y

Tìte autèc grfontai:

acosωt− bsinωt = ccosωt+ dsinωt (2.18)

asinωt+ bcosωt = csinωt− dcosωt (2.19)

T¸ra ergazìmaste wc ex c: pollaplasizoume th sqèsh (2.18)·cosωt, th sqè-sh (2.19) ·sinωt kai prosjètoume tic dÔo sqèseic kat mèlh.

acos2ωt− bsinωtcosωt = ccos2ωt+ dsinωtcosωtasin2ωt+ bsinωtcosωt = csin2ωt− dsinωtcosωt

⇒

a(cos2ωt+ sin2ωt

)= c

(cos2ωt+ sin2ωt

)⇒ a = c

16

Antikajist¸ntac sthn parapnw sqèsh ta a,c paÐrnoume thn pr¸th exÐswshkÐnhshc sto peristrefìmeno sÔsthma suntetagmènwn.

x− 2yω − ω2x = (1− µ)x1 − xr3

1

+ µx2 − xr3

2

(2.20)

Gia na proume kai th deÔterh exÐswsh kÐnhshc ergazìmaste wc ex c : polla-plasizoume thn sqèsh (2.18)·−sinωt, th sqèsh (2.19)·cosωt kai prosjètoumekai pli kat mèlh.

−asinωtcosωt+ bsin2ωt = −csinωtcosωt− dsin2ωtasinωtcosωt+ bcos2ωt = csinωtcosωt− dcos2ωt

⇒

b(cos2ωt+ sin2ωt

)= −d

(cos2ωt+ sin2ωt

)⇒ b = −d

Antikajist¸ntac ta b,d sth sqèsh (28) tìte ja èqw kai thn deÔterh exÐswshkÐnhshc sto peristrefìmeno sÔsthma suntetagmènwn:

y + 2xω − ω2y = −y(

1− µr3

1

+µ

r32

)(2.21)

Sunep¸c to sÔsthma twn exis¸sewn sto peristrefìmeno sÔsthma ja eÐnai:

x− 2yω − ω2x = (1− µ)x1 − xr3

1

+ µx2 − xr3

2

y + 2xω − ω2y = −y(

1− µr3

1

+µ

r32

)Sthn parapnw sqèsh emfanÐzontai ta r1 kai r2 ta ìpoia wc apostseic pa-ramènoun analloÐwtec apì ton metasqhmatismì strof c

r21 = (x1 − x)2 + y2 r2

2 = (x2 − x)2 + y2 (2.22)

Lambnontac upoyin tic parapnw sqèseic kai to gegonìc ìti oi jèseictwn dÔo prwteuìntwn swmtwn sto peristrefìmeno sÔsthma eÐnai x1 = µkai x2 = −(1 − µ) antÐstoiqa tìte gia to periorismèno sÔsthma twn tri¸nswmtwn èqoume to paraktw sÔsthma exis¸sewn kÐnhshc:

x− 2yω − ω2x = −(1− µ)x− µr3

1

− µx+ 1− µr3

2

(2.23)

y + 2xω − ω2y = −y(

1− µr3

1

+µ

r32

)(2.24)

ìpou r21 = (x− µ)2 + y2 kai r2

2 = (x+ 1− µ)2 + y2.

17

2.2.4 To olokl rwma tou Jacobi,oi kampÔlec mh-

denik c taqÔthtac kai ta shmeÐa isorropÐac

An jewr soume thn sunrthsh

U =ω2

2(x2 + y2) +

1− µr1

+µ

r2

(2.25)

tìte oi exis¸seic (2.23),(2.24) mporoÔn na grafoÔn sthn paraktw morf :

x− 2yω =∂U

∂x(2.26)

y + 2xω =∂U

∂y(2.27)

Pollaplasizontac t¸ra thn sqèsh (2.26)·x, thn sqèsh (2.27)·y kai sth su-nèqeia prosjèsoume kat mèlh tic dÔo sqèseic èqoume:

xx+ yy =∂U

∂xx+

∂U

∂yy =

dU

dt(2.28)

Oloklhr¸nontac th sqèsh (2.28) wc proc to qrìno paÐrnoume thn paraktwsqèsh:

1

2(x2 + y2) = U − C ⇒ x2 + y2 = 2U − CJ (2.29)

ìpou CJ mia stajer olokl rwshc. 'Omwc x2 + y2 = u2 eÐnai h taqÔthtatou trÐtou s¸matoc sto peristrefìmeno sÔsthma suntetagmènwn. Sunep¸cmporoÔme na gryoume:

u2 = 2U − CJAn t¸ra sth sqèsh (2.29) antikatast soume to U apì th sqèsh (2.25) aut grfetai:

CJ = ω2(x2 + y2) + 2

(1− µr1

+µ

r2

)− x2 − y2 (2.30)

H posìthta CJ = 2U − u2 eÐnai mia stajer thc kÐnhshc kai ètsi h parap-nw sqèsh onomzetai kai olokl rwma stajer tou Jacobi (merikèc forèconomzetai kai olokl rwma thc enèrgeiac).

Apì th sqèsh (2.30) u2 = 2U −CJ prokÔptei pwc to dexÐ mèloc prèpei naeÐnai jetikì kai ètsi èqoume:

2U − CJ ≥ 0

18

H parapnw sqèsh mac dÐnei thn epitrept perioq kÐnhshc gia kpoia orismènhtim tou CJ . Mlista oi kampÔlec thc morf c

2U − CJ = 0

lègontai kai kampÔlec mhdenik c taqÔthtac zero velocity curves (ZVC).Gia na broÔme ta shmeÐa isorropÐac sto periorismèno prìblhma twn tri¸n

swmtwn jètoume stic sqèseic (2.26),(2.27) x, y, x, y = 0 ìpote to prìblhmaangetai sthn lÔsh tou paraktw sust matoc grammik¸n exis¸sewn:

∂U

∂x=∂U

∂r1

∂r1

∂x+∂U

∂r2

∂r2

∂x= 0

∂U

∂y=∂U

∂r1

∂r1

∂y+∂U

∂r2

∂r2

∂y= 0

ApodeiknÔetai ìti uprqoun telik 5 shmeÐa isorropÐac sto sÔsthma, gnwstkai wc shmeÐa Lagrange. Ta trÐa apì aut brÐskontai epnw ston xona Ox(L1, L2, L3) en¸ ta upìloipa dÔo (L4, L5) brÐskontai stic korufèc twn dÔoisìpleurwn trig¸nwn m1L4,5m2. Ta suggrammik shmeÐa (L1, L2, L3) eÐnaigrammik astaj gia kje tim tou m. AntÐjeta ta trigwnik shmeÐa (L4, L5)eÐnai grammik eustaj gia µ ≤ 0.0385.

Sq ma 2.3: ShmeÐa Lagrange kai kampÔlec mhdenik c taqÔthtac (ZVC) gialìgo maz¸n µ = 0.2

19

2.3 H Prosèggish Hill

2.3.1 Metafor tou sust matoc suntetagmènwn

Apì to periorismèno prìblhma twn tri¸n swmtwn an stic exis¸seic kÐnhshctou trÐtou s¸matoc (sqèseic (2.23),(2.24)) jèsoume ω = 1 èqoume :

x− 2y = x− (1− µ)x− µr3

1

− µx+ 1− µr3

2

(2.31)

y + 2x = y

(1− 1− µ

r31

− µ

r32

)(2.32)

ìpou r21 = (x−µ)2 +y2 kai r2

2 = (x+1−µ)2 +y2. Knoume t¸ra ton paraktwmetasqhmatismì:

ξ = x+ 1− µ⇒ x = ξ + µ− 1 & η = y

Oi exis¸seic (2.31),(2.32) t¸ra grfontai:

¨ξ − 2 ˙η = ξ + µ− 1− (1− µ)ξ − 1

r31

− µ ξr3

2

(2.33)

¨η + 2˙ξ = η

(1− 1− µ

r31

− µ

r32

)(2.34)

ìpou t¸rar2

1 = (x− µ)2 + y2 ⇒ r21 = (ξ − 1)2 + η2

r22 = (x+ 1− µ)2 + y2 ⇒ r2

2 = ξ2 + η2

Sto peristrefìmeno sÔsthma suntetagmènwn tou periorismènou probl matocta prwteÔonta s¸mata brÐskontai stic jèseic

P1(x1 = µ, y1 = 0) & P2(x2 = −(1− µ), y2 = 0)

antÐstoiqa. Oi jèseic twn prwteuìntwn swmtwn sto nèo sÔsthma suntetag-mènwn ja eÐnai:

ξ1 = x1 + 1− µ = µ+ 1− µ = 1 & η1 = y1 = 0 → P ′1(1, 0)

ξ2 = x2 + 1− µ = −(1− µ) + 1− µ = 0 & η2 = y2 = 0 → P ′2(0, 0)

ParathroÔme loipìn ìti to kèntro tou sust matoc suntetagmènwn èqei meta-ferjeÐ sto deÔtero prwteÔon s¸ma en¸ to pr¸to èqei metaferjeÐ se monadiaÐaapìstash apì to kèntro.

20

2.3.2 Metasqhmatismìc KlÐmakac

Sth sunèqeia jewroÔme ton paraktw metasqhmatismì klÐmakac:

ξ = ξ/µa ⇒ ξ = ξµa

η = η/µa ⇒ η = ηµa

En t¸ra stic sqèseic (2.33),(2.34) efarmìsoume ton parapnw metasqhmati-smì kai diairèsoume me µa èqoume:

ξ − 2η = ξ + (µ− 1)µ−a − µ−a (1− µ)(µaξ − 1)

r31

− ξµ

r32

(2.35)

η + 2ξ = η

(1− 1− µ

r31

− µ

r32

)(2.36)

Ekfrzoume t¸ra kai tic apostseic r1, r2 sto nèo sÔsthma:

r21 = (ξ − 1)2 + η2 ⇒ r2

1 = (ξµa − 1)2 + η2µ2a = ξ2µ2a − 2ξµa + 1 + η2µ2a ⇒

r1 = ((ξ2 + η2)µ2a − 2ξµa + 1)1/2 (2.37)

r22 = ξ2 + η2 ⇒ r2

2 = (ξ2 + η2)µ2a ⇒

r2 = ((ξ2 + η2)µ2a)1/2 (2.38)

An anaptÔxoume se seir Taylor th sunrthsh (x + 1)k/l gÔrw apì to mhdènkai krat soume mìno ìrouc pr¸thc txhc tou x èqoume ìti (x+1)k/l ∼= 1+ k

lx.

Efarmìzontac thn sqèsh aut gia thn sqèsh (2.37) gia to r1 tìte èqoume:

r1 = 1 +1

2(ξ2 + η2)µ2a − ξµa

ParathroÔme ìmwc epÐshc pwc an a > 0 tìte ta ìria twn apostsewn kaj¸cto µ teÐnei sto mhdèn ja eÐnai:

limµ→0

r1 = limµ→0

(1 +1

2(ξ2 + η2)µ2a − ξµa) = 1

limµ→0

r2 = limµ→0

(((ξ2 + η2)µ2a)1/2) = 0

21

Antikajist¸ntac t¸ra tic sqèseic (2.37),(2.38) stic exis¸seic kÐnhshc ja è-qoume:

ξ − 2η = ξ + (µ− 1)µ−a − µ−a (1− µ)(µaξ − 1)

((ξ2 + η2)µ2a − 2ξµa + 1)3/2︸︷︷︸Critical Terms

− ξµ1−3a

(ξ2 + η2)3/2

(2.39)

η + 2ξ = η

1− 1− µ((ξ2 + η2)µ2a − 2ξµa + 1)3/2︸︷︷︸

Critical Term

− µ1−3a

(ξ2 + η2)3/2

(2.40)

2.3.3 Exis¸seic tou kuklikoÔ probl matoc Hill

To epìmeno b ma eÐnai na proume thn oriak diadikasÐa µ → 0. Wstìso pa-rathroÔme ìti stic exis¸seic (2.39),(2.40) gia mhn katal xoume se tetrimmènhlÔsh ja prèpei na epilèxoume thn stajer a Ðsh me a = 1/3. Epilègoume dhla-d o metasqhmatismìc klÐmakac (µ1/3) na eÐnai thc txhc megèjouc thc aktÐnacHill gia to deÔtero s¸ma. H fusik shmasÐa aut c thc eklog c eÐnai ìti meautìn ton metasqhmatismì megenjÔnoume sthn perioq gÔrw apì ton plan thìpou to trÐto s¸ma ja kineÐtai upoqrewtik wc dorufìroc tou. Sthn exÐswsh(2.39) faÐnontai oi krÐsimoi ìroi gia thn oriak diadikasÐa. AutoÐ mporoÔn nagrafoÔn wc ex c:

(µ−1)µ−a−µ−a (1− µ)(µaξ − 1)

((ξ2 + η2)µ2a − 2ξµa + 1)3/2=

(µ− 1)

µa+

(µ− 1)(µaξ − 1)

µar31

=

=µ− 1

µa

(r3

1 − 1 + µaξ

r31

)=µ− 1

µa

(r3

1 − 1

r31

+µaξ

r31

)=

=µ− 1

µar3

1 − 1

r31

+(µ− 1)ξ

r31

(2.41)

An proume thn oriak diadikasÐa µ → 0 gia th sqèsh (2.41) to ìriogrfetai:

limµ→0

µ− 1

r31

· limµ→0

r31 − 1

µa+ lim

µ→0

(µ− 1)ξ

r31

Ac upologÐsoume t¸ra to kje ìrio qwrist:

22

• Gia to pr¸to ìrio èqoume:

limµ→0

µ− 1

r31

=limµ→0(µ− 1)

(limµ→0 r1)3=−1

1= −1

• Gia to deÔtero ìrio ja anaptÔxoume to r31 sÔmfwna me ton tÔpo pou

prokÔptei apì thn anptuxh kat Taylor thc sunrthshc (x + 1)k/l hìpoia gÔrw apì to mhdèn gÐnetai (x + 1)k/l ∼= 1 + k

lx. 'Etsi to r3

1 jaeÐnai:

r31 = 1 +

3

2(ξ2 + η2)µ2a − 3ξµa

Sunep¸c to ìrio ja eÐnai:

limµ→0

r31 − 1

µa= lim

µ→0

1 + 32(ξ2 + η2)µ2a − 3ξµa − 1

µa= −3ξ

• Gia to trÐto ìrio ja eÐnai:

limµ→0

(µ− 1)ξ

r31

= ξlimµ→0(µ− 1)

(limµ→0 r1)3= −ξ

H telik tim tou orÐou ja eÐnai:

limµ→0

µ− 1

r31

· limµ→0

r31 − 1

µa+ lim

µ→0

(µ− 1)ξ

r31

= (−1) · (−3ξ) + (−ξ) = 2ξ

An antikatast soume thn tim tou orÐou sth sqèsh (2.39) kai jèsoume epÐshca = 1/3 tìte paÐrnoume thn pr¸th exÐswsh kÐnhshc gia to kuklikì prìblhmaHill.

ξ − 2η = ξ + 2ξ − ξ

(ξ2 + η2)3/2

Sth sqèsh (2.40) faÐnetai epÐshc o krÐsimoc ìroc kaj¸c to µ → 0. JaupologÐsoume loipìn to ìrio tou:

limµ→0

1− µ((ξ2 + η2)µ2a − 2ξµa + 1)3/2

= limµ→0

1− µr3

1

= 1

Antikajist¸ntac thn tim tou orÐou kai jètontac a = 1/3 sthn (2.40) paÐr-noume kai th deÔterh exÐswsh gia to kuklikì prìblhma Hill:

η + 2ξ = η

(1− 1− 1

(ξ2 + η2)3/2

)23

Epomènwc to sÔsthma exis¸sewn kÐnhshc gia to kuklikì prìblhma Hill jaeÐnai:

ξ − 2η = 3ξ − ξ

(ξ2 + η2)3/2(2.42)

η + 2ξ = − η

(ξ2 + η2)3/2(2.43)

2.3.4 To olokl rwma tou Jacobi,oi kampÔlec mh-

denik c taqÔthtac kai ta shmeÐa isorropÐac

An jewr soume thn sunrthsh

Ω =3

2ξ2 +

1

ρ(2.44)

ìpou ρ = (ξ2 + η2)1/2 tìte oi exis¸seic kÐnhshc mporoÔn na grafoÔn me thmorf :

ξ − 2η =∂Ω

∂ξ(2.45)

η + 2ξ =∂Ω

∂η(2.46)

Pollaplasizontac th sqèsh (55)·ξ,th sqèsh (56)·η kai prosjètontac katmèlh èqoume:

ξξ − 2ηξ = ∂Ω∂ξξ

ηη + 2ξη = ∂Ω∂ηη

⇒

ξξ + ηη =∂Ω

∂ξξ +

∂Ω

∂ηη =

dΩ

dt(2.47)

Oloklhr¸nontac th sqèsh (39) wc proc to qrìno paÐrnoume thn paraktwsqèsh:

ξ2 + η2 = 2Ω− CHìpou CH mia stajer olokl rwshc. 'Omwc ξ2 + η2 = u2 eÐnai h taqÔthtatou trÐtou s¸matoc sto peristrefìmeno sÔsthma suntetagmènwn. Sunep¸cmporoÔme na gryoume:

u2 = 2Ω− CH

24

An t¸ra sth sqèsh (40) antikatast soume to Ω apì th sqèsh (54) aut grfetai:

CH = 3ξ2 +2

(ξ2 + η2)1/2− (ξ2 + η2) (2.48)

H pìsìthta CH = 2Ω−u2 eÐnai mia stajer thc kÐnhshc kai ètsi h sqèsh (42)onomzetai kai olokl rwma stajèra tou Jacobi gia thn prosèggish Hill.Apì thc sqèsh (42) u2 = 2Ω−CH prokÔptei pwc to dexÐ mèloc prèpei na eÐnaijetikì kai ètsi èqoume:

2Ω− CH ≥ 0

Sq ma 2.4: Oi kampÔlec mhdenik c taqÔthtac (ZVC) sto epÐpedo (ξ,η) kaioi epitreptèc perioqèc kÐnhshc(me leukì) gia timèc tou oloklhr¸matoc touJacobi: (a) CH = 6, (b) CH = 4.326749, (c) CH = 4, (d) CH = 1.5

H parapnw sqèsh mac dÐnei thn epitrept perioq kÐnhshc gia kpoiaorismènh tim tou CH . Mlista oi kampÔlec thc morf c

2Ω− CH = 0

25

lègontai kai kampÔlec mhdenik c taqÔthtac zero velocity curves (ZVC).Gia na broÔme ta shmeÐa isorropÐac sto kuklikì prìblhma Hill ja prèpei

stic sqèseic (55),(56) na jèsoume ξ, η, ξ, η = 0. Tìte ja eÐnai:

Ωξ = ξ(3− 1

ρ3) = 0 Ωη = − η

ρ3= 0

Apì tic dÔo autèc sqèseic prokÔptoun ta shmeÐa isorropÐac

L1(−3−1/3, 0) & L2(3−1/3, 0)

H tim tou oloklhr¸matoc tou Jacobi gia ta shmeÐa isorropÐac eÐnai

CH = 34/3 ∼= 4.326749

2.4 H qamiltonian tou kuklikoÔ probl -

matoc Hill

Mia llh endiafèrousa prosèggish sto kuklikì prìblhma Hill eÐnai ìti toolokl rwma tou Jacobi mporeÐ na proèljei apì aÔto tou periorismènou pro-bl matoc. 'Opwc eÐdame se prohgoÔmenh pargrafo to olokl rwma tou Jacobisto periorismèno prìblhma twn tri¸n swmtwn, an jèsoume ω = 1, eÐnai:

CJ = −x2 − η2 + x2 + y2 + 2

(1− µr1

+µ

r2

)ìpou r2

1 = (x− µ)2 + y2 kai r22 = (x+ 1− µ)2 + y2.

O metasqhmatismìc Hill apoteleÐtai ìpwc eÐdame kai parapnw apì miametafor tou sust matoc suntetagmènwn sto deÔtero s¸ma kai èna metasqh-matismì klÐmakac, dhlad eÐnai o ex c:

x = ξµ1/3 + µ− 1, y = ηµ1/3, x = ξµ1/3, y = ηµ1/3

An efarmìsoume to metasqhmatismì Hill sto olokl rwma tou Jacobi ja è-qoume:

CJ = −ξ2µ2/3 − η2µ2/3 + (ξµ1/3 + µ− 1)2 + ηµ2/3+

+2(1− µ)√

(ξµ1/3 − 1)2 + ηµ2/3+

2µ√(ξ2 + η)µ2/3

26

An anaptÔxoume t¸ra thn parapnw sqèsh se seir Taylor wc proc to µ1/3

ja proume:

CJ = 3 +

(−ξ2 − η2 + 3ξ2 +

2√ξ2 + η2

)· µ2/3 +

(−4 + 2ξ3 − 3ξη2

)· µ+

+

(2ξ4 − 6ξ2η2 +

3η4

4

)· µ4/3 +O

[µ5/3

]JewroÔme t¸ra ìti CH = (CJ − 3)µ−2/3 kai paÐrnoume to ìrio tou CH kaj¸cto µ→ 0 ìpote katal goume sthn paraktw sqèsh:

CH = −ξ2 − η2 + 3ξ2 +2√

ξ2 + η2(2.49)

ParathroÔme ìmwc ìti h parapnw sqèsh eÐnai Ðdia me th sqèsh (2.48), toolokl rwma tou Jacobi dhlad gia to kuklikì prìblhma Hill.

An tèloc jewr soume ìti h qamiltoninh tou kuklikoÔ probl matoc HilleÐnai HH = −CH/2 kai knoume ton kanonikì metasqhmatismì

q1 = ξ, , q2 = η, p1 = ξ − η, p2 = η + ξ

katal goume sthn ex c èkfrash gia thn qamiltonian 2

HH(q1, q2, p1, p2) =1

2

(p2

1 + p22

)− 1√

q21 + q2

2

+ q2p1 − q1p2 − q21 +

1

2q2

2 (2.50)

Me thn kanonik morf thc qamiltonian c mporoÔme plèon na doÔme kajarìti to kuklikì prìblhma Hill eÐnai to prìblhma twn dÔo swmtwn tou Kepler(1

2(p2

1 + p22) − 1/(q2

1 + q22)1/2), diataragmèno apì th dÔnamh Coriolis (q2p1 −

q1p2) kai thn epÐdrash tou prwteÔontoc pou brÐsketai se peirh apìstash(−q2

1 + 12q2

2). Sunep¸c to montèlo tou Hill eÐnai mia pr¸th prosèggish stoperiorismèno prìblhma twn tri¸n swmtwn katllhlh gia na melet sei kaneÐcth perioq gÔrw apì to deÔtero s¸ma.

2Ο χαμηλτονιανός φορμαλισμός παρουσιάζεται εδώ μόνο για λόγους πληρότητας. Στην

μελέτη του προβλήματος θα χρησιμοποιήσουμε τις εξισώσεις κίνησης (2.42,2.43).

27

Keflaio 3

Epifneiec Tom c Poincare

3.1 MejodologÐa

To kuklikì prìblhma Hill èqei dÔo bajmoÔc eleujerÐac prgma pou shmaÐnei ìtio q¸roc twn fsewn eÐnai tetradistatoc (dÔo jèseic ξ,η kai oi antÐstoiqecormèc ξ, η). Wstìso, ìpwc eÐdame prohgoumènwc, uprqei èna olokl rwmathc kÐnhshc, to olokl rwma tou Jacobi (CH) pou periorÐzei thn kÐnhsh. 'EtsimÐa apì tic metablhtèc, ac poÔme h η, mporeÐ na ekfrasteÐ san sunrthshtwn llwn tri¸n. Sunep¸c eÐnai arketì na anaparistoÔme tic troqièc tousust matoc se èna trisdistato q¸ro (ξ, η, ξ). An t¸ra jewr soume miaepifneia tom c, ac poÔme thn η = 0 kai proume thn akoloujÐa tom¸n miactroqic me thn epifneia tom c proc thn Ðdia kateÔjunsh (η > 0) pargoumemia apeikìnish sto epÐpedo (ξ, ξ) h opoÐa onomzetai tom Poincare.

Sq ma 3.1: H epifneia tom c Poincare

28

H epifneia tom c apoteleÐ èna apì ta shmantikìtera ergaleÐa gia th me-lèth autìnomwn dunamik¸n susthmtwn 2 bajm¸n eleujerÐac, kai autì giatÐmac prosfèrei thn epopteÐa tou q¸rou twn fsewn. EpÐshc den uprqei sh-mantik ap¸leia plhroforÐac kaj¸c oi shmantikìterec idiìthtec miac troqicantikatoptrÐzontai stic idiìthtec tou sunìlou twn shmeÐwn thc epifneiac to-m c. Mlista ta stajer shmeÐa thc apeikìnishc antistoiqoÔn se periodikèctroqièc tou sust matoc kai h eustjeia touc eÐnai Ðdia me thn eustjeia twnperiodik¸n troqi¸n stic opoÐec antistoiqoÔn.

Sq ma 3.2: Upologistik diadikasÐa eÔreshc epifneiac tom c Poincare

29

3.2 Epifneiec tom c tou probl matoc Hill

Skopìc mac eÐnai h dunamik anlush tou probl matoc. EÐdame dh ìti oiepifneiec tom c eÐnai èna polÔ kalì ergaleÐo gia th melèth aut . Sth su-nèqeia meletoÔme ton q¸ro ton fsewn qrhsimopoi¸ntac epifneiec tomèc giadiforec timèc thc enèrgeiac (CH) kai sqolizoume ta qarakthristik touc.

3.2.1 Epifneiec tom c gia CH ≥ 4.5

Sto sq ma (3.3) parajètoume thn epifneia tom c gia tim thc enèrgeia CH =5. ParathroÔme se aut n ìti ìla ta shmeÐa brÐskontai pnw se analloÐwteckampÔlec oi opoÐec periblloun ta dÔo stajer shmeÐa thc apeikìnishc. AutantistoiqoÔn se dÔo oikogèneiec periodik¸n troqi¸n tic f kai g. H kÐnhsh eÐnaikanonik se ìlo to q¸ro to fsewn prgma pou mporeÐ na exhghjeÐ eÔkola : odorufìroc paramènei polÔ kont ston plan th ètsi ¸ste se pr¸th prosèggishèqoume kÐnhsh sÔmfwna me to prìblhma twn dÔo swmtwn. Oi analloÐwteckampÔlec pou brÐskontai gÔrw apì thn oikogèneia g antistoiqoÔn se kanonikèctroqièc en¸ antÐstoiqa autèc gÔrw apì thn oikogèneia f se andromec. Giatimèc thc enèrgeiac CH > 5 oi epifneiec tom c paramènoun poiotik Ðdiec,sunep¸c ja strèyoume to endiafèron mac se mikrìterec timèc.

Sq ma 3.3: Epifneia tom c Poincare gia CH = 5

30

3.2.2 Epifneiec tom c gia 4.5 ≥ CH > 4.326

PhgaÐnontac loipìn proc mikrìterec timèc enèrgeiac endiafèronta prgmataarqÐzoun na sumbaÐnoun. Gia timèc thc enèrgeiac CH ≤ 4.5 oi periodikèc tro-qièc thc oikogèneiac g gÐnontai astajeÐc. Thn Ðdia stigm dÔo nèec eustajeÐcperiodikèc troqièc thc oikogèneiac g′ emfanÐzontai. ParathroÔme loipìn miatupik diakldwsh diqlac.

To stajerì shmeÐo g thc epifneiac tom c qarakthrÐzetai plèon wc uper-bolikì kai gÔrw tou emfanÐzetai ìpwc parathroÔme omoklinikì qoc. 'Omwc hdiataraq eÐnai emfan c se ìlo to dexÐ tm ma thc epifneiac tom c. Oi anal-loÐwtec kampÔlec èqoun d¸sei th jèsh touc se nhsÐdec eustjeiac oi opoÐecperibllontai apì qaotikèc perioqèc (sq ma 3.4). 'Oso mei¸noume thn enèr-geia to eÔroc twn qaotik¸n perioq¸n auxnetai. Tèloc parathroÔme ìti oianalloÐwtec kampÔlec suneqÐzoun na uprqoun gÔrw apì thn oikogèneia ftwn andromwn troqi¸n.

Sq ma 3.4: Epifneia tom c Poincare gia CH = 4.4

31

3.2.3 Epifneia tom c gia CH ≤ 4.326749

H tim thc enèrgeiac CH = 4.326749 eÐnai h tim thc enèrgeiac gia thn opoÐaoi kampÔlec mhdenik c taqÔthtac ftnoun sta shmeÐa isorropÐac L1, L2. Giamikrìterec timèc thc enèrgeiac h kentrik perioq tou q¸rou twn fsewn eÐnaiplèon anoiqt wc proc thn dieÔjunsh x kai o dorufìroc mporeÐ na diafÔgei(blèpe sq ma 2.4). Autì ja sumbeÐ sÐgoura gia tic qaotikèc troqièc diìtièqoun thn idiìthta na katalambnoun ìlo to diajèsimo q¸ro.


Sto Sq ma (3.5) blèpoume thn epifneia tom c akrib¸c prin anoÐxoun oikampÔlec mhdenik c taqÔthtac. H qaotik troqi èqei katalbei ìlo to du-natì diajèsimo q¸ro. ParathroÔme epÐshc tic nhsÐdec eustjeiac gÔrw apìtic eustajeÐc periodikèc troqièc thc oikogèneiac g′. Sta ìria tic teleutaÐacanalloÐwthc kampÔlhc gÔrw apì tic nhsÐdec eustjeiac parousizontai qao-tikèc troqièc oi opoÐec paramènoun se aut thn perioq gia arketì qrìno prinqajoÔn kai autèc sthn qaotik perioq .

32

Sq ma 3.6: Oi nhsÐdec eustjeiac gÔrw apì tic eustajeÐc periodikèc troqiècthc oikogèneiac g′.

Oi troqièc thc oikogèneiac g′ gÐnontai astajeÐc gia timèc enèrgeiac CH ≤4.27143. Sto sq ma 3.7 parathroÔme pwc h perioq sto dexÐ mèroc thc epif-neiac tom c èqei adeisei, prgma pou shmaÐnei pwc gia ìlec autèc tic arqikècsunj kec o dorufìroc diafeÔgei.


33

'Opwc parathroÔme apì tic parapnw epifneiec tom c oi analloÐwtec kam-pÔlec suneqÐzoun na uprqoun gÔrw apì thn oikogèneia andromwn troqi¸n fgia ìlo to fsma energei¸n pou melet same, mlista den uprqoun emfaneÐcqaotikèc perioqèc anmesa touc. Oi analloÐwtec autèc kampÔlec suneqÐzounna uprqoun gÔrw apì thn oikogèneia f gia osod pote mikrèc timèc thc e-nèrgeiac. H perioq pou katalambnoun surrikn¸netai kaj¸c elatt¸netai henèrgeia dÐnontac th jèsh thc se mia perioq diafug c gia to dorufìro.

Sta parapnw anafèrame pwc oi analloÐwtec kampÔlec suneqÐzoun na u-prqoun gia osod pote mikrèc timèc thc enèrgeiac. Autì den eÐnai apolÔtwcakribèc diìti gia dÔo timèc thc enèrgeiac oi kampÔlec autèc exafanÐzontai.Sugkekrimèna gia CH = 0.015388 kai CH = −1.411618 h oikogèneia f dia-staur¸netai me mia llh oikogèneia periodik¸n troqi¸n thn g3 me thn opoÐabrÐskontai se suntonismì 1/3. Sta dÔo shmeÐa aut lambnei q¸ra èna endia-fèron fainìmeno to opoÐo onomzetai “squeezing” (Sq ma 3.8).


34

Keflaio 4

Periodikèc Troqièc

Se autì to keflaio ja melet soume tic periodikèc troqièc tou sust matoc.Arqik parajètoume thn aparaÐthth jewrhtik jemelÐwsh kai thn mejodologÐapou qrhsimopoi same. Sthn sunèqeia parousizoume to dÐktuo twn apl¸nsummetrik¸n troqi¸n tou probl matoc kaj¸c kai meletoÔme thn eustjeiatouc. Merikèc oikogèneiec megalÔterhc pollaplìthtac sqolizontai epÐshcgiatÐ ja mac qreiastoÔn sto epìmeno keflaio.

4.1 MejodologÐa

4.1.1 Periodikèc troqièc

Oi exis¸seic kÐnhshc tou kuklikoÔ probl matoc Hill mporoÔn na grafoÔn wcex c :

ξ = F1(ξ, η, ξ, η)

η = F2(ξ, η, ξ, η)

An jewr soume t¸ra arqikèc sunj kec ξ0, η0, ξ0, η0 mia lÔsh mporeÐ na grafeÐwc

ξ(ξ0, η0, ξ0, η0; t)

η(ξ0, η0, ξ0, η0; t)

H lÔsh aut onomzetai periodik ìtan plhreÐ tic paraktw sqèseic:

ξ(ξ0, η0, ξ0, η0; t+ T ) = ξ(ξ0, η0, ξ0, η0; t)

η(ξ0, η0, ξ0, η0; t+ T ) = η(ξ0, η0, ξ0, η0; t)

35

ìpou T h perÐodoc thc troqic.

4.1.2 Summetrikèc periodikèc troqièc

ParathroÔme ìti oi diaforikèc exis¸seic tou probl matoc paramènoun anal-loÐwtec apì to metasqhmatismì

ξ → ξ, η → − η, t→ −t

Autì shmaÐnei pwc an hξ(t), η(t)

eÐnai lÔsh tou sust matoc tìte kai h

ξ(−t),−η(−t)

eÐnai epÐshc lÔsh tou sust matoc. Sunep¸c, an mia troqi xekin sei kjetaston xona ξ (ξ0 = 0) kai tèmnei xan ton xona ξ kjeta (ξ = 0) tìte h troqiaut eÐnai kleist , dhlad periodik . Oi arqikèc sunj kec gia summetrikècperiodikèc troqièc eÐnai thc morf c

ξ0, η0 = 0, ξ0 = 0, η0

Dhlad oi mh mhdenikèc arqikèc sunj kec eÐnai (ξ0, η0). Oi periodikèc troqiècìmwc genik den eÐnai apomonwmènec, an koun se oikogèneiec kat m koc twnopoÐwn h perÐodoc diafèrei. Mia oikogèneia summetrik¸n periodik¸n troqi¸nmporeÐ na anaparastajeÐ apì mia suneq kampÔlh sto epÐpedo (ξ0, η0). Gialìgouc eukolÐac, antÐ gia thn metablht η mporoÔme na upologÐsoume to o-lokl rwma tou Jacobi (CH) kai ètsi na parast soume t¸ra tic summetrikècperiodikèc troqièc sto epÐpedo (ξ0, CH).

4.1.3 Exis¸seic metabol¸n

Oi exis¸seic metabol¸n apoteloÔn to grammikopoihmèno sÔsthma twn exis¸-sewn thc kÐnhshc. Oi exis¸seic kÐnhshc tou probl matoc mporoÔn na graftoÔnsth morf :

ξ = pξ

η = pη

pξ = f1(ξ, η, pξ, pη)

pη = f2(ξ, η, pξ, pη)

36

JewroÔme t¸ra mia periodik troqi ~ξ(t). 'Estw t¸ra mia diataragmènh troqi

~ξ′(t) = ~ξ(t) + δ~ξ(t)

ìpou oi metabolèc δ~ξ(t) (δξ(t), δη(t), δpξ(t), δpη(t)) prokÔptoun wc lÔseic tougrammikopoihmènou sust matoc twn exis¸sewn kÐnhshc:

δξδηδpξδpη

=

0 0 1 00 0 0 1∂f1∂ξ

∂f1∂η

∂f1∂pξ

∂f1∂pη

∂f2∂ξ

∂f2∂η

∂f2∂pξ

∂f2∂pη

·

δξδηδpξδpη

Gia to kuklikì prìblhma Hill oi exis¸seic metabol¸n paÐrnoun th morf :

δξδηδpξδpη

=

0 0 1 00 0 0 1

3 + 2ξ2−η2

(ξ2+η2)5/23ξη

(ξ2+η2)5/20 2

3ξη(ξ2+η2)5/2

3 + 2η2−ξ2(ξ2+η2)5/2

−2 0

·

δξδηδpξδpη

H lÔsh tou sust matoc twn exis¸sewn metabol¸n mporeÐ na grafeÐ wc:

δ~ξ(t) = ∆(t)δ~ξ(0)

Kai gia qrìno Ðso me thn perÐodo thc periodik c troqic t = T :

δ~ξ(T ) = ∆(T )δ~ξ(0)

O pÐnakac ∆(T ) onomzetai monìdoromoc pÐnakac kai mporeÐ na upologisteÐupologistik oloklhr¸nontac tic exis¸seic metabol¸n gia qrìno Ðso me thnperÐodo thc periodik c troqic kai gia arqikèc apoklÐseic:

δ~ξ0 =

1000

δ~ξ0 =

0100

δ~ξ0 =

0010

δ~ξ0 =

0001

O monìdromoc pÐnakac eÐnai sumplektikìc sunep¸c oi idiotimèc tou eÐnai mi-gadikoÐ suzugeÐc arijmoÐ kai epÐshc sqhmatÐzoun zeugria antÐstrofwn. 'Etsiuprqoun 4 idiotimèc me thn idiìthta :

λ1 · λ2 = 1 λ3 · λ4 = 1 (4.1)

37

H orÐzousa tou monìdromou pÐnaka ja eÐnai :

det(∆(T )) = λ1 · λ2 · λ3 · λ4 = 1

H parapnw idiìthta mporeÐ na qrhsimopoihjeÐ gia ton èlegqo twn upologi-sm¸n mac.

4.1.4 Eustjeia periodik¸n troqi¸n

H eustjeia thc periodik c troqic exarttai apì tic idiotimèc tou monìdromoupÐnaka. 'Opwc eÐdame parapnw oi idiotimèc tou monìdromou pÐnaka ikanopoioÔnthn sqèsh (4.1). 'Omwc epeid oi exis¸seic metabol¸n pou epilÔoume anti-stoiqoÔn se mia periodik troqi h idiotim λ1 eÐnai monda kai telik ja eÐnaiλ1 = λ2 = 1. H eustjeia loipìn kajorÐzetai apì tic mh mhdenikèc idiotimècλ3, λ4. To Ðqnoc tou monìdromou pÐnaka ja eÐnai:

trace∆(T ) = λ1 + λ2 + λ3 + λ4

Kai an jèsoumeK = trace∆(T )− 2

tìte oi idiotimèc ja eÐnai rÐzec thc exÐswshc

λ2 +K · λ+ 1 = 0

H eustjeia t¸ra exarttai apì thn tim tou K to opoÐo onomzetai kai deÐkthceustjeiac. 'Eqoume loipìn tic akìloujec peript¸seic gia to K:

• |K| < 2 : oi idiotimèc λ3, λ4 eÐnai migadikèc suzugeÐc pnw ston monadiaÐokÔklo. Se aut thn perÐptwsh h periodik troqi eÐnai eustaj c.

• |K| = 2 : oi idiotimèc λ3, λ4 eÐnai Ðsec me th monda kai èqoume krÐsimheustjeia.

• |K| > 2 : oi idiotimèc λ3, λ4 eÐnai pragmatikèc kai èxw apì ton monadiaÐokÔklo. Se aut thn perÐptwsh h periodik troqi eÐnai astaj c.

38

Sq ma 4.1: Oi pijanoÐ sunduasmoÐ gia tic idiotimèc λ3, λ4 : i) Na brÐskontaipnw sto monadiaÐo kÔklo (eustjeia) ii,iii)Na brÐskontai sto ±1 (krÐsimheustjeia) iv,v) Na eÐnai pragmatikèc kai na brÐskontai ektìc tou monadiaÐoukÔklou (astjeia).

4.1.5 EÔresh periodik¸n troqi¸n

'Opwc anafèrame kai sto prohgoÔmeno keflaio oi periodikèc troqièc anti-stoiqoÔn se stajer shmeÐa thc tom c Poincare. JewroÔme loipìn èna shmeÐothc tom c P (ξ0, ξ0) kai upologÐzoume to epìmeno shmeÐo sthn epifneia tom c.Gia na eÐnai periodik h troqi ja prèpei P (ξ0, ξ0) = P ′(ξ(t∗), ξ(t∗)), ìpout∗ o qrìnoc pou qreisthke gia na persei h troqi apì thn tom . Ja prè-pei t¸ra na diorj¸soume tic arqikèc mac sunj kec ètsi ¸ste se qrìno t∗ naepanèljoume sto Ðdio shmeÐo.

ξ(t∗, ξ0 + ∆ξ, ξ0 + ∆ξ) = ξ0 + ∆ξ

ξ(t∗, ξ0 + ∆ξ, ξ0 + ∆ξ) = ξ0 + ∆ξ

Gia mikr ∆ξ,∆ξ mporoÔme na anaptÔxoume se anptugma Taylor to pr¸tomèloc opìte ja èqoume:

ξ(t∗, ξ0, ξ0) +∂ξ

∂ξ0

∆ξ +∂ξ

∂ξ0

∆ξ = ξ0 + ∆ξ

ξ(t∗, ξ0, ξ0) +∂ξ

∂ξ0

∆ξ +∂ξ

∂ξ0

∆ξ = ξ0 + ∆ξ

kai epilÔontac wc proc ∆ξ,∆ξ paÐrnoume:(∆ξ

∆ξ

)=

(∂ξ∂ξ0− 1 ∂ξ

∂ξ0∂ξ∂ξ0

∂ξ

∂ξ0− 1

)−1(ξ0 − ξξ0 − ξ

)(4.2)

To shmeÐo P (ξ0 + ∆ξ, ξ0 + ∆ξ) proseggÐzei to zhtoÔmeno shmeÐo. H para-pnw diadikasÐa epanalambnetai xekin¸ntac t¸ra apì to diorjwmèno shmeÐo.

39

H epanalambanìmenh diadikasÐa stamatei ìtan ikanopoihjeÐ h sqèsh

∆ = |ξ0 − ξ(t∗)|+ |ξ0 − ξ(t∗)| < AkrÐbeia Periodik c Troqic

Sq ma 4.2: Upologistik diadikasÐa eÔreshc periodik¸n troqi¸n.

H parapnw mèjodoc apoteleÐ mia Newton-Raphson diadikasÐa eÔreshc ri-z¸n. H mèjodoc sugklÐnei an xekin soume arkoÔntwc kont sthn periodik troqi. Oi merikèc pargwgoi pou emfanÐzontai sto sÔsthma (4.2) upologÐ-

40

zontai arijmhtik. Gia pardeigma h ∂ξ∂ξ0

ja eÐnai:

ξ0 −∆ξ0

olok rwsh mèqri t∗−→ ξ1

ξ0 + ∆ξ0

olok rwsh mèqri t∗−→ ξ2

→ ∂ξ

∂ξ0

≈ ∆ξ

2∆ξ0

=ξ2 − ξ1

2∆ξ0

Gia tic summetrikèc periodikèc troqièc (ξ = 0) eÐdame ìmwc pwc den eÐnaiapomonwmènec all sqhmatÐzoun oikogèneiec sto epÐpedo (ξ0, CH). Ac upojè-soume loipìn ìti br kame mia summetrik periodik troqi me ton algìrijmopou perigryame parapnw. Metabloume lÐgo thn enèrgeia kai san arqi-k sunj kh qrhsimopoioÔme thn troqi pou upologÐsame. 'Etsi mporoÔme nabroÔme mia geitonik troqi sthn arqik . SuneqÐzontac th diadikasÐa gia toapaitoÔmeno eÔroc tim¸n thc enèrgeiac ja proume olìklhrh thn oikogèneiaperiodik¸n troqi¸n.

An h periodik troqi antistoiqeÐ se èna k-periodikì shmeÐo thc tom cPoincare, tìte to k onomzetai pollaplìthta thc troqic. 'H enallaktiksÔmfwna me ton Stromgren (1935) k-periodik onomzoume mia troqi poutèmnei ton xona ξ 2 · k forèc. O upologismìc k-periodik¸n troqi¸n mporeÐna gÐnei me ton algìrijmo tou sq matoc (4.2) en efarmìsoume thn routÐnaTom Poincare k forèc.

4.2 DÐktuo summetrik¸n periodik¸n tro-

qi¸n

Apì th jewreÐa problèpetai pwc se tètoiou eÐdouc sust mata, ìpwc to kukli-kì prìblhma Hill, uprqoun apeÐrwc pollèc oikogèneiec periodik¸n troqi¸n.Gia ton lìgo autì ja arkestoÔme sth melèth twn oikogenei¸n apl-periodik¸nkai summetrik¸n periodik¸n troqi¸n tou sust matoc. EpÐshc ja melet soumekai merikèc oikogèneiec troqi¸n me megalÔterh pollaplìthta pou parousi-zoun idiaÐtero endiafèron.

4.2.1 Oikogèneiec a kai c

Oi oikogèneiec a kai c xekinoÔn apì ta shmeÐa isorropÐac Lagrange L1 kaiL2. MporoÔme na parathr soume pwc oi exis¸seic kÐnhshc paramènoun anal-loÐwtec apì to metasqhmatismì (ξ = −ξ,η = −η). 'Etsi gia kje periodik

41

troqi uprqei kai h summetrik thc wc proc thn arq tou sust matoc sun-tetagmènwn. Oi troqièc thc oikogèneiac a eÐnai oi summetrikèc twn troqi¸nthc oikogèneiac c. 'Olec oi periodikèc troqièc twn dÔo oikogenei¸n eÐnai aplperiodikèc kai brèjhke ìti eÐnai astajeÐc.

Sq ma 4.3: Oikogèneiec summetrik¸n periodik¸n troqi¸n tou kuklikoÔ pro-bl matoc Hill.

4.2.2 Oikogèneia f

H oikogèneia f xekinei apì tic troqièc twn andromwn dorufìrwn gia todeÔtero prwteÔon. Oi troqièc thc oikogèneiac f eÐnai apl periodikèc kai eÐnaisummetrikèc wc proc thn arq twn axìnwn. 'Etsi h summetrik thc oikogèneiacf eÐnai h Ðdia h f . Apì th melèth thc eustjeiac thc oikogèneiac prokÔpteiìti ìlec oi andromec periodikèc troqièc thc oikogèneiac f eÐnai eustajeÐc.

42

4.2.3 Oikogèneia g

H oikogèneia g anaparist troqièc orjìdromwn dorufìrwn gia ton plan th.Ta mèlh thc eÐnai apl periodikèc troqièc summetrikèc wc proc thn arq twnaxìnwn. Oi periodikèc troqièc thc oikogèneiac g eÐnai eustajeÐc apì meglec ti-mèc thc enèrgeiac mèqri kai thn krÐsimh troqi (ξ0 = 0.283350, CH = 4.49999).Gia mikrìterec timèc thc enèrgeiac ta mèlh thc oikogèneiac eÐnai plèon astaj .

Sq ma 4.4: Merikèc troqièc thc oikogèneiac f (arister) kai thc oikogèneiacg (dexi) sto q¸ro morf c. Me kìkkino qr¸ma faÐnetai h tim thc enèrgeiacCH gia kje troqi.

4.2.4 Oikogèneia g′

H oikogèneia g′ diakladÐzetai apì thn krÐsimh troqi thc oikogèneiac g (ξ0 =0.283350, CH = 4.49999). Se aut thn troqi èqoume mia tupik diakldwshdiqlac. H oikogèneia g′ apoteleÐtai apì dÔo kldouc pou perièqoun aplperiodikèc troqièc pou eÐnai summetrikèc metaxÔ touc wc proc thn arq twnaxìnwn. Oi troqièc autèc eÐnai eustajeÐc gia timèc enèrgeiac mèqri kai CH =4.27143. Gia mikrìterec timèc thc enèrgeiac kai oi dÔo kldoi thc oikogèneiacgÐnontai astajeÐc.

43

Sq ma 4.5: Merikèc troqièc thc oikogèneiac g′ sto q¸ro morf c. Me kìkkinoqr¸ma faÐnetai h tim thc enèrgeiac CH gia kje troqi.

4.2.5 Oikogèneiec g3 kai Hg

H oikogèneia g3 apoteleÐtai apì tripl periodikèc summetrikèc troqièc. 'Olecoi troqièc thc oikogèneiac eÐnai astajeÐc. H oikogèneia diastaur¸netai dÔo fo-rèc me thn oikogèneia f se timèc enèrgeiac CH = 0.015388 kai CH = 1.411618.Se aut ta shmeÐa oi troqièc thc oikogèneiac g3 perigrfontai wc troqiècthc oikogèneiac f grammènec treic forèc kai emfanÐzetai epÐshc to fainìmeno“squeezing” (Sq ma 3.8).

Sq ma 4.6: Merikèc troqièc thc oikogèneiac g3 sto q¸ro morf c. Me kìkkinoqr¸ma faÐnetai h tim thc enèrgeiac CH gia kje troqi.

44

Sq ma 4.7: Merikèc troqièc thc oikogèneiacHg sto q¸ro morf c. Me kìkkinoqr¸ma faÐnetai h tim thc enèrgeiac CH gia kje troqi.

H oikogèneia Hg apoteleÐtai apì tetrapl periodikèc summetrikèc troqièc.H oikogèneia diastaur¸netai me thn oikogèneia f gia tim thc enèrgeiac CH =0.823630. Se autì to shmeÐo h troqi thc oikogèneiac Hg perigrfetai wcmia troqi thc oikogèneiac f grammènh tèsseric forèc. To endiafèron me thnoikogèneia Hg eÐnai ìti perièqei periodikèc troqièc oi opoÐec eÐnai eustajeÐc.

45

Mèroc II

To Elleiptikì Prìblhma Hill

46

Keflaio 5

Exis¸seic KÐnhshc

Sthn pargrafo aut ja doÔme pwc mporeÐ kaneÐc na katal xei stic exis¸-seic kÐnhshc tou elleiptikoÔ probl matoc Hill xekin¸ntac apì tic exis¸seictou elleiptikoÔ periorismènou probl matoc twn tri¸n swmtwn. H diadikasÐaeÐnai anlogh me aut pou akolouj same kai sthn perÐptwsh tou kuklikoÔprobl matoc Hill.

5.1 Genikì prìblhma twn tri¸n swmtwn

Ja xekin soume apì to genikì prìblhma twn tri¸n swmtwn. 'Estw loipìnta trÐa s¸mata S1,S2,S3 me mzec m1,m2,m3 antÐstoiqa. En jewr sou-me to peristrefìmeno sÔsthma suntetagmènwn me kèntro to barÔkentro twnswmtwn S1,S2 tìte h Lagkranzian tou sust matoc ja eÐnai:

L =m1

2 · (1− µ)(x2

1 + x21θ

2)+

+m3

2 ·M(m1 +m2)[x2

3 + y23 + (x2

3 + y23)θ2 + 2θ(x3y3 − y3x3)]+

+G(1− µ)m1m2

x1

+Gm1m3[

(x3 − x1)2 + y23

]1/2 +Gm2m3[(

x3 + m1

m2x1

)2

+ y23

]1/2(5.1)

ìpou µ = m1/(m1 +m2)1,M = m1 +m2 +m3 kai G h stajer thc pagkìsmiacèlxhc.

1η γνωστή από το κυκλικό πρόβλημα παράμετρος

47

An jèsoume t¸ra G = 1 kai M = 1, oi diaforikèc exis¸seic pou prokÔptounapì thn (5.1) eÐnai:

x1 − x1θ2 + (1− µ)3 (1−m3)

1

x21

− (1− µ)m3x3 − x1[

(x3 − x1)2 + y23

]1/2 +

+(1− µ)m3

x3 + µ1−µx1[

(x3 − µx1)2 + y23

]3/2 = 0 (5.2)

x3−2y3θ−y3θ−x3θ2+µ

x3 − x1[(x3 − x1)2 + y2

3

]3/2 +(1−µ)x3 + µ

1−µx1[(x3 − µx1)2 + y2

3

]3/2 = 0

(5.3)

y3+2x3θ+x3θ−y3θ2+µ

y3[(x3 − x1)2 + y2

3

]3/2 +(1−µ)y3[

(x3 − µx1)2 + y23

]3/2 = 0

(5.4)

5.2 Periorismèno elleiptikì prìblhma

twn 3 swmtwn

Apì to prìblhma twn dÔo swmtwn gia ta dÔo prwteÔonta gnwrÐzoume pwcisqÔei h sqèsh:

θ =Pθµ

1−µx21

(5.5)

H stroform tou sust matoc diathreÐtai stajer . Gia thn upologÐsoumemporoÔme na jewr soume ta prwteÔonta me arqikèc jèseic pnw ston xonax kai arqikèc taqÔthtec kjetec. EpÐshc jewroÔme θ(0) = ω = 1 kai x1(0) =x10 kai h sunolik stroform Pθ tou sust matoc ja eÐnai:

Pθ = P1 + P2 = µx10u10 + (1− µ)x20u20 (5.6)

Oi arqikèc taqÔthtec ja eÐnai:

u10 = ω ·R = ωx10 = x10 u20 = ω ·R = ωx20 = x20

'Omwc epÐshc ja eÐnai:

µx10 = (1− µ)x20 ⇒ x20 =µ

1− µx10

48

Kai h sunolik stroform eÐnai:

Pθ = µx210 + (1− µ)

(µ

1− µ

)2

x210 =

µ

1− µx2

10 (5.7)

Sq ma 5.1: Arqikèc sunj kec twn prwteuìntwn gia ton upologismì thc stro-form c Pθ

Antikajist¸ntac thn stroform apì th (5.7) sthn sqèsh (5.5) ja èqoume:

θ =x2

10

x21

(5.8)

EpÐshc ja eÐnai :

θ = − 2Pθµ

1−µx31

x1 = −2x210

x31

x1 (5.9)

An t¸ra jewr soume ìti to trÐto s¸ma èqei amelhtèa mza se sqèsh me taprwteÔonta, stic exis¸seic kÐnhshc (5.2,5.3,5.4) mporoÔme na jèsoume m3 = 0kai antikajist¸ntac ta θ, θ apì tic sqèseic (5.8,5.9) antÐstoiqa katal goumesto paraktw sÔsthma exis¸sewn gia to elleiptikì periorismèno prìblhmatwn tri¸n swmtwn:

x1 − x410x−31 = − (1− µ)3 x−2

1 (5.10)

x3 − 2x210x−21 y3 − x4

10x−41 x3 + 2x2

10x−31 y3x1 =

µ(x1 − x3)

ρ313

− µx1 + (1− µ)x3

ρ323

(5.11)

y3 + 2x210x−21 x3 − x4

10x−41 y3 + 2x2

10x−31 x3x1 = −µy3

ρ313

− (1− µ)y3

ρ323

(5.12)

ìpou stic parapnw sqèseic eÐnai

ρ213 = (x3 − x1)2 + y2

3 ρ223 = (x3 − µx1)2 + y2

3

49

5.3 To Elleiptikì Prìblhma Hill

Gia na katal xoume stic exis¸seic kÐnhshc tou elleiptikoÔ probl matoc Hillja akolouj soume thn Ðdia diadikasÐa pou qrhsimopoi same kai sto kuklikìprìblhma. Ja efarmìsoume, dhlad , to metasqhmatismì Hill (x3 = x1 +µ1/3ξkai y3 = µ1/3η) stic exis¸seic kÐnhshc tou elleiptikoÔ probl matoc twn tri¸nswmtwn kai met ja proume thn oriak diadikasÐa µ→ 0. H diadikasÐa aut ja epanalhfjeÐ gia kajemi apì tic exis¸seic kÐnhshc (5.10),(5.11),(5.12).

Sqèsh (5.10)

To ìrio thc sqèshc (5.10) kaj¸c to µ→ 0 eÐnai

x1 − x410x−31 + x−2

1 = 0 (5.13)

Sqèsh (5.11)

Xekinme loipìn apì th sqèsh (5.11)

x3 − 2x210x−21 y3 − x4

10x−41 x3 + 2x2

10x−31 y3x1 =

µ(x1 − x3)

ρ313

− µx1 + (1− µ)x3

ρ323

Ja efarmìsoume to metasqhmatismì Hill,(x3 = x1 + µ1/3ξ kai y3 = µ1/3η)

x1 + µ1/3ξ − 2x210x−21 µ1/3η − x4

10x−41 (x1 + µ1/3ξ) + 2x2

10x−31 µ1/3ηx1 =

=µ(x1 − x1 − µ1/3ξ)

ρ313

− µx1 + (1− µ)(x1 + µ1/3ξ)

ρ323

(5.14)

An antikatast soume t¸ra to x1 apì th sqèsh (5.10) kai diairèsoume thparapnw sqèsh me µ1/3 ja èqoume:

−(1− µ)3

x21µ

1/3︸︷︷︸Crit. 1

+ξ−2x210x−21 η−x4

10x−41 ξ+2x2

10x−31 x1η = − µξ

ρ313︸︷︷︸

Crit. 2

− x1

ρ323µ

1/3︸︷︷︸Crit. 3

− (1− µ)ξ

ρ323︸︷︷︸

Crit. 4(5.15)

'Omwc epÐshc ja eÐnai:

ρ213 =

(x1 + µ1/3ξ − x1

)2+ η2µ2/3 = µ2/3

(ξ2 + η2

)(5.16)

ρ223 =

(µ1/3ξ + (1− µ)x1

)2+ µ2/3η2 =

50

µ2/3(ξ2 + η2) + 2µ1/3(1− µ)x1ξ + (1− µ)2 x21 (5.17)

ParathroÔme ìtilimµ→0

ρ323 = x3

1

Sth sqèsh (5.15) èqoun shmeiwjeÐ oi krÐsimoi gia thn oriak diadikasÐaµ→ 0. Sth sunèqeia ja upologÐsoume ta ìria twn krÐsimwn ìrwn.

• Gia ton krÐsimo ìro 2 èqoume:

− µξρ3

13

= − µξ

µ (ξ2 + η2)3/2= − ξ

ρ3

ìpou ρ2 = ξ2 + η2

• Gia ton krÐsimo ìro 4 èqoume:

limµ→0−(1− µ)ξ

ρ323

= − ξ

x31

• Tèloc gia touc krÐsimouc ìrouc 1 kai 3 ja eÐnai:

−(1− µ)3

x21µ

1/3+

x1

ρ323µ

1/3=x3

1 − (1− µ)3 ρ323

µ1/3ρ323x

21

(5.18)

An anaptÔxoume to ρ323 sÔmfwna me to anptugma Taylor

(x+ a)3/2 ≈ a3/2 +3

2a1/2x

jètontac x = µ2/3(ξ2 + η2) + 2µ1/3(1 − µ)x1ξ kai a = (1− µ)2 x21 ja

eÐnai:

ρ323 = (µ2/3(ξ2 + η2) + 2µ1/3(1− µ)x1ξ + (1− µ)2 x2

1)3/2 ≈

≈ (1− µ)3 x31 +

3

2(1− µ)x1µ

2/3(ξ2 + η2) + 3(1− µ)2x21µ

1/3ξ (5.19)

An antikatast soume to anptugma tou ρ323 sth sqèsh (5.18) ja èqoume

tou paraktw ìrouc

limµ→0

x31 − (1− µ)6x3

1

µ1/3ρ323x

21

= limµ→0

x31(1− (1− µ)6)

µ1/3ρ323x

21

=

51

limµ→0

x1µ(−6 + 15µ− 20µ2 + 15µ3 − 6µ4 + µ5)

µ1/3ρ23

= 0

limµ→0

−32(1− µ)4x1µ

2/3(ξ2 + η2)

µ1/3ρ323x

21

= 0

limµ→0

−3(1− µ)5x21µ

1/3ξ

µ1/3ρ323x

21

= limµ→0

−3(1− µ)5ξ

ρ323

= −3ξ

x31

Sunep¸c ja eÐnai:

limµ→0

x31 − (1− µ)3 ρ3

23

µ1/3ρ323x

21

= −3ξ

x31

(5.20)

Antikajist¸ntac ta ìria twn krÐsimwn ìrwn sthn sqèsh (5.15) paÐrnoume thnparaktw exÐswsh kÐnhshc gia to elleiptikì prìblhma Hill.

ξ − 2x210x−21 η + ξ(ρ−3 − 2x−3

1 − x−410 x

41) + 2x2

10x−31 x1η = 0 (5.21)

Sqèsh (5.12)

Xekin¸ntac apì th sqèsh:

y3 + 2x210x−21 x3 − x4

10x−41 y3 + 2x2

10x−31 x3x1 = −µy3

ρ313

− (1− µ)y3

ρ323

Efarmìzoume kai pli to metasqhmatismì Hill x3 = x1 + µ1/3ξ kai y3 = µ1/3η

µ1/3η + 2x210x−21 (x1 + µ1/3ξ)− x4

10x−41 µ1/3η − 2x2

10x−31 x1(x1 + µ1/3ξ) =

−µµ1/3η

ρ313

− (1− µ)µ1/3η

ρ323

(5.22)

Sthn parapnw sqèsh ekteloÔme tou arijmhtikoÔ upologismoÔc kai diairoÔmeme µ1/3 ètsi ¸ste katal goume sth sqèsh:

η + 2x210x−21 ξ − x4

10x−41 η − 2x2

10x−31 x1ξ = −µη

ρ313︸︷︷︸

Crit. 1

− (1− µ)η

ρ323︸︷︷︸

Crit. 2

(5.23)

Oi krÐsimoi ìroi gia thn oriak diadikasÐa µ→ 0 èqoun shmeiwjeÐ sthn para-pnw exÐswsh. Ja upologÐsoume touc ìrouc autoÔc.

52

• O pr¸toc krÐsimoc ìroc ja eÐnai:

−µηρ3

13

= − µη

µ (ξ2 + η2)3/2= − η

ρ3

ìpou ρ2 = ξ2 + η2

• O deÔteroc krÐsimoc ìroc ja eÐnai:

limµ→0−(1− µ)η

ρ323

= − η

x31

An antikatast soume sth sqèsh (5.23) ja èqoume kai thn teleutaÐa exÐswshkÐnhshc tou elleiptikoÔ probl matoc Hill

η + 2x210x−21 ξ + η(ρ−3 + x−3

1 − x410x−41 )− 2x2

10x−31 x1ξ = 0 (5.24)

5.3.1 Exis¸seic KÐnhshc

'Eqontac efarmìsei loipìn mia diadikasÐa parìmoia me aut pou efarmìsamekai gia to kuklikì prìblhma Hill katal goume sto sÔsthma twn diaforik¸nexis¸sewn kÐnhshc gia to elleiptikì prìblhma Hill:

x1 − x410x−31 + x−2

1 = 0 (5.25)

ξ − 2x210x−21 η + ξ(ρ−3 − 2x−3

1 − x−410 x

41) + 2x2

10x−31 x1η = 0 (5.26)

η + 2x210x−21 ξ + η(ρ−3 + x−3

1 − x410x−41 )− 2x2

10x−31 x1ξ = 0 (5.27)

ìpou ρ2 = ξ2 + η2.Stic exis¸seic kÐnhshc tou elleiptikoÔ probl matoc Hill mporoÔme na ka-

tal xoume kai apì thn qrono-exart¸menh sunrthsh qmilton:

HeHill =1

2(p2ξ + p2

η)−1√

ξ2 + η2+ x2

10x−21 (ηpξ − ξpη)− x−3

1 (ξ2 − η2

2) (5.28)

ìpou x1 eÐnai mia periodik sunrthsh tou qrìnou , lÔsh thc diaforik c exÐ-swshc (5.25) kai

pξ = ξ − x210x−21 η, pη = η + x2

10x−21 ξ (5.29)

53

H exÐswsh (5.25) perigrfei thn elleiptik kÐnhsh tou s¸matoc S1 meekkentrìthta e kai perÐodo T , ta opoÐa sundèontai me thn arqik sunj kh x10

me tic sqèseic:x3

10 = 1 + e (5.30)

T = 2π

[1 + e

(1− e)3

]1/2

(5.31)

Oi dÔo parapnw sqèseic isqÔoun ìtan gia t = 0 ta dÔo prwteÔonta s¸matabrÐskontai sto perÐkentro. Oi antÐstoiqec sqèseic gia thn perÐptwsh tou a-pìkentrou brÐskontai an antikatast soume to e = −e. Oi dÔo diaforetikècpeript¸seic, dhlad to prwteÔon s¸ma S1 na xekinei apì to perÐkentro apì to apìkentro, odhgoÔn se diaforetik sumperifor tou sust matoc. Giato lìgo autì ja qrhsimopoioÔme thn ekkentrìthta e sto disthma (-1,1), je-wr¸ntac pwc oi arnhtikèc timèc anafèrontai sthn perÐptwsh tou apìkentrou.

54

Keflaio 6

Periodikèc Troqièc

Sthn prohgoÔmenh pargrafo eÐdame pwc mporeÐ kaneÐc na katal xei stic exi-s¸seic kÐnhshc tou elleiptikoÔ probl matoc Hill. Se aut thn pargrafo jaarqÐsoume thn melèth tou probl matoc, xekin¸ntac apì tic periodikèc troqièctou sust matoc. 'Opwc eÐdame kai sthn perÐptwsh tou kuklikoÔ probl ma-toc oi periodikèc troqièc eÐnai idiaÐtera shmantikèc gia polloÔc lìgouc. Seìla ta montèla dunamik¸n susthmtwn h gn¸sh thc jèshc kai thc eustjeiactwn periodik¸n troqi¸n mac dÐnoun shmantikèc plhroforÐec gia thn dunamik tou sust matoc. GÔrw apì eustajeÐc periodikèc troqièc uprqoun eustajeÐcperioqèc sto q¸ro twn fsewn, en¸ gÔrw apì astajeÐc periodikèc troqièc è-qoume thn emfnish qaotik c kÐnhshc. Oi periodikèc troqièc loipìn apoteloÔnkat kpoio trìpo to skeletì tou q¸rou twn fsewn kai h melèth touc jamac bohj sei na katalboume kalÔtera to montèlo mac.

6.1 MejodologÐa

An stic exis¸seic kÐnhshc tou elleiptikoÔ probl matoc Hill jèsoume x10 =1, tìte apì tic sqèseic (5.30,5.31) prokÔptei ìti h kÐnhsh tou prwteÔontocs¸matoc S1 ja eÐnai kuklik (e = 0) me perÐodo T = 2kπ. Mlista anjèsoume stic exis¸seic kÐnhshc tou elleiptikoÔ probl matoc x1 = x10 = 1tìte autèc paÐrnoun thn morf

ξ − 2η +ξ

ρ3− 3ξ = 0 (6.1)

η + 2ξ +η

ρ3= 0 (6.2)

55

pou den eÐnai llec apì tic gnwstèc mac dh apì to keflaio 2, exis¸seic toukuklikoÔ probl matoc Hill.

'Eqontac melet sei to kuklikì prìblhma Hill (keflaia 3,4) èqoume upolo-gÐsei èna sÔnolo apì oikogèneiec periodik¸n troqi¸n oi opoÐec sthn perÐptwshpou x10 = 1 e = 0 apoteloÔn kai periodikèc troqièc tou elleiptikoÔ pro-bl matoc Hill. To endiafèron ìmwc eÐnai na prosdiorÐsoume poiec apì ìlec ticperiodikèc troqièc miac oikogèneiac tou kuklikoÔ probl matoc suneqÐzoun nauprqoun gia e 6= 0, apoteloÔn dhlad arqikèc troqièc gia ton entopismì nèwnoikogenei¸n periodik¸n troqi¸n sto elleiptikì prìblhma Hill. To parapnwprìblhma eÐnai gnwstì wc sunèqish periodik¸n troqi¸n kai gia thn perÐptwshthc sunèqishc apì to kuklikì sto elleiptikì prìblhma Hill èqei deiqjeÐ ìtisto elleiptikì prìblhma suneqÐzontai oi troqièc tou kuklikoÔ probl matoctwn opoÐwn h perÐodoc plhroÐ thn sunj kh:

Tc = 2kπ Tc = 2p

qπ ìpou k, p, q : akèraioi (6.3)

Sunep¸c troqièc tou kuklikoÔ probl matoc Hill me perÐodo 2kπ apoteloÔnshmeÐa tom c oikogenei¸n tou kuklikoÔ kai tou elleiptikoÔ probl matoc kaimporoÔn na suneqistoÔn wc proc thn ekkentrìthta dÐnontac mac oikogèneiecperiodik¸n troqi¸n sto elleiptikì prìblhma Hill. Oi oikogèneiec periodik¸ntroqi¸n mporoÔn na parastajoÔn se èna trisdistato q¸ro (ξ0, η0, e) sansuneqeÐc kampÔlec. Sthn bibliografÐa èqoun brejeÐ kai parousizontai 3 oi-kogèneiec periodik¸n troqi¸n sto elleiptikì prìblhma Hill, ìmwc ìlec eÐnaiastajeÐc.

6.2 Eustjeia Periodik¸n Troqi¸n

JewroÔme t¸ra mia periodik troqi ~ξ(t) tou elleiptikoÔ probl matoc Hill

kai mia mikr diataraq δ~ξ, ètsi ¸ste:

~ξ′(t) = ~ξ(t) + δ~ξ(t)

ìpou oi metabolèc δ~ξ(t) (δξ(t), δη(t), δpξ(t), δpη(t)) prokÔptoun wc lÔseic tougrammikopoihmènou sust matoc twn exis¸sewn kÐnhshc:

δξδηδpξδpη

=

0 0 1 00 0 0 1∂f1∂ξ

∂f1∂η

∂f1∂pξ

∂f1∂pη

∂f2∂ξ

∂f2∂η

∂f2∂pξ

∂f2∂pη

·

δξδηδpξδpη

56

Apì thn parapnw sqèsh katal goume sto sÔsthma twn diaforik¸n exis¸se-wn metabol¸n:

δξ = δpξ (6.4)

δη = δpη (6.5)

δpξ =

(2ξ2 − η2

(ξ2 + η2)5/2+ 2x−3

1 + x410x−41

)·δξ+

(3ξη

(ξ2 + η2)5/2− 2x2

10x−31 x1

)·δη

+2x210x−21 · δpη (6.6)

δpη =

(3ξη

(ξ2 + η2)5/2+ 2x2

10x−31 ξ1

)· δξ +

(2η2 − ξ2

(ξ2 + η2)5/2− x−3

1 + x410x−41

)· δη

−2x210x−21 · δpξ (6.7)

H lÔsh tou sust matoc twn exis¸sewn metabol¸n mporeÐ na grafeÐ wc:

δ~ξ(t) = ∆(t)δ~ξ(0)

Kai gia qrìno Ðso me thn perÐodo thc periodik c troqic t = T :

δ~ξ(T ) = ∆(T )δ~ξ(0)

O pÐnakac ∆(T ) onomzetai monìdromoc pÐnakac kai ìpwc eÐdame h kai sthn pa-rgrafo (4.1.4) h eustjeia thc periodik c troqic exarttai apì tic idiotimèctou. O monìdromoc pÐnakac eÐnai sumplektikìc kai oi idiotimèc tou sqhmatÐ-zoun zeÔgh antÐstrofwn λ, 1/λ, µ, 1/µ. H periodik troqi eÐnai eustaj c enkai mìno en kai oi tèsseric idiotimèc brÐskontai epnw sto migadikì kÔklo.OrÐzoume touc suntelestèc eustjeiac a1, a2 apì tic sqèseic

a1 = −(k1 + k2) a2 = 2 + k1k2

ìpou k1, k2 oi deÐktec eustjeiac:

k1 = λ+ 1/λ k2 = µ+ 1/µ

H periodik troqi eÐnai eustaj c ìtan isqÔoun oi paraktw anisìthtec

a21 − 4a2 + 8 > 0, k2

1 < 4, k22 < 4

57

Sq ma 6.1: Oi ept perioqèc eustjeiac tou Broucke sto epÐpedo (a1, a2). Hperioq 1 eÐnai eustaj c,sthn perioq 2 emfanÐzetai migadik astjeia, en¸ oiperioqèc 3 èwc 7 eÐnai astajeÐc.

58

O Broucke(1969) èdeixe pwc uprqoun 7 diaforetikoÐ tÔpoi eustjeiacpou antistoiqoÔn se 7 diaforetikèc perioqèc sto epÐpedo (a1, a2) oi opoÐecorÐzontai apì thc eujeÐec grammèc

a2 = 2a1 − 2

a2 = −2a1 − 2

kai thn parabol a2

1 − 4a2 + 8 = 0

Upologistik oi suntelestèc eustjeiac mporoÔn na upologistoÔn apì tastoiqeÐa tou monìdromou pÐnaka ∆(T ) apì tic paraktw sqèseic

a1 = −trace∆(T )

a2 = b12 + b13 + b14 + b23 + b24 + b34

ìpou

bij =

∣∣∣∣ ∆ii ∆ij

∆ji ∆jj

∣∣∣∣6.3 Nèec Oikogèneiec Periodik¸n Troqi¸n

tou ElleiptikoÔ Probl matoc Hill

Stic prohgoÔmenec paragrfouc parajèsame tic basikèc gn¸seic gia thn eÔ-resh periodik¸n troqi¸n kaj¸c kai gia ton prosdiorismì thc eustjeic touc.Mèqri stigm c sthn bibliografÐa mporeÐ kaneÐc na brei treic oikogèneiec pe-riodik¸n troqi¸n tou elleiptikoÔ probl matoc Hill, tic opoÐec parousizoumesunoptik ston paraktw pÐnaka.

PÐnakac 1: Oi arqikèc sunj kec twn periodik¸n troqi¸n tou kuklikoÔprobl matoc apì tic opoÐec suneqÐzontai oi gnwstèc apì th bibliografÐa

oikogèneiec tou elleiptikoÔ probl matoc1

Oikogèneia Nèa OnomasÐa ξ0 η0 Suntonismìc

a a1/1e 0.066501 5.526301 1/1

g′ g′1/1e 0.142591 3.650216 1/1

f2 f1/2e -0.797043 2.196153 1/2

1Για τη νέα ονομασία βλέπε στην επόμενη παράγραφο

59

'Eqoume loipìn dÔo oikogèneiec pou suneqÐzontai apì astajeÐc periodikèctroqièc twn oikogenei¸n a kai g′ me perÐodo 2π kai mia oikogèneia pou sune-qÐzetai apì eustaj periodik troqi thc oikogèneiac f me perÐodo π. 'Opwcanafèrame kai oi treic parapnw oikogèneiec periodik¸n troqi¸n den perièqouneustajeÐc periodikèc troqièc. 'Enac apì touc skopoÔc thc ergasÐac aut c eÐ-nai na elègxoume an uprqoun eustajeÐc periodikèc troqièc sto elleiptikìprìblhma Hill. Qrhsimopoi¸ntac tic eustajeÐc oikogèneiec tou kuklikoÔ pro-bl matoc prospaj same na upologÐsoume nèec oikogèneiec tou elleiptikoÔprobl matoc kai melet same thn eustjeia touc.

6.3.1 Oikogèneia f

'Olec oi periodikèc troqièc thc oikogèneiac f tou kuklikoÔ probl matoc eÐnaieustajeÐc all kamÐa troqi den èqei perÐodo pou eÐnai akèraio pollaplsiotou 2π. Gia to lìgo autì qrhsimopoioÔme gia tic troqièc thc th mèjodo twnpollapl¸n diaklad¸sewn. JewroÔme dhlad mia periodik troqi grammènhparapnw apì mÐa for ètsi ¸ste h perÐodoc thc na eÐnai Ðsh me 2π, kpoiopollaplsio tou 2π. 'Etsi mporoÔme na suneqÐsoume orismènec apì tic tro-qièc autèc sto elleiptikì prìblhma. Mia apì tic oikogèneiec tou elleiptikoÔprobl matoc, pou proèrqetai apì thn periodik troqi thc oikogèneiac f thcopoÐac h perÐodoc eÐnai Ðsh me π eÐqe brejeÐ se prohgoÔmenh ergasÐa kai eÐnaiastaj c (PÐnakac 1).

Sq ma 6.2: H perÐodoc TC sunart sei thc arqik c sunj khc ξC0 gia tic troqiècthc oikogèneiac f tou kuklikoÔ probl matoc.

60

Gia thc troqièc tou elleiptikoÔ probl matoc ja qrhsimopoi soume tonsumbolismì f qe , o opoÐoc shmaÐnei ìti prìkeitai gia mia oikogèneia tou elleipti-koÔ probl matoc (deÐkthc e), pou proèrqetai apì thn sunèqish miac troqicthc oikogèneiac f tou kuklikoÔ probl matoc thc opoÐac h perÐodoc eÐnai Ðshme TC = q · 2π. 'Etsi me ton sumbolismì autì h oikogèneia pou anafèrame

parapnw ja eÐnai h f 1/2e .

Ston paraktw pÐnaka parajètoume tic arqikèc sunj kec (ekkentrìth-ta=0) gia merikèc apì tic nèec oikogèneiec pou upologÐsame kaj¸c kai thsqèsh metaxÔ thc periìdou thc troqic tou kuklikoÔ probl matoc apì thnopoÐa suneqÐzontai kai thc periìdou tou elleiptikoÔ probl matoc2. Tèloc,shmei¸noume an suneqÐzontai wc eustajeÐc (s) astajeÐc (u) sthn perÐptwshtou perihlÐou (p) kai tou afhlÐou (a) antÐstoiqa.3

PÐnakac 2: Oi periodikèc troqièc thc oikogèneiac f tou kuklikoÔprobl matoc (ekkentrìthta=0) pou apoteloÔn shmeÐa diakldwshc gia nèec

oikogèneiec tou elleiptikoÔ probl matoc

Oikogèneia ξ0 η0 Suntonismìc Eustjeia

f2/3e -1.0658883521 2.5223785325 2/3 a:u p:s

f1/2e -0.7970426447 2.1961553754 1/2 a:u p:u

f2/5e -0.6570502814 2.0809419653 2/5 a:s p:u

f1/3e -0.5661433175 2.0360618179 1/3 a:u p:u

f2/7e -0.5010288513 2.0235566448 2/7 a:s p:u

f1/4e -0.4516045408 2.0282343263 1/4 a:u p:u

f2/9e -0.4125814140 2.0427638542 2/9 a:s p:u

f1/5e -0.3808637845 2.0631950413 1/5 a:u p:u

f1/6e -0.3321860326 2.1134958027 1/6 a:u p:u

Na shmei¸soume ìti gia tic oikogèneiec pou proèrqontai apì periodikèctroqièc tou kuklikoÔ probl matoc me perÐodo TC = 1

n·2π h pollaplìthta twn

troqi¸n touc eÐnai n, en¸ gia thc oikogèneiec pou proèrqontai apì troqièc meperÐodo TC = 2

2n+1· 2π h pollaplìthta eÐnai 2n+ 1.

2Να θυμίσουμε πως η περίοδος του ελλειπτικού προβλήματος είναι 2π

3Αναφερόμαστε σε μικρές τιμές της εκκεντρότητας. Κατά μήκος της οικογένειας, φυσικά,

η ευστάθεια μπορεί να αλλάξει.

61

Sq ma 6.3: Oi nèec oikogèneiec periodik¸n troqi¸n tou elleiptikoÔ probl -matoc Hill. Digramma thc arqik c jèshc ξ0 sunart sei thc ekkentrìthtac.

Sq ma 6.4: Nèec oikogèneiec periodik¸n troqi¸n tou kuklikoÔ probl matocHill. Digramma thc arqik taqÔthtac η0 sunart sei thc ekkentrìthtac.

62

Exairetikì endiafèron parousizei h perÐptwsh thc oikogèneiac f 2/3e . H

sugkekrimènh oikogèneia,ìpwc mporeÐ na parathr sei kaneÐc sta sq mata (6.3kai 6.4), faÐnetai na termatÐzetai apìtoma. 'Omwc sthn pragmatikìthta autìpou sumbaÐnei eÐnai ìti aut h oikogèneia periodik¸n troqi¸n en¸netai omal

me mia llh oikogèneia periodik¸n troqi¸n, thn g32/1e . H oikogèneia periodik¸n

troqi¸n g3 tou kuklikoÔ probl matoc apoteleÐtai apì troqièc pollaplìthtac3. H troqi thc oikogèneiac me perÐodo Ðsh me 4π suneqÐzetai sto elleiptikì

prìblhma (oikogèneia g32/1e ) kai en¸netai telik me thn oikogèneia f 2/3

e . Kaj¸cplhsizoume proc thn ènwsh sthn perÐptwsh tou afhlÐou, en¸ oi periodikèc

troqièc thc oikogèneiac g32/1e eÐnai arqik astaj c, gÐnontai diadoqik migadik

astajeÐc kai telik eustajeÐc.

Sq ma 6.5: H perÐptwsh twn oikogenei¸n f 2/3e kai g3

2/1e pou en¸nontai omal.

Digramma thc arqik c jèshc ξ0 sunart sei thc ekkentrìthtac.

PÐnakac 3: Arqikèc sunj kec gia thn periodik troqi tou kuklikoÔ

probl matoc apì thn opoÐa suneqÐzetai h oikogèneia g32/1e


g32/1e -0.9771636484 2.4485020552 2/1 a:u p:u

63

Sq ma 6.6: H perÐptwsh twn oikogenei¸n f 2/3e kai g3

2/1e pou en¸nontai omal.

Digramma thc arqik taqÔthtac η0 sunart sei thc ekkentrìthtac.

Sq ma 6.7: Trisdistato digramma twn oikogenei¸n f 2/3e kai g3

2/1e pou en¸-

nontai omal.

64

Sq ma 6.8: Merikèc periodikèc troqièc thc oikogèneiac f 2/3e sto q¸ro morf c

(ξ, η). Sto kèntro tou sust matoc suntetagmènwn brÐsketai o plan thc en¸h troqi sto q¸ro morf faÐnetai ìpwc ja thn èblepe ènac parathrht c poubrÐsketai pnw ston plan th.

65

Sq ma 6.9: Merikèc periodikèc troqièc thc oikogèneiac f 1/2e sto q¸ro morf c

(ξ, η). Sto kèntro tou sust matoc suntetagmènwn brÐsketai o plan thc en¸h troqi sto q¸ro morf faÐnetai ìpwc ja thn èblepe ènac parathrht c poubrÐsketai pnw ston plan th.

66

6.3.2 Oikogèneia g

Sthn pargrafo aut ja doÔme poiec apì tic troqièc thc oikogèneiac g toukuklikoÔ probl matoc suneqÐzontai sto elleiptikì prìblhma. Kai pli h su-nèqeia gÐnetai mìno gia troqièc tou kuklikoÔ probl matoc oi opoÐec èqounperÐodo thc morf c:

TC =p

q· 2π

Sto paraktw digramma faÐnetai pwc metablletai h perÐodoc twn periodik¸ntroqi¸n tou eustajoÔc kldou thc oikogèneiac g.

Sq ma 6.10: H perÐodoc TC sunart sei thc arqik c sunj khc ξC0 gia tictroqièc thc oikogèneiac g tou kuklikoÔ probl matoc.

PÐnakac 4: Oi periodikèc troqièc thc oikogèneiac g tou kuklikoÔprobl matoc (ekkentrìthta=0) pou apoteloÔn shmeÐa diakldwshc gia nèec

oikogèneiec tou elleiptikoÔ probl matoc.


g1/5e 0.2865970951 1.6624741053 1/5 a:u p:u

g2/11e 0.2745383199 1.7019353947 2/11 a:s p:u

g1/6e 0.2633214371 1.7433315064 1/6 a:u p:u

g1/7e 0.2434191393 1.8272090087 1/7 a:u p:u

67



68

6.3.3 Oikogèneia g′

Sthn pargrafo aut ja doÔme poiec apì tic troqièc thc oikogèneiac g′ toukuklikoÔ probl matoc suneqÐzontai sto elleiptikì prìblhma. Sto paraktwdigramma faÐnetai pwc metablletai h perÐodoc twn periodik¸n troqi¸n toueustajoÔc kldou thc oikogèneiac g′. ParathroÔme pwc lìgw tou ìti h oiko-gèneia g′ apoteleÐtai apì dÔo kldouc, gia kje tim thc periìdou TC uprqoundÔo periodikèc troqièc thc oikogèneiac me aut thn perÐodo, mÐa me megalÔterharqik sunj kh ξC0 kai mÐa me mikrìterh. 'Etsi gia kje suntonismì èqoumedÔo pijanèc troqièc pou mporeÐ na suneqÐzontai sto elleiptikì prìblhma. Giana eÐnai safèc apì poia troqi tou kuklikoÔ probl matoc proèrqetai h k-je oikogèneia tou elleiptikoÔ probl matoc ja qrhsimopoioÔme to prìshmo +dÐpla apì ton suntonismì ìtan anaferìmaste sthn troqi me th megalÔterharqik sunj kh ξC0 kai to prìshmo - gia thn troqi me th mikrìterh arqik sunj kh ξC0 antÐstoiqa.

Sq ma 6.13: H perÐodoc TC sunart sei thc arqik c sunj khc ξC0 gia tictroqièc thc oikogèneiac g′ tou kuklikoÔ probl matoc.

69

Se treic apì tic tèsseric peript¸seic oikogenei¸n tou elleiptikoÔ pro-bl matoc pou upologÐsame eÐqame periodikèc troqièc pou emfanÐzoun grammik eustjeia. EpÐshc endiafèron parousizei to gegonìc ìti entopÐsame kai pe-riodikèc troqièc pou eÐnai migadik astajeÐc, gegonìc pou den parathr jhkese kamÐa apì tic troqièc pou upologÐsthkan apì tic llec oikogèneiec toukuklikoÔ probl matoc.

PÐnakac 5: Oi periodikèc troqièc thc oikogèneiac g′ tou kuklikoÔprobl matoc (ekkentrìthta=0) pou apoteloÔn shmeÐa diakldwshc gia nèec



g′1/5−e 0.2309914125 2.0826409888 1/5- a:s p:u

g′1/5+e 0.3416892711 1.3124472634 1/5+ a:u p:s

g′2/9+e 0.4236607443 0.9200063968 2/9+ a:u p:u

g′1/4+e 0.4809637383 0.7033778696 1/4+ a:u p:s


70


6.3.4 Oikogèneia Hg

H teleutaÐa oikogèneia tou kuklikoÔ probl matoc pou perièqei eustajeÐc pe-riodikèc troqièc eÐnai h oikogèneia Hg. H oikogèneia perièqei ìpwc eÐdame kaisthn pargrafo (4.2.5) tetrapl periodikèc troqièc, wstìso uprqoun dÔ-o mèlh thc me perÐodo Ðsh me 2π pou suneqÐzontai sto elleiptikì prìblhma.Kai pli ja qrhsimopoi soume ta prìshma + kai - dÐpla apì ton suntonismìgia na dhl¸soume apì poia troqi tou kuklikoÔ probl matoc suneqÐsthke hoikogèneia tou elleiptikoÔ probl matoc.

PÐnakac 6: Oi periodikèc troqièc thc oikogèneiac g tou kuklikoÔprobl matoc (ekkentrìthta=0) pou apoteloÔn shmeÐa diakldwshc gia nèec



Hg1/1−e -0.5699057100 1.8724003896 1/1- a:u p:u

Hg1/1+e -0.3576613454 2.2355759920 1/1+ a:u p:u

71

Sq ma 6.16: H perÐodoc TC sunart sei thc arqik c sunj khc ξC0 gia tictroqièc thc oikogèneiac Hg tou kuklikoÔ probl matoc.


72


6.3.5 Sumpersmata

UpologÐsame mia seir apì nèec oikogèneiec tou elleiptikoÔ probl matoc H-ill. Pollèc apì tic periodikèc troqièc twn oikogenei¸n aut¸n eÐnai eustajeÐc,gegonìc pou den eÐqe parathrhjeÐ stic oikogèneiec pou eÐqan upologisteÐ stopareljìn. Tèloc endiafèron parousizei to gegonìc ìti oi oikogèneiec pouproèrqontai apì troqièc thc oikogèneiac g′ perièqoun periodikèc troqièc poueÐnai migadik astajeÐc.

73

Keflaio 7

Qrtec Eustjeiac

Kat th melèth tou kuklikoÔ probl matoc Hill qrhsimopoi same tic epifneiectom c Poincare. H qr sh touc ìmwc eÐnai dunat mìno se autìnoma sust mata2 bajm¸n eleujerÐac. To elleiptikì prìblhma Hill eÐnai dÔo bajm¸n eleuje-rÐac all den eÐnai autìnomo. 'Etsi gia thn diexodik melèth tou q¸rou twnfsewn ja prèpei na katafÔgoume se llec mejìdouc melèthc. MÐa apì autèceÐnai oi qrtec eustjeiac, oi opoÐoi eÐnai disdistatec apeikonÐseic perioq¸ntou q¸rou twn fsewn stic opoÐec shmei¸noume me diaforetikì qr¸ma tic ka-nonikèc kai tic qaotikèc perioqèc. H dikrish metaxÔ twn dÔo diaforetik¸neid¸n kÐnhshc gÐnetai qrhsimopoi¸ntac touc deÐktec qaotikìthtac. Oi deÐktecqaotikìthtac (ìpwc o mègistoc ekjèthc Lyapunov (L.C.I),o F.L.I, o R.L.I,oG.A.L.I. k.a.) apoteloÔn ergaleÐa thc mh grammik c dunamik c pou mporoÔn namac deÐxoun me arket axiopistÐa an mia troqi eÐnai kanonik qaotik . 'EtsisqhmatÐzontac èna plègma me arqikèc sunj kec kai elègqontac me kpoio apìtouc deÐktec qaotikìthtac to eÐdoc twn troqi¸n kataskeuzoume èna qrtheustjeiac, pou se sunduasmì me th melèth twn periodik¸n troqi¸n, mporeÐna mac bohj sei na sqhmatÐsoume mia oloklhrwmènh poyh gia thn dom touq¸rou twn fsewn.

7.1 O deÐkthc qaotikìthtac F.L.I

Sthn paroÔsa ergasÐa epilèxame na qrhsimopoi soume wc ergaleÐo gia tonèlegqo thc kanonikìthtac miac troqic ton gr goro deÐkth Lyapunov (F.L.I: Fast Lyapunov Indicator). OrÐzoume ton F.L.I wc th mègisth tim tou lo-

74

garÐjmou tou mètrou tou dianÔsmatoc apìklishc δ~ξ(t) wc proc to qrìno t:

F.L.I = sup

log∣∣∣δ~ξ(t)∣∣∣t

(7.1)

ìpou to dinusma δ~ξ(t) (δξ(t), δη(t), δpξ(t), δpη(t)) prokÔptei wc lÔsh twn e-xis¸sewn metabol¸n tou probl matoc. O deÐkthc F.L.I parousizei saf¸cdiakrit sumperifor gia kanonikèc kai qaotikèc troqièc, gegonìc pou ekme-talleuìmaste gia na tic diaqwrÐsoume. 'Etsi se mia kanonik troqi o deÐkthcF.L.I paramènei stajerìc se sqetik mikrèc timèc (<5) se sqèsh me to qrìno.Antijètwc se qaotikèc troqièc auxnetai apìtoma me to qrìno kai fjnei seidiaÐtera meglec timèc (>10).

Sq ma 7.1: O deÐkthc F.L.I gia mia qaotik kai mia kanonik troqi antÐstoiqa.Sto sq ma faÐnetai h diaforetik sumperifor tou se kje mia perÐptwsh.

7.2 Qrtec eustjeiac kat m koc twn oi-

kogenei¸n tou kuklikoÔ probl matoc

Se aut thn pargrafo ja melet soume diforec perioqèc tou q¸rou twnfsewn tou elleiptikoÔ probl matoc Hill kataskeuzontac qrtec eustjeiacgÔrw apì tic eustajeÐc oikogèneiec tou kuklikoÔ probl matoc. San arqikèc

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mac sunj kec (ξC0, ηC0) ja proume autèc twn eustaj¸n periodik¸n troqi¸ntwn oikogenei¸n f ,g kai g′ kai ja tic oloklhr¸soume gia diforec timèc thcekkentrìthtac, qrhsimopoi¸ntac san montèlo to elleiptikì prìblhma Hill. Meautì ton trìpo mporoÔme na elègxoume mèqri poiec timèc thc ekkentrìthtactou plan th oi eustajeÐc periodikèc troqièc tou kuklikoÔ probl matoc HilldiathroÔn thn eustjei touc.

Oi paraktw qrtec eustjeiac apoteloÔntai apì èna disdistato plègma,ston orizìntio xona tou opoÐou topojetoÔme tic arqikèc sunj kec twn perio-dik¸n troqi¸n thc ekstote oikogèneiac tou kuklikoÔ probl matoc kai stonkatakìrufo xona thn ekkentrìthta tou plan th. Oloklhr¸nontac tic arqi-kèc sunj kec katal goume se difora eÐdh troqi¸n. Me kìkkino qr¸ma stoucqrtec faÐnontai oi kanonikèc troqièc, me mple oi qaotikèc troqièc, me prsinoqr¸ma anaparistoÔme tic troqièc kat tic opoÐec o dorufìroc diafeÔgei apìto sÔsthma kai me sièl tic troqièc kat tic opoÐec o dorufìroc sugkroÔetaime ton plan th.

Sq ma 7.2: O Qrthc eustjeiac kat m koc thc oikogèneiac f tou kuklikoÔprobl matoc.

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Sq ma 7.3: O qrthc eustjeiac kat m koc thc oikogèneiac g tou kuklikoÔprobl matoc.

Sq ma 7.4: O qrthc eustjeiac kat m koc thc oikogèneiac g′ tou kuklikoÔprobl matoc.

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7.3 Diagrmmata Henon

Sthn perÐptwsh tou kuklikoÔ probl matoc Hill eÐdame ìti oi epifneiec tom cPoincare mac bohjne arket gia na melet soume to q¸ro twn fsewn. An kaieÐnai idiaÐtera qr simh san mèjodoc parousizei to meionèkthma ìti ja prèpeina sqedizoume diaforetikì digramma gia kje tim thc stajerc tou Jacobi(CH). 'Etsi gia na èqoume mia genikìterh eikìna tou sust matoc mporoÔme naqrhsimopoi soume thn epifneia tom c pou orÐzetai apì

η = 0, ξ = 0, η =

√3ξ2 +

2

ξ− CH (7.2)

Me autìn ton trìpo kje troqi mporeÐ na anaparastajeÐ sto epÐpedo (CH ,ξ0)me èna shmeÐo. H parapnw skèyh ègine gia pr¸th for apì ton M.Henon, oopoÐoc sto digramma twn periodik¸n troqi¸n tou kuklikoÔ probl matoc Hill(sq ma 4.3) shmeÐwse ta ìria twn eustaj¸n perioq¸n gÔrw apì autèc. Gia tolìgo autì tètoiou eÐdouc diagrmmata onomzontai diagrmmata Henon.

Sto digramma Henon, ìpwc kai se kje epifneia tom c, den mporoÔn naanaparastajoÔn troqièc oi opoÐec den tèmnoun thn epifneia. H axÐa enìc dia-grmmatoc Henon prokÔptei apì thn parat rhsh ìti oi perissìterec eustajeÐctroqièc pernoÔn kont sto shmeÐo η=ξ = 0 periodik. Oi troqièc oi opoÐecden emfanÐzontai sto digramma Henon eÐnai eÐte troqièc pou periorÐzontai sekpoiec suntonismènec nhsÐdec, oi opoÐec apoteloÔn èna mikrì posostì touq¸rou twn fsewn, eÐte eÐnai troqièc diafug c, oi opoÐec den mac endiafèrounètsi kai alli¸c.

To digramma Henon eÐnai ousiastik ènac qrthc eustjeiac pou apote-leÐtai apì èna disdistato plègma ston orizìntio xona tou opoÐou èqoume thnarqik sunj kh ξ0 kai ston kjeto thn tim thc paramètrou CH . Oi upìloipecarqikèc sunj kec gia kje troqi dÐnontai apì tic sqèseic (7.2) en¸ to eÐdoctwn troqi¸n mporeÐ na melethjeÐ me kpoio deÐkth qaotikìthtac. Tètoiou eÐ-douc qrtec mporoÔme na ftixoume gia diforec timèc thc ekkentrìthtac thctroqic tou plan th gÔrw apì ton lio qrhsimopoi¸ntac to elleiptikì prì-blhma Hill. Sthn sunèqeia ja melet soume difora diagrmmata Henon poukataskeusame.

7.3.1 Digramma Henon gia ekkentrìthta mhdèn

To digramma Henon gia ekkentrìthta mhdèn, apoteleÐ ousiastik to digram-ma Henon gia to kuklikì prìblhma Hill kai eÐnai to digramma pou parousÐase

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o Henon sthn arqik tou ergasÐa. Sto digramma autì mporoÔme na uperjè-soume kai to dÐktuo twn periodik¸n troqi¸n tou kuklikoÔ probl matoc Hill.

Sq ma 7.5: Digramma Henon gia ekkentrìthta mhdèn.

Sto sq ma (7.5) parathroÔme exairetik sumfwnÐa metaxÔ thc grammik cmejìdou melèthc tou probl matoc, mèsw twn periodik¸n troqi¸n, kai thcmh grammik c, mèsw tou qrth eustjeiac. Diapist¸noume ton shmantikìrìlo pou paÐzoun oi periodikèc troqièc, kaj¸c faÐnetai na apoteloÔn twnskeletì tou q¸rou twn fsewn, gÔrw apì ton opoÐo organ¸nontai ta diforaeÐdh kin sewn. 'Etsi èqoume kanonikèc troqièc gÔrw apì eustajeÐc periodikèctroqièc kai qaotikèc troqièc gÔrw apì tic astajeÐc periodikèc troqièc touprobl matoc.

Mlista melet¸ntac to parapnw digramma Henon katafèrame na ento-pÐsoume kai mia nèa oikogèneia periodik¸n troqi¸n tou kuklikoÔ probl matocHill. Sto sq ma (7.5) mporeÐ kaneÐc na parathr sei sthn ktw arister gw-nÐa mia sten z¸nh eustjeiac parllhlh me th z¸nh eustjeiac gÔrw apìthn oikogèneia f . PaÐrnontac arqikèc sunj kec mèsa sthn sten aut perio-q , entopÐsame mia nèa oikogèneia periodik¸n troqi¸n. Oi periodikèc troqiècthc oikogèneiac èqoun pollaplìthta 5 kai mlista h oikogèneia sunant thn

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oikogèneia f se mia periodik troqi ìpou grfetai 5 forèc h troqi thc oiko-gèneiac f . Kti anlogo sunèbaine kai me thn oikogèneia g3, gia to lìgo autìja onomsoume thn oikogèneia aut g5. Oi periodikèc troqièc thc oikogèneiacg5 eÐnai ìlec eustajeÐc.

PÐnakac 7: H oikogèneia periodik¸n troqi¸n g5 tou kuklikoÔ probl matocξC0 ηC0 CH

-2.8383840000 5.5207434854 -5.60471114-2.6043935095 5.1055746325 -4.95036256-2.4002834794 4.7499935174 -4.44512116-2.2022949752 4.4139907340 -4.02486117-2.0024977656 4.0907751979 -3.70569714-1.9074484460 3.9498638666 -3.63782471

'Ena llo endiafèron sumpèrasma pou prokÔptei apì to sq ma (7.5) eÐnaiìti uprqoun eustajeÐc perioqèc, stic opoÐec ja mporoÔse na brÐsketai k-poioc dorufìroc, arket èxw apì thn aktÐna Hill tou plan th. Ed¸ ja prèpeina orÐsoume wc dorufìro enìc plan th èna mikrì s¸ma tou opoÐou h apìsta-sh apì ton plan th den uperbaÐnei potè to meglo hmixona tou plan th (ap).Autìc o orismìc, pou dìjhke apì ton Fabrycky (2008), faÐnetai aplìc kailogikìc kai apokleÐei apì dorufìrouc s¸mata ta opoÐa brÐskontai se Trwi-kèc troqièc (Trojan orbits) kai s¸mata pou brÐskontai se petaloeideÐc troqièc(horseshoe orbits). Qrhsimopoi¸ntac ton parapnw orismì gia ton plan thkai melet¸ntac to digramma Henon gia to kuklikì prìblhma Hill diapist¸-noume ìti eÐnai dunatìn na uprqoun eustajeÐc dorufìroi se apìstash polÔmegalÔterh apì thn aktÐna Hill tou plan th1.

7.3.2 Diagrmmata Henon gia touc meglouc pla-

n tec

EÐdame sthn prohgoÔmenh pargrafo ìti gia plan tec se kuklik troqi mpo-roÔn na uprxoun dorufìroi se meglh apìstash apì autoÔc. Se aut n thnpargrafo ja elègxoume ti gÐnetai sthn perÐptwsh pou o plan thc den kineÐ-tai se teleÐwc kuklik troqi gÔrw apì ton astèra all h troqi tou eÐnaièkkentrh. 'Etsi ja kataskeusoume diagrmmata Henon gia diforec timèc

1Υπενθυμίζουμε ότι στο πρόβλημα Hill η ακτίνα Hill βρίσκεται σε απόσταση ahill =

3−1/3 ∼= 0.6933 από τον πλανήτη

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thc ekkentrìthtac tou plan th. San timèc thc ekkentrìthtac epilèxame autèctwn meglwn planht¸n tou hliakoÔ mac sust matoc DÐa, Krìnou, OuranoÔkai Poseid¸na.

Sq ma 7.6: Diagrmmata Henon gia ekkentrìthta Ðsh me aut tou DÐa gia ticpeript¸seic tou perihlÐou kai tou afhlÐou. O sunduasmìc twn dÔo peript¸-sewn ja mac d¸sei to telikì digramma Henon gia kje plan th.

Gia kje mia ekkentrìthta ìmwc èqoume dÔo diaforetikèc peript¸seic, oplan thc na brÐsketai sto af lio o plan thc na brÐsketai sto peri lio. Sekje mia apì autèc tic peript¸seic kataskeuzoume to antÐstoiqo digrammaHenon. O plan thc ìmwc, lìgw twn diataraq¸n pou dèqetai apì touc lloucplan tec, xekin¸ntac apì mÐa apì thc dÔo arqikèc katastseic ,ac poÔme peri- lio, ja kalÔyei telik ìlo to fsma twn arqik¸n tou katastsewn, dhlad kai aut n tou afhlÐou. To fainìmeno autì onomzetai mÐxh twn fsewn kaiprèpei na to proume upoyin mac gia na pargoume mia pio realistik eikìnatou probl matoc. Autì pou ja knoume loipìn gia na katal xoume ston teli-kì digramma Henon gia kje plan th eÐnai na krat soume san kanonikèc mìnotic troqièc oi opoÐec eÐnai kanonikèc kai sta dÔo diagrmmata Henon (af liokai peri lio) gia thn ekkentrìthta tou ekstote plan th.

Sthn sunèqeia parajètoume ta diagrmmata Henon gia touc meglouc pla-n tec, ìpwc aut prokÔptoun apì thn diadikasÐa pou perigryame parapnw.Parìmoia doulei eÐqe gÐnei kai apì touc Shen kai Tremaine. Sthn ergasÐatouc kataskeÔasan diagrmmata Henon qrhsimopoi¸ntac ìmwc san plaÐsio er-gasÐac to prìblhma twn N-swmtwn gia ton 'Hlio kai touc meglouc plan tec.Gia sÔgkrish parajètoume kai ta apotelèsmat touc.

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Sq ma 7.7: Digramma Henon gia ekkentrìthta Ðsh me aut tou DÐa.

Sq ma 7.8: Digramma Henon gia ton DÐa ìpwc upologÐsthke apì touc Shenkai Tremaine.

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Sq ma 7.9: Digramma Henon gia ekkentrìthta Ðsh me aut tou Krìnou.

Sq ma 7.10: Digramma Henon gia ton Krìno ìpwc upologÐsthke apì toucShen kai Tremaine.

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Sq ma 7.11: Digramma Henon gia ekkentrìthta Ðsh me aut tou OuranoÔ.

Sq ma 7.12: Digramma Henon gia ton Ouranì ìpwc upologÐsthke apì toucShen kai Tremaine.

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Sq ma 7.13: Digramma Henon gia ekkentrìthta Ðsh me aut tou Poseid¸na.

Sq ma 7.14: Digramma Henon gia ton Poseid¸na ìpwc upologÐsthke apìtouc Shen kai Tremaine.

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7.3.3 Sumpersmata

ParathroÔme pwc ta apotelèsmat mac tairizoun , poiotik toulqiston, sepolÔ meglo bajmì me ta apotelèsmata thc ergasÐac twn Shen kai Tremai-ne. Epibebai¸noume loipìn to gegonìc ìti mporoÔn na uprxoun dorufìroi seandromec troqièc stouc meglouc plan tec tou hliakoÔ mac sust matoc seapostseic polÔ makrÔtera apì thn aktÐna Hill tou plan th. O Poseid¸naceÐnai o plan thc me th megalÔterh z¸nh eustjeiac, kai ra o kalÔteroc upo-y fioc gia thn eÔresh enìc tètoiou dorufìrou. O DÐac kai o Ouranìc èqounkai autoÐ eustajeÐc perioqèc pou ja mporoÔsan na filoxenoÔn andromouc ma-krinoÔc dorufìrouc all to eÔroc twn perioq¸n eÐnai arket mikrìtero. Tèloco Krìnoc faÐnetai na èqei periorismènec eustajeÐc perioqèc èxw apì th sfaÐraHill tou. Parìmoia prgmata parat rhsan kai oi Shen kai Tremaine sthnergasÐa touc. Katafèrame loipìn qrhsimopoi¸ntac to elleiptikì prìblhmaHill kai upojètontac ìti èqoume mÐxh twn fsewn na anapargoume ta apote-lèsmata tou pl rec montèlou twn N-swmtwn. Ja perÐmene kaneÐc h akrÐbeiatou elleiptikoÔ probl matoc Hill na eÐnai ikanopoihtik mìno se perioqèc po-lÔ kont ston plan th all ta apotelèsmata deÐqnoun pwc h prosèggish maceÐnai ikanopoihtik akìmh kai se apostseic thc txhc twn merik¸n aktÐnwnHill.

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BibliografÐa

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