Luận văn Lý thuyết thế vị ứng dụng trong trọng lực học và từ trường

LÝ THUYẾT THẾ VỊ ỨNG DỤNG TRONG TRỌNG LỰC HỌC VÀ TỪ TRƯỜNG TRẦN HOÀI NHÂN Trang nhan đề Mục lục Mở đầu Chương1: Lý thuyết thế vị cho trường trọng lực và từ trường. Chương2: Bài toán xác định nguồn trọng lực và nguồn từ. Kết luận Tài liệu tham khảo

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J2LLijPLomL lnf!£ uj 61 q ~L ']{)oOirnJuin B~ngcachxemme)tngu6ntunhu'la me)tt~pcaclu'ongqI'ctuhaytutich,ta nh~ndu'Qcnhi~ucachkhacnhaud~nh~ndu'Qcacmahlnhxa'pXlchongu6ntu. Volume of l'-tagnctizalion Vohtll'\C and Sur'facc Charge ~-Q$ L/ \/OIUlllC and Surface C:::lttTr~nt$ Poisson's Relation ('--?\ (~l~=~~~)-'f ~-~.:-/ $ D Hinh 2.5: Bondt;mgphanb6'cuasf!tuh6a. f)£ngthlic(2.12)ho~c(2.13)coth~tinhtoantu'ongminhchonhITngv~th~ cohlnhdangdongianvacoth~dungchophu'ongphapthu~n.Khi do,me)tngu6n tucoth~du'QchiathanhN ph~ndongianvakhidocactinhtoandu'Qcthvchi~n tu'ongtv nhu'd6ivoi(2.2).Bathanhph~ncuatutru'ongtrdthanh B,J = N IMjbjj, j=l (2.14) trongdo, Bj la tutru'ongt<;=tidi~mquailsatthlij, Mj la de)IOncuade)tuhoacua ph~nthlii, vabijla tUtru'ongt<;=tidi~mquailsatthli j gayrabdiph~nthli ivai de) tuhoadonvi bo. 1J f - 1 = - CmV'P M.V'Q-dv , 0 r 1 (2.15) NSucacph~nchiadunho,chungco th~du'Qcxemnhu'code)tuhoad~u. TrangthvctS,caca phaiconhITnghlnhdangdongian.Ch~ngh<;=tnnhu'hlnhhQp chITnh~tho~ccacc~plu'ongcvc,d~d8danghontrongvi~ctinhtoanbi~uthlic (2.14). "!!.llt/-I'LlJillL /},f!R uj 62 q v4n 7IJoa1(],lJUYL Nhu'dad~c~ptrongcacph~ntru'ac,tichphankh6icua (2.12)cothedu'QC chuyenthanht6ngcuatichphanffi~tva tichphankh6ib~ngcachapdl,mgh<% thlicvecto V.(~A)=V~.A+~V.A vaapdl,mgdinhly Divergence.The'tuco d~ng v = CmfM,VQ!dv = CmfM.nds - CmfV.M dv R r sr Rr =Cm fQs ds + CmfQv dv. s r R r (2.16) Cactichphantrong(2.16)coclingd~ngnhu'the'ha"pd~n,vaid~ilu'Qngvo hu'angQs va Qv tu'dnglingla bieudi6nchotutichtrenb~ffi~tvaphiatrong cuav~tthe.Ne'udQtuhoad~u,thltichphanthlihaicua(2.16)tri<%ttieu,vathe' vidu'Qchobdi v = C fM.nm -ds. s r (2.17) Ne'udQtuhoad~u, v~thecothehoantoandu'Qcbieudi6nbdis1,1'phanb6 cuacactutichtrenb~ffi~tcuav~the.S1,1'bi€u di6nnayd~ntaiffiQtso'cacthu~t toan hi<%uq a,ffiQtso'trongcacthu~toandosedu'Qcd~c~pd ph~nsau. 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Cac kh6i hlnhhQpchITnh~tclingdu'c)cdungtrongphu'dngphap nghichd~tinhtrljctie'pvectd M tudQdi thu'ongtru'ongtoanph~ndod~cdu'c)c. Ch6ngcaebanmong Talwani(xem[1])dffsadl,mgtichphanbQi(2.12)d~du'arathu~ttoantinh toantUtru'ongsinhbdi ngu6ntu co hlnhd~ngtuy '1.Phu'dngphap nay tu'dngtlj nhu'phu'dngphapcuaTalwanivaEwingtrongtinhtoandQdi thu'ongtru'onghap d§ncuangu6nbachi€u dffd€ c~pd ph~ntren.Ngu6ntudu'c)cxapXlbdich6ng cacbanmong,m6ibanmongl~idu'c)cxapXlbdimQtdagiac.TudotichphanbQi (2.12)du'c)cchuy~nthanhtichphantheox,y vat6ngtheoz. Plouff md rQngphu'dngphap cua Talwani (xem [1]) b~ngcach thay the' ch6ngvo h~ncacbanmongbdi mQtt~nghU'uh~ncacbanmongvoi dQdaynho. Phu'dngphapnaybi~udi~nmQtcachchinhxachdnmohlnhcuangu6ntu.Ngoai fa, ngu'oita condungphu'dngphapxapXl cuaTalwaniva Plouff chobai toan al2 = MxFy + MyFx' al3 = MxFz + MzFx' a23 = MyFz + MzFy, J2I1ijl'L mi1L fJ,f!£ ill 66 g,4,L '7{)oairmullL nguejctinhbathanhphc1ncuadQtllhoatn!ctie'ptlldQdithuongtruongloanphc1n dod~cduejc. Caekh6idadi~n Ne'udQtll hoala d~u,mQtngu6ntll co th~duejcmahlnhbdicactUtich lIen b~m~tcuangu6ntll. Dt!avaod~cdi~mnay,Bott,Barnett,Hanssenva Wang (xem [1]), duara phuongphapxa'"pXl ngu6ntll bdi mQtkh6i .dadi~nma m6im~tcuano la mQtdagiacph~ng.(hlnh2.8) y z Hinh 2.8. MQtngu6ntitv6'ihlnhd(;mgtuyydu(femohlnh nhumQtkhatdadi?nv6'ieaemijtLaeaedagiaephdng. Phuongtrlnh (2.17)chota the'tll sinhbdi mQtngu6ntll co dQtll hoa d6ng nha't,voi tll truongduejcchobdi - fM.n-B = - VpV = Cm zrds. s r (2.20) £)~dongian,xettruonghejpg6ctQadQduejcd~tt~idi~mquailsatP. Khi b~ m~tcuangu6ntll duejcthaythe'bdiN m~tdagiacph~ng,(2.20)trdthanh N - ; B = CmL(M.n) f2ds, i=\ s r, (2.21) J2JltP' ImlL tltf!£ £lj 67 g nil, 7{;oiiir1lJuU, vai Sj lami[ttthli i, n)avectophaptuy€nhu'angrang oaitu'ongling. (a) A :;;;X 1\ . -/~ Vr - n;j I..P I IJ p/ P , con1cr j+ I z (0) ~ c I \ I A \ I" I" \ ",\1--- - D '-pl Alca AlJCD '" Arca ABP' +Area BCP' +ArcoilCOP' - Area ADP' Hinh 2.9. 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J!.tUpL IJmL thf!£ nj 68 g ~L ,,;/6oai(J{J;,mt " PH IJ It.(-.r-- c:rncr .J Iv corncr j+ I f~' Hinh 2.10: Mr;ittamgidctranghf trl:lct(JadQmaiu,v,w. ChQng6ctQadQt<;tidi~mPi, tn;lctQadQmoiu,v va w songsongvoi Pij, Ijj va n( hinh2.10).Trangh<%trt;lctQadQmoinay,cacdlnhcuami€n tamgiacco tQadQla (0, 0, di), (uij,Vij,dj)va (Ujj,Vj,j+l,dj) voi dj 1akhmlngcachtli'P tOiPj'. Tich phanla'ytrenmi€n tamgiactrdthanh ~ij - fr- -dsr2 sij U f ijV fb U-p .. +vf +d,~- IJ IJ 1 lJ d d- 3 U v, OVa (u2+v2+dnZ Doc lu'Qngrichphankeptren,tadu'Qc - [ a [ r+p J 1 r +V ] - [ 1 ( r+p J]~ij = Pij -J1+a2 log ldJ - 2"logr-v - Ijj -J1+a2 log ldJ [ ]J IV=V' , I - dj a(r-Idjl) 1.]+ - n; jdJarClan( r+a2Id;1 - IV-Vii (2.23) vOi va ViP , Vi.j+lU= va Vb = - Uij Ujj J2114nmln lhLfl!uj 69 q ~t 'j6oai {J(JuHl , . v ~ 2 2 J 2 2 d2VOl a = - P = u.. +V r = u.. +V + ., 1J' 1J I . Uij Ne'uP n~mtrongm~tph~ngchuami€n tamgiac, (2.23)trdnensuybie'nva truonghQpnayduQcxU'ly b~ngcachbodi mQtph~nhlnhqU(;ltnhobaaquanh di~mPi' rakhoimi€n tamgiacdangxet,ke'tquathuduQcla - [ a ( ( r+p ) 1 r +V J] ~ij = Pjj l+log - - -log- ~1+a2 28 2 r - V - Ijj [ 1 ( 1+log r +p J] IV=Vi,.i+1 ~1+ a2 28 I , Iv=vij (2.24) trongdoE lamQts6nhotuyy. 2.1.2.2. Xa'pXlmohinhhaichi~u Nguoi ta sU'd\lngkhai ni~mb€ m~ttiI'tich d~xay d1!ngmQts6 thu?ttoan choma hlnhV?tth~hai chi€u va thaythe'cac d(;lngthie'tdi~nc~tngangcuaV?t th~bdicacdagiacN-dlnh.Ne'uV?tth~la tiI'hoad€u, dQtiI'hoacoth~duQcthay the'bdicactUtichtrenb€ m~tcuano(hlnh2.11). x x CB -~~~\~ z z Hinh 2.11.Xdpxl cilamQtvt;'itthi hai chiiu v6'icae ddih?p vahc;mtittich. Khi do,bai toanduQcchuy~nthanhvi~ctinhtoanl1!ctiI'cuaN daiph~ngtiI' tich,mdrQngva h(;lntheohuang+yva -yo TiI' truongcuamQtV?ttiI'd€u vai th~ tichR vab€ m~tS duQcchobdi (2.20), J2tttJn olin tJlf!£ u] 70 g r&L 'Jf5oO1 (j{JUUL - f M.n- B = Cm~rds. s r Ding thlicnayWongtl!nhuchotrUonghfipd~ncuamQtvor6ngclInghlnhd(;lng S a(S) - g = -y ~rds, s r vai a(S) la m~tdQm~trendonvi di<%ntich,trongdotittruongd~uc;;uamQtv~t th~titvaith~tichR coth~du<;jctinhb~ngcachthaythe'-M.~ choa(S) va Cm choy. (a) (b) ,. ~x (c) \0 ~ Hinh 2.12. (a)Ddi ndmngangmiJr(3ngvahc;mrheatrl!cy. (b) thanhv~tthi dili vahc;mrheatrl!cy cdtm(itphdngx,z t(li (x' , z') X6tmQtdaihypphing,n~mngangvaim~tdQb~m~tla a, mdrQngvoh(;ln theohuang+yva-y, codQrQngcuadaitit(Xl,z') tai (X2,z') (hlnh2.12a).MQt thanhph~ndx cuadaihypc~tm~tphingX, z t(;li(x', z') va Wongduongvai thanhv~tth~dai vo h(;lndQctheotrl;lcy co kh6i lu<;jngtrendonvi chi~udai A=adx(hlnh2.12b).Nhudffd~c~pd ph~ntruac,thanhv~tth~ co truonghfip d~nquailsatt(;lig6ctQadQla g = -2YA~r J!.llijn 'Jan tnLfl!uJ 71 g P4.n7!6oiL1rnJuiI" " - = 2YA x l+z'k X'2 '2'+z (2.25) Truongha'pd§n cuadai h(fpn~mngangdu<;5ctinhb~ngcachd~tA=crdx trong(2,25)vala'ytichphantheox, Xz , gx = 2ycrf '2X '2 dx'x +z XI r ~ = 2ycrlog~, rl Xz dx' g = 2ycrz'f---;z- '2z X +z XI = 2ycr(81-82), g [ - r - ]= 2ycrilog r~+k(81- 82) , (2.26) trongd6ri var2la khoangcach tUdi~mP taic(;tnh1va2,81va82 la cacg6c giuatn:lcx va cacduongn6ic(;tnh1,c(;tnh2 tuongung(hlnh2.12b).£)~ngthuc (2.26)la truongha'pd§ncuamQtdaih(fpn~mngang.T6ngquat,vaiba'tky dai h(fpnao,chungtaquaydaih(fpmQtg6ctuyyvavaihaivectodonvi Ii va ; lin IU<;5tlaphapvectovavectochIphuongcuadaih(fptuongung(hlnh2.12c). - - Vec to s luonc6huangsongsongvai daih(fptITc(;tnh1tai c(;tnh2.Vec to n vuongg6cvaidiEh(fpvadu<;5cdinhhuang nX = sz, nz = -sx' Thanhphincuatruongha'pd§ntheohuang~ van du<;5cchoboi (2.26) r gs = 2ycrlog~, rl gn = - 2ycr(81- 82), £llIpl (WI flu.a uj 72 g ~I '7()tJitirnluitrl Thanhph~ntheohuangxvaz duQc hoboi gx = l.g = sxgs+nxgn = sxgs+szgn [ - r - ]= 2ycrSxlog r~-sz(8t -82) . gz = k.g, = szgs+nzgn = szgs- sxgn [ - r - ]= 2ycrSzlog~-sx(8t-82) . D~ngthuctrencho ta h/c ha'pdfincuaffiQtdeliv~tth€ hyp dai vo h(;ln.D€ chuy€n v€ truonghQptUtruong,chungta chi c~nd<1ity =Cmvacr=-M.n B, = -2Cm(M.~{S,lOg~ - S,(8,-82)} (2.27) - [ - r - ]Bz = -2Cm(M.n) szlogr~ - sx(8\-82) . (2.28) D~ngthuc(2.27) va (2.28)bi€u di€n choh;tctu cuaffiQtdelihyptu tich, chungcoth€ duQCsadlJngN l~nd€ tinhtminh;tctucuaffiQtv~th€ hlnhlangtrlJ N-dlnh N B = I(rn\x+kB\z)'i=\ (2.29) J2tLlpL tJiUL thLJ-eu] 73 q P4n '7{JoOi(](J,iYL vai B1xva BIz la cacthanhph~ncuaB gayra bdi m~t1.Tru'ongtoanph~ndu'Qc tinhbdi N ~T = I(FxBlx +FzBlz)'1=1 (2.30) vai Fx va Fz la thanhph~ntheohu'angx, z cuatru'ongxungquanb,khangbi nhi~ulo<;ln.Ma hlnhhaichi~ud~dangxaydvnghdnnhi~usovai mahlnhba chi~u,chungthu'ongdu'QCsadl;lngm6ikhidi~uki~ndiav~tly chophep. 2.2. Phuongphap nghjch Trongph~ntru'ac,taco cacquailh~giuangu6nva tru'ongthe'sinhrabdi ngu6n.Vdi R la vungchuangu6n,rex, y, z) la di~mquailsatd~tngoaiR, Q (x', y', z') la di~mla"ytichphann~mtrongR, r la vec td du'Qcdinhhu'angtu Q Wi P. Ta co tru'ongha"pd~ntheophu'dngth~ngdung g(P) = J z-z' - y p(Q)~dv, R r (2.31) tutru'ongtheophu'dngth~ngdung Bz(P) = 8 J 1 - Cm- M(Q).V -dv8z Q rR = CmfMr~Q)[3;(Z-ZI)-rkJdv, R (2.32) dQdi thu'ongtoanph~nxa"p Xl - J 1 ~T(P) = -CmF.VpM(Q)VQ-dv R r = C fM(Q) [3(F.;);- FJdv,m r3R (2.33) .l2tltPl lfflfil tJllJP ilj 74 q ~l '7fJLJmf'J1Jtml trongd6,p(Q)vaM(Q) l~nluQtla phanb6m~tde)vade)tli'h6acuangu6n.Vec todonvi F lahuangcuatli'truongkhongbinhi~u,vectodonvi k duQcdinh huangth~ngdung.H~so"y vaCmlacach~ngso"dfid~c~ptrongchuang1. Phuongtrlnh(2.32)va(2.33)c6dc.lUgchungla fer)=fs(Q).G(P,Q)dv. R (2.34) Chungta sexemxetdangdi~udinhhuangcuade)tli'h6atrongme)twi truonghQp. Gia sU'de)tli'h6aduQchoantoancamungboi truongxungquanh. Lucd6,de)tli'h6aseg~nnhutheome)thuangduynha"tne-ude)camlad~nghuang va khonggiantli'dangxetkhongquaIOn.Trongnhfi'ngtruonghQpnay,trong (2.32)va (2.33),tac6th€ d~tM(Q)=M(Q)M vachuy€nvectodonvi vaotrong da"ungo~c.Cacphuongtrlnhtli'(2.31)de-n(2.33)c6dc.lUgt6ngquat fer) =fs(Q).\V(P,Q)dv, R (2.35) trongd6fer) la giatri truongthe-tC;liP, s(Q)motame)tdC;liuQngv~tly (phanb6 m~tde)ho~cde)tli'h6a)tC;liQ. Phuongtrlnh(2.35)la me)tphuongtrlnhFredholm, trongd6\V(P,Q)vaG(P,Q)duQCgQilacachamGreen. Phuongtrlnh(2.35)th€ hi~nra slj khacnhaucuaphuongphapthu~nva phuongphapnghich.Pheptinhloanthu~n h~mml;lcdichtinhloanfer) tli'cac hams(Q),\V(P,Q)vavungngu6nR chotruac.Gia tri fer)duQchoanloanduQC xacdinhtUcacthongtinv~hams(Q),\V(P,Q)va vungngu6nR. N6i khacdi, phuongphapthu~nc6 duynha"tnghi~m.Tuy nhien,c~nchuy r~ngm~cdu phudngphapthu~nluauchoduynha"tnghi~mv~m~tloanhQC,nhungmohlnh cacngu6ntli'hayngu6nha"pd~nxaydljngtheophuongphapthu~nthlkhongduy nha"t.NguQcIC;li,trongphuongphapnghich,hamfer) chotruacboi cacphepdo J2tl4rl luin flute uj 75 q v&l 7f5omrn.Juirl d(,lctrongtht!cte',tudo tatlms(Q)hayR. BfLitmlnxacdinhs(Q)la bili loan ngl1(Jctuytntinh,trongkhibaitoanxacdinhR labililoanngl1(Jcphituytn. Xet bai toantuye'ntinh.D~ngthuc (2.35)coth€ vie'tl(,lidu'aid(,lngma tr~n N fj =:L>j\Vjj, j=l i =1,2,...,L (2.36) Khi L>N, ngu'oitadungphu'ongphapblnhphu'ongt6i thi€u d€ tlm N gia tri cuaSj' Di~unaykhongdonghlndokhokhand:1utieng~pphaila nghi~mcua bai toanthu'ongkhongduynhc1t.Ngayca khi ta bie'tf(P) chinhxac,ta clIng khongth€ xacdinhnghi~ms(Q)mQtcachduynhc1t.St!duynhc1tcuabaitoanco th€ du'Qcxacdinhbdi st!t6nt(,lihaykhongnghi~mkhongt:1mthu'onga(Q) trong Ja(Q)\V(P,Q)dv= O. R Khi phu'ongtrlnhnayconghi~mkhongt:1mthu'ong,thlnghi~ms(Q),ne'u co, la khongduynhc1t.Lop tc1tca a(Q)du'QcgQila nhancua\V(P,Q) va vung ngu6nR. D€ kh3:cphl;!cst!khongduynhc1tnghi~m,ngu'oitathu'ongtlmcachdongian hoacac gia thuye'tv~ngu6n.Ch~ngh(,lnta co th€ gia sa dQtu hoacuangu6nla d~uho~cngu6ndu'QcmdrQngvoh(,lntheomQthu'ang.Chuydingnhil'nggiadinh nay thu'ongcho phepgiambot s61u'Qngcuacac nghi~md€ It!achQn.MQtcach tie'pc~nkhacla HmcachxacdinhmQtsO'd~cdi€m, khiac(,lnhcuangu6nchung chomQinghi~m.Ch~ngh(,lnhu'HmcachxacdinhdQsaut6idacuangu6n. 2.2.1. Bai tminngu'Qctuye'ntinh Theo (2.35),tru'onghc1pd~nhay ttttru'ongla phl;!thuQctuye'ntinhvaom~t ""21tiJ-fILmYL thf!£ .Ill 76 g ~L 76o.itifllJuirL dQhaydQtuh6avadod6,baitmlnxacdinhphanb6m~tdQhaydQtuh6atudfi' ki~ncuatnI'ongha'pd~nhaytu tru'ongsinhra boi ngu6ndu'<jcgQiIa bai toan ngu'<jctuye'ntinh. Phu'dngtrlnh(2.4)matathanhph~ntheophu'dngth~ngdungcuatru'ongha'p d~ncuamQtv~tth€ voim~tdQd€u. Ne'uhlnhd(;lngcuav~th€ du'<jcbie'thltich phantrong(2.4)c6 th€ du'<jctinhboi phu'dngphaptie'n0 ph~ntru'oc.DQ di thu'ongha'pd~ndu'<jcdot(;liN di€mroinhaula gi =P\Vj i =I,2,...,N , trongd6 h~ngsO'P c6 th€ du'<jcxac dinhboi pheph6i guytuye'ntinhddngian. (xem[1]). Chungta c6 th€ xetbai toanphuct(;lphdnb~ngcachchiav~tth€ thanhcac ph~nnhohdnva sadl:mgphu'dngphapblnhphu'dngt6i thi€u d€ tinhtoanphanb6 m~tdQcuatungph~n.Cachtie'pc~nnayclingdu'<jcxemxettrongph~nsaucho tru'ongtutru'ong. 2.2.1.1. D{)tit'hoacuam{)tfing TrongmQts6mahlnhtlmdQtuh6acuadaybi€n b~ngphu'dngphapngu'<jc, Bot,BottvaHutton,EmiliavaBodvarsson(xem[1])giadinhr~ngdQtuh6ala haichi€u, trongd6mahlnhngu6ntunaybaag6mmQtdaycaca dagiac(hlnh 2.13),voicaca du'QCs~pxe'pl(;lisaDchodinhcuam6ia tu'dngungvoidinhcua , tangtu. Trong (2.36),dQdi thu'ongtru'ongtoanph~nt(;lidi€m i du'Qcvie'tl(;lithanh N ~Tj = LMj\Vjj j=l i =I,2,...,L, (2.37) trongd6Mj la dQIOncuadQtuh6aaj, \Vjjla dQdi thu'ongtru'ongtoanph~nt(;li J2tt4rt tJiit't Ihf!£ uJ 77 g PAn ';/{;tffli(}{J,.tin di~mi gayraboi0j voidOtithoadonvi.vt tniicua (2.37)baag6mto~mbOcac d<;lilU<;5ngdfidodU<;5c.Ma tr~n\!Iijcoth~du<;5cHnhtoanvoi (2.30).Tit do,chIco duynha'tmOtd<;lilu<;5ngchuaxacdinhtrong(2.37)la N giatri cuadOtithoa,va n€u N <L, nhii'ngd<;lilU<;5ngayco th~du<;5ctinhb~ngphuongphapblnhphuong t6i thi~u(xem[1]). .6Ti ...... Hinh 2.13: Mo hinhnguf/caatangtadaybiin.Dr)dj thuimgtruimg formphdn,11;duf/cdophia trenhangcaacac0, miJrr)ngvoht;mvuonggac wJi m{itgilly.MtJi 0 La tahaadiu vaidr)tahaaMj. Voi mOtdOSailchotruoccuat~ngtit,BottvaHutton(xem[1])khaosattinh 6ndinhcuaphuongphapnghich.Nhuminhhatronghlnh2.14,hd~~Ti=O0 kh~pnoi,tritt<;linhii'ngdi~mdon,noi~Ti=1nTvadung(2.37)d~tinhdOtithoa. ... «1) 1- 0 -. --- 0 [IIIITIII1TTT I I I I IU.l.lITilIIIII]IITIIJ = 2 "V II "" J (b) s= ~ c£ .!2 .~ 0 a:;c:: OJ) ~ 0/\-\/=3 " -'~i~----- -3 Hinh 2.14: nghifmkhongtJndjnhciladr)tahaacilatangta (a) dr)dj thuimgtruimg loan phdn Lakhong tra tc;zicac diim d(/fl. Tdng Latgp h(fpcac 0 vai dr)rr)ng W va d{ittc;zi dr)sauD. Dr)daycilatangLa2W. (b) Ke'tquaciladr)tahaakhiD/W=3. R6rang,mOtnghi~mth\fcchoMj gayradOdi thuongd~cbi~tnaylaxa'pXl J21u!-1'LtJiiI't Ihq.£ ui 78 grAn 7!fJoiIirmuHt khongvoi mQij. D~thaythe'hQtimnghi~mdQtuhoakhongth\ickhacnhau (hinh2.14)valu'Qngbie'nd6iph\!thuQcvao dQsaucuat~ngvadQrQngcuacac 0 (hinh2.15). BottvaHuttonducke'theokinhnghi~mr~ngne'udQrQngcua0nhohondQ mQthayhail~nsovoidQsaucuat~ng,thinghi~mngu'Qccua (2.37)1akhong6n dinh. J§- .~3 - ::;;N '::::: 2 .. :!::! ~ ~ i 1- J2o~. :! :." D/W Rink 2.15:Duongcongbiiu diJn chocaegiatrj thudu(1cilacaethinghi~m otaiJ hinh2.15.Th1!chi~nbiJi BottvaHutton[38J, Ly do cua tinh khong6n dinh nay co th~tha'ydu'oidc,mgd<;lis6. Vie't l<;li (2.38) M6i cQtcuamatr~n\Vij bi~udiencho dQdi thu'ongtru'ongtoanph~ndQc theomQtprofintrenmQt0 don,la 0j. Ne'udQrQngcua0 la nhosovoi dQsau,thi profintren0donj sera'tgi6ngvoiprofintren0j+1ho~c0j-l. Noi cachkhac,dQ rQng0 nhothi caccN g~nnhaucuamatr~n\Vijla g~ngi6ngnhau.Haytheo (2.37) d d<;lngma trn, ta co - T1 \VII \VII ... \VII M1 T2 \VII \V12 ... \VIN M2 T3 = \V21 \V22 ... \V2N M3 TL \VLJ \Vu ... \VLN MN j;!1lJ/1LOii1L Ihf!£ uj 79 q,d,L 760-m{J{J,mL ngonngud£.lis6thlmatr~nla xffu.:£)6hi6udu'<;5csv anhhu'angcuatru'ongh<;5p nayde'nnghi~mMj, taxetmQttru'ongh<;5pdongiancua (2.38), Zl =al1X + a12Y, Zz=aZlx+allY, ~ trongd6Zl,zzlad£.liu'<;5ngdodu'<;5c,al1,a12,azJ,azzlacacd£.liu'<;5ngdffbie'tvax,y lacacgiatric~nxacdinh.Nhu'chiratronghlnh2.16,haid~ngthlicnayxacdinh haidu'ongth~ngtrongm~tph~ngx,y,voi nghi~m(xo,Yo)chinhla tQadQgiao di6mcuachung. ( _\. \.'"I (0) 'Y x x xo Hinh 2.16: Saisf)'cuaZlhoi;icZ2tamchocaeduimgthllngdickchuyfntenhoi;icxuf)'ng (a)Ntu caeduimgldn nettflOvoinhaum(Jtgacl6'n,thisai sf;'nhocuaZl vaZ2khong lamanhhuangnhi~udennghi?m.(b)Ntuhaiduimgtagdnsongsongvoinhau,thisaiso' nhocuaZlvaZ2seanhhUClngrfitl6n dennghi?m. Cacsais6trongphepdoZlva Zzgayrasvdichchuy6nsongsongcuacac du'ongth~ngbi6udi€n. Ne'uhai du'ongth~ngt£.lOvoi nhaumQtg6cIOn(hlnh 2.16(a» thl sv dichchuy6ntrenkh6nganhhu'angnhi~ude'nvi~cxac dinh nghi~m(xo,Yo).Tuynhien,ne'uhaidu'ongth~ngla g~nsongsongvoinhau(hlnh 2.16(b»sais6dlinhocuaZlhayZzsegayrasail~chdangk6trongvi~cxacdinh nghi~m(xo,Yo),vanghi~mlakhong6ndinh. Hai du'ongse g~nsongsongvoi nhaune'uall ~ a21 . Du'oid£.lngt~ngtli' a12 a22 J2tUP'Loiit'Lfhf!£ ul 80 goPAn7txfflirmw.n (hlnh2.13)tlnhtr~ngnayseKayrane'ugiatri tru'ongt~idi€m i gayrabdi0j la g~ngi6ngvoigiatrjtru'ongt~idi€m i gayrabdi0 0+1)vane'ugiatrj tru'ongt~i di€m i gayrabdi0j lag~ngi6ngvoigiatritru'ongt~idi€m (i+l) gayrabdi0j. Phu'ongtrlnh(2.38)chila tru'ongh<JpmdrQngN chi~ucuatru'ongh<Jpkhao satneutren.Hangva CQtcua\!fijt~othanhcachamdutrOll.Tu do, cacpheptinh thu~nchocac~Tj tuMj choke'tquala mN hamtrolltrongkhi cacpheptinh ngu'<Jcl~ithlkhong.Honfilla,dQsauhoncuat~ngtucom6ilienh~voidQrQng cuaO.Ne'udQrQngcua0 la ra'tnhosovoidQsaucuat~ng,matr~n\!fijtrdthanh xa'uvamQtslj thayd6inhocua~TisegayramQtsais61Ontrongcactinhroan Mj. 2.2.1.2. Xacdinhhu'ongcuade)tithoa De)tITboa d~u Trongnhi~utru'ongh<Jpdiacha'tquantn;mg,ngu'oitacoth€ giasudingv~t th€ la tuhoad~umalingdl;lngthu'ong~pcuabairoanngu'<Jcd~ngnay13bai roanxacdinhdQtuhoacuamli bi€n. Khi do,cacphepdo d~cv~dQsauva dQdi thu'ongtru'ongroan ph~nthu'ongdu'<Jcthljc hi~ntren m~tbi€n. Nui bi€n nay thu'ongdu'<Jcgiasula tuhoad~uvadu'<JCgiOih~nbdihait~ng: t~ngtrenvat~ng du'oi. Phu'ongtrlnh(2.33)du'<Jcvie'tl~id~ng - R a 1 a 1 a 1 ] ~T(P)=-CmF.Vp Mx(Q)--+M y(Q)--+Mz(Q)-- dv. ax'r ay'r az'r Khi M lad~u,taco ~T(P) = M,[-CJVP 1:'1V]... .I211fjrLtJiiI'L tJu,£ uj 81 q ni.n 7J{;oQ1(J{J,jyL + M,[ -CJ\7p I~'1v] + Mz [ -CmF.Vpf~~v ]8z'rR ilT(P) = Mx~x(P) + MA/P) + Mz~z(P) ~ (2.39) voi ~X, ~Y' ~z la s6h,;mgtichphantuongling,la'ytheod~nghinhhQccuanui bi~n.Gia sa dQdi thuongcua truongto~mphftnduQCdo t~iN vi tri khacnhau, (2.39)co th~duQcvie'tl~i(j d~ngmatr~n ilT, ilT2 ~lx ~2x ~lY ~2Y ~lz [ Mx ] ~2z M. y I Mz~Nz (2.40)= ilTN ~Nx ~NY Ba CQtcuamatr~n ~jjtrong (2.40)bi~udi€n cho dQdi thuongtruongtoaD phftnt~icacvi tri khacnhau,giasadQtuhoadonvi theocachuang x, y, z tuong ling.M6i thanhphftncuamatr~nco th~duQctinhb~ngcachsad\lngphuong phapthu~n,nhungd~lamduQcdi€u naydoihoiphaicoslfxa'pxi hinhd~ngv~t th~theocacmahinhdongian,ch~ngh~nnhula mQtt~pcackh6ihQpchii'nh~t, mQtch6ngcacbanmong,haymQtch6ngcactftng.Lucdo, (2.40)choN phuong trinhtheo3 bie'nmatagQila bathanhphftncuadQtuhoavachungcoth~duQc tinhb~ngky thu~tbinhphuongt6i thi~u(xem[1]);tlic la, timMx,My,Mz saD cho d?i IU<;1ng N 2 "'" ' 2 E = L.i(ilTj -ilTj) i=l la nhonha't,voi ilT' j, i=1,2, ...,N la cacgiatridQdi thuongdoduQc.Honnii'a, ~I2Il1P'LmYL fhf!£ vj 82 q nin 7It5oidrmum b~ngcachtrudi truongdiaphuongF(P), (2.39)trdthanh ~T(P) = Mx~x(P)+ My~/P) + Mz~z(P)- F(P) Cac d<:;tiluQngkhacnhauco th€ duQcdnh d€ daubgia kha nangtrungkhit giua ma hlnh don gian nay voi du li~u do duQc. Truong ph~n du, e = ~T - ~T' vathamsf)I I I N LI~Til r - i=l- N Lied i=l duQcdungd€ daubgiasvsaikhaccuamahlnh. D()tit boakhongd~u Trongth1!ct€, conhi~uIi dod€ dftnd€n d('>tuhoala khangd~u.Nui bi€n coth€ duQchlnhthanhlientl,lcclingnhucacthanhph~nhoahQCvakhoangcha't khangd6ngnha'tcuanuibi€n gayranhungthayd6idangk€ cuatruongtu. M('>tsf)nghiencUudia ly chophepta phanngu6ntu thanhcacph~nroi nhau,m6iph~nla d('>tuhoad~u.Khi do,nguoitavftncoth€ dungphuongphap blnhphuongtf)ithi€u d€ Hmd('>tuhoatrongm6iph~n. M('>tcachti€p c~nkhacduQcd~xua'tbdi Parkeret al. (xem[1])b~ngcach thayvi~cxemxetd('>tuhoathanhhaithanhph~nd6ngnha'tvakhangd6ngnha't, saDcho thanh ph~nkhang d6ng nha'tnh6 d€n milc co th€. Y tudngnay duQctom Hitnhusail D('>dithuongtruongtoanph~ncuanuibi€n duQc hobdi (2.33) - f 1 ~T(P) = -CmF.Vp M(Q),VQ-dv R r Duoi d<:;tngdongianhon J2um'LtJiULlime uj. " . 83 g uin 7HJo-ai(llJULI'L ~Ti = - JM(Q).Oj(Q)dv R (i =1,2,..., L), (2.41) trongdochIsfSi d~chIcacdi~mquailsatroi nhau.HamM(Q) trongdAngthuc trenla mQthamvecto.T~pta'tCelcachamtithoanayt~onenmQtkhonggian Hilbertvoh~nchi€u.HamGreenGj(Q)trong(2.41)gifSngvoivectotQadQtrong khonggian3 chi€u, va duQCgQila hamtQadQ.Tich trongcuahaithanhph~n A(Q)vaB(Q)cuakhonggianHilbertnaydu<Jchobdi (A,B) = JA(Q).B(Q)dv R tuongtt,I'nhutichvohuangcuahaivecto.Tit do, (2.41)du<Jcvie"tduoid~ngtich vohuang ~Tj = (M,OJ, (2.42) va ~Tj trdthanhhamtuye"ntinhcuaM(Q). "DQIon"cuamQtthanhph~nduQc dobdichuffn 1 IIAII = (A,A)2 1 = [pAl'dvr vadQsaikhaccuahaithanhph~nduQc hobdi IIA- BII. TrongphuongphapcuaParkeret aI.,dQtit hoaM(Q) trong (2.41)duQc phantichthanhhaithanhph~nd6ngnha'tvakhongd6ngnha't, M(Q) = Mo + MN(Q), voiMola mQtvectoh~ngsfS.Phuongtrlnh(2.42)trdthanh ~Tj = (Mo,GJ + (MN,GJ (i=I,2,...,L) (2.43) Thanhph~nM(Q) cogiatrjnhonha'tla MN(Q)la dQtithoag~nd€u voiba't kyMochotruoc. J2Wp'Lo£ULl~ uj 84 g ,,4,'L7I{joOit'IlhtiIL Xettruongh<jpModffbie't.Lucdo,thanhph~nMN(Q)cochufinnhonha'tvai di~uki~nthoa (2.43)vaimii=I,..,L.,khita'tcacacd(,\ilu<jngd~ubie'tngo(,\itru MN(Q).Parkeretal. chIrar~ngdQtuhoanaycoth~du<jcbi~udi€n b~ngkhai tri~nhamGreen, L MN(Q) = IajG/Q). j=1 (2.44) Ta't ca cach~sO'khaitri~naj 0=1,2,...,L) d~udu<jcxacdinh.The' (2.44) VaG(2.43),tadu<jc L ~Ti = (Mo,G)+ Ia/Gj,G) j=1 L = (Mo G. ) + "aT.., I L.J J 1J j=1 (2.45) tich trong lij duila matr~nGram.Ta'tca cacd(,\ilu<jngtrong (2.45)d~u dffxacdinh, L h~sO'khaitri~nIa xacdinhmQtcachduynha'tbdi Cj lakhongsuy bie'n.Vec toh~ngMola chuaxacdinhtronglingdl;lngdQtuhoanuibi~n,vala d(,\ilu<jngcgntinh. Bai toand~trala : TIm Mo vaMN(Q)d~ MN(Q)cochufin nhonha't.MothuQckhonggiancon3 chi~ucuakhonggianHilbert,va co th~ du<jcvie'tduaid,;mgtudngtvnhu(2.44) 3 Mo = IPkXk k=1 (2.46) vai Xk, k = 1,2,3 ladQtuhoaddnvi theo3huangtrvcgiao.The't6ngtrenVaG (2.45)tadu<jc 3 L ~Ti = IPk(Xk,G) + Ialij k=1 j=1 (2.47) Tu do,chungtasetlmcacthamsO'PI. P2,P3,aI. a2,..., aL d~ ,.e,'.tP'LomL 1Ju,.el{j 85 q uin ';J()OO1fllhtin I ( L L J 2 11M-Moll = ~ajGj' ~akGk (2.48) la nhonha't,vaigiathuye'trangbuQcla (2.47).Ta'tcacacd<;lilu<;1ngtrong(2.47) va(2.48)d~udffxacdinh,tnIL+3thamsf)~J,~2,~3,aJ, a2,..., aL.Parkeretal. [210] dua ra cach xac dinh nhungtham sf) tren b~ngphuongphap nhan tU' Lagrange.Vai (L+3)thamsf)nay,huangtuh6ad~u,du<;1cchobdi (2.46).ChuY dog, ne'udQtu h6a thvcsv d~u, (2.48)tri<$ttieu, (2.47) gian lu<;1cthanhd<;lng (2.39). 2.2.2. Bili toaDngtiqcphituye'n. Truongthe'd ve'traicua(2.35)la mQthamtuye'ntinhcuasvphanbf)khf)i lu<;1nghaycha'tu.Ch~ngh<;ln,Svga'pdoidQIOncuadQtuh6a,lamga'pdoidQ IOn cuadQdi thuongtruongtoanphgn,trongkhiSvga'pbadQtuh6aselamga'p badQIOncuadQdi thuongtru'ongtoanphgn.MQtcacht6ngquat,mQth<$dU<;1c gQila tuye'ntinhne'un6thoaman:Ne'ufl(P) la truonggayrabdisvphanphf)i ngu6nSI(Q), fir) la tru'ongayrabdisvphanphf)ingu6nS2(Q),thltruonggay rabdisvphanphf)iasl(Q)+bs2(Q)chidongianla truongafl(P)+bf2(P).(a,b la h~ngsf)) Di~unayse khongdungchocacthongsf)khaccuangu6n.Truongthe' khonglahamtuye'ntinhcuadQsail,dQday,hayhlnhd<;lngcuangu6n.Ta'tcacac thamsf)trenduCJchuatrong\V(P,Q)vatronggiOih<;lncuatichphantrenth€ tich R d (2.35).PhuongphapngU<;1cduaracachdnhtoancacthams6tren,vadU<;1c gQila cacphuongphapkhongtuye'ndoh,nhungtrongthvcte'cacphuongphap khongtuye'ntinhd§nde'nsvdongianh6acacgiathuye't,madi~ud6l<;liamcho baitoantrdnentuye'ntinh. J2tltpL om'LIhLf-R-uj 86 g ~L ';J(Joai{J{hmL 2.2.2.1. ffinh d~ngngu6n Caephu'dngphapl~p Nhudad~c~pd ph~ntruac,mohinhphuongphapthu~nduQcxaydl!ng theo3 buacxU'ly. DQdi thuongduQctinhtumohinhvaduQCsosanhvOidQdi thuongquansatduQC.Mo hinhseduQcdi~uchinhvahoanthi~ntheo~sl!sosanh. BabuacxU'ly seli[Lpdi li[Lpl~ichode'nkhimohinhmaithoamanyellc~udi[Ltfa. ChungtasexemxetchungtheophuongphapnguQc. c; ---r--r, CO]J . . -, . L_J .: " II - '.', ',' ',_11\ Hinh 2.17..Ma hinhthittdi~nngangcuaLuuvifCtramtich,trongphu(jflgphapcuaBatt. Luu vifcdu(Jcgia thuyttLamiJrQngvah(;lnvadu(Jcchia thanhcackhoihQpchilnh(jt, mJi khathQptrenmQttru(mgdilm. MQtvi dl,1duQcduarabdiBottd~danhgiad~ngthie'tdi~nngangcuahtu vl!ctr~mtich.Trangphuongphapnay,htuvl!cduQcgiathuye'tlamdrQngvoh~n theomQthuangvacom~tdQd~u.Ltfuvl!cduQcchiathanhN kh6ihQpchii'nh~t, mdrQngvoh~ntheohuangsongsongvai Itfuvl!Cva mdrQngtheochi~uSailtj, j=1,2, ...,N (hinh2.17).Chi co N truongdi~mgi (i=1,2, ...,N) dctheocacprafin vuonggocvai Itfuvl!cduQcxet,va m6i truongdi~mco tamphiatrenm6ikh6i. Gia thuye'tm6ikh6ila mQtphie'nvoh~ntheochi~ungang.Truongha'pdfincua phie'nvoh~nduQcchobdi g = 21typtk, vai p,t la m~tdQvadQdaycuaphie'nvoh~n,k lavectodonvi theohuang J2tltPL miLLlltf!£ £{j 87 g 1'&11';J{)oiLi(/(j,nLL th£ngdung.Luc d6,dQdaycuam6iphie'nvoh(;lndu<;5cdoboi gj (I) - - , tj - 27ty~p j =1,2, ..., N. ChI s6trenchIs61:1nl~p.Nen phuongphap3buocsed~nWi Yi<%cchlnhsaa dQdaycuakh6i.Cacbuocdu<;5ctie'nhanhnhusan,yoi k d€ chI s61:1nJ~p 1. Truong gjk) du<;5ctinht(;lim6idi€m quailsatgayra boi tit.cacackh6i, giasar~ngdQdaydu<;5ctinhtheophepl~pke'truoc. 2. Ph:1ndu gj - gjk) du<;5ctinh t(;lim6i di€m quail sat. 3. Sv xip Xl phie'nYOh(;ln du<;5csa d\lllg cho toi khi uoc lU<;5ngdu<;5cdQ day maioSv hi<%uchlnhcho m6i kh6i du<;5ctinh duOigia thuye'tdng m6ikh6ila mOtphie'nYOh(;lnyoimOtdQdaytinhtheoph:1ndu,tucla dQdaymoicuaphie'nYOh(;lndu<;5ctinhtheocongthuc ( (k)) t~k+l)= gj - gj +t~k) J 27ty~p J Ba buoctrendu<;5cl~pdi l~pl(;lichode'nkhi mohlnhmoi la thoamanyell c:1u. Cordellva Hendersondahoanthi<%nphuongphapnayb~ngcachsad\lllg dli li<%ud <;5cdo(ho~cnQisuy) trenmOtm(;lngluoicackh6ichlinh~t,lucd6cac ngu6ndu<;5cxemxettrongkhonggian3 chi€u. Cac ngu6ndu<;5cmohlnhnhumOt b6 cackh6i chli nh~t,m6i kh6i c6 truonghip d~nrieng(hlnh2.18).DQ daycua kh6i tj, j =1,2, ...,N du<;5cdinhnghIac6 lien quailWi mOtb€ m~tthamchie'u,d6 la b€ m~tc6 th€ du<;5cxemch£ngh(;lnnhum~tdlnh,ho~cm~tday cuatoanbQ cac kh6i. Tu'ongtv nhuphuongphapBott,dQday kh6i band:1udu<;5cuoc lu<;5ng b~ngcachgiathuye'tm6ikh6i la mOtphie'nYOh(;ln.Do d6,ty s6 J2tltP'L (JaIL /hf!£ uj 88 g uin 360mrmuitt t~k+l) ~ --L- = (k) t~k) gjJ du'Qcsu d\lngd€ di€u chInhl<;LidQday cuaphie'nvo h<;Lnxa'pxl. Phu'ongphap3 bu'ocsetinhtoan,sosanh,di€u chInht<;Lim6ibu'ocl~p. R.:.:::nccsurfacc; Hinh 2.18: mahinh3chducilaphU{fflgphapZ{IpdoCordellva Henderson d~xuat. DI) day cila khaT-phf:lthul)cvao b~m{Itthamchitu chung. TruOngh5pdJn quansatdu(Jcdo trenml)tZullichilnh{jt. M9t cachtie'pc~nhoikhacdu'Qcdu'arabdi JachensvaMoring.Gi6ngnhu' haiphu'ongphaptren,hQu'oclu'Qngcacd<;Lnghtuvf!cvoim~tdQtr~mtichthffp, nhu'nghQcotinhde'nkhanangm~td9thayd6icuat~ngn€n. Phu'ongphapnay chiaphepdotru'ongha'pd~nthanhhaiph~n: thanhph~ngayrabdibanthanhill vf!cvathanhph~ngayra sf!bie'nd6im~tdQcuat~ngn€n. Xet g bi€u di~ncho tru'onghffpd~ndu'Qcquailsat,saukhidalo<;Libocactru'ongdiaphu'ong.D~tg= gb+gd,vOigbla d9di thu'onghffpd~ngayrabdit~ngn€n vagdla dQdi thu'ong gayrabdicaclOptrffmtichm~td9tha'p.Phu'ongphapdu'Qctie'nhanhtheocac bu'ocsau: 1. Bu'ocl~pthunha'tgiasur~nggbdu'QcxacdinhChId nhungvi tridinh xutrent~ngcosdtr6ilenvatinhtoanmQtb€ m~tnhanvoinhungdu J.!.tU!1'LtJiit'L fJu,£ uj 89 grAn '3tJoa1rmum li<%unay,nhuchIrabaiduongnetg~ch(hlnh2.19).Dieunayl~pnen xa'pXlthlinha'tgb(l)chotruongha'pd~ncuat~ngcosagb. 2. gdduQCxa'pxIl~n thlinha'tbai pheptrUcuatruongha'pd~nquailsatg cho gb(l).Ph~ndu mOi gd(l)duQCsa d\lngd~Omxa'pXl thli nha'tcho chieusaucuat~ngcosasad\lngphepxa'pXlphi€n banvQh~ntuong ttfnhuphuongphapBott. 3. Hi<%ulingha'pd~ncuahillvtfccoth~duQctinhbaimQts6phuongphap khacnhau.La'ytruongha'pd~nt~ngcdsatrltchok€t quaOmduQccho taphepxa'pXlk€ ti€p cuatruongha'pd~nt~ngcdsa gb(2). Iteration1 \ /\ CdGravityV Observ Iteration1 , / I / , Hinh 2.19..ChiaphdnducuatntilnghapdJn thanhhaiphdn,thanhphdngayra biJisf!thayd6imtJtdQtrongtangcosiJvathanhphdngayrabiJihtuvf!C.Cacnot chambiiu thichophepdotrenphdntr8ileucuatangcosiJ,cachlnhtrimlanlim trenbi m(ittramrich. Ba buGCnayduQcl~pdi l~pl~ichod€n khi nghi<%mla thoamanyell c~ud~t fa. H<%qua la : OmduQChlnhd~ngcuaIttuvtfctr~mtichm~tdQtha'pva truong ha'pd~ncuat~ngcosakhongchillanhhuangcuaIttuvtfc. .1211f/l'LomL tluJ-e .uj 90 g"&L dVomrnJuuL Tuye'ntinhhoahili tminphituye'n. M~cdli tntongthe'phl;!thuOcphi tuye'nvao cacthongsO'tv nhiencua ngu6n,nhungSvphl;!thuOcnayla "gffntuye'ntinh"khichiconhii'ngthayd6inho trencacthongsO'.Ch~ngh(;ln,truongthe'cuamOthlnhkhO'idagiacphil thuOcvao tQadOcuacacdinhcuadagiactheohamarctangva logarit(xem (2.23),(2.27), (2.28)),tucla truongthe'lamQthamphituye'ncuatQadOcacdinhdagiac,mO'i quailht%phituye'n aycoth€ lamchotuye'ntinhb~ngcachthayd6ichutit cac thongsO'.Vi dl;!truongha'pd~nva tITtruonggay ra bdi mOtt~pcac hlnhkhO'ida giac co th€ du<;1ckhai tri€n theochu6iTaylor dva tren thay d6i tQadOcua da giac.Ne'usVthayb6i tQadOla nho,thlchu6iTaylorco th€ du<;1crutgQn,ham phl;!thuOcvaosvthayd6inaydodotrdnentuye'ntinh.MOtthu~toandu<;1cd€ xua'tnh~mxacdinht~pcachlnhkhO'idagiacto'tnha'tlingvdimOtdOdi thuong chotrudc,vdi cacd(;lngthie'tdit%ngangcuacachlnhkhO'idu<;1cchinhl~pbdi IU<;1nghotrongphuongphapblnhphuongto'ithi€u tuye'ntinh. Cachtie'pc~nnaydU<;1cmotatheocacd(;lngkhacnhauchocadOdi thuong ha'pd~n(xemCorbato,AI-Chalabiva Colestrong[1])va dOdi thuongtIT(xem Johnson,McGrathva Hood,Rao va Babutrong[1]).Cac d€ c~psail tomt~t phuongphapd€ xua'tbdiJohnson. Phuongtrlnh (2.2) va (2.30)cling ca'pthanhphffnha'pd~ntheo phuong th~ngdungvadOdi thuongtrtiongloanphffntuongling cuamOthlnhkhO'ivo h(;ln mdrOngtheomOthuangvdim~tdOhaydOtIThoad€u vacothie'tdit%nladagiac N-dinh(hlnh2.20).D~tAi la mOttrongL phepdoroi r(;lCcuadOdi thuongha'p d~nhaydOdithuongtIT. Ai = A(x~,ZI' X2'Z2'..., XN' ZN'Xi' z) Jl.tuJ-rLoibL I~ uj 91 g nhL JfJoai (J{JuUL = A(Xi,Zi'W) i =1,2,...,L vai (Xi,Zj)la vi tricuaphepdothui. (x\' Z'i) la tQadQcuaN dlnhcuadagiac, vaduQckyhi~ubdimQtmang2Nchi€u w.Honnlia,coth~xemcacdlnhcuada giaccoth~dichchuy~ntheohuangtuyy. ~B ~B .\" x .: Rink 2.20.. Xapxl cilamQtngu6nhaichdu blingmQthinhkhatmiJrQngvah{lnvcJithief difn nganghinhdagiac. Vecf{JM va BfahinhchituciladQtah6avatatnt(rngxung quanhfenm{itph/lngx, z. GQiAi va Ai tuongungIa dQdithuongquailsatvadQdithuongdoduQct~i mQtdi~mquailsat.ChungtaseHmvectoWsaDchoblnhphuongsais6 L 2 E2 = 2:[Ai -Ai(W)] i=l (2.49) la nhonha't.Bdi Ai la mQthamphi tuye'ncuaw, chungtakhongth~sadvng phuongphapblnhphuongt6ithi~u.Thaythe',chungtased6icacthanhph§ncua WmQtluQngnho.Ne'uthayd6itrongWla nho,Ai sela hamg§ntuye'ntinhd6i vai nhlingsl,thayd6inay. Chu6iTaylorla congcvthichhQpd~thl,tchi~ncongvi~ctren.Vi dV,giatri cuahamf(x,y)coth~duQcngo~isuytheo(x+~x,y+~y)bdichu6i .£tapLoillL thf!£-uj 92 g PAn7/f;oidfJlJuin f(x +~x,y+~y) = f(x,y) + 8f(x,y)~ + 8f(x,y)~y + ... ax ay voicacs6h~ngb~ccaDhondtiQclo~ibobdi~x va~y dtiQCla'ydunho. (k) 9 - ? ( "" " ) ? Ttiongtl!,chungta xem w Ia biendiencua xI' zl' X2'Z2'..., xN' ZN 0 phepl~pthilk.Lucdo,khaitri€n chu6iTaylorcuadQdi thtiongt~idiemi la 2N 8 Ai(W(k+l») ~ Ai(W(k») + L-Ai(w(k»)~W~) m=lawm (i =1,2, ...,L) (2.50) voi ~W(k)= W(k+l)- W(k) ChuY/ r~ng A. (W(k+l») la mothamtuye'ntinhcuam m m . I . ~w~)voi m=1,2,...,2N.Tilc lachungtadiituye'ntinhhoabairoanphituye'n. Thay the'(2.50)vao (2.49)tadtiQc L [ 2N ] 2 E2 =L Ai -Ai(w(k») - L~Ai(W(k»)~W~) i=1 m=l8wm Bi€u thilc Ai(W) dii dtiQcsuyra cl;1th€ tu (2.22)va (2.30), chungtaco th€ Hm bi€u thilcd~ohamriengcuachung.Do do,chotrtiocmQtmangkhdidfiucua thams6 W(k)thl cac iin s6cfinHmtrong trenla ~Wffi(k) , m=1,2, ...,2N. B€ Hm caciin s6naysaDcho E2la nhonha't,chungtatinhcacd~ohamriengcuaE2 rheaWj,j=I,2,...,2N, vaxeth~ t [ Ai - Ai(W(k») - I:~Ai(W(k»)~W~) ] . [ ~Ai(W(k») ] =0 (j =1,2, ..., i=l m=lawm awj 2N.) ? trongdochungtal~itie'ptl;1Clo~ibocacs6h~ngb~ccaD.d d~ngmatr~ntaco 2N a. ="G .~W(k)J ~ ffiJ m m=l (2.51) J2tUP'LtnUL lluJ-e uj 93 f1,d,L 7HJoa1t1lJuin voi aj =I[Ai -Ai(W(k»)J~Ai(W(k») i=\ 8wj (2.52) Gmj =I [ ~Ai(W(k»)~Ai(W(k») ]i=\ 8wm 8wj (2.53) cacbu'ocdu'<Jctie"nhflllhnhu'sau 1. ChQn mQt t~p cac gia tri ban dffu cua tQa dQ cac dlnh W = (x\, ZIpx'z,z'z,...).Gia sar~ngtfitcacacdlnhcuadagiac1a co th6dichchuy6ntuyy. Trongthl;tcte"thl tased6hffuhe"tcacdlnh cuadagiac1ac6 dinh,chidichchuy6nmQtvai dlnhtrongquatrlnh tinhloan. 2. T' h A .( (k») , 8 A .( (k») ,.. - 1 2 L ' - 1 2 2NIn I W va - I W VOl 1- , ,..., va m- , ,..., . 8wm 3. Tinhaj va Gmjtheo(2.52)va(2.53). 4. TImmatr~n ghichdaotrong(2.51)d6tinh~W(k). 5. Bi€u chlnhWchophuh<Jpvoi yell cffud~tfa. Bu'oc2 du'<Jcl~pdi l~pl~ichode"nkhi nghi~mOmdu'<Jc1ahQitv, tucla, cho de"nkhi E2du'<Jcnit gQntOimQtmuctheoly thuye"tchIra hay~Wm,m =1,2,...,2N aunho. v~uth~compact Bfitky sl;tphanph6ikh6i1u'<Jnghaychfittii'cuatru'onghfipd~nhaytru'ong tii'tu'ongling,du'<JcOmd6thoaman(2.35).Phu'ongtrlnh(2.35)la rangbuQcchu ye"u.Chungtacoth6du'athemvaomQtvairangbuQckhac.Vi~cthemvaocac ./211ijPLtJ-ii#L1Ju,e uJ 94 g ~L ']{)oa;r1lJuYL rangbuQcdalamql'cti~uhoahaycvcd(;lihoamQtvai thuQctinhvahu'angcua .ngu6n.Ch~ngh(;ln,LasvaKubikdamatavi~clamsaod~tlmmQtv~th~vai th~tichnhonha'thoamancacrangbuQc.H9 chia ngu6nthanhN kh6ichlinh~t, haiho:)cbachi~u.llinh 2.21chotru'ongh<jp2chi~u,giclsadingm~tdQcuam6i kh6i la d6ngnha't.Cho bi~ttru'ongha'pdin dodu'<jct(;liL vi tri roi r(;lc\rangbuQc (2.35)du'Qcvi~tl(;li(j d(;lng N gj =L\VjjPj + ej, j=l (i =1,2, ...,L) (2.54) g; 7 /' --...." '" ,,'" -. ." ""-~ -. "'.-'" ""Oo-- _..- -..r_- - =+-++=--+3:- ,-J - ==£:1::~-:._- . ,. ~.:-->-+-g=t-._' ::-'-~~--::t:--,:::1:- - =i=i:::.::1=:J::j.:: p~./ ",/ Hinh 2.21..Mi~nngu8ndu(lcchiathanhcaekhatchflnh(tt,veJiso'Lu(lngcaekho'ivu(lt quaso'Lu(lngcaephepdo.CJ day,xetngu8nLahai chiiu, vai ej, i =1,2, .."L, la cacsaiso'cuam6iphepdo.N~uL >N, chungtacoth€ sad1,mgphu'dngphapblnhphu'dngt6i thi~ud~tlmm~tdQchom6ikh6i,di~u naydadu'Qcd~c~ptrongcacphgntru'achotutru'ong.Vai L <N va doihoi phgnkhackhangcuami~nngu6nnhod~nmilccoth~,mi~nla svdanhgiahlnh d~ngcuav~th~compactla thoamanL phepdotru'ongha'pdin noitren. N~ud:)t 2 { o I' Pj1m = E~O pf+E 1 (Pj=0) (pj :;t:0) - 1:::, -'-- .-4 £'UpL IJiUL IhLfl! uj 95 gnin 7tJoairnltan thlth~tich(haydi~ntich)cuav~th~duQcchobdi N p~ V = ~VlimL ~ 'E~O . 1P. +EJ= J vai~V la th~tichcuamQtkh6irieng(ho~cdi~ntichcuamQthlnhchii'nh~trieng trongtruonghQphaichi€u).Thu~tminhIedodbihoiphaiQfcti~uhaad<;liluQng vahuang N p2 L q =L 2 j + Lwie~, j=l pj +E i=l (2.55) trongdoL bi~uthucrangbuQccua (2.55),vaiEduQcchQndunhovaWi, i=1,2, ...,L lahamnhi€u. Dodo,nghi~mOmduQclac1;l'cti~uhoacath~tichcuav~th~vat6ngtrQng cuablobphuongphgndu. N giatq cuaPj coth~b5ngkhongho~ckhackhong. Caca khackhongtrongmi€n ngu6nclingcffpmQtuacluQngcuahlnhd<;lngcua v~th~compactthoamancacphepdotruonghffpdftn. 2.2.2.2. Dt)saungu6n CacphuongphapdaubgiadQsitucuangu6ntuhayngu6nhffpdftncoth~ duQcchiathanhhailOp:MQtlopphuongphapphilotichmQtdQdi thuongdon,co l~p. Lop phuongphapcon l<;liphilo dch profin cuanhi€u ngu6n.MQt vai vi d\l cuam6ilo<;liseduQcnghiencUud phgnduaiday. PhuOngphapPeters Ne'udQdi thuongtUduQCgayra bdi mQtv~tth~hai chi€u theochi€u dQc, co dQtu hoad€u theochi€u dQCva dQsitulOn,hie do dQsitucuav~tth~co th~ duQctinhxa'pXlb5ngphuongphapd6thi (hlnh2.22). J211fj.lL tJiUL thf!£- £Ii 96 grAn ';j{joQir1lJuUL Ve haidu'ongth~ngsongsong voih~s6gocb~ngYzgradientclfcd~icua dQdi thu'ong,mQtdu'ongtie'pxucvoidi€m clfc cuadQdi thu'ong,du'ongconl~i tie'pxucvoi di€m clfc ti€u cuadQdi thu'ong.Khmlngcachtheophu'ongn~m nganggiuahaidu'ongth~ngla ty l~thu~nvoidQsancuav~tth€. H~s6tyl~la 1.2chonhungv~tth€ mongva la 2.0chonhungv~tth€ day;giatq 1.6thu'ong, duQCsadl;lngchungchocacv~th€ khac. " ,,'I "x " /~I" "...r-.. " //J / / / / / / ,- / ,- //~ // / J ' / . / J / b // b / ' a // 20" ;' x =1.2 d if body very thin x =2.0 d if body very thick x =J.6 d for intermediate thickness Rinh 2.22: Minh hQachophu(JflgphdpPeters Phu'ongphapPeterdoihoinhi6uslfdongianhoacacgiathuye'tv6v~th€. Tuynhienphu'ongphapthlfchi~nd€ dang,chicc1ngia'"yvevabutchIchungtaco th€ daubgiasoluQcvanhanhchongdQsancuangu6ntrongnhi6utr~ngthaidia ch§tkhacnhau. DQsaucticd~i ChungtaxetmQtvaiphu'ongphapthu~nti~n,HnhloannhanhchongdQsan clfcd~icuacacngu6n,du'Qcphattri€n bdiSmith,Bottva Smith(xem[1]).Cac phu'ongphapnaydlfatrend~ohamb~cnh§t,b~chai,b~cbacuadQdi thu'ong h§pd§nvadQdi thu'ongtu,du'QcdodQctheoprofin(hlnh2.23).Chungd~cbi~t r§t hi~uquaVI chungkh6ngdoi hoi gia thuye'tnaov6 hlnhd~ngcuav~tth€ ngu6n. J21lf/l'L milL f~ ui 97 g Nin ';/{Jo.QirnJuuL .A(:\")",.....- \ d .::-\ (:.:, I ~ \ d:t::: I""<%.. L I dA(_:) IrL>.: '..a... --0---- Hinh 2.23.. DQsauqtc d(licilangu8ndlfatrend(lohambljcmQt,bljchai,bljcbaciladQ di thuCln.Profile A(x) bilu ddn dQdi thuClngtithaydQdi thuClnghapJan. BQ saugiai h(;lntheoSmith, Bott va Smithdu'<Jctomta:tnhu'du'Oidaycho ca dQdi thu'ongtUva dQdi thu'ongha"pdftn.Trong cac ba"td&ngthucnay A(x) bi~udi€n la profincuadQdi thu'ongha"pdftnho~cdQdi thu'ongtu.Trongtru'ong h<Jptru'ongha"pdftn,A(x)bi~udi€n thanhphftnha"pdftnth&ngdung.Trongtru'ong h<Jptu tru'ong,A(x) bi~udi€n cho thanhphftncua tu tru'ongsongsongvai ; = (;x,;y,;z)(ch&ngh(;lnA(x)la dQdi thu'ongtru'ongtoanphftn,; la vecto theohu'angcuatru'ongKungquanh),thams6dladQsaucuangu6n.Ky hi~u A =A(x), A' = dA(x) A" = d2A(x) Alii - d3A(x) dx ' dX2' - dX3 chIs6"max"d~chIgiatri IOnnha"tthudu'<JcdQctheotr\lCx. Thams6/)'wla ml'a dQIOncuaffiQtdQdi thuongd6iKung,tucla,khoangcachgiuavi tridQdi thuong d(;ltgiatri qtcd(;livavi tridQdi thu'ongd(;lt'12giatriqtcd(;li.Gia triqtcd(;licua ffi~tdQhaydQtuhoaduQckyhi~uboiPmaxhayMmax.Cach~thucsauduQcsuy ratrongh~EMU Vi) di thliunghill'dlin3chilu ,.8ujn oiULIhf!P uj 98 g,d,L 'dtJoaifllIuin TruiJngh(1pchung d :s;5.40 YPmax I A" I max d2 :s;6.26 YPmax I A"' I max TruiJngh(lpm(itd()hoimtoimduang(ho(ichoimtoimamntudauthayd6ithich h(lp) A d :s;1.5jA1' '\Ix d2 < A- -3- A" ' '\Ix E {A"<O} A d :s;0.86~ A' max d :s;2.70 YPmax I A" I max d2 :s; 3.13 YPmax I A"I I max EJi)dj thlldnghapdan2 chiiu M(itd()hoimtoimduangho(ichoimtoimam. d < A - jA1' '\Ix d2 < A- -2- A" ' '\Ix E {A"<O} d :s;0.65 Amax IA'lmax '£'upL oillL 1Ju,.euj 99 g ~L 71&a1t1lJuin d ~ f'j,w (dQdtthuangdoLxang) 81)dj thlilingtit3 chi€u Khongcosf!rangbuQcnaochodQtithoa 1-2 -2 -2- M d ~ 6.28(4rx +3ry + 3rz)21Afaxmax 1 2 -2 -2 -2 - M d ~ 9.73(3rx+2ry + 2rz)2, rxA" max DQtahoasongsongvacoClAngchdu tc;zimQidilm 1-2 -2 -2 - M d ~ 3.14(4rx+3ry + 3rz)21A'C:x 1 2 -2 -2 -2 - M d ~ 4.87(3rx+2ry + 2rz)21 rxA" max r vaM cohuangthangdang M d ~ 5.18 max IA'lmax d2 ~ 6.28 Mmax I A" I max r vaM cohuangthangdang,tc;zimQidilm M coClAngchiiu M d ~ 2.59~ IA'lmax d2 M~ 3.14 max IA"lmax 81)dj thlilingtit2 chi€u JlJltpt luU'L Ihlf1!-uj 100 gNin 7IJoairmHut Kh6ngco sZ;trangbuQcnaochodQtithoa I d ~ 8(;~+ ;~)2Mmax IA'lmax I 2 -2 -2 - M d ~ 9.42(rx + rz)2 1 m l ax A" max M songsongvacungchiiu tt;limQidilm I -2 -2 - M d ~ 4(rx + rz)2~ IA'lmax I 2 -2 -2 - M d ~ 4.7l(rx + rz)2 1 m l ax A" max PhuongtrinhEULER Cac phu'ongphapu'oc1u'<;1ngdQsaud~c~pd trenla thichh<;1pchodQdi thu'ongayrabdicacngu6ndon,col~p.MQtlOpcackythu~tkhacxemxetdQdi thu'ongha'pd§:nhaydodi thu'ongtitgayrabdinhi~ungu6ntu'ongd6idongian. Ch~ngh<;tn,phu'ongphapu'oclu'Qngvi tri cuamQtv~tth€ dongian(donqfc, lu'ongclfc,...)titmQtvaiphepdocuatru'ongtitho~ctru'onghffpd§:n,coth€ ap dl;lngVaGmQtprofindaib~ngcachchiaprofinthanhcac0 cuacacphepdolien tl;lc,m6i0 la mQtu'oc1u'<;1ngdonchovi tri ngu6n. Phu'ongtrlnhEulerd d<;tngt6ngquatdu'<;1cchobdi LVf =-nf Cachamf thoaphu'ongtrlnhEulerdu'<;1cgQila thufinnhfftb~cn,ne'uchung clingthoaLaplace,chungcoth€ du'<;1cbi€u di~ntrongtQadQcfiunhu'lat6ngcua cachamdi~uboam~tcfiu.£)<;tohamcuabfftkyhamthufinnhfftnaGclinglaham J2,ItPL tJdIL Ihq.e uj 101 g nin 7J{}oQ1rnJuln thu~nha't.Ch£ngh(;ln,la'yd(;lohamriengtheox cahaivficua Euler a a a a -[r.Vf] = -f +r.V-f = -n-fax ax ax ax r.V [ ~f ] = -(n+1)~fax ax vad§:ntoi :x f la thu~nha'tb~c(n+1) D€ tha'y,ham! =Vr thoaphu'dngtrlnhEulervoin=1.Do dothficuamQt cha'tdi6m(haymQthinhc~ud6ngnha't)la thoaphu'dngtrinhEuler.Bdivi tru'ong thfi cua cac lo(;lingu6ndi6mkhac(c~plu'ongqic ...)chuad(;lohamcua Vr nen chungcGngthoa phu'dngtrlnh Euler. DQ di thu'ongtru'ongtoan ph~ncua c~p lu'ongqI'ctu(ho~chinhc~uvoi dQtuhoad6ngnha't)du'<;1cchobdi - 1 ~T =Cmb'v(m.-)r voi b la vectdddnvi songsongvoitru'ongxungquanh,vam la momenlu'ong cl;l'c.D€ tha'yding~T thoamanphu'dngtrlnhEulervoi n=3.Thams6n trong phu'dngtrlnhEulerdu'<;1cgQila chIso'e[{utruehay tYl~suygiam.Bang2.1du'a rachIso'diutruechotunglo(;lingu6nkhacnhau n Ki~ungu6n 1 Du'ongkh6i 2 Du'onglu'ongcl;l'c 2 Cha'tdi6m(hinhc~uvoi m~tdQd€u) 3 Lu'ongcl;l'cdi6m(hinhc~uvoi dQtuhoad€u) Bang2.1.."chislYdiu true"cilacaelo(lingu8nhapdJn vangu8ntitkhacnhau. Phu'ongtrinhEulerdfidu'<JCsad\lngbdimQts6tacgiadungd€ phantich caedQdi thu'ongtuvadQdi thu'ongha'pd~n.Ch~ngh(;ln,xetdQdi thu'ongtru'ong toaDphelntrenmQtv~tth€ dongiannhu'hlnhceluhayhlnhtr\l.B~t~Tila di€m thlii cuamQtnghiencUututrenmQtv~th€ dongian,voidi€m dot(;li(x,y,z) va tamcuav~tth€ t(;li(xo,Yo,zo).The'vao Euler tadu'<Jc r x- x ] [ ~~Ti' ~~Ti' ~~Ti ] y-y~ =n~Ti ax Oy oz z-zo Gia sa chungta dfi do du'<Jcho~ctinhdu'<JcgradientcuadQdi thu'ong tru'ongtoaDpheln.Phu'ongtrinhtrenchicob6nffnlaXo,Yo,Zovan, baffndelucho tavi tri cuav~tth€. Chungtaco th€ thudu'<Jcnhi€u theoyell Celli,b~ngcachvie't nayt(;licacvi tridokhacnhau a -~Tiax a -~T2ax a -~T1 Oy a -~T2 Oy a -~T1oz a -~T2oz r x- xo l [ Sf, ]~=:: =n !1~: vasad\lngphu'ongphapblnhphu'ongt6ithi€u d€ tlmnghi~m.Ne'uvi tricuav~t th€ du'<Jcbie't,chungtacoth€ tlm nvabie'tdu'<Jcd(;lngngu6n(xemBarongo[1]). M~t khac,ne'uchungta nghi ngo d(;lngv~tth€ ngu6n,chungta co th€ chnn thichh<Jpvagiai tlmvi tri v~tth€. M~cduphu'ongtrinhEulerclingca'pmQtcachhfi'uhi~ud€ xacdinhvi tri cuav~tth€ ngu6n19tu'dng,nhu'v~tth€ ngu6nd(;lngCelli,d(;lnghlnhtr\l,nhu'ng phu'ongphapnayco mQtvai h(;lnche'khi apd\lngchocacd(;lngv~tth€ ngu6n khac.Trongcactru'ongh<Jpnay,n coth€ khongphaila h~ngs6d6ivoidQsauva J2lLi):n tJiUL fJu,.e uj 103 g'nilL '7l5oidrmuYL vi tri cuav~tth~ngu6n,bCiivi f khongchila dt;lohamcua1/rmala tichphan trentoanbQphanb6 ngu6n.Ravat(xem[1])chi ra r~ngphuongphapnayap dlJngdu<;1ckhi dQdi thuongco h~so'suygiamty l~voi khoangcachtinhtu ngu6n.

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