Luận văn Thiết lập hàm lagrange - Hamilton cho mô hình sóng biển

Cllil1Uf001 3 THIE T LAp. "' HAM LAGRANGE - HAMIL TON "' , CHO BAI TOAN SONGNudc TRONG TRUONG. . Xiiy d.,nghamDangItrQ'ngcuasongnmyctrQngtrtrO'ng ./ Hamd(mgnangvathi nangeilasongnlfaetr9ngtrlfCrng ./ Hamnangllf(Yflgeilasongnlfaetr9ngtrlfCrng . Xiiy d.,ngbai tminbiensongntrcYctrQngtrtrO'ngfir ham LagrangecuaLuke ./ Thiit l(iplC;Zibai loantichamLagrangeeilaLuke ./ H? thuGgiifahamLagrangevahamHamilton ./ Caem?nhdJ ./ Dtnhhlfangehomohinhsongnlfaetr9ngtrlfCrng 36 CHUaNG3 THIET L~PLAGRANGE- HAMIL TON CHO BAI ToAN SONGNUOC TRONG TRUONG. Vi~cxemnangluqngnhulam9thamHamilton h~mxacdinhchuySnd9ngcuah~ duqctrinhbaykhanhi~ua cactaili~ucO'hQc[8],[9],[11],[12].V~nd\lngy tuang naytrongvi~ckhaosatbaitoansongnuactrQngtruemgclingduqcnh~cd€n ([13], [17],[18]).Trongchuangnay,m\lCtieuchinhlakiSmchungl~inhfrngk€t quama trong[13]dad~tra duaid~ngnhfrngbaitoanma.Tir vi~cxaydvnghamnang luqngtoanph~nsongnuactrQngtruemgtheohaibi€n chinht~c([5],[7]chuad~ c~p)d€n vi~ckh~ngdinh m5i tUO'llgquail gifracac phuO'llgtrinh chuySndQng Hamiltonvai hai di~uki~nbiencuabai toanbien songnuac trQngtruemg.Vi~c chungminhhamLagrangedoLuked~xuangb~nghi~ucuad9ngnangvath€ nang lanetmaimatrong[13]chuatrinhbay,trongkhi doytuangnayh~unhuduqcbi€t d€n trongcactai li~uCo'hQc.GQilanetmaivi hamLagrangedoLuked~xuang duqcxaydvngdvatrenphuO'ngtrinhBemouillichukhongduqcxaydvngtiTth€ nangva d9ngnang.Kh~ngdinh "haiphuongtrinhchuyinG(Jngcuah? ham Hamiltonchinhla GiJuki?nbienG(JnghQCvaG(JnghrchQccuabailoanbiensong nuactrQngtrz((Yng"khongchichoth~ybaitoansongnuactrQngtwemgcothSxem nhulam9th~hamHamiltonmaconduaranhfrngdinhhuanggiaiquy€tbaitoan i119tcachd~danghO'nnhud~c~ptrongph~ncu5ichuangnay. 3.1xA Y DVNG HAM NANG LU<}NGSONGNUaC TRQNG TRUONG Nangluqngsongla t6ngcuad9ngnangvath€ nang.D9ngnangcoduqcladosv chuySnd9ngcuacach~tlongriengbi~trongkh~pch~tlongconth€ nangphatsinh dotrQngtamcuach~tlongn~mcaohO'ng5cth€ nang(chQndaylamg5cth€ nang). 3.1.1D{jngniingsongntl'O'ctr!mgtrU'O'ng GQiS ladi~ntfchhinhthangcongvaduqcki hi~ulami~nS nhuhinhve3.1.D9ng nanglasvchuySndQngcuacach~tlongriengbi~trongmi~nS t~onen. 37 z Mng ccngbe~ z =1;(X,O 1 h ::;;"21J(x,t) x Hinh 3.1 MiJn xacajnhcho thi nangvaa(JngnangciLasongtr9ngtruong Xet m('>th~tlongcokh6i1uqngdm chuy~nd('>ngtrongh~tn;tcOxz,t~ivi tri (x,z) cov~nt6c6thaidi~mt 1ftv=vex,z,t).Khi do,d('>ngangcuano1ft~v2dm . Kh6i 1uQ'ngh~tlong dm=p dxdzxacdinhtheodO'nvi di~ntichph~ngtrenOxzvft v~nt6c v=(vx'Vz) cod('>IOnduQ'cxacdinhb6i v=~v; +v; ::;;-JU2+W2. Tir do,d('>ngnangcuah~t ! p (U2 +w2).dx.dz=!p [ (\1 t/J)2 + (Ot/J)2 ] dxdZ 2 2 & (3.1) Suyfa,d('>ngnangtrongvungdi~ntichS 1ft ED::;;ff!p [ (\1t/J)2 + (Ot/J)2 ] dXdZ s 2 Oz ::;;!p ff[ (\1t/J)2+ (Ot/J)2] dXdZ 2 s oz (3.2) TheoPL3, chQnT ==S, bienL 1ftcacdo~nth~ngOC,CB,BA vft1do~ncongOA fO(\1t/J)2 +(Ot/J)2) dXdZ::;; ft/J ot/J d(J"sJ~ oz L On (3.3) Suyra (3.2)~ ED ::;;!p ft/Jot/Jd(J" 2 L On 1 f( ot/J ot/J ) -7 ~ED::;;2PL t/Jax,t/Joz nd(J" (trongdo n 1ftphilpvectO'dO'llvi ngofticuabienL) (3.4) 38 Tinh tichphantrong(3.4)trenbienL 1. Tren do<;lnth~ngCB: vectaphapdO'llvi ngoai n=(0,-1), hanmlado 8cp=0 chonentichphanb~ngO. 8z 11. Hai do<;lnth~ngbellAB, OC: vectaphapdO'llvi ngoailfin luqt trenAB va oc la n=(-1,0) va n=(1,0); Do giftthiStchuySnd9ngsongla tufin hoantheokhonggianvathaigiannengiatri hamcpt<;li2 diSmtUO'llgung tren2 c<;lnhAB vaOC lab~ngnhau,ddndSnt6ng2tichphanb~ngO. Tren do<;lncongAO: do<;lncongAO co phuangtrinhla z =1](x,t),phap111. vectadanvi ngoaiduQ'cxacdinhbai ~ [ 81] 2 ] -Yz 81] n= l+(ax) (- ax' 1)=N(1]) (3.4) ED =~p S(cp8cp ,cp 8CP)N(1])da2 AD ax 8z (3.5) M~tkhac,phuangtrinhthams6do<;lncongAO la { X =s v6i 0::::;s::::;L z =1](s,t) D~tcP=cp(s,z=1](s,t),t),1f/=If/(s,t)=cPva G(1])If/(s,t)= ( 81f/,8CP(S,z,t) 1 I N(1]) [ 1+(81])2 ] Yz 8s 8z Z=T/(s,t) 8s B~ngcachthams6hoatichphanduang,khido (3.5) ED =~p flf/(S,t) ( 81f/ , 8CP(S,z,t) 1 I N(1]) [ l +(81])2 ] Yzds 2 0 8s 8z Z=T/(s,t) 8s 1 L ED =- p SIf/(x,t)G(1])If/(x,t)dx 2 0 (3.6) - ~ L ( - 81f/ 81] +8CP(X,z,t) 1 J dx ED - 2P flf/ ax ax 8z Z=T/(x,t)0 (3.7) 39 3.1.2Thi niingsongnlfUCtr{JngtrU'ung Trang miendi~ntich S, v~chhatduangth~ngdung,vuonggoc vai Ox va mien di~ntich dS giai h~nbai hatduangth~ngdungnaydQrQngphuO'ngngang1adx (phc1ng~chcheotranghinh3.1).Khi domien dS co di~ntich I]dx, dftnd€n kh6i 1uQ'ngcua no 1adm=P I]dx. Vac1uqngdQcaotn;mgtamcuadS nay(sovai g6c th€ nang)1ah=h(x,t)=~I], 1licdoth€ nangtrendS 1admg h=~P g 1]2dx.2 2 Suyfa,th€ nangtrenmienS L 1 Er =f- P g 1]2dx 02 (3.8) NgoalcachchQng6cth€ nang,th€ nangcoth~duQ'cxacdinhbaidinhnghlasau Er =P gffzdas (3.9) Co2cachd~xacdinhth€ nangnay Cach1:tinhtnJcti€p tichphanhatlap L!1)(X,t) JIfz d(J =~l _[zdzdx 1L =2f(1]2- d2\px 0 LI f 2 1 2=- I] dx--Ld 20 2 Cach2:dungcongthucGreen (3.10) ffzd(J =~ff~(Z2)d(Js 2soz =~ff(~(Z2)-~(a) ) d(J 2s oz Ox =!f(Z2dx+adz)(dungcongthucGreen) 2L 1rf 2 =-'jz dx 2L (3.11) 40 (trongdoL =OCuCB u BAuAO) Tinhtichphantrong(3.11) i. Trendo~nth~ngth~ngOC vaduangth~ngBA: haiphuO'ngtrinhthams6 do~nth~ngOCvaBA l~nluQila { X =0 { X =L z =s,- d ~s ~0 va z =s,- d ~s ~0 Suyra fZ2dx=0 OCuBA (3.12) 11. Trendo~nth~ngCB: phuO'llgtrinhthams6do~nth~ngAC la { X =s, 0~s~L z=-d Suyra fZ2dx=-d2L CB (3.13) 111. (luuyphuO'ngcuavectO'doovi ngoaihu6TIgxu6ng) TrenduangcongOA:phuO'llgtrinhthams6do~ncongOA la { x=S,O~S~L z =ry(s,t) L Suyra fZ2dx=fry2ds OA 0 (3.14) Tir (3.12),(3.13),(3.14)thi 1L 1 (3.11)Q HzdCJ=-fry2dx--Ld2 s 20 2 K~tqua(3.15)trungv6'i(3.10)chothftyhaieachtinhtoanlachinhxac.Thaychung vao(3.9),th~nangcuami€nS la (3.15) 1 [ L JET =2-pg - d2 + Jry2dX ~(-~nd'L H~nf~2dxJ s6 h~ngd~utrongv~phaicua(3.14)bi€u thith~nangcuanu6'ctInhnenseb6qua (3.16) no.Nhuv~y,th~nangcuami€nS la 41 1 L ET =- pg f172dx 2 0 (3.17) 3.1.3Ham niingIU'{1ngsongnU'uctr(JngtrU'irng NfmgluQ'llgto~mph~nsongnuactrQngtruemgtrongmi~nS la E =ED +ET =~P ff[(\l </J)2+ (a</J)2] dXdZ+pg ffzda-2 s & s Ma mi~nS co phmmgngangtinh trenmQtchu kY song,nfmgluQ'llgto~mph~nla (3.18) 1 L 1 L E =- pflf/(X,t)G(17)If/(X,t)dx+- gpf172(X,t)dx 2 0 2 0 (3.19) 3.2XAY D\J'NG BAI ToAN BIEN SONG NUaC TRQNG TRUONG Tit HAM LAGRANGE CUA LUKE 3.2.1Thiit l{iphai loansongnU'uctr(JngtrU'irngtir hamLagrangecuaLuke TrongmohinhsongnuactrQngtruemg,apsu~tt~inhftngdi~mtrongmi~nkhflOsat thayd6itirb~m~thoangxu6nglapday.D\fatreny~ut6nay,Lukechor~ngcoth~ bi~utrungchuy~nd('>ngsongcuakh6inuacb~nghamLagrangemahamnaydugc Kayd\fngd\fatreny~ut6apsu~thayd6itir lapdaylenb~m~t.f)~cbi~thall,ap su~tnaydugcxacdinhtirphuangtrinhBemouilli,phuangtrinhdugcKayd\fngtir cacphuangtrinhthuyd('>ngEulermadffdugcd~c~p6chuang1. TirphuangtrinhBemouilli(1.6),apsu~tsongnuacdugcxacdinhb6i p(X,z,t)=_p { a</J+gz+~ ( (a</J)2+(a</J)2 )}at 2 ax az HamLagrangedoLuked~nghi (3.20) 1] L(x,t) = fp(x,z,t)dz -d 'I f { a</J 1 a</J 2 1 a</J 2 } L(x,t)=-p -+gz+-(-) +-(-) dz -d at 2 ax 2 az (3.21) 42 V6'ihamLagrangephia tren,Lukedffduaranguyenly biend6i "tEchphiinham Lagrangetheotrentn,lcnlimngangx vatheothaigiant fam9ts6c6iljnhkhicae bufnljJ va 1] thayilc5i".DlJa trennguyenly nay,baitmlnbiensongnu6'ctrQng truemgcoth€ duqcxiiydlJilg. f)~t S = f fLCx,t,ljJ,lJ)dtdx (3.22) X I (trongdo, S =SCljJ,lJ)),khi do(3.22)duqcvietl~inhusail SCljJ,lJ) =- fffp { OljJ+gz+.!. [ COljJ)2 +COljJ)2 ]} dzdtdx xl-d at 2 Ox OZ ~ S(~+(j~)~- ffpJ{a(~;~) +gz+~[(a(~a:~))'+(a(~;~)J'J} dzdldx ~-p f ff { OljJ+gz+.!. ( OljJ ) 2 +.!. ( OljJ ) 2 +oC8ljJ)+oljJ oC8ljJ)+oljJ OC8ljJ) } dzdtdx x t -d ot 2 Ox 2 oz ot Ox Ox oz oz =S(ljJ)-pfff { oC8ljJ)+OljJ ( OC8ljJ) ) +OljJ ( OC8ljJ) )} dZdxdt x I -d ot Ox Ox oz oz (bodi s6h~ngchua8ljJb~chai) (3.23) ~ 8 S =-p fff{ oC8ljJ)+OljJ ( OC8ljJ) ) +OljJ ( OC8ljJ) )} dzdtdx x I -d ot ox Ox oz oz M~tkhac i. ~(:~)=:(a~))+~(~~)=:(a~))+~(~~) ii. ~ ( oljJ8ljJ ) =OljJ ( OC8ljJ) ) +8ljJ ( O2ljJ ) =OljJ ( OC8ljJ) ) +8ljJ ( O2~ JOZoz oz oz OZ2 oz oz oz Chonen '1 { 0 0 ( oljJ ) 0 ( oljJ ) ( O2ljJ 02ljJ )}8 S =-p J J f -(8ljJ) +- -8ljJ +- -8ljJ -8ljJ -+- dzdxdtt x -d ot ax ax oz oz ax2 OZ2 =-pJ Jf { ~(8ljJ)+~ ( OljJ8ljJ ) +~ ( oljJ8ljJ )}dZdxdt+J Jf { P8ljJ ( O2~+o2f )} dzdxdt tx-d ot ax ax oz oz tx-d ax oz 43 '1 -> '1 { (82rjJ 82rjJJ}=-p ffJV'Cc5rjJB) dzdxdt+ffJ pc5rjJax2+8Z2 dzdxdt (trongd6 13=[1, 8rjJ, 8rjJ] )ax 8z (3.24) * Khaosatriengv6'is6hangd~utrong(3.24) Mi~nkhaosatbaitoanbiens6ngnu6'ctrQngtruangnhuhinh3.1(Mi~nbi gi6'i h~nb6i2bienbellOC,AB, biendayCB vabientrenIaduangcongOA) i. BiendayCB: phapvectO'dO'llvi ngoaicuado~nCB Ia Ii =[0,0,-1]. Khi , --> --> [ arjJ arjJ] arjJdo, B.n = 1, -, - [0, 0,-1]= --;ax az az ii. HaibienbellOCvaAB: phapvectO'dO'llvi ngoaicuado~nOCvaAB I~n Iuqt Ia iiI=[0,-1,0] va li2=[0,1,0]. Khi d6,gia tri lIngv6'im6ido~n , -->-> ,--> ->;. ,arjJ, arjJ tuO'ngung B.nl va B.n2 IanIuqtIa - ax va ax; iii. BientrenduangcongOA: phapvectO'dO'llvi ngoaicuaduangcongOA --> ( a a J -h [ a a ] c6d~ngn= 1+(~)2 +(~)2 - ~,-~, 1 .Khi d6,at ax at ax B.~=[I, ~, ~:](I+(~~)2+(~)2f'l - ~>~:'I] Dungdinhnghlatichphanduang,(3.24)duQ'cbi~nd6iti~p (3.24)""OS =PUo~~~d"dl+p{U,~ :~d"dl- LL o~: d"dl} , v " v- ' (AI) (A2) +p ffc5rjJ(81] +8rjJ81]- 8rjJ)dCTdt+fff{Pc5rjJ(82~ +82f]}dZdxdtt tren 8t ax ax 8z t x -d ax 8z , v ' , v- ~ ' (A3) (A4) Thea nguyen1:9bi~nd6i, c5S =0,V c5rjJ,phuO'ngtrinhchuy~ndQngtrongm6ido~n thaigianseIa 44 1. (A 4) =0q :~+~:f=0 V(x,z)ES, dayla phuO'ngtrinhchudc;tocuabai toclnbiensongnuactn;mgtruang. 11.(A1)=0q ~~=0 V(x,z)thuQcbienduai,dayla di~uki~nkhongco chuy~ndQngsongbiendaytrongbaitoansongnuactrQngtruang. iii. (A2)=Of f orjJarjJdCJdt= f f orjJarjJdCJdt V(x,z) n~m6 bien 1 va 2 b"' l ax b"' 2 axt Ion t len tuO'ng(rngtrencungvi tri z , dayladi~uki~ntufmhoancuahamthSrjJ. iv. (A3)=0q a7] + arjJ a7] =afjJ V(x,z)n~mtrenbienz =7](x,t),dayladi~uat axax az ki~ndQnghQctrongbaitoansongnuactrQngtruang. v. Di~uki~ndQnghJchQccuabaitoansongnuactrQngtruanglaphuO'ngtrinh Bernouillinhungdi~uki~nnaydffduqcdungchovi~cxacl~phamLagrange bandfmmaLliked~xufLt. 3.2.2H~thircgiil'ahamLagrangevahamHamilton TJ f{ arjJ 1 2 1 arjJ 2}(3.21)L= -p-+pgz--p(VrjJ) --pC-) dz-d at 2 2 az =Y:- p arjJ J dZ+ fpgzdz_!p TJr((\7rjJ)2+(arjJ)2)dZ-~l at -d 2 -~l az 1 1 'IarjJ 1 K ad. )=-pgd2 __pg7]2 - P f-dz--p (\7rjJ)2+(-'[:~ydz2 2 -d at 2 -d OZ 1 d2 'IfarjJ { I 2 P I 2 orjJ 2] }=- pg - p - dz - - pg 7] +- (\7rjJ) +(- ) dz2 -dat 2 2 -d OZ A 1 d2 'IfarjJ 1 2 P R 2 orjJ 2 ] Vc;tyL =-pg - p -dz--pg7] -- (\7rjJ)+(-) dz 2 -dat 2 2 -d OZ '-v--' ~ ' y I , , A B V T , v ' C (3.25) * Nhanxet 45 +s6h?ngA trong(3.25)1ah~ngs6vakh6ngph\!thu9cvaocacbi€nd9ngh,rc hQcliencoth~b6quas6h?ngnay. + s6 h?ngC trong(3.25)1at6ngcuath€ nang(kihi~u1aV) vad9ngnang(ki hi~u1aT) trenm9tdanvi di~ntichtinhtheophuangngangx.Nhuv~ys6h?ng C 1anang1uQ'ngtoanph~nH tinhtrenm9tdanvi di~ntichtheophuangngang. +s6h?ngB trong(3.25): d ( 1J(X,t) J 1J(x,t) Do - fq/(x,z,t)dz = fq/t(x,z,t)dz+q/(X,17(x,t),t)17t(x,t) dt -d -d 1J(x,t) d ( 1J(X,t) J => fq/t(x,z,t)dz=- fq/(x,z,t)dz -q/(X,17(X,t),t).17t(X,t) -d dt -d (3.26) s6 h?ngd~utrong(3.26)coth~b6quavi no1atichphantrenbientheothaigian vakh6ngcoph\!thu9Cbi€n d9ngh,rchQc. Tirdo,(3.25)~ L =P q/(x,z=17(X,t),t)17t- H ~ L =P If/ 17t- H (3.27) trongdo, 171abi€n chinht~cthunh~t,If/ 1abi€n chinht~cthu haiva (3.27)1ah~ thuclienh~gifrahamHamiltonvahamLagrange. 3.2.3Cacm~nhilJ * Khainiemhebaotofm"M(jth?alf(YcgQila baataimniu ttitcacacIl,fclaca(jng tenmQiphdnfirtrangh?aJu congu6ng6cfirhamthi". V6'isongnu6'cthih,rctac d9ng1atrQngh,rclienh~1abaotoan. Khi do,hamHamiltonva hamLagrangeseduQ'cxay dl,l'ngdl,l'atrend9ngnangva th€ nangcuah~. H=E=Ef)+ET (3.28) (3.29)L =ED- ET Mfnh ilJ 3.1:hamLagrangetheaLukela hi?uciLaar)ngnangva thi nangtrenm(jt aO'nvi di?nItchtinhtheaphlfO'ngngangx. 46 Chungminh - Truaeh~t,n~uxettrenmQtdonvi di~ntiehtheophuongngang,vaih~baD tofm,hamHamiltonvahamLagrangeIAnlugtduQ'exaedinhb6'i H*=T+V (3.30) (3.31)L*=T-V (dAu* dungd~phanbi~tvai hamLagrangeeuaLuke) - K~ti~p,trongeachI~pbaitoans6ngnuaetirnguyenIy bi~nd6ieuaLuke trongph~ntruae,thi orp l - orp l 01] - orp- orp01] trenz =7](x,t)~ 1]/=OZz='I Oxz='IOx1],- oz Oxox (3.32) - Thay(3.31),(3.30)va(3.32)vaotrong(3.27),thi 1 ( orp l 01] orp l J dpem~T=-PIf/ -- -+- 2 OxZ=17Ox ozZ=17 (3.33) Do T ladQngnangtrenmQtdonvi di~ntiehlAytheophuongngangx, lAy tiehphan(3.33)theotrenmQtdonvi ehi~udais6ng,v~traieua(3.33)sela nangluQ'llgdQngnangtrenmi~ndi~ntiehS ;Khi d6 1 L ( orp l 01] +orp l J dx (3.33)~ ED ="2 P JIf/ - OxZ=17Ox ozZ=17 1 L ~ ED =- P flf/ G(1])If/dx 2 0 R5rang(3.34)luandungvi n61a(3.6)nhutrongph~ntren(dpem). (3.34) Mfnh ilJ 3.2 HamLagrangeL =L(1],1]pt)cocacphuongtrinhLagrangexudtphattunguyenIf; lac GrlngClfCtiJu la 1 8L If// =-- (A) i P 81]1 8L If/=-- P 81]/ HamHamiltonH =H(1],If/,t)cocacphuangtrinhHamiltonla (3.35) (3.36) 47 18H 1]1=-- (B) i P 81f/18H If/ = --- . 1 P 81] Niu hamHamilton va hamLagrange thoa (3.27)thi (A) (B) (3.37) (3.38) Chungminh (~) ChocacphuangtrinhLagrangenhutrongh~(A) Do (3.36)(If/= 2- 8L ) nen 1]1ph\!thuQcvao If/ P 81]1 (3.27)H(1],If/,t) = PIf/1]1 -L(1],1]l't) L~yd~oham(3.39)Iftnluqttheohaibi~nIf/va 1] (3.39) 8H 8 81f/=81f/(PIf/ 1]1-L(1],1]I't)) 81]t 8L 81]1 = P 1]1+P If/ 81f/ - 81]1 81f/ = P 1]1(do3.36) ~ 1] =2-8H1 - P 81f/ 8H 8 81] = 81](PIf/1]t -L(1],1]l't)) 8L 81] = -P If/I (do 3.35) 18H (d ' ~ h "' )~ If/I =--- leup at11m P 81] (~) ChocacphuangtrinhHamiltonhutrongh~(B) Do 3.37(1]1=! ~H), If/ ph\!thuQcvaobi~n1]1;1]ph\!thuQcvao t. P ulf/ (3.27)L =PIf/1]t -H(1],If/,t) (3.40) 48 L~yd~oham(3.40)l~nlugttheohaibi~n1]va 1]/ 5L 5 51]=51./P If/1]/- H(1],If/,t») 5H 51] =-Plf/I (do3.38) 1 5L ~ If/I =- P 51] 5L 5 51]/=51]/(PIf/1]/-H(1],If/,t») 51f/ 5H 51f/ =PIf/ +P 1]151]/- 51f/ 51]/ =PIf/ (do3.37) 1 5L (d ' ;. h"' )~ If/ =-- leup attIm P 51]1 Y nghlacuam~nhd~3.2laphuangtrinhchuySndOngtheoLagrangecothSduqc xacdinhphuangtrinhchuySndOngill hamHamilton.Nhuv~y,v6'ihamHamilton H =H (If/,17), h~phuongtrinhchuySndOngdola 10H 17 =-- 1 P Olf/ 10H If/ =--- 1 P 017 Chungtasechirar~ng:"TantqihamHamilton,haiphuongtrinhchuydnt/(jngxac t/inhtith?Hamiltonlahaiphuongtrinhmotachuydnt/(jngsongtrenbJ mijt". DtJatheohamth~v~nt6cvaphuangtrinhdichchuySnb~m~trongbaitoanbien songnu6'ctn;mgtruang,haibi~nchinht~cd~nghi (3.41) 1]=1](x,t) If/=If/(x,t)=rjJ(x,z=1](x,t),t) (3.42) (3.43) V6'ih~hamHamiltonla H =H(1],If/),b~ngcachchQnngo~cPoissonthichhqp,hai phuangtrinhchuySndOngHamiltonduqcxacdinhdu6'id~ng 49 {'7t~['7,H]Ij/t - [Ij/,H] (3.44) (3.45) V6i [,] 1ango~cPoisson. Mfnh dJ 3.3:haiaiJu ki?nbientrencuabailoansongmractr9ngtruongtuong auongvaihaiphuongtrinhchuyinat5ngHamilton,tucla (3.44)q 0'7+VcpV'7 = ocpvaot oz ocp 1 2 1(Ocp) 2 (3.45)q-+-(Vcp) +- - +g'7=Oot 2 2 oz Chung minh Bu6c 1chQnngo~cPoisson Ngo~cPoisson d8nghi [F,G]=~(6F 6G - 6G 6F Jp 6'7 61j/ 6'7 61j/ trongdo F =F('7,Ij/),G =G('7,Ij/),v6i : lad;;tohamFrechetsuyrQng; (3.46) Suyra ['7,H]=~ ( 6'76H - 6H 6'7 J =~6H p 6'761j/ 6'761j/ P 61j/ [1j/,H]=~ ( 61j/6H - 6H 6Ij/ J =_! 6H p 6'761j/ 6'761j/ P 6'7 R5rang1a(3.47)va(3.48)trimgv6i(3.37)va(3.38). (3.47) (3.48) Bu6c2xacdinhhamHamilton Vi baitoansongnu6ctrQngtruemgla mQth~baotoan,hamHamiltonduQ'c chQnse1ahamnangluQ'ngsongnu6ctrQngtruang. Dvatheo(3.19),hamHamiltond~nghi 1 L 1 L H(Ij/,'7)=- p flj/(x,t) G('7)Ij/(x,t)dx+- gp f'72(x,t)dx2 0 2 0 (3.49) 50 Bu6c 3 chungminh (3.44) 817+V f/JV 17=8f/J 8t 8z 1 L .:. Tir ET ="2gp f172dx,cho thdy ET khongph\!thuQcIf/0 =>8ET =0 81f/ (3.50) 1 L .:. ED(If/)=-P flf/(X,t)G(17)If/(X,t)dx . 2 0 1 L ED(1f/ +81f/) =- P f(1f/+81f/)G(17)(If/+8'1')dx 2 0 lL lL lL =-P flf/Glf/dx+-p f8If/Glf/dx+-p fIf/G8If/dx+9(81f/2) 20 20 20 L ~ ED (If/) +p f81f/(GIff)dx 0 L =>8ED= f81f/(p G Iff)dx 0 '" 8ED=PGIf/- 81f/ (3.51) ... 8H 8E 8E (3.50). - =---12.+ I... - 81f/ 81f/ 81f/ (351)P G(17)If/ (3.52) .:. G(17)1f/= (8f/J(~z't)1 ' 8f/J(X,z,t) 1 I N(17) [ I+(817)2 ] li Z='1(x,t)8z Z='1(X,t) ax = ( 8f/J(X,z,t) 1 ' 8f/J(X,z,t) 1 J( - 817, 1 )ax z='1 8z z='1 ax = ( _8f/J(X,z,t) 1 817 +8f/J(X,z,t) 1 Jax z='1ax 8z z='1 (3.53) 51 1 5H 8rjJ817 8rjJ .:. Tir (3.52)va (3.53)suyratrenz=17(X,t):p51f/=- axax+8z 15H Han mla 171= [17,H]=p 51f/ Suy ra 171=- 8rjJ817+8rjJ (di~uphai chungminh)axax 8z Bu6c4chungminh(3.45)~ 8rjJ+~(vrjJ?+l ( 8rjJ ) 2 +g17=0 8t 2 2 8z , 1 L 5E 8 ( 1 2 ).:. Tu Er =- gpf172dx,suyra L =- - gP17 =g P 172 0 517 817 2 5Er =>-=g P17 517 (3.54) 1 L .:. ED(17)=-P flf/(X,t) G(17)If/(x,t)dx 2 0 1 L ED (17 + 517) =- P fIf/ * G( 17+ 517)1f/* dx 2 0 / [ 2 ( J 2 1 I] L 1 8rjJ 1 8rjJ 8rjJ 8178rjJ dx ~ ED(ry)+lOry p 2(aJ.J - 2 azl,~, + az ,., ax ax ,~, L [( IJ 2 ( IJ 2 1 I] =>5E - f517P ~8rjJ - ~8rjJ +8rjJ 8178rjJ dx D - 0 2 axZ=1] 2 8zZ=1] 8zZ=1] axaxZ=1] [ [ J 2 ( J 2 ] 5ED 1 8rjJ 1 8rjJ 8rjJ 8178rjJ "" Jry ~ p 2: ",1, - 2: &J" +az I,~Ox Ox I", ~ ~p[[:IJ' +[:IJ']- ~~IJ- ::1,.,+:IJ (3.55) . 5H 5ED 5Er... -=-+- 617 617 517 52 (3~4)~P [( 01 1 J 2 + ( 01 1 J 2 ] - 01 1 ( - 01]01 1 +01 1 ] +gp1] (3.56) (3.55) 2 oXZ=T/ OZZ=T/ OZZ=T/ axaxZ=T/ OZZ=T/ .:. Tir (3.56) sur ra tren Z =1](x,t) ~5H=~ [( 01 ) 2 + ( 01 ) 2 ] - 01 ( - 01]01+01 ) +g1] P 51] 2 ox oz oz oxax oz 1 5H Han mla 1jI/= [1jI,H] =- P 51] Sur ra 1jI/=-~ [( 01 ) 2 + ( 01 ) 2 ] +01 ( - 01]01+01 ) - g1] 2 ox OZ oz axox oz (3.57) 01 01]+01 .:. M~tkhac,do 1]/=- axax oz Tir ljI(x,t)=1(x,z=1](x,t),t) oljl 01 0101]~-=-+-- ot ot OZot Q oljl =01 +01 ( - 0101]+01 )ot ot oz axax oz (3.58) .:. Tir (3.57)va(3.58) Surra 01+~(Y'1)2+~ ( 01 ) 2+g1]=O(di€uphaichungminh) ot 2 2 OZ 3.2.4lJjnh !urungchomohinhsongnlructrfJngtrtdlng Thayki hi~uhamHamiltonbing hamnangluQ'ng 1 5E 1]=-- (3.41) Q ~ / P 51j11 5E 1jI/=--- P 51] (3.59) Baitoanhamth~:xacdinhhamth~v~nt6cljJth6a 53 J2rjJ /).rjJ+- =0 JZ2 <jJ(x,z,t)=Ij/(X,t) a<jJ=0 aZ tren z =1](X,t) tren z =-cl Khi d6,n~ugiaiduQ'cbaitminhamth~thi d1;lngphuO'ngtrinhbem~tduQ'Cxacdinh dlJatheohaicachsau Cach1Tir d1;lnghamth~<jJ,giairaduQ'chamn[mgluQ'ngE, k~thQ'pv6'i(3.59), d1;lngcuahaiham1]va If seduQ'cxacdinh. Cach2 Xet trenbienz =1](x,t) . ! a<jJ VIj/ =V<jJ+-V1]az Ij/(x,t) =<jJ(x,z=1](x,t),t) ~ alj/ = a<jJ+a<jJa1] at at azat ! a<jJ V<jJ=VIf/--V1]az ~" a<jJ= alj/ - a<jJa1] at at az at { V<jJ=VIj/-(V1] ~ a<jJ= alj/ - a1] (v6'i t; = ~<jJ(x,z ol- ) at at t; at z-r/(x,t) . a1] a<jJ a1]-+ V<jJv1] =- ~ - = -V<jJV1] +t; at az at ~ a1]=-(VIj/ -t; v 1])V1]+t; at ~1]t =-VIj/V1]+t;V1]V1]+t; ~1]t =-VIj/V1]+t;(1+V1]V1]) a<jJ 1 2 1(a<jJ) 2 -+-(v <jJ)+- - +g1]=0at 2 2 az . 54 ( 8'1f 81] ) 1 ( )2 1 2Q --t;- +- VljJ +-t; +g1]=O8t 8t 2 2 8'If 1( )212 81]Q-=-g1]-- V'If-t;V1] --t; +t;-8t 2 2 8t 1 1 1 Q'lft =-g1]__(V'If)2+t;'11]'1If__t;2(V1])2__t;2 -t; V'IfV1]+t;t;+t;2(V1])22 2 2 1 2 1 2 2 1 2 Q'If/ =-g1]-2('1 'If) +2t; ('11])+2"t; Q'If/ =-g1] -lev 'If)2+It;2(1+(V1])2)2 2 t 1]/ =-V 'IfV 1]+t;(1+'11]'11]) Suyra 1 1 trenz =1](x,t) 'Ift = - g 1]- - V 'IfV 'If +- t;2(1+V 1]'11])2 2 (3.60) Nhu v~y,n~uxacdinhduQ'cv~nt6ctheochi~uz, t; =~ljJ(X,z,t) 1 ' r6i thay 8z Z=1J(x,t) vao(3.60),giaih~(3.60)thixacdinhduQ'c1]vaIf/.