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WuChong-shi 1 §31.1 • ! #$%& ’()*+, ’-.&/0 1$23# ’/0,45-•67 ’ #89:)*+x(:;=?@),AB’C DB&yEFGH-yI(x, ’ J(y=f(x)-•Kx,yLMNDBOPQy(x) RSTL=ZCds=Zx1x0q1+y02dx.UV y(x)WX LYWX ZL,’C[\]%& ’y(x)^_*-L‘ ’y(x)F?,.a[\bc I( bc-defghijklmno-gp qrstuvfwx ywstz{|}~ vf x wx - | f -•KG](=B ’ @,)A B& ’y(x) D B&’J[y]EFGH IJ[y](y(x), -. , ’ Z ,:; y(x) B:, ¡¢£ ⁄¥ƒD§¤,'“«’-.‹,y(x)I(1› ’-fifl fl–†‡· ¶-•‚„” »|…‰x ¿…‰ `´•ˆ˜•‚¯” ˘˙¨»m …˚¸ f y(x) ˝˛ˇ—…‰ J[y]-(| …˚¸ f) - J[y]` y(x)´¸f y(x) Æ - , ª1$# a ‹, # ŁØŒº ] J[y]=Zx1x0F(x,y,y0)dx:;, RŁ,F# ,æ+, ’ ƒD§¤,'“ «’-ı *+ ’#' ’u(x,y) (J[u]=ZZSF(x,y,u,ux,uy)dxdy,WuChong-shi§31.1łøœß 2 RŁux≡∂u/∂x,uy≡∂u/∂y-G] &)*+, ’ Y1$D ,:;- 31.1 31.1ı 31.16 B& (x0,y0) LMPQy(x) ) (x1,y1) 6 , ?T=Z(x1,y1)(x0,y0)dsp2g(y0−y)=Zx1x0p1+y02p2g(y0−y)dx#y(x), -. *+ ’y(x)B: (x0,y0)‘(x1,y1)-( GalileoGalilei m) 31.2 , !-K N#$% &’,B( )( ,*=12ρΔx∂u∂t2,+*=12TΔx∂u∂x2,RŁu(x,t)# , ,-. ρ# ,Q/T T#0 -.‹ ,Hamilton +S=Zt1t0dtZx1x012hρ∂u∂t2−T∂u∂x2idxY#-.u(x,t), -L=Zx1x012hρ∂u∂t2−T∂u∂x2idxI(Lagrange+(Lagrangian) ^1 ’12hρ∂u∂t2−T∂u∂x2iI(Lagrange+/T-WuChong-shi 3 §31.2 23•47 5C6• ›5C,7 ¢£6•89:B Db ’5C,/0-67 ’f(x)x0 ›5;C #8xx0 =R?|x−x0|ε @Df(x)≥f(x0);^ı @Df(x)≤f(x0), I ’f(x)x0 ›5AC- ’f(x)x0 ›5C(5;B5A),7 ¢£#) ,«’(0 f0(x0)=0.•1$ X‹,CD:; ,5C-E*+ ’(y(x) J[y]›5;CF, ;#GG]5C ’y(x)=RE?F,*+ ’y(x)+δy(x) @DJ[y+δy]≥J[y].67 ’y(x)+δy(x) B& ’y(x),E?F 8,#G1.|δy(x)|ε˜2.D H |(δy)0(x)|ε-. ,δy(x)I( ’y(x),IJ-•1$KL ’5C7 ¢£,«%MD «% ›5C,7 ¢£-WNWOPQRS: 6TU,*+ ’V W:,X& y(x0)=a,y(x1)=b,Zδy(x0)=0,δy(x1)=0.TU ,YCJ[y+δy]−J[y]=Zx1x0hF(x,y+δy,y0+(δy)0)−F(x,y,y0)idx, ’,* δy(x) &; 1$Z1 ’5C ’? Taylor[\ ]# DJ[y+δy]−J[y]=Zx1x0hδy∂∂y+(δy)0∂∂y0iF+12!hδy∂∂y+(δy)0∂∂y0i2F+···dxWuChong-shi§31.2łøœ]^ 4 =δJ[y]+12!δ2J[y]+···,RŁδJ[y]≡Zx1x0h∂F∂yδy+∂F∂y0(δy)0idx,δ2J[y]≡Zx1x0hδy∂∂y+(δy)0∂∂y0i2Fdx=Zx1x0h∂2F∂y2(δy)2+2∂2F∂y∂y0δy(δy)0+∂2F∂y02(δy)02idx _# J[y],B‘* ‘'‘* -.‹a GfiflJ[y]bcde·fghi fifl·jkIJl0 δJ[y]≡Zx1x0hδy∂F∂y+(δy)0∂F∂y0idx=0.ZNª Ł,m'n o X pq ¡¢£ DδJ[y]=∂F∂y0δy x1x0+Zx1x0hδy∂F∂y−δyddx∂F∂y0idx=Zx1x0h∂F∂y−ddx∂F∂y0iδydx=0. ]δy,A R 1$a ∂F∂y−ddx∂F∂y0=0..&CrI(Euler–LagrangeCr # J[y]›5;C,7 ¢£,s ª-Bt!u .#B&'“ s Cr-•‚ J[y]vwx fyz jkde{|} ~ ˇ— z f˙ -«%Euler–LagrangeCr N * D,B& , GKφ(x)#x,§¤ ’ η(x)ƒD§¤,'“«’ ¥η(x) x=x0=η(x) x=x1=0 G]A η(x) Zx1x0φ(x)η(x)dx=0V3 7Dφ(x)≡0- 31.3K D+ Ł q=q(t) t0,q(t0) t1,q(t1) ,Hamilton +#S=Zt1t0L(t,q,˙q)dtRŁq‘˙q# ,5; ‘5;+ L=T−V#*T‘+*VFY I(Lagrange+-WuChong-shi 5 Hamilton ØŒ j ( ¡·)¢£⁄¥ƒ §¤ ·('“« ‹›fifl·)⁄¥ –†‡Sbce-· NM,¶•1 +S›5C,7 ¢£, ª‘s ª _#δS=Zt1t0h∂L∂qδq+∂L∂˙qδ˙qidt=0‘∂L∂q−ddt∂L∂˙q=0.9:,D+ Ł %Lagrange+L,ƒ‚ ª „”» ‘Newton …, …Cr‰ B‹-»¶• J[y]=Zx1x0F(x,y,y0)dx,Xa ¿, `´ -F Ł,F=F(x,y0)WU y. ,Euler–LagrangeCr#ddx∂F∂y0=0,6$ Z1$a ,ˆ˜ ∂F∂y0= +C.F Ł,F=F(y,y0)WU x¯˘˙¨ ddxhy0∂F∂y0−Fi=y00∂F∂y0+y0ddx∂F∂y0−∂F∂yy0−∂F∂y0y00=−y0h∂F∂y−ddx∂F∂y0i,6$ . ,Euler–LagrangeCrY1$Dˆ˜ y0∂F∂y0−F= +C. .&˚ H ¸31.3Ł ı Lagrange+LWU t D˙q∂L∂˙q−L= +C,.#£‡ ˝- M˛ˇ' ’,´ -KD' ’u(x,y),(x,y)∈S — N1$:; J[u]=ZZSF(x,y,u,ux,uy)dxdy.WuChong-shi§31.2łøœ]^ 6 V : u(x,y)S, ¡ΓN,’C9: Zu ΓW:.ˆ8 V J[u+δu]−J[u]=ZZSF(x,y,u+δu,(u+δu)x,(u+δu)y)dxdy−ZZSF(x,y,u,ux,uy)dxdy=ZZShδu∂∂u+(δu)x∂∂ux+(δu)y∂∂uyiFdxdy+12!ZZShδu∂∂u+(δu)x∂∂ux+(δu)y∂∂uyi2Fdxdy+···,]# J[u]›5C,7 ¢£# ,B‘* (0 δJ[u]=ZZShδu∂F∂u+(δu)x∂F∂ux+(δu)y∂F∂uyidxdy=ZZSh∂F∂u−∂∂x∂F∂ux−∂∂y∂F∂uyiδudxdy+ZZSh∂∂x∂F∂uxδu+∂∂y∂F∂uyδuidxdy=0. ªZZS∂Q∂x−∂P∂ydxdy=ZΓPdx+Qdy,›Q=∂F∂uxδu,P=−∂F∂uyδu,*ZNM,˚ (δJ[u]=ZZSh∂F∂u−∂∂x∂F∂ux−∂∂y∂F∂uyiδudxdy+ZΓh−∂F∂uxdx+∂F∂uydyiδu.· ¡¢£ u ΓW: 1 δu Γ=0,Nª m'n,Q (0 6$δJ[u]=ZZSh∂F∂u−∂∂x∂F∂ux−∂∂y∂F∂uyiδudxdy=0.WuChong-shi 7 δu,A R 1$«%NM,1 ’B:(0 ∂F∂u−∂∂x∂F∂ux−∂∂y∂F∂uy=0,.#' ’´ J[u]=ZZSF(x,y,u,ux,uy)dxdy›5C,7 ¢£,s ª(Euler–LagrangeCr)- .&˚ H ¸31.2Ł , !N a +S=Zt1t0dtZx1x012hρ∂u∂t2−T∂u∂x2idx›5C,7 ¢£∂2u∂t2−Tρ∂2u∂x2=0,. #m13 «%, , Cr-k … o f w Æ| Æ ª v| ~ Æ f ª …| 0-ŁØfiflceŒº l fl flæ ·fiflceŒº- d ı f ~ł ıøœ ßf- M$B ’(¸ ˚B * , ¢ D -1.ˆ8 ]* #G ’y , ])*+x 6$ * ‘s Bs 1 ˜ δdydx=d(δy)dxZδy0=(δy)0.2.* Y#B&QR δ(αF+βG)=αδF+βδG,RŁα‘β# ’-3. 1$a ’ ,* D Gδ(FG)=(δF)G+F(δG).4.* ‘ (s , )Y1$ ˜ δRbaFdx=Rba(δF)dx..º ªX ,: 3‘’‘Z12%-WuChong-shi§31.2łøœ]^ 8 5. ’,* RD ‘s ‰ º d δ !#$δF(x,y,y0)=∂F∂yδy+∂F∂y0δy0.%&’($)*F+, -.$/01y + $23+4x/5+, !67$895:; (∂F/∂x)δx =!%?@ $ABCD7E5FG HIJKL01 MN!•OPCQ R0STUV$WX;R0YST Z [\] Euler–Lagrange^_‘$ab Wcd de[\fge ^_$hijgkST01!•b ’($Euler–Lagrange^_l/R0YST Z [\$m5/n Z [\!Wcd de[\f$Euler–Lagrange^_ ej5opq$rsl/ST01 tuv!Jwxp(y)qe/ g ST01$ab zp{|7}~!• g01ST MNp $}~ ^ i !•p / 6gk e 01 R0T$ R0ST d |7 !% ^ 5 $ : K @!• p ^ / @R0 + δ2J$# 9 6gk e$R0 + Y ( )T$ e PST01$R0YS ( )!% ^ AB ¡$¢# + P0$ b £⁄¥ƒ§ + !• ¤UV''“«~ ‹%›/Wcd fifl[\f$Euler–Lagrange^_lipqe$ $–†‡·1 ¶•‚“j $ R0 STpd„W$”»$