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WuChong-shi ( ) 1 ( )§32.1 !#$%&’()*+,-•./0%&’f(x,y)123)*(45678df=∂f∂xdx+∂f∂ydy=0.9:dx,dy;1=0%&’f(x,y)3)*(4567?@AB∂f∂x=0,∂f∂y=0.•C/DE0%&’()*+,10%&’(67)*+,1FGHI67g(x,y)=C#Jf(x,y)()*-KL1GMNO1@PHI67QRy=h(x)1STUVf(x,y)W(y-KX1OY67)*+,Z[\:%&’f(x,h(x))(]^)*+,123)*(4567Z8∂f∂x+∂f∂yh0(x)=0.•_‘KabcC/DdeQ-9:Ofghi5jklmy=h(x)(nop1qri5lmdydx≡h0(x).KX1sth4(Gu$’vw#xh@y)JRy=h(x)1Z@z{_HI67|}∂g∂xdx+∂g∂ydy=0,~qJRdydx=−∂g/∂x∂g/∂y,‘8F@ OY0%&’3)*(4567AB∂f∂x−∂f∂y∂g/∂x∂g/∂y=0. 1 - 1 1 ¡¢£¢⁄¥-WuChong-shi§32.1ƒ§¤'“«‹ 2 •G›fiW1fl fiLagrange–†‡· e$%&’(67)*+,-¶•1_‘Of(GHI67g(x,y)=C#J&’f(x,y)()*+,1Z@‚„Lagrange”»λ1q…‰a (0%&’¿h(x,y)=f(x,y)−λg(x,y). x`y´B8ˆa˜¯˘˙1KX1Ka0%&’3)*(4567Z8(¨ ´R1UVλ1KZy\:Of˚R(4567)∂(f−λg)∂x=0,∂(f−λg)∂y=0.P¸@JRx=x(λ),y=y(λ), ˝HI67W1…RLagrange”»λ(’*1Z@JR@y()*˛(x,y)-ˇ— 1 -ˇ— 1 ¡ Æ Lagrange ª - G˝ &(67)*+,-•c5J &J[y]=Zx1x0F(x,y,y0)dxG Ł67y(x0)=a,y(x1)=bØHI67J1[y]≡Zx1x0G(x,y,y0)dx=C#()*1N@…‰J0[y]=J[y]−λJ1[y], δy´B8˜¯(1N &J0[y]G Ł67#3)*(4567Z8∂∂y−ddx∂∂y0(F−λG)=0.P¸|}Œº Ł67ØHI67145L 1Z@JRLagrange”»(*λ=λ0 )*&’y=y(x,λ0)1Øæ ( &J0[y](67)*- 32.1J &I[y]=Z10xy02dx¿ ı łøœß łLagrange ı WuChong-shi ( ) 3 G Ł67y(0)/Ł,y(1)=0`HI67Z10xy2dx=1#()* - fiOf Y(Lagrange”» 1@ ˝4567∂∂y−ddx∂∂y0 xy02−λxy2=0,Fddxxdydx+λxy=0.(#)¸ŒºØ ( Ł67F Ba *+,12( *λi=μ2i,μi8 &’J0(x)( iak ˛1i=1,2,3,···k Z8Lagrange”»1q)*&’Z8æ ( &’yi(x)=CJ0(μix). ˙C@PHI67…R-9:C2Z10xJ20(μix)dx=C22J21(μi)=1,=C=√2J1(μi).KX1ZJR )*&’yi(x)=√2J1(μi)J0(μix).P‘Lagrange”»(‚„1GEuler–LagrangeŒºR … ˙1` Ł67 G1Z B *+,-q!: *+,12(Q1 *` &’1/#$a-K$/ˆa+,i5%&-F a+,1K#$a &’’8)*&’-K@~#f(˘}()´R-P Ł67ØP¸* (δy x=0/Ł,δy x=1=0.@JRI[y](+˘}δI[y]=2Z10xy0(δy)0dx,„q@JRI[y](0+˘}δ2I[y]=2Z10x δy02dx0.WuChong-shi§32.1ƒ§¤'“«‹ 4 9: &I[y](0+˘},3k*1=K-)*&’./ &3)0-F 0a+,81K#a *k xZ8 &()*-K89:1 Œº(#)”)*&’y(x)112}1Z/λZ10xy2dx=−Z10y xy00dx=−y·xy0 10+Z10xy02dx=Z10xy02dx,34HI671Zy ˝λ=Z10xy02dx.561789 1:;= ?@A BC1 DE FGHIJKL;MN OPQR BSTU@A-VW1= ?@A1XYZ[K@A(Isoperimetricproblem)-WuChong-shi ( ) 5 §32.2\]^_‘abcdef bc g]hi= P? j8 kl (Euler–Lagrangemn) Xklmnopklmn1qr Ms tHu£1¡vwXklmnopklmn Ms@Axyz= ?@A1{j8 |}~ M (Lagrange ª)1qr tHu£1¡vwklmn ?@A-:; q @A ˇU klmn Ms@Ao ?@A Z= ?o ?@A1o 1ˇU klmn Ms@Ao ?@A l - 32.2AR |}Œº *+,ddxp(x)dydx+q(x)y(x)=f(x),x0xx1,(#)y(x0)=y0,y(x1)=y1(z)( &wp1F Ræ ( &12G Ł67(z)#3)*(4567F:(#)- S &)*4567(|}wpZ8Œº(#)1 1KaŒº…· Zx1x0ddxp(x)dydx+q(x)y(x)−f(x)δy(x)dx=0. G(+,Z85 Op \B 2}(˘}1K_‘ 2} 2&’( 0 ¡¢8£¨ › (1Zx1x0q(x)y(x)δy(x)dx=12δZx1x0q(x)y2(x)dx,Zx1x0f(x)δy(x)dx=δZx1x0f(x)y(x)dx.⁄¥ q(x)rf(x) ƒy(x) l§¤ 1VW1' l“«|1q‹› X -_‘ 2&’W( ¢1@}fi2}1Zx1x0ddxp(x)dydxδy(x)dx=p(x)dydxδy(x) x1x0−Zx1x0p(x)dydxd(δy)dxdx=−Zx1x0p(x)dydxδdydxdx=−12δZx1x0p(x)dydx2dx,flWfi˝ δy(x) x0=δy(x) x1=0- Of(bc ·1Z ˝Zx1x0ddxp(x)dydx+q(x)y(x)−f(x)δy(x)dx=−δZx1x0(12p(x)dydx2−q(x)y2(x)#+f(x)y(x))dxWuChong-shi§32.2– †‡· ¶•‚„”‹¶•¤ »… 6 =0.KZ‰ 1Œº(#)…Z8 &J[y]=Zx1x0(12p(x)dydx2−q(x)y2(x)#+f(x)y(x))dx3)*(4567- 32.3AR¿|}Œº…Q+,∇2u(r)+k2u(r)=−ρ(r),r∈V,u(r) Σ=f(Σ)(˘}wp- @ `´ˆ¶32.2(˜ 1¯˘2}ZZZV∇2u+k2u+ρ(r)δudr,_‘ 2&’W(Tˆ¢1/ZZZVk2uδudr=12δZZZVk2u2dr,ZZZVρ(r)δudr=δZZZVρ(r)udr._‘ 2&’W( ¢1Ni5 fiGreen ˙pØ Ł67δu(r) Σ=01ZZZV∇2uδudr=ZZΣδu∇u·dΣ−ZZZV∇u·∇ δudr=−12δZZZV ∇u2dr.9¸1MŒºZ[\:δZZZV12 ∇u2−k2u2−ρudr=0.K‰ 1M·(…Q+,Z¨ ‘G Ł67u(r) Σ=f(Σ)#J &ZZZV12 ∇u2−k2u2−ρudr()*+,- 32.4AR¿|}Œº( *+,∇2u(r)+λu(r)=0,r∈VWuChong-shi ( ) 7 u(r) Σ=0(˘}wp- ˚ 1@ +,´B8¶32.3(¸ vw-9¸1¸ *+,Z¨ ‘ &J[u]=ZZZVn∇u(r)2−λu(r)2odrG Ł67u(r) Σ=0#()*+,-fl„˝1 *λ´B8Lagrange”»1 1Ka &)*+,?¨ ‘ &J[u]=ZZZV∇u(r)2drGOY Ł67`HI67( &’(˛\67)J1[u]≡ZZZVu(r)2dr=1#(67)*+,-hˇeQ1K- &’k Z8 &()*&’1q *k 8 &()*-P‘ &J[u](0+˘}δ2J[u]=2ZZZV∇ δu(r)2dr,:k1=1 &()*8)0*-K-)0*W(—0 1 SZ8 *+,(—0 *-WuChong-shi§32.3Rayleigh–Ritz† 8 §32.3Rayleigh–Ritz^ •˘} G e W( fi1@}:ˆa 5(Œf-•d fi8!: e (nY -–@fiHamiltonMe fl E ( Y Æ( ˛ ˛ ······)(ª 1–@fiFermatMe Y G W( Ł1ØŒGŁfO(º ` 1–x@fi˘}( Y æt| »(ª 1¨¨-G e (K-} W1 ı ª ( d¸…wp( 1¶ ł’@nY: ( &)*+,-˘} (Kd fi1ø/œ5(e&‰-2@/ß flÆł Q Ł(ª 1@/ß flŒ ł~ l( e ( -•˘} ( 0d fiN8 R2(›fi * 2:JQø ( e+, d ( - ß L S8 ‘/fi|}ŒºV AK- e+,1 r/ ’(+, y !#$$%&’()* + ,-. /01 23456 7 )* .+89:;;= ?@!LX=λρX,α1X(a)+β1X0(a)=0,α2X(b)+β2X0(b)=0. ABCDEFG(H:IJK )LM-NO P QRSTUVWX JG?@Y?Z +:[\]&^;\]&DLM-NO6_‘ ab;\]&Dcde7LM-NOf . g& GLM-NO +;=hi j5k lmno Z p 6qrs&’(?@ t uvw+xyzX00(x)+λX(x)=0{|}~ NO :n [ sin√λxYcos√λx ;= UVWX .FG?@ uvw+ ;= ?@! { q2] 5+,-.2 5 ?@ )*N.+•7Rayleigh–RitzN.)* ?@! / D^WuChong-shi ¡¢£⁄(¥)ƒ9§–¤E'?@!“« ‹Z WX›@! –QR ;J Z fifl# '!–“«Z WX›@!+•%e†‡ Z fifl( ·?@!)D ¶ •‚1C]„”»… ‰ )?@ … @+•¿ 7 `´ 2De†‡;ˆ˜¯ Z fifl( 1D;Z ˘˙) ;N¨ ˚¸ ;N¨–&»”» ‰˝”»… ‰ ’?@ )*@+•{2e Z ˘˙˛:?Z ˇe —e/ ? e E¿ 1Y 1 ?Z Gt + 32.5 ?@!1xddxxdydx+λy(x)=0,y(0):V,y(1)=0 } ?@+ {?@! 32.1 32.1#lm Æ+ ª D‹ZI[y]=Z10xy02dx UVWXy(0):V,y(1)