变分法讲义_第四章

1oÙ Euler-Lagrange § §1 Euler-Lagrange1§ f(x,y,z),fy(x,y,z),fz(x,y,z) C(R3)§ K¼ F(y) = Z b a f(x,y(x),y0(x)dx = Z b a fy(x)dx ½ÂD= C1a,b =Y y,v Y F(y;v) = Z b a (fyy(x)v(x) + fzy(x)v0(x)dx 5. ?f(x,y,z) = f(x,y)عz§Y=D= Ca,b F(y;v) = Z b a fy(x,y(x)v(x)dx ·K. eé,:y0Y,F(y0;v) = 0,v D0Y§K d dxfzy(x) = fyy(x) x (a,b) £Euler-Lagrange1§¤ F(y0;v) = fzy(x)v(x)|b a, v Y y² Ún2 d dxfzy(x) = fyy(x)§ ©ÜÈ© FLª“ ½Â. ¡y D¼F7:§e d dxFz = fy ½¡yf7:¼ê“ 1 21§A«A2 §2 1§A«A A. f(x,y,z) = f(z)§ dfy 0 fzy(x) const = f0(y0(x) AO/§ y0(x) const½´Euler-Lagrange§)“ 1. Ρþ᧠x = cos y = sin z = z() J(z) = R1 0 px02() + y02() + z02()d = R1 0 p1 + z02()d D = z C10, z(0) = 0, z(1) = z1 fz= z 1+z2 = const,z = const, z0() = const, z = z1 1(Ú) B. F(x,y,z) = f(x,z) fz(x,y0(x) = fzy(x) const?5¼êÒؽ´) 2. ¥¡þ᧠L() = R Z 1 0 q 1 + 02()sin2d f(,0) = f(,0) = R p 1 + 02sin2 f0= Rsin2 p 1 + 02sin2 R 0sin2 p 1 + 02sin2 const 0 0 C. f(x,y,z) = f(y,z) d dxf(y(x),y 0(x) = fyy0 + fy0y00= fyy0+ (fy0y0)0 (fy)0y0 d dx(f y 0fy0) = y0(fy0)0 fy) = 0 f y0fy0 const 21§A«A3 3. f(y,z) = y2(1 z)2 fz= 2y2(z 1) y2(1 y0)2 y0(2y2(y0 1) = C y2y02= y2 C Pu = y2 C u02= 4u euk“: u 0 y0 C euÓ:§u 0 u0= ±2u u0 = ±1 u = (x + C1)2y2= C + (x + C1)2 eD = y C21,1, y(1) = 0, y(1) = 1 f3Dþ7:¼êy0= q (x + 1)(x 1 2) / C 21,1§Ã“ ?y0´f3D = y C1a,b, y(1) = 1, y(2) = 3 2þ7:¼ê“ 4. ü¯K7:¼ê T(y) = 1 2g Z x1 0 p1 + y02(x) y(x) dx D = y C10,x1, y(0) = 0, y(x1) = y1, y(x) 0, x (0,x1), Rx1 0 1 y(x)dx 0§y16 0 ? =k±(0,0)©:§L(x1,y1):Ó“(!) y² ey1= 0§ Kr = x1 2=“ ey16= 0 x y = sin 1 cos + 2 0 0 0+ ( sin 1 cos )0= 2(1 cos) sin (1 cos)2 = cos 2 (sin 2) 3(tan 2 2) 0 !1§ ¦x1 y1 = 1sin1 1cos1 r = x1 1sin1 ?0 0, y(0) = 0, y(x1) = y1 0 Rx1 0 1+y02(x) y(x) dx x F(y) = R1 0 y xdx y²FÃ7:¼ê“ y²y0= x´4?¼ê“ ?yF(y0;v) = 0 v Dad(y0) §4 gdà:¯KI§ g,. D = y C1a,b, y(b) = b1 x = a:´gd.§ ½=y(a)?¿“ F(y) = Z b a fy(x)dx y D Dad(y) = v C1a,b, v(b) = 0 Ïd§ ey0 D´F3DþÛÜ4:§ K d dxfy 0y0(x) = fyy0(x) x a,b F(y0;v) = Z b a fyv + fy0v0= fy0v|b a v C1a,b AO/§ ?v Dad(y)§ 0 = F(y0;v) = fy0y0(a)v(a) v = b x fy0y0(a) = 0g,. y0(b) = b1r5. Ó§ éuD = y C1a,b, y(a) = a1þ4¼ê½÷v fy0y0(b) = 0 y0(a) = a1 AµBernoulleü T(y) = 1 2g Z x1 0 p1 + y 02 ydx 5p¼êmþ¼EULER-LAGRANGE§10 D = y 0,x1|y(x) 0, x (0,x1 y(0) = 0, y C10,x1, Z x1 0 p1 + y 02 ydx . 0 = fy0y(x1) = y0(x1) py(x 1) p1 + y02(x 1) Ïdù´±(0,0)©:§ $:3x = x1þÓ“ SK ¦ÑF3DþU4¼ê“ 1. F(y) = R 2 0 y2(x) y02(x) D1= y C10, 2 y(0) = 0, D2 = y C10, 2 y(0) = 1 2. F(y) = R1 0 cosy0(x) D = y C10,1 y(0) = 0 §5 p¼êmþ¼Euler-Lagrange§ f(x,y,z,w) C(a × b × R × R × R) 3 fy,fz,fw F(y) = Rb a f(x,y(x),y0(x),y00(x)dx = Rb a fy(x)dx F½ÂC2a,b y,v C2a,b : F(y;v) = F(y + tv) t |t=0 = Z b a fyy(x)v(x) + fy0y(x)v0(x) + fy00y(x)v00(x) ¦F3D = y C2a,b|y(a) = a1, y(b) = b1, y0(a) = a2, y0(b) = b2þÛÜ4? :y0U÷v7“ Dad(y) = v C2a,b|v(a) = v0(a) = v(b) = v0(b) = 0 0 = F(y;v)v Dad(y0) Pg(x) = Rx a fyy(x)dx C1a,b, g0(x) = fyy(x) f(y;v) = Rb afy0y(x) g(x)v 0(x) + fy00y(x)v00(x)dx Ph(x) = Rx a fy0y(x) g(x)dx C1a,b, h0(x) = fy0y(x) g(x) 5p¼êmþ¼EULER-LAGRANGE§11 F(y;v) = Rb afy00y(x) h(x)v 00(x)dx = 0 v Dad(y0) dÚn4§ c1,c2¦ fy00y(x) h(x) = c1x + c2 fy00y(x) = h(x) + c1x + c2 Ïdfy00y(x) C1a,b d dxfy 00y(x) = h0(x) + c1= fy0y(x) g(x) + c1 Ïd d dxfy 00y(x) fy0y(x) = g(x) + c1 C1a,b d dx d dxfy 00y(x) fy0y(x) = fyy(x) ef C2(a,b × R3)§ K fyy(x) d dxfy 0y(x) + d2 dx2 fy00y(x) = 0 x a,b ½Â. f C1(a,b × R3)§ey C2a,b§÷v d dx d dxfy 00y(x) fy0y(x) + fyy(x) = 0 x a,b K¡y´f7:¼ê“ g,.II ¦F3D1= y C2a,b|y0(a) = a2, y(b) = b2þ4¼êy0U÷v7 µ Dad(y0) = v C2a,b|v0(a) = v(b) = 0 D1 0 = v C2a,b|v(a) = v(b) = v0(a) = v0(b) = 0 Ïd§ y0÷vEuler-Lagrange§§ =y07´f7:¼ê“ Ï § v Dad(y0) 0 = F(y0;v) = Z b a fyv + fy0v0+ fy00v00 = Z b a fyv + (fy0 d dxfy 00)v0 + fy00v0|b a = Z b a (fy d dx(fy 0 d dxfy 00)v + (fy0 d dxfy 00)v|b a+ fy00v 0|b a = fy0y(a) d dxfy 00y(x)|x=av(a) + fy00y(b)v0(b) v = (x a)2(x b) fy00y(b) = 0g,. v = (x a)2(x a ab 2 ) fy0y(a) d dxfy00y(x)|x=a = 0g,. 6þ¼êmþ¼EULER-LAGRANGE§12 §6 þ¼êmþ¼Euler-Lagrange§ F(Y ) = Rb a f(x,Y (x),Y 0(x)dx = Rb a fY (x)dx f,fY,fZþ3a,b × Rn× RnþëY fY= fy1 . . . fyn , fZ= fz1 . . . fzn , Y 0(x) = y0 1(x) . . . y0 n(x) D= Y (C1a,b)n Y,V DY (x) = y0 1(x) . . . y0 n(x) , V (x) = v0 1(x) . . . v0 n(x)