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1、1o Euler-Lagrange 1 Euler-Lagrange1 f(x,y,z),fy(x,y,z),fz(x,y,z) C(R3) K F(y) = Z b a f(x,y(x),y0(x)dx = Z b a fy(x)dx D= 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)zY=D= Ca,b F(y;v) = Z b a fy(x,y(x)v(x)dx K. e,:y0Y,F(y0;v) = 0,v D0YK d dxfzy(x) = fyy(x) x (a,b)
2、Euler-Lagrange1 F(y0;v) = fzy(x)v(x)|b a, v Y y n2 d dxfzy(x) = fyy(x) FL“ . y DF7:e d dxFz = fy yf7:“ 1 21AA2 2 1AA A. f(x,y,z) = f(z) dfy 0 fzy(x) const = f0(y0(x) AO/ y0(x) constEuler-Lagrange)“ 1. 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,
3、 z(1) = z1 fz= z 1+z2 = const,z = const, z0() = const, z = z1 1() B. F(x,y,z) = f(x,z) fz(x,y0(x) = fzy(x) const?5) 2. 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+ (fy0
4、y0)0 (fy)0y0 d dx(f y 0fy0) = y0(fy0)0 fy) = 0 f y0fy0 const 21AA3 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 eu“:u 0 u0= 2u u0 = 1 u = (x + C1)2y2= C + (x + C1)2 eD = y C21,1, y(1) = 0, y(1) = 1 f3D7:y0= q (x + 1)(x 1 2) / C 21,1“ ?y0f3
5、D = y C1a,b, y(1) = 1, y(2) = 3 27:“ 4. K7: 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 0y16 0 ? =k(0,0):L(x1,y1):“(!) y 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
6、 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 yF7:“ yy0= x4?“ ?yF(y0;v) = 0 v Dad(y0) 4 gd:KI g,. D = y C1a,b, y(b) = b1 x = a:gd. =y(a)?“ F(y) = Z b a fy(x)dx y D Dad(y) = v C1a,b, v(b) = 0 d ey0 DF3D4: K d dxfy 0y0(x) = fyy0(x) x
7、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. uD = y C1a,b, y(a) = a14v fy0y0(b) = 0 y0(a) = a1 ABernoulle T(y) = 1 2g Z x1 0 p1 + y 02 ydx 5pmEULER-LAGRANGE10 D = y 0,x1|y(x) 0, x (0,x1 y(0) = 0, y C10,x1, Z x1 0 p1
8、+ y 02 ydx . 0 = fy0y(x1) = y0(x1) py(x 1) p1 + y02(x 1) d(0,0): $:3x = x1“ SK F3DU4“ 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 pmEuler-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
9、 a fy(x)dx FC2a,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) = b24? :y0Uv7“ 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
10、 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) 5pmEULER-LAGRANGE11 F(y;v) = Rb afy00y(x) h(x)v 00(x)dx = 0 v Dad(y0) dn4 c1,c2 fy00y(x) h(x) = c1x + c2 fy00y(x) = h(x) + c1x + c2 dfy00y(x) C1a,b d dxfy 00y(x) = h0(x) + c1= fy0y(x) g(x) + c1 d d dxfy 0
11、0y(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 . f C1(a,b R3)ey C2a,bv d dx d dxfy 00y(x) fy0y(x) + fyy(x) = 0 x a,b Kyf7:“ g,.II F3D1= y C2a,b|y0(a) = a2, y(b) = b24y0Uv7 Dad(y0) = v C2a,b|v0(a) = v(b) = 0 D1 0 = v
12、 C2a,b|v(a) = v(b) = v0(a) = v0(b) = 0 d y0vEuler-Lagrange =y07f7:“ 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) fy
13、00y(b) = 0g,. v = (x a)2(x a ab 2 ) fy0y(a) d dxfy00y(x)|x=a = 0g,. 6mEULER-LAGRANGE12 6 mEuler-Lagrange F(Y ) = Rb a f(x,Y (x),Y 0(x)dx = Rb a fY (x)dx f,fY,fZ3a,b Rn RnY 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)
