x,u ( x ) ,u ( x ) @ y @ y @ L @ L

J

C¹([a, b],R) C²([a, b],R)

C¹ C² [a, b] R a

b

H = h2C²([a, b],R), h(a) =h(b) = 0 S= y2C²([a, b],R), y(a) =y0, y(b) =y1 .

S

8u2S,kuk=kuk1+ku⁰k1

S H

S

L2C⁰([a, b]⇥R²,R)

J :u2S7 ! Z b

a

L(x, u(x), u⁰(x)) dx.

@L

@y

@L

@y x,u(x),u⁰(x)

f Rⁿ R n k|fk|

f

k|fk|=supn |f(x1, ..., xn)|

kx1k^Rn...kxnk^Rn,(x1, ..., xn)2Rⁿ {0}o k.k^Rn Rⁿ

J

J J

S dJ

J dJ

J x2[a, b] (y, z)2R²7 !L(x, y, z) C² @L

@y

@L

@z [a, b]⇥R²

J u 2 S

h2H

dJ(u)(h) = Z b

a

⇣@L

@yh(x) +@L

@zh⁰(x)⌘ dx u2S h2H

J(u+h) J(u) = Z b

a

L(x, u(x) +h(x), u⁰(x) +h⁰(x)) L(x, u(x), u⁰(x)) dx.

x2[a, b]

K(x) =L(x, u(x) +h(x), u⁰(x) +h⁰(x)) L(x, u(x), u⁰(x)).

x2[a, b] (y, z)2R²7 !L(x, y, z) x2[a, b]

K(x) = @L

@yh(x) +@L

@zh⁰(x) +p

h²(x) +h⁰²(x)"x(p

h²(x) +h⁰²(x))

"x lim

y!0"x(y) = 0

dJ(u) : H ! R

h 7!

Z b a

⇣@L

@yh(x) +@L

@zh⁰(x)⌘ dx

h2H h h⁰ @L

@y

@L

@z [a, b]⇥R² x2[a, b]7 ! @L

@yh(x) +@L

@zh⁰(x) [a, b]

dJ(u) H

J k·k

Z b a

ph²(x) +h⁰²(x)"(p

h²(x) +h⁰²(x)) dx=khk (khk)

kulimk!0 (kuk) = 0

L C²

x2[a, b]

H(x) =

✓h(x) h⁰(x)

◆

U(x) =

✓u(x) u⁰(x)

◆

x2[a, b]

|p

h²(x) +h⁰²(x)"x(p

h²(x) +h⁰²(x))|⇣h²(x) +h⁰²(x) 2

⌘supn

k|d²L(U(x)+tH(x))k|, t2[0,1]o x2[a, b]

h²(x) +h⁰²(x)2khk².

d²L(U+.H) : (t, x)2[0,1]⇥[a, b]7 !d²L(U(x) +tH(x))

[0,1]⇥

[a, b] M > 0

(t, x)2[0,1]⇥[a, b]

k|d²L(U(x) +tH(x))k|M

Z b a

ph²(x) +h⁰²(x)"x(p

h²(x) +h⁰²(x)) dx (b a)khk²M

J

L u2 S J

Z b a

⇣@L

@yh(x) +@L

@zh⁰(x)⌘ dx= 0 h2H

dJ

u v u < v ]u, v[ a b

x2]u, v[ (x)>0 x2R ]u, v[ (x) = 0

h2 H g [a, b]

R Z b a

g(x)h(x) dx= 0 g [a, b]

u v

u < v (x) =

((x u)²(v x)² x2]u, v[

0 x2R ]u, v[

H

u v R (a) = (b) = 0

x2[a, b] (x)>0

x2]u, v[

(x) (u)

x u = (x u)(v x)²

0(u) = 0 ⁰(v) = 0

0(x) =

(2(x u)(v x)²+ 2(x u)²(v x) x2]u, v[

0 x2R ]u, v[

g

[a, b] R h2H

Z b a

g(x)h(x) dx= 0 g

c2[a, b] g(c) g(c)>0 g

]u, v[ c g(x)>0 x2]u, v[

(x) > 0 x2]u, v[ (x) = 0 x2R ]u, v[

Z b a

(x)g(x) dx= Z v

u

(x)g(x) dx >0 x2[a, b] g(x) = 0.

J

x2[a, b] (y, z)2R² 7 !L(x, y, z) C² @L

@y [a, b]⇥R²

@L

@z 2C¹([a, b],R)

h2H u2S

Z b a

@L

@yh(x) +@L

@zh⁰(x) dx= 0

u2S x2[a, b]

@L

@y d dx

@L

@z = 0

dJ

h2H u2S

@L

@z 2C¹([a, b],R) Z b

a

@L

@zh⁰(x) dx=h@L

@zh(x)ib a

Z b a

h(x) d dx

@L

@z dx

h(a) =h(b) = 0 dJ

h2H u2S

dJ(u)(h) = Z b

a

(@L

@y d dx

@L

@z)h(x) dx

h2H u2 S dJ(u)(h) = 0 x2[a, b] @L

@y d dx

@L

@z = 0.

x2[a, b] g(x) =@L

@y d dx

@L

@z.

J J

E f

C¹ E R f x2E

x f df(x) = 0.

(E,k.k^E) f

C¹ E R x2E f f

C¹ f h2E

limt!0

f(x+th) f(x)

t =df(x)(h) h2E

f(x+th) f(x) =tdf(x)(h) +t"(t)

tlim!0"(t) = 0

x f

>0 y2E f(x) f(y)<0 kx yk^E<

h 2 E |t|khk < f(a+th) f(a) < 0 tdf(x)(h) +t"(t)<0

t 0 t >0 df(x)(h)0

df(x)(h) 0 t 0 t <0

h2E df(x)(h) = 0 J

J

x2[a, b] (y, z)2R² 7 !L(x, y, z) C¹ @L

@y [a, b]⇥R²

@L

@z 2C¹([a, b],R)

u2S J

u

@L

@y d dx

@L

@z = 0.

u2 S J u

J dJ(u) = 0

J

J

(a, y0) (b, y1)

y=u(x) x

[a, b] u S

u : [a, b] ! R² x 7!

✓u(x) x

◆

u2S

u2S L( u) =

Z b

a k u⁰(x)k^R2dx

k.k^R2 R²

J:u2S 7 ! Z b

a

p1 +u⁰²(x) dx.

(y, z) 2 R² L(y, z) =p

1 +z² L

d dx

⇣ u⁰ p1 +u⁰²

⌘= 0

u⁰⁰= 0.

S x 2 [a, b] 7 ! u(x) = y0 y1

a b x+ay1 by0

a b u J

J

J

J

J

J

x 2 [a, b] (y, z) 2 R² 7 !

L(x, y, z) C³ @²L

@y²,@²L

@z²

@²L

@y@z [a, b]⇥R²

u2S (h, k)2H²,

d²J(u)(h, k) = Z b

a

⇣@²L

@y²h(x)k(x)+ @²L

@y@z(h⁰(x)k(x)+k⁰(x)h(x))+@²L

@z²h⁰(x)k⁰(x)⌘ dx

J(u+h) J(u) = Z b

a

L(x, u(x) +h(x), u⁰(x) +h⁰(x)) L(x, u(x), u⁰(x)) dx.

x2[a, b]

K(x) =L(x, u(x) +h(x), u⁰(x) +h⁰(x)) L(x, u(x), u⁰(x)).

x2[a, b] (y, z)2R² 7 !L(x, y, z)

C² K x2[a, b]

K(x) =@L

@yh(x) +@L

@zh⁰(x) +@²L

@y²h²(x) + 2@²L

@y@zh⁰(x)h(x) +@L

@zh⁰²(x) + (h²(x) +h⁰²(x))"(h²(x) +h⁰²(x))

" lim

y!0"(y) = 0

d²J(u) : H⇥H ! R

(h, k) 7!

Z b a

⇣@²L

@y²h(x)k(x) + @²L

@y@z(h⁰(x)k(x) +k⁰(x)h(x)) +@L

@zh⁰(x)k⁰(x)⌘ dx H

J k·k

J C² u J

d²J(u)

u2S J

h2 H dJ(u)(h) = 0 u J

J

t2R h2H

J(u+th) J(u) =d²J(u)(th, th) +kthk²"(kthk)

zlim!0"(z) = 0

u J

u ⌘ > 0 k.k B(u,⌘) =

{y2S,ku yk<⌘} y2B(u,⌘) J(u) J(y)0

t |t|khk<⌘

J(u+th) J(u) =t²d²J(u)(h, h) +t²khk²"(|t|khk)>0

t >0 |t|khk<⌘ d²J(u)(h, h) +khk²"(|t|khk)>0

t 0 d²J(u)(h, h) 0 d²J

J

J J u 2 S

J C² k >0

h2H d²(u)(h, h)> kkhk²

u J

J t2R h2H

J(u+th) J(u) =t²d²J(u)(h, h) +kthk²"(kthk)

zlim!0"(z) = 0

J

t2R h2H

J(u+th) J(u)> t²khk²⇣

k+"(kthk)⌘

zlim!0"(z) = 0 > 0 kthk< "(kthk) ^k₂

kthk<

J(u+th) J(u) k

2t²khk² 0

x 2[a, b] 7 ! u(x) = y0 y1

a b x+ ay1 by0

a b u

h2H d²J(u)(h, h) =

Z b a

h⁰²(x) (1 +u(x)⁰²)(p

1 +u(x)⁰²)dx

J

J

u2S h2H

d²J(u)(h, h) = Z b

a

@²L

@z²

2(x) dx

C¹ [a, b] R u h

@²L

@z² [a, b]

x

L C²

L x2[a, b]

@²L

@z² >0

@²L

@y@z C¹ [a, b]

d²J u2S h2H

d²J(u)(h, h) = Z b

a

⇣@²L

@z²h⁰²(x) +B(x)h²(x)⌘ dx

B =@²L

@y² d dx

@²L

@y@z

u2S h2H d²J(u)(h, k) =

Z b a

⇣@²L

@y²h²(x) + 2@²L

@y@z(h⁰(x)h(x)) +@²L

@z²h⁰²(x)⌘ dx

@²L

@y@z C¹ [a, b]

x2 [a, b] 2 @²L

@y@z(h⁰(x)h(x)) = @²L

@y@z d dxh²(x)

Z b a

2h(x)h⁰(x)@²L

@y@zdx=h

h²(x)@²L

@y@z i^b

a

Z b a

h²(x) d dx

@²L

@y@zdx h(a) =h(b) = 0

Z b a

2h(x)h⁰(x)@²L

@y@zdx= Z b

a

h²(x) d dx

@²L

@y@zdx

L

y²= @²L

@z²(B+y⁰) B= @²L

@y² d dx

@²L

@y@z

w

u2S h2H

d²J(u)(h, h) = Z b

a

@²L

@z²

⇣h⁰(x) +w(x)

@²L

@z²

h(x)⌘2

dx u 2 S h 2 H

Z b J

a

d dx

⇣w(x)h²(x)⌘

dx w C¹

[a, b]

Z b a

d dx

⇣w(x)h²(x)⌘

dx=w(a)h²(a) w(b)h²(b) = 0

h(a) =h(b) = 0

d²J(u)(h, h) =d²J(u)(h, h) + Z b

a

d dx

⇣w(x)h²(x)⌘ dx

d²J(u)(h, h) = Z b

a

⇣@²L

@z²h⁰²(x) +B(x)h²(x) + d dx

⇣w(x)h²(x)⌘⌘

dx

B =@²L

@y² d dx

@²L

@y@z

w

y²= @²L

@z²(B+y⁰)

f : R⇥[a, b] ! R

(t, x) 7! t²

@²L

@z²

B(x)

x C¹ t

w x2[a, b]

@²L

@z²h⁰²(x) +B(x)h²(x) + d dx

⇣w(x)h²(x)⌘

=@²L

@z²

⇣h⁰(x) +w(x)

@²L

@z²

h(x)⌘2

w

@²L

@z²

n2N

U ✓Rⁿ I R f :I⇥U !Rⁿ

(t, y)2I⇥U f(t, y) =A(t)y A2C⁰(I, Mn(R))

y⁰=f(t, y)

L C² @²L

@z²

@²L

@y@z C¹

d dx

⇣@²L

@z²v⁰⌘ ⇣@²L

@y² d dx

@²L

@y@z

⌘v= 0

L C² @²L

@z²

@²L

@y@z C¹

v [a, b]

[a, b]

w= v⁰ v

@²L

@z² C¹

v

c2]a, b] a v(a) =v(c) = 0

[a, b]

U ✓ Rⁿ f : [a, b]⇥U ! Rⁿ

⌦=U ⇥[a, b]

y⁰=f(t, y)

: ⌦ ! U

(y0, t) 7! y(t) (y0, t)2⌦ (y0, t) =y(t)

(y⁰=f(t, y) y(a) =y0

f

L C²

@²L

@z²

@²L

@y@z C¹

a ]a, b]

u [a, b]

a [a, b]

x=u y=u⁰ 8>

>>

><

>>

>>

: x⁰=y y⁰ =

⇣@²L

@y² d dx

@²L

@y@z

⌘x y d dx

@²L

@z²

@²L

@z²

⌦=R²⇥[a, b]

R² (v(a), v⁰(a))

t2[a, b] ((1,0), t)

p : R² ! R

(x, y) 7! x t2[a, b] v(t) =p( ((0,1), t)) v

a ]a, b] t 2]a, b] v(t) 6= 0

p p

⌘ > 0 (x0, y0) 2 R² k(x0, y0) (0,1)k1 < ⌘ t 2 [a, b] p( ((x0, y0), t)) 6= 0

R² (x, y)2R² k(x, y)k1=max{|x|,|y|}

(x0, y0)2R^⇤⇥R

|x0|<⌘ |y0|<⌘ ˜v=p( ((x0, y0), t)) [a, b]

u2S d²J(u)

u2S L C²

@²L

@z²

@²L

@y@z C¹

a u2S d²J(u)

a

[a, b]

J

u2S J h2H

J(u+h) J(u) = 1

2d²J(u)(h, h) + Z b

a

⇣⌘(h)h⁰²(x) + (h)h²(x)⌘ dx

⌘ 0 khk 0

u2S J

h2H ✓2]0,1[

J(u+h) J(u) =dJ(u)(h) +1

2d²J(u+✓h)(h, h) = 1

2d²J(u+✓h)(h, h) u

1

2d²J(u+✓h)(h, h)

J

L C²

@²L

@z²

@²L

@y@z C¹ u2S

a ]a, b] u

J

>0 u2S

h2H

K( , h) =d²J(u)(h, h) Z b

a

h⁰²(x) dx= Z b

a

(@²L

@z² )h⁰²(x) +B(x)h²(x) dx

K K( , h) =

Z b a

l(x, h(x), h⁰(x)) dx

l : [a, b]⇥R² ! R

(x, t, w) 7! (^@_@z^2L2 )w²+B(x)t² l

x 2 [a, b] l

@²l

@w² = 2⇣@²L

@z²

⌘

@²l

@t@w = 0 C¹ [a, b]

K( )

d dx

⇣@²l

@w²v⁰⌘ ⇣@²l

@t² d dx

@²l

@t@w

⌘v= 0

d dx

⇣@²L

@z²

⌘u⁰ Bu= 0

x2[a, b]

@²L

@z² >0

⌘ >0 x2[a, b] @²L

@z² >⌘ x2[a, b] @²L

@z² >0 <⌘

a ]a, b]

= 0

K v : (x, ) 2 [a, b]⇥]0;⌘[7 ! v(x, )

x2]a, b] x

a ]a, b]

(x,0) v ">0 >0

2]0,⌘[ < |v(x, ) v(x,0)|<" v(x,0)6= 0 v(x,0)>0 "= ^v(x,0)₂

<

v(x, )>v(x,0) 2 >0 a ]a, b]

K a ]a, b]

h2H d²K(h)

>0 h2H

1

2d²J(u)(h, h)>

2 Z b

a

h⁰²(x) dx

R= Z b

a

⌘h⁰²(x) + h²(x) dx

⌘ 0 khk 0

>0 >0 & >0

(khk< |⌘|<

khk<& |⌘|<

max=max( ,&) khk< max

khk< max

|R|<

Z b a

h⁰²(x) +h²(x) dx x 2 [a, b]

h²(x) =⇣ Z ^x

a

h⁰(t) dt⌘2

(x a) Z x

a

h⁰(t)²dt(x a) Z b

a

h⁰²(x) dx.

Z b a

h²(x) dx (b a)² 2

Z b a

h⁰²(x) dx

khk< max

|R| ⇣

1 + (b a)² 2

⌘ Z ^b

a

h⁰²(x) dx 2 Z b

a

h⁰²(x) dx

= 2(1 +^{(b a)}₂ ²)

khk< max

J(u+h) J(u) =1

2d²J(u)(h, h) + Z b

a

⌘h⁰²(x) + h²(x) dx

>2 Z b

a

h⁰²(x) dx 2

Z b a

h⁰²(x) dx= 0

u J

a

J a [a, b]

[a, b]

u (c1, c2)2R²

u1 = @u

@c1

u2 = @u

@c2

C²

[a, b] L C²

(u1, u2)

u1 u2

u

d dx

@L

@z

@L

@y = 0

@

@c1

⇣ d dx

@L

@z

@L

@y

⌘= 0

L C²

@

@y

⇣@L

@y

@u

@c1

+@L

@z

@u⁰

@c1

⌘= @

@y

⇣@L

@c1

⌘= @²L

@c1@y d

dx

⇣@

@z

⇣@L

@y

@u

@c1 +@L

@z

@u⁰

@c1

⌘⌘= d dx

@

@z

⇣@L

@c1

⌘= @

@c1

d dx

@L

@z

@

@y

⇣@L

@y

@u

@c1

+@L

@z

@u⁰

@c1

⌘ d dx

⇣@

@z

⇣@L

@y

@u

@c1

+@L

@z

@u⁰

@c1

⌘⌘= @

@c1

⇣ d dx

@L

@z

@L

@y

⌘= 0

u⁰₁= d dx

@u

@c1

= @u⁰

@c1

@²L

@y²u1+ @²L

@y@zu⁰₁ d dx

⇣@²L

@z@y +@²L

@z²u⁰₁⌘

= 0

d dx

⇣@²L

@z²u⁰₁⌘ ⇣@²L

@y² d dx

@²L

@y@z

⌘u1= 0

u2

u1 u2

(u1, u2)

↵u1+ u2 (↵, )2R² c

a ↵u1(a) + u2(a) =↵u1(c) + u2(c) = 0 u1(a)u2(c) =u2(a)u1(c)

L C² @²L

@z²

@²L

@y@z C¹

u1= @u

@c1

u2= @u

@c2 C² [a, b]

c a [a, b]

u1(a)u2(c) =u2(a)u1(c)

x2[a, b]7 !u(x) =y0 y1

a b x+ay1 by0

a b

c a [a, b] c

u1(a)u2(c) =u2(a)u1(c) A= ^y0_{a b}^y1 B =^ay_{a b}^{1 by0}

x2[a, b]

u1(x) = @u

@A(x) =x u2(x) = @u

@B(x) = 1

c=a a

]a, b] u

z2R7 !L(z) =p 1 +z²

m

~ m

F =Fx~x+Fy~y+Fz~z

F~ w=Fxdx+Fydy+Fzdz

m F~

F~

w

Ep C¹ R³ R

w= dEp

Ep

dEp gradE~ p

R³ 5.2

F~ = gradE~ p

~ x

w=Fxdx Fx= d

dx(Ep)x

Ep x

M m

t1 t2

x:t 2[t1, t2]7 !x(t) S m

A= Z t2

t1

1

2m(x⁰(t))² Ep(t, x(t), x⁰(t)) dt A

1 2mx⁰² M

Ep

⇣1

2m(x⁰(t))² Ep(t, x(t), x⁰(t))⌘ dt

t1 t2

M

~x M

g