8/12/2019 Variational Principles
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y = y (x) l (2a,a] y (a) = 0
(a, 0) l
f(x) = 0 f(x) x y I[y]
I[y]
A=
aa
y (x) dx
L=
aa
1 + y2dx
A[y] L [y]
L= L [y]
y
Rn
Cn
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x R x + x9 =b b R
x+x9
f(x) = x2
2 + x10
10 bx f (x) = x+ x9 b
f (x) = 0 x f
f + x f
f(x) 1 f f(0) = 0 < 1
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f : Rn R x=
xjej = (x1, , xn) ej = (0, , 1, , 0)
x = x2j1/2 L : Rn R L (V + W) = L (V) +L (W) , R
V,W Rn
L (x) =
xiL (ej) =
Ljxj = L x
L= (L1, , Ln) = (L (e1) , , L (en)) f : Rn R x L
x
f(x + v) f(x) Lv= o (v)
> 0 >0
0< v < = |f(x + v) f(x) Lv| < v
n= 1
v = tej
f x
limt0
f(x + tej) f(x)t
Lej =Lj L
f x fxj L
f
L= fx1
,
, f
xn= f(x)
Rn
f x Rn
L= f(x)
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Rn
f
f : R2 R f
f(x, y) =
xy
x2+y2 x2
+ y2
= 00 x= y = 0
C1 (Rn;R) Rn
Rn
Cr (Rn;R) r
f(x) f(y) y Rn
f(x) = 0
f C1
y
x
f
f
f < 0
f : Rn R
2f
xixj
f
y r By(r) ={a Rn :|y a| < r}
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m m Aij A > 0
vTAv=i,j
Aijvivj >0
v =0 Rm A 0
i,jAijv
ivj
i,jAijvivj
A= A
vAv > 0
A
f C2 (Rn) f(x) = 0
x Aij =
2fxixj
Aij = 2fxixj
x
x0
x0 f
f C2 (R) f (x0) = 0 f (x0)> 0 f
x0 x0
Rn n 2 C2 (Rn)
R R f(x0) = f(y) (x0, y) f f(x0) f (x0) x0
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S Rn x,y S (0, 1)
x + (1 )y S
f : Rn R
f((1 )x+ y) (1 ) f(x) + f(y)
(1 )x+y f f : D R D Rn f D
Ef= {(z, x) : z f(x)} R1+n
f : R2 R Ef f
R2 R g (s) = f(x + sv)
f
f (f)
f C1 (Rn)
f
f(y) f(x) + f(x) (y x) x y [f(x) f(y)] (x y) 0
(i) = (ii) H(t) = (1 t) f(x) + tf(y) f((1 t)x + ty) 0 H(0) = 0 H(0)
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0
H(0) = limt0+
H(t) H(0)t
= f(x) + f(y) (y x) f(x) 0
(ii) = (i)
f(y) f(z) + f(z) (y z)f(x) f(z) + f(z) (x z)
(1 t) f(y) + tf(x) (1 t + t) f(z) + f(z)[(1 t) (y z) + t (x z)]= f(z)
z= (1 t)y + tx
(ii) = (iii)
f(y) f(x) + f(x) [y x]f(x) f(y) + f(y) [x y]
(iii) = (ii)
f f f
f C1 (Rn) x x f f
f(x) = b f(x) b x
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f C1
f(x) = b
f(x) f(y) = 0
f C2 (Rn)
f 2fxixj 0 x
2f
xixj >0 x
= f
f(x, y) = x4 + y4
(i) =
f(x) f(y) = [f(u)]xy
= [f(y + t (x y))]10
= 10
d
dtf(y + t (x y)) dt
2ijf 0
[f(x) f(y)] (x y) = 10
d
dtf(y + t (x y)) (x y) dt
=i
10
d
dt
xif(y + t (x y)) (xi yi) dt
= i
10
2
xjxif(y + t (x y)) (xi yi) (xj yj) dt
0
(i) =
(ii) =
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P= (P1, , Pn)
S(P1, , Pn) =
Piln Pi
0 Pi 1 i Pi = 1 (1 t)p +tq p,q {1, 2, , n} t
[0, 1] (1 t)pi+ tqi [0, 1] i
[(1 t)pi+ tqi] = (1 t) + t= 1
S
p
2S
PiPj =
2SPs
1
2SPsn
= 1P1
1Pn
p (p lnp) = 1 lnp
2
p2(p lnp) = 1p
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C (x (t) , y (t)) = (cos t, sin t) (t) =f(x (t) , y (t)) =
(sin t)2
t= 2 ,32
ddt = 0
ddt
= fx (t)y (t)
= 0
x
y
C
ddt = 0 f C
f x x+ x
x f fx> 0 f(x + x) =f(x) + f x +O
x2
> f(x) x f(x)
g= 0 f= constant
+ f
g = 0
f g= 0
f, g C2 (Rn) g (x)= 0 x C= {x Rn :g (x) = 0}
f
|C x0
[f(x) g (x)]x0
= 0
g (x) =0
C
x0
C x= v (t1, , ts)
x0= v
t01, , t0s
C1 C
f(x0) = maxxC
f(x)
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(t1, , ts) = f(v (t1, , ts))
t01, , t0s
tj (t01
,
,t0s)
= 0
f(x0)
v
tj
(t01, ,t0s)
= 0
f
f(x0) g (x0)
f(x0) = g (x0)
f(x0) g (x0) = 0h (x, ) = 0
h (x, ) = f(x) g (x) h f
h= f
(x, y)
(x, y)
A = 4xy
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x2 + y2 1 = 0
h (x,y,) = A g= 4xy x
2 + y2 1h = 0
h
x = 4y 2x= 0
h
y = 4x 2y= 0
h
= x2 + y2 1 = 0
y = 12x x= 12y = 2
4y 4x= 0 x= y = 12
x= 0
y = 1 f
S(p) = pilnpi pi= 1 p [0, 1]
h = pilnpi pi 1h
pi= lnpi 1 = 0
pi p1 = p2 = =pn pi=
1n
S
f|C x0 f, g C2
Hij =
2h
xixj
x0
=
2 (f g)
xixj
x0
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Hij 0 Hij x0 Hij
x0
2
tjti=
tj
f(v (t)) v
ti
= f(v (t)) 2v
tjti+
tj
f
xk
vkti
= f(v (t)) 2v
tjti+
2f
xlxk
vltj
vkti
2
tjti=g
2v
tjti+
2f
xlxk
vltj
vkti
v (t) C
g (v (t)) = 0
g (v (t))
xk
vkti
= 0
g (v (t))
xk
2vktjti +
2g
xlxk
vltj
vkti = 0
2
tjti=
2g
xlxk+
2f
xlxk
vltj
vkti
= 2h
xlxk
vltj
vkti
(t)
h
v
ti
yTHy 0 {y: g (x0) y = 0}
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11 22 + + + + +
x+ y x2 +y2 = 1 h =
x + y x2 + y2 1 h
x= 1 2x
h
y = 1 2y
2hxixj
=2 0
0 2
(x,y,) =
1
2,
12
, 1
2
,
1
2, 1
2, 1
2
A = 4xy x2 + y2 = 1
h= 4xy x2 + y2 1 h
x= 4y 2x
h
y = 4x 2y
2h
xixj=
2 4
4 2
(x,y,) =
1
2,
12
, 2
,
1
2, 1
2, 2
, (0, 1, 0) , (1, 0, 0)
4 44 4
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8
1
1
0
1
1
x y
8
2gxixj
= 0
g
f
f
g
2g
xixj = 0
f
f
f(t)
f()
f(t) f()
f x
x
df /dx
x f
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f : R R f
f (p) = supx
[px f(x)]
f(x)
f (x)> 0 x
p (x) dfdx
f
p
x (p) g (p) = f(x (p)) f
p
f
g (p)
g (p) = d
dpf(x (p))
= x (p) f (x (p))= x (p) p
x (p)
x
h (p) = x (p)p f(x (p))h (p) = x (p)p + x (p) g (p)
= x (p)
h (p) q(p) =
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h (p) = x (p)
h (p (q)) = h (p (x))
= xp (x) f(x)p (q) q h (p (q)) = p (x) x [xp (x) f(x)]
= f(x)
f(x) + f (p) = xp
x = x (p) p = p (x) x p
f (p) = x (p)p f(x (p)) f (p) = supx[xp f(x)] x (p)
d
dx[xp f(x)] = p f (x) = p p= 0
sup
f (x) = p
xp f(x) x p f (x)< 0 sup
f
f : Rn R f
f (p) = supx
[p x f(x)]
y= f(x) = ax2 a >0
f (p) = supx
px ax2
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f(x) = ax2
a > 0
d
dx
px ax2= p 2ax= 0
x = p/2a
y = px y = f(x)
f
f (p) = p2
2a a p
2
4a2 =
p2
4a
f (y) = supp
yp p
2
4a
= ay2
f f
f
f(x) = ax2 a
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f (p) 1 y p
f (p)
t (0, 1) x
t (p1x f(x)) + (1 t) (p2x f(x)) = (tp1+ (1 t)p2) x f(x)
t supx
(p1x
f(x)) + (1
t)sup
x
(p2x
f(x)) = tf (p1) + (1
t) f (p2)
tf (p1) + (1 t) f (p2) (tp1+ (1 t)p2) x f(x)
p1 p2 tp1+ (1 t)p2
tf (p1) + (1 t) f (p2) f (tp1+ (1 t)p2)
f
f C2 (R) f (x) c > 0 f
f =f
f f (x) =p
x f (p) p px
f(x)
f (x) c
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y= f(x) p
y f(X(p)) = p [x X(p)]y = px [pX(p) f(X(p))]
= px f (p)
f(z) pz f (p) z z = X(p) p z
f(z) pz f (p) p p= f (z) f z
f (z) = supp
[zp f (p)]
= f(z)
f (p) y f (p)
p
f (p) C1
f (p)
f (x) c > 0 c x
f(x) = ex
supx
[px f(x)] = supx
[px ex]
p < 0 px x ex 0
f f (x)> c
f g
g (p) = supx
[p x f(x)] p x f(x)
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x
p
x
f(x) + g (p) p x
x p
L
L = T V=
T =T(x) = 12mx
x V =V(x)
L (x, x) = 12
mx x V (x)
qi
x
x
L (x,p) = supx
[p x L (x, x)]
x
xj[p x L (x, x)] =pj mxj = 0
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p= mx
L (x,p) = p pm
1
2mp p V (x)
=
1
2mp p + V(x)
H (x,p) = L (x,p)=
1
2mp p + V (x)
= T+ V
= +
pi
xj =H
pj pj = H
xj
N
U=U(S, V)
V S
= dq = TdS
= dU = = TdSpdV=
U
S
V
dS+ U
V
S
dV
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T = U
S
V
p = U
VS
T
V
S
= pS
V
U
F =F(T, V) = infS
[U(S, V) ST]
U(S, V)
S
S
T = U
S
V
S= S(T, V)
F(T, V) = U(S(T, V) , V) T S(T, V)dF = dU TdS SdT
= (TdSpdV) TdS SdT= pdV SdT
p = FV
T
S = FT
V
p
T
V
= S
V
T
S= S(T, V)
T = U
S
V
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S 2U
S2
V
>0
cV
V
cV =T S
T
V
= TTS
V
= T2US2
V
U S
E
S
T0S T0
S= kB
ipilogpi kB
pi
dq= TdS
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V R C V
V = C(R) R R x0
x0 :f f(x0) R
V ={f C: f(x + 2) = f(x) x} 2 sin x V
I0[f] =
20
[f(x)]2 dx
I1[f] =
20
[f(x)]
2+ [f (x)]2
dx
h (x)
0
x
v
hv(t) = h (x0+ tv) t= 0
I[f] I[f+ t] (x)
f |t|
d
dtI[f+ t]
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f I
I0[f]
ddt
I0[f+ t] = ddt
20
[f(x) + t (x)]2 dx
=
20
d
dt[f(x) + t (x)]
2dx
=
20
2 (x) [f(x) + t (x)] dx
(x)
t= 0
d
dtt=0
I0[f+ t] = 20
2fdx
DI0[f] =
20
2fdx
v h (x) v x
h= 0 x
h (x)
f, g = 20
f(x) g (x) dx
f, g = 20
f(x)g (x) dx
DI0[f] = 2f,
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h (x)
I0f
= 2f
,
I[f]
DI[f] ddt
t=0
I[f+ t]
I
f,
I/f I[f]
R C
L x L (x) R C y y,x =L (x) DI[f]
I
f,
I
fdx
x0 x0
Dx0[f] = d
dt
t=0
x0[f+ t]
= d
dt
t=0
(f(x0) + t (x0))
= (x0)
= x0[]
(x0)
f dx= (x0)
(x0) /f
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(x x0) g (x) dx= g (x0)
(x0)
f (x x0)
f I1[f] =
f2 + (f)2
dx
I1[f] =
20
[f(x)]
2+ [f (x)]2
dx
2
DI1[f] = d
dt
t=0
I1[f+ t]
= d
dt
t=0
20
[f+ t]2 + [f+ t]2
dx
=
20
(2f + 2t + 2f+ 2t) dxt=0
=
20
(2f + 2f) dx
20
fdx = [f]20 20
fdx
= 20
fdx
f (2) (2) = f (0) (0)
DI1[f] =
20
(2f 2f) dx
= 2
0
(2f
2f) dx
I1f
= 2f+ 2f
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f
f
y (x) f(x,y,y)
V =
y (x) C2 [a, b] : y (a) = , y (b) = [a, b] I :V R
I[y] =
ba
f
x,y,
dy
dx
dx=
ba
f(x,y,y) dx
f(x,y,y)
I
y =
f
y d
dx
f
y
(x) C2 [a, b]
(a) = (b) = 0
y+ t V
DI[y] = d
dt
t=0
I[y+ t]
= d
dt
t=0
ba
f(x, y+ t,y+ t) dx
f
DI[y] = b
a f
y
(x,y,y) + f
y (x,y,y) dx
DI[y] =
ba
f
y d
dx
f
y
dx +
f
y
ba
=
ba
f
y d
dx
f
y
dx
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(a) = (b) = 0
I
y =
f
y d
dx
f
y
I
y =fy fyx fyyy fyyy
h (x) = 0
f I/y = 0 y
ba
f(x) (x) dx= 0
(x)
(x) = 0 x [c, d] (a, b)
f f C[a, b] f 0 [a, b]
[a, b]
f
f >0 f 0 f
f > 0 |f(x) | < /2 x |x x0| <
f(x) 2
x (x0 , x0+ )
[a, b]
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c da b
(x) =
e
1/(x21)2 x2
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supp = cl {x: (x) = 0}
[a, b]
(a, b)
supp [c, d] (a, b)
c d
f dx
(a, b)
f Ck Ck (x) (a) = (b) = 0 = (x a) (x b) f
ba
fdx=
ba
(x a) (x b) f2dx= 0
f 0 (a, b) k 1f 0 [a, b]
f I[f]
y
I/y 0
f
y d
dx
f
y
= 0
y (x)
I[f]
I[y] =ba
f(x,y,y) dx
f
y d
dx
f
y
= 0
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I[y]
(a, )
(b, )
y= y (x)
I[y] =
ba
1 + y2dx
f=
1 + y2
fy
ddx
fy
= ddx
y1 + y2
= 0
f
y
fy = y=
y0 = cx + d
= ( )
(b a) (x a) +
f = f(y)
f (y) = 1/ (1 + y)3/2
>0
f(y)> f(y0) + fy(y0) [y
y0]
y=y0 y=y0
I[y] =
ba
f(y) dx > ba
[f(y0) + fy(y0) [y
y0]] dx
I[y0] + ( ) ba
[y y0] dx= I[y0]
y
y0
f x y
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r (u) = 10+u
u
c (t) = 169 (t 12)2
t
V (t) t
I[V] =
c (t) r (u) dV
=
c (t) r (u)
dV
dtdt
=
c (t) r (u) u dt
t V u
t x V y u y
f
V d
dt
f
u
= 0 d
dt(c (t) [r (u) + r (u) u]) = 0
c (t) [r (u) + r (u) u] =
V
169 (t 12)2 [10 + u + u] = A
u = A
2[169 (t 12)2] 5
= A/2
[13 + (t 12)] [13 (t 12)] 5
V(t) = B arctanht
12
13 5t + C
V(0) = B arctanh
1213
+ C= 0
V (24) = B arctanh
12
13
120 + C= 100
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C = 110
B = 110
arctanh 1213
V(t) = 110arctanh
t1213
arctanh
1213
5t + 110 u= V (t) 0
24
t0
50
100
V
I[V]
I[V] = 24200
3 + 13
log5
V1(t) = 100t/24
r (u) = 10 + u 14.17
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g (x) = sin(nx)
I[u] =
1
2(u)2 + u2 gu dx
2 u
u C
([, ])
f(x,u,u) = 1
2
(u)2 + u2
gu
f
u = u g
fu
= u
f
u d
dx
f
u
= u g du
dx
= u g u
= 0
2
u= u0+ t 2
u0
u0 u0+ sin (nx) = 0
u0 = A cosh x + B sinh x +sin(nx)
1 + n2
ex ex u C
([, ]) u cosh x
sinh x
u0 =sin(nx)
1 + n2
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I[u0+ ] =
1
2
(u0+
)2 + (u0+ )2
g (u0+ )
dx
= I[u0] +
[u0+ u0 g] dx +
1
2
2
+
2dx
u0
(u0 + u g) = 0
I[u0+ ] =I[u0] +
1
2
2 + 2
dx
I[u0+ ] I[u0] C
([, ]) 0 I[u0+ ]> I[u0]
u0 =sin(nx)
1 + n2
I
u0 2
v w= u0v w 2 w w = 0
0 =
w (w+ w) dx= [ww]+
1
2
w2 + w2
dx
w
w w 0 v= u0
C
(un)
unk u0 V J
J(u0)
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I[u] =
f(x, u,u) dV
u= u (x) Rn
u=
u
x1,
u
x2, , u
xn
x= (x1, x2, , xn) dV dx
f
xi u/xi
f=fx1, x2, , xn, u,
u
x1
, u
x2
,
,
u
xn
I[u] =
1
2|u|2 g (x) u
dV
Rn
DI[u] = ddtt=0
I[u + t]
= d
dt
t=0
1
2|u + t|2 g (x) (u + t)
dV
=
(u g) dV
=
I
u dV
u dx=
u
dS
2u dV]
= 0
DI[u] =
2u g dV
I
u= 2u g
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u u
2u= g
u
g (x)
g (x)
I[u] =
1
2|u|2 g (x) u
dV
I[u] =
f(x, u,u) dV
f
u
nj=1
xj
f
pj(x, u,u)
= 0
pj =u/xj
DI[u] = d
dt
t=0
I[u + t]
= d
dt
t=0
f(x, u + t,u + t) dV
=
f
u+
nj=1
f
pj
xj
dV
=
f
u+ f
p dV=
f
u f
p
dV
=
f
u f
p
dV
fp =
, fpj ,
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f
u f
p 0
R2
1
2
u
t
2
x
t
2dxdt
x= (t, x) p= (ut, ux)
f=1
2u2t u
2x
t
(ut) x
(ux) = utt+ uxx= 0
E B
I[y] =
aa
y (x) dx
J[y] =
aa
1 + y2 dx= L
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[y, ] = I[y] + (J[y] L)=
aay+ 1 + y
2
L
2a dx
y
y =
f
y d
dx
f
y
= 1 d
dx
y
1 + y2
x y
1 + y2 =c
y1 + y2
= x c
y2 = [(x c) /]21 [(x c) /]2
dy =
[(x c) /]
1 [(x c) /]2
dx
x= c + sin y= y0 cos
(x c)2 + (y y0)2 =2
x (t) =
(x (t) , y (t)) x (t)
R
n
f(t,x (t) , x (t)) dt
f
xk d
dt
f
xk
= 0 j = 1, 2, , n
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x (t) R2
A=1
2
(xy yx) dt
L=
x2 + y2
12 dt
[x, t] =
1
2(xy yx) + x2 + y2 12 dt
f
x d
dt
f
x
=
1
2y d
dt
1
2y+
x
x2 + y2
= y y (yx xy)(x2 + y2)
32
=y
x2 + y2 32 (yx xy)
(x2 + y2)
32
= 0
f
y d
dt
f
y
= 1
2x d
dt
1
2x +
y
x2 + y2
= x x (xy yx)(x2 + y2)
32
=x x2 + y2 32 (xy yx)
(x2 + y2)32
= 0
x = 0 y = 0
x2 + y2
32 (yx xy) = 0
x2 + y2 32 (xy yx) = 0
(yx xy)(x2 + y2)
32
= 1
y
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x
x2 + y2
=1 (y y0)
yx2 + y2
= 1 (x x0)
1 = 2 (y y0)2 + 2 (x x0)2
2 = (x x0)2 + (y y0)2
(yx xy)(x2 + y2)
32
= 1
x (t) 1
J[y] = 0 = 1, , N
= I[y] +
J[y]
v: R3 R3
v (x) = 0 x
I[v] =
1
2|v|2 v f
dV
v (x) = 0
v=
vi
xj
i,j=1,2,3
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vi v
vi
|v|2
|v|2 = i,j=1,2,3
v
i
xj
2=
3i=1
|vi|2
[v, ] =
1
2|v|2 v f (x) v
dV
(x) x R3 (x) v
v dx=
[]
v dx +
v
dS
0 v |x|
[v, ] =
1
2|v|2 v f+ [ (x)] v
dV
d
dt
t=0
[v + tw, ] =
(v: w fw + [ (x)] w) dV
d
dt
t=0
1
2|v + tw|2 = d
dt
t=0
i,j
1
2
vi
xj+ t
wi
xj
2
=i,j
wi
xj
vi
xj
= v: w
i
vi
widx=
i wi
2vidV
Dw =
2v f+ w dV
2v + = f
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v= 0
2= f
v v
= 0
I
2v + = f
2 = f
O|v|2
(x)
f
y
fy ddx
(fy) = 0
f=f(x, y) y
fy =
f=f(y, y) x
yfy f=
fy = 0
d
dx(fy) = 0
fy =
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d
dx[yfy f] = yfy+ y d
dx(fy) df
dx
= yfy
+ y d
dx(fy
) yfy yfy
= y
d
dx(fy) fy
= 0
d
dx[yfy f] = 0 fx
d
dx[yfy f] + fx = 0
f y f y
t
x
dH/dt= 0
i = r
(x1, y) (x2, y)
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x (x1+ x2) /2
(x1, y) (x2, y) (a, 0)
T(a) =1
c (x1 a)2
+ y21/2
+ (x2 a)2
+ y21/2
T (a) = x1 a
(x1 a)2 + y21/2 + x2 a
(x2 a)2 + y21/2
sin i = sin r
x1 < a < x2
y
c= c (y)
y = y (x)
T =
ba
1 + y2
c (y) dx
=
ba
f(y, y) dx
fy ddx
(fy) = 0
V(x)
F= V (x)
md2x
dt2 = V(x)
S[x] =
1
2m |x|2 V(x)
dt
=
L (x, x) dt
L
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f
xi d
dt
f
xi
= V
xi d
dt(mxi) = 0
md2x
dt2 =
V
L
V x F
f
xi=mxi
V
x Lx
L = x mx
12
m |x|2 V(x)
= 1
2m |x|2 + V (x)
=
L
H
L x
H =p x L (x, x)
p
p=Lx
H = x Lx
L =
L x
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L=
ds
s
(a, )
(b, )
y= y (x)
L=
b
a
1 + y2dx
x (t)
L [x] =
ba
x dt
x
f
xj=
xjx2j1/2 =
d
dt
x
x
= 0
L =
j
dxjdt
21/2
dt
=
j
dxjd
21/2
d
=(t) (t)> 0
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j
dxjd
1/2
=
I[x] =
1
2x2 dt=
{ } dt
x= 0, x=
m= 1 V = 0
C=
(x,y,z) : x2 + y2 =R2, < z <
x= R cos y= R sin z = z
R z
t = (t) z= z (t)
x2 =
ds
dt
2=
dx
dt
2+
dy
dt
2+
dz
dt
2
x2 =R sin()
2+
R cos()2
+ z2
= R22 + z2
C
x (t) = (R cos (t) , R sin (t) , z (t))
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I[x] =1
2
R22 + z2
dt
I
= 0 ddt
R2
= 0
I
z = 0 d
dt(z) = 0
=
z =
g (x, y) = x2 + y2
R2 = 0
(t)
[x, ] =
1
2x2 (t) x2 + y2 R2dt
=
1
2
x2 + y2 + z2
(t) x2 + y2 R2dt
2x ddt
(x) = 0 x + 2x= 0
2y ddt
(y) = 0 y+ 2y= 0
0 ddt
(z) = 0 z =
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(t)0 x y x2 + y2 R2 = 0
xx + yy = 0xx + yy+ x2 + y2 = 0
(t) x= 2x y= 2y
2x2 2y2 + x2 + y2 = 02R2 = x2 + y2
(t) = 1
2R2 x2 + y2
0
(t) = 2 R
x + 2x= 0, y+ 2y= 0
xx + yy = 01
2[xx + yy] = 0
x2 + y2 =
22
x = R cos(t + )
y = R sin(t + )z = at + b
y (x)
y (x)
(0, 0) (X, Y)
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y Y >0 v=
x2 + y21/2
1
2mv2 =mgy
v=
2gy
I[y] =
ds
u =
x2 + y2
1/2
2gy dt
= 1
2g
X0
1 + y2
1/2
y dx
x
yfy f= y2
y (1 + y2)
(1 + y2)y
=C
y2 1 + y2 = Cy (1 + y2)1 = C2y
1 + y2
y1/2dy
(1 c2y)1/2 =
1
c
dx=
x
c
u= y1/2 dy/du= 2u
y1/2dy
(1 c2y)1/2 =
2u2du
(1 c2u2)1/2
u = 1csin2
y= 1c2sin2 2 dy/d= 1c2sin 2cos 2
y1/2dy
(1 c2y)1/2 =
1csin
2 1c2sin 2cos 2d
cos 2
= c3
sin2
2d
= 1
2c3( sin )
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y = 1
c2sin2
2=
1
2c2(1 cos )
x = 1
2c2
(
sin )
(0, 0) (X, Y) Y 0 (0, 0)
I[y]
I[y] =
ba
f(x,y,y) dx
f f(y)
I[y] f (x)> 0
h C2 (Rn) > 0 >0 h (x + x) h (x) h (x) x
1
2
ni,j=1
2h
xixjxixj
x2 x <
h (x) = 0
Aij = 2h
xixj
x
h (x + x)> h (x)
x x
x h (x) = 0 Aij
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I[y] C1 (a) =
(b) = 0 x f
f(x, y+ , y+ ) = f(x,y,y) + fy+ fy
+ 12
2fyy + 2fyy+ 2fyy
+O
[|| + ||]3
f (x,y,y)
> 0 >0
O
||2 + ||2
max[a,b]
(|| + ||)<
I[y+ ] =I[y] + DI[y] +1
2D2I[y] + O
ba
||2 + ||2
dx
DI[y] D2I[y]
D2I[y] =
ba
2fyy+ 2
fyy+ 2fyy
dx
||
||C1 = max[a,b]
(|| + ||)
y C1 I[y] I[y+ ] I[y] ||C1 y 0
||
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I [y] 10
2 102 dx
(x) = sin x (x) = cos x
10
2 cos2 x 10sin2 x dx = 2
2 10
2 0 c
I[y+ t]
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D2I
I
Aij = 2f
xixj
A
f A
vTAv> 0 v
|v|2 vTAv
|v|2
,
vTAv
|v|2 [ , ]v =0 vTAv [
,
]v: |v| = 1
v
D2I[y] M[] =ba
R (x) (x)2 dx
M[] = C
M[]
v R (x)
R= 1
D2I
M =
ba
P(x) 2 + Q (x) 2
dxb
a (x)2 dx
L []
= ddx
(P ) + Q=
L []
n
Ln= nn
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n c > 0 n
D2I c ba
(x)2
dx
ba
2 + 2
dx