Variational Principles

8/12/2019 Variational Principles

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2/61


3/61

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


4/61

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


5/61

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)


6/61

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}


7/61

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


8/61

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)


9/61

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


10/61

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) =


11/61

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


12/61

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)


13/61

(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


14/61

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


15/61

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}


16/61

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


17/61

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


18/61

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) =


19/61

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


20/61

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


21/61

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


22/61

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)


23/61

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


24/61

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


25/61

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


26/61

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


27/61

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]


28/61

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,


29/61

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


30/61

(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


31/61

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


32/61

(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]


33/61

c da b

(x) =

e

1/(x21)2 x2


34/61

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


35/61

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


36/61

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


37/61

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


38/61

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


39/61

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)


40/61

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


41/61

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 ,


42/61

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


43/61

[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


44/61

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


45/61

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


46/61

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


47/61

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 =


48/61

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)


49/61

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


50/61

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


51/61

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


52/61

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))


53/61

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 =


54/61

(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)


55/61

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 )


56/61

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


57/61

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

||


58/61


59/61

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]


60/61

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


61/61

n c > 0 n

D2I c ba

(x)2

dx

ba

2 + 2

dx

Documents

Variational Principles