Учебное пособие 800365
.pdf6 δJ(y, γδy) = γδJ(y, δy)#
G 3 δJ(y, δy) ! 3/ 4 6 5
! ! J(y) L L y# H /
4 6/ @ 5
#
4 & 7 J 2 2 y = y(x) - 5 6 &
- 2 2 77 δJ(y, δy) ≡ 0 δy#
? 3 δJ ! J3(y)7
x |
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J = J(y + δy) − J(y) = x1 |
2 (y(x) + δy)dx − |
x1 |
2 y(x)dx = |
x2
=δy(x)dx = δJ(y, δy),
x1
x2
δy(x)dx 3 / 3 δy/
x1
@ J#
( ! 8 & /:
E 3 3 @ @ ! 7
! !
b
J(y) = F (x, y(x), y (x))dx, (3)
a
! 3 F (x, y, y ) 3 3 5
3 3 / 3 5
! / 3 G 3 3 ! J(y)/ 5
D[a, b] 3 3 3 !
y(x)/ [a, b] 3 !
7
y(a) = y1, y(b) = y2. |
(4) |
H 3 3 ! 3 3
4B6/ ! δy(x) = y2(x) − y1(x), y1, y2 D(a, b) !
[a, b] @ 7 δy(a) = δy(b) = 0# ? G 3 δy (x) = y2(x) − y1(x) = (δy) (x) @ 4 ! 6 #
H ! F (x, y, y ) 3
3 3 3/ @ 3
! 3 4,67
F= F (x, y + δy, y + δy ) − F (x, y, y ) ≈ dF (x, y, y ) =
=Fx(x, y, y )Δx + Fy(x, y, y )δy + Fy (x, y, y )δy =
=Fy(x, y, y )δy + Fy (x, y, y )δy ,
x = x − x = 0#
R @ ! J(y) L L y(x)
7
b
δJ(y, δy) = (Fy(x, y(x), y (x))δy(x) + Fy (x, y(x), y (x))δy (x)) dx.
a
2 3 3 /
(Fy (x, y(x), y (x)))x =
= Fy x(x, y(x), y (x)) + Fy y(x, y(x), y (x))y (x) + Fy y (x, y(x), y (x))y (x),
3
b
Fy (x, y(x), y (x))δy (x) dx = (Fy (x, y(x), y (x))δy(x)|ba−
a
b
−(Fy x(x, y(x), y (x))+
a
+Fy y(x, y(x), y (x))y (x) + Fy y (x, y(x), y (x))y (x))δy(x)dx,
3 / δy(a) = δy(b) = 0# R
δJ(y, δy) = a b Fy(x, y(x), y (x)) − (Fy x(x, y(x), y (x))+ |
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+ Fy y(x, y(x), y (x))y (x) + Fy y (x, y(x), y (x))y (x)) |
δy(x)dx. |
(5) |
? G 3 δJ(y, δy) 3 |
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3 5
7
b b b b b
(u + v)dx = udx + vdx, γudx = γ udx.
a a a a a
" 3 +#,/ @ y(x) G 3 3 5
! J(y) 3 / δJ(y, δy) δy# H
3 δy(x) 3 4+6 3 / 5
δJ(y, δy) 3 / 5
/ 3 δy(x) 3
3 4+67 Fy(x, y(x), y (x)) −(Fy x(x, y(x), y (x)) + Fy y(x, y(x), y (x))y (x)+ +Fy y (x, y(x), y (x))y (x)) = 0.
H 3 3/ 3 @ 3 #
, y - 7 J(y) =
= ab F (x, y(x), y (x))dx& y D(a, b)& 7 y
77 0 5 E 6
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(6) |
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? G 3 3/ - 7
D[a, b]&
77 5K6& 0
0 5L6#
4P6 / @ / ! 5
/ @ 5
/ 3 5
4B6# E 4P6/ @ 3 4B6/
- #
F G 3 3 3 3 3 5 3 3 / / 3 +#&/ -
- #
($ " 7 #
( ) ! 3 3 3 ! 5
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1 + y (x)2dx, |
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J1(y(x)) = x1 |
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/ / ! F (x, y, y ) = |
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x y# |
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? G 3 4P6 |
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Fy y (x, y, y )y (x) = 0. |
(7) |
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H Fy y (x, y, y ) = |
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> 0 y / 4-6 G 5 |
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(1 + y 2)3/2 |
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y = 0# 2 / 3 @
y(x) = C1x + C2# H G 3 3 / @
(x1, y1), (x2, y2)# R / 3 3 3 ! J1 5 @ / 3 3 3 / 3
4 6 # ? G 3 3 5
G 3 3 3 3 ! #
( ) " 3 (x1, y1)
OY / 3 3 @ G 3 5
7 LH 3 3 3 3 m
3 ! y = y(x)
y(0) = 0 y(a) = H/ a > 0, H > 0 4 6#L
? (x, y(x)) 5
mgy(x)/ G mv2/2# R 5
x# " |
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v(x) = ds(x)/dt = |
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(x) dx/dt |
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3 v(x) = 2gy(x)/ v(x) ! |
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dt = |
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v(x) |
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1 + y (x) dx |
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y(x)/ |
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J(y) = |
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(8) |
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2gy(x) dx, |
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F (x, y, y ) = |
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1 2gy |
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9 7 F x& 5K6
- #
4 - y &
(y Fy (y, y ) − F (y, y )) = 0# . 0 & &
y Fy (y, y ) − F (y, y ) = C1. |
(9) |
3 3 3 3
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− |
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1 + y 2√2gy |
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/ /
1 = C2. (1 + y 2)2gy 1
) G 3 τ
3 y = r(1 − cos τ )/ r = 1/(4gC12)# H
3
(1 + y 2) = |
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y 2 = |
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+ cos τ |
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sin2 τ |
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− cos τ |
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− cos τ )2 |
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− cos τ |
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= ±1 − cos τ |
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1 − cos τ |
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dx = |
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cos τ )dτ. |
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sin τ |
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R x = ±r(τ − sin τ ) + C2#
) 4'/'6 C2 = 0# H
3 G 3 3 3
x = r(τ − sin τ ) y = r(1 − cos τ ), τ 0. |
(10) |
3 ! / 5
r/ @ OX# ? 3
3 3 G 3 3 # ? 3 r
3
r(τ − sin τ ) = a, r(1 − cos τ ) = H, 0 < τ < 2π.
F # +#, ! r/
#
E # +#,# " 3 G 3
? 3 ) 3 3/ G 5
3 3 ! f (M) 3 @ Ω/
3 g(M) = h#
R / G 3
N Ω/ G grad f (N) Ω/ 3 5
/ N / 3 / 3 5
f (N)#
F Ω / h ! g(M)#
R grad g(N) ! / /
grad f (N)# 9 & λ& grad f (N) =
= λ grad g(N)#
R 7 - 7 f (M) &
g(M) = h& -
7 E F (M, λ) = f (M) − λ(g(M) − h)# λ
E #
R / ! 3 G 5 3 3 ! J1(y) = ab F (x, y(x), y (x))dx 3 ! /
3/ 3 3 ! 3
b
J2(y) = G(x, y(x), y (x))dx,
a
3 7 λ&
0 J1 J2 -
y(x) #
( ) ) 3 5
! F (x, y, y ) = y G(x, y, y ) = |
1 + y 2/ |
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G 3 4 3 3 |
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4P6 4-66 |
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1 = λ |
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= k. |
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+ y (x)2)3/2 |
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F 3 y(x)
x# R / 3 G 3 J G
/ 3 1/k# H 3 5
3/ ) @
/ 3 @ 3 3 @ 5
# A / @
/ @ 3 4 # +#&6#
E # +# E )
( ( " #
H 3 +#& G 3 !
! # 5
3 3 3 5
3 4 3 3 6#
* / @ 3 3 3
G 3 3 ! # W 3 @ / 5
/ ! G #
F 3 3 3 3 # E ! # G 3 3 G 3 @ 3 y(x) = ϕ(x, c1, c2, ..., cn)/ 5
@ 3 4 G ! 5
6# ? G 3 3 / 3 3
c1, c2, ..., cn ! y(x) = ϕ(x, c1, c2, ..., cn) 3 5
3 y(a) = y1, y(b) = y2#
b
? y(x) 3 J(y) = F (x, y, y )dx 5
a
x J(y) @ 3 n 3 7
J(y) = Q(c1, c2, ..., cn)# 3 G 3 3 ! 5
G 3 3 ! Q n 5
c1, c2, ..., cn# $ 3 3 3 G 3
3 # ? 3 3/ 5 3 / 5
3 3 y(x) = ϕ(x, c1, c2, ..., cn) 3 3
3 / G 3 3 ! Q#
? 3 3 3 !
1
I(y) = (y 2 + y2)dx, y(0) = 0, y(1) = 1, F (x, y, y ) = y 2 + y2.
0
A 3
y = ϕ(x, c1, c2) = x + c1x(1 − x) + c2x2(1 − x).
/ y 3 3# ?
3 I(y) 3
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I(ϕ) = |
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c1c2 |
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c2. |
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" / G 3 3 3
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15c1 |
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−473 = −0 |
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2 = −43 = −0 163 |
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R c = |
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, 145, |
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3 y1 = x − 0, 146x(1 − x) − 0, 163x2(1 − x)#
) 3 3 / 3 3
3 3 y = x + c1x(1 − x)# 2 3 G 3
3 y2 = x − 0, 227x(1 − x)#
" @ ! I(y) 5
y − y = 0/ 3 y = ex−e−x
e−e−1 # "
0.1, 0.2, ..., 0.9 /
y(x) = y1(x)/ y2
0.27 y(0.2) = 0, 171/ y2(0.2) = 0, 164/
#
" @ 3 3 3 3 5
! / @ ! 3 7
I = |
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∂x |
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∂y |
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+ 2u dx dy, |
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∂u |
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∂u |
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u ! −1 x, y 1#
A 3 u = C(1−x2)(1−y2)#
H
I(C) = 25645 C2 + 329 C
I (C) = 0 C = −5/16# R / @ 5
u = −165 (1 − x2)(1 − y2) 4 5
3 / / 3 36 3 ,/+ ^#
( ; 2 + - # - " #
3 3 / G / 3 / G 3 5
! 3 3 3 5
x1(t), x2(t), ..., xn(t)/ 3 t 5
n ! 3 37
dxi |
= fi(x1 |
, x2, ..., xn; u1, u2, ..., um), i = 1, 2, ..., n. |
(11) |
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3 3 3 3 3 3 x1(t), x2(t), ..., xn(t) 3 / / G / 5
/ ! ! 3 @ # #/ 3 3
t 3 3#
4,,6/ 3 3 xi/ m
3 @ 3 u1(t), u2(t), ..., um(t)#
D
t0 t tF m ! # u1(t), u2(t), ..., um(t) &
0 xi(t), i = 1, 2, ..., n 77
5??6 7
5 & & &
00 # #6=
tF
x0(tF ) = f0(x1, x2, ..., xn; u1, u2, ..., um)dt. (12)
t0
- 0
&
Qj (u1(t), u2(t), ..., um(t)) dj , j = 1, 2, ..., k, |
(13) |
0 # ( &
x1, x2, ..., xn t = t0 t = tF
&
#
( ) 4 6#
H 3 4 6# " @
3 3 3 3 3 7 5
x1(t) x2(t)# /
dx1(t) |
= x2(t), |
dx2(t) |
= u(t), |
(14) |
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dt |
dt |
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u(t) J /
# H 0#
? x1(0) x2(0)# H / 3 3 3 5
u(t)/
x1(tF ) = 0, x2(tF ) = 0 4 6 3
3 3 tF # ? G 3 u(t)
|u(t)| 1#
3
"""E @ 3 @ 3 3 #
%&'& (#
" G 3 ! 3
@ 3 3 ! 4,,6/ 4,&6/ 4,O6
@ 7
,6 ! !
H(x1, x2, ...xn; p0, p1, p2, ..., pn, u1, u2, ..., um) =
= p0f0(x1, x2, ..., xn; u1, u2, ..., um)+
+p1f1(x1, x2, ..., xn; u1, u2, ..., um) + p2f2(x1, x2, ..., xn; u1, u2, ..., um)+
... + pnfn(x1, x2, ..., xn; u1, u2, ..., um),
p0, p1, p2, ..., pn 4 6 3 :
&6 3 3 5
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dpi |
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∂H |
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= − |
, i = 0, 1, 2, ..., n; |
(15) |
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dt |
∂xi |
O6 3 3 3 ! H 3 3 3 3 u1, u2, ..., um/
3 3 4,O6/
M(x1, ..., xn; p0, p1, ..., pn) = |
max H(xi; pi; uj ). |
(16) |
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u1,...,um |
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H / xi (t), i = 0, 1, 2, ..., n, uj (t), |
j = 1, 2, ..., m J 3 5 |
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/ @ @ 3 ! pi (t), i = 0, 1, 2, ..., n/
@ 4,+6/
p0(t) = −1, |
(17) |
t0 t tF |
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H(x1, ..., xn; p1, ..., pn; u1, ..., um) = M(x1, ..., xn; p0, p1, ..., pn) = 0. |
(18) |