Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2

2t = t1 C2 = 0 x = 0 y = 0

x = C21 (t1 − sin t1) y = C21 (1 − cos t1),

C1

2

(x1, y1)

x1

v[y(x)] = (y 2 + 2yy − 16y2) dx.

x0

x1

v[y(x)] = (xy + y 2) dx.

x0

1

v[y(x)] = (y2 + y 2 − 2y sin x) dx.

0

x1

v[y1, . . . , yn] = F (x, y1 , . . . , yn, y1, . . . , yn)dx

x0

	y1(x0) = y10,			y2(x0) = y20, . . . , yn(x0) = yn0,
	y1(x1) = y11,			y2(x1) = y21, . . . , yn(x1) = yn1,
				yj (x)(j = 1, . . . , n)
	v[y1, y2, . . . , yn] = v˜[yi]			n	y1, . . . , yn
					yi(x) :
	Fyi		d	Fy = 0	(i = 1, . . . , n).
		− dx
				i
	i = 1, . . . , n				yi

n

2n x, y1, y2, . . . , yn

y(x) z(x)

v[y(x), z(x)] = x1F (x, y, z, y , z )dx

x0

y(x0) = y0, y(x1) = y1, z(x0) = z0, z(x1) = z1,

y = y(x), z = z(x),

Fy − dxd Fy = 0, Fz − dxd Fz = 0.

v(x, y, z)

B(x1, y1)	A(x0, y0)
	T
	y = y(x) z = z(x)

T = x1 1 + y 2 + z 2 dx. v(x, y, z)

x0

√

∂v

1+y 2+z 2

+

d

y

= 0

∂y

v2

dx

√1+z

2

+z

2

∂v

√1+y22+z 2

+ d

= 0

v y

∂z

v

dx

v√1+y

2

+z

2

x1

v[y(x)] = F (x, y, y , . . . , y(n))dx,

			x0
F Cn+2
		y(x0) = y0,	y (x0) = y	, . . . , y(n−1)(x0) = y(n−1)				,
			0				0
		y(x1) = y1,	y (x1) = y	, . . . , y(n−1)(x1) = y(n−1).
			1				1
		(n − 1)			C	2n	y = y(x) C2n y¯(x)
		y(x, α) = y(x) + α[¯y(x) − y(x)]			y(x, α) = y(x) + αδy.
α = 0 y(x, α) = y(x)			α = 1 y(x, α) = y¯(x)
		v[y(x)]			y = y(x, α)			α = 0
	∂		α
		v[y(x, α)]α=0 = 0
	∂α

∂α

v

δv

δv =

∂

x1 F (x, y(x, α), . . . , y(n)(x, α))dx

=

x

1

x0

α=0

x0

+ . . . + Fy(n)

δy

=

Fyδy + Fy δy + Fy δy

(n)

dx.

x1

x1

d

x1

−

Fy δydx = [Fy δy ] x0

dx

Fy δydx,

x0

x0

x1

+

,

x1

x1

2

x0

x0

d

d

x1

Fy δy dx =

Fy δy

x0 − %

dx

Fy δy& x0 +

dx2

Fy δydx,

n + 1

n

x1

x1

d

x1

x0

Fy(n) δy(n)dx = Fy(n) δy(n−1)

− %

Fy(n) δy(n−2) &x0

+ . . .

x0

dx

x1

dn

+(−1)n

Fy(n)δydx.

dxn

x0

x = x0

x = x1

δy = δy =

= . . . = δy(n−1) = 0

x1

d

d2

dn

δv = Fy −

δydx.

Fy +

Fy + . . . + (−1)n

Fy(n)

dx

dx2

dxn

x0

δv = 0

δy

d

d2

dn

Fy −

Fy

+

Fy +

. . . + (−1)n

Fy(n) ≡ 0.

dx

dx2

dxn

y = y(x)

d

d2

dn

Fy −

Fy

+

Fy +

. . . + (−1)n

Fy(n) = 0.

dx

dx2

dxn

2n

2n

l

v[y(x)] = ( 12 μy 2 + ρy)dx,

−l

y(−l) = 0, y (−l) = 0, y(l) = 0, y (l) = 0.

			ρ =			μ =
	d2			ρ
ρ +		(μy ) = 0	yIV = −		,
	dx			μ

y = −24ρμx4 + C1x3 + C2x2 + C3x + C4.

y =	−ρ	(x4 − 2l2x2 + l4)	y = −	ρ	(x2 − l2)2.
	24μ			24μ

x

v[y(x), z(x)] = F (x, y, y , . . . , y(n), z, z , . . . , zm)dx,

x0

y(x) z(x) y(x) z(x)

		d					dn
Fy −				Fy + . . . + (−1)n				Fy(n) = 0
		dx					dxn
F	d		F		+ . . . + ( 1)m		dm		(m) = 0.
								F
z − dx				z	−	dxm z

m

x1

v[y1, . . . , ym] = F (x, y1, . . . , y1(n1), . . . , ym, . . . , ym(nm ))dx.

x0

							yi(x)
		d		+ . . . + ( 1)ni	dni		(ni) = 0, i = 1, . . . , m .
Fyi			Fy			F
	− dx
			i	−	dxni yi

v[z(x, y)] =	F	x, y, z, ∂x ,			∂y dxdy,
			∂z		∂z
	D
C	D					z(x, y)
˜
C

∂x∂z = p; ∂z∂y = q F C3(D), z C2(D).

				z(x, y, α) = z(x, y) + αδz,
δz = z¯(x, y) − z(x, y),				α = 0			z = z(x, y)
z = z¯(x, y).				α = 1		z(x, y, α)
								v
		∂		α				α = 0
	∂α		v[z(x, y, α)] α=0= 0					δv
		∂α						=
δv = ∂			F (x, y, z(x, y, α), p(x, y, α), q(x, y, α))dxdy
			D				α=0

=[Fz δz + Fpδp + Fqδq]dxdy,

D

z(x, y, α) = z(x, y) + αδz,

p(x, y, α) =

∂

z(x, y, α) = p(x, y) + αδp,

∂x

q(x, y, α) =

∂

z(x, y, α) = q(x, y) + αδq.

∂x

∂

[F δz] =

∂

[F ]δz + F δp,

∂

[F δz] =

∂

[F

]δz + F δq,

∂x

p

∂x

p

p

∂y

q

∂y

q

q

(Fpδp + Fqδq)dxdy =

%∂x {Fpδz} +

∂y {Fqδz}& dxdy−

∂

∂

D

−

D

%∂x {Fp} + ∂y {Fq}& δzdxdy,

∂

∂

D

∂

{Fp}

x

∂x

y =

z, p q x

∂

{Fp} = Fpx + Fpz

∂z

+ Fpp

∂p

+ Fpq

∂q

,

∂x

∂x

∂x

∂x

y

∂

{Fq} = Fqy + Fqz

∂z

+ Fqp

∂p

+ Fqq

∂q

.

∂y

∂y

∂y

∂y

∂x +

∂y

dx dy = - N dy − M dx,

∂N

∂M

D

C

%∂x {Fpδz} + ∂y {Fqδz}&dx dy = - (Fp dy − Fq dx)δz = 0.

∂

∂

D

Γσ

C

δ

˜

C

[Fpδp + Fqδq] dx dy = − %∂x {Fp} +

∂y {Fq}&δz dx dy,

∂

∂

D D

(Fz δz + Fpδp + Fqδq) dx dy = 0

D

(Fz − ∂x {Fp} − ∂y {Fq})δz dx dy = 0.
	∂				∂
D
δz			δz			z = z(x, y)		δz		C = 0
			∂			∂	{Fq} ≡ 0.
Fz −				{Fp} −
		∂x				∂y
z(x, y)
Fz −			∂	{Fp} −		∂	{Fq} = 0.
		∂x				∂y

v[z(x, y)] = σ	. ∂x			+	∂y		/ dx dy,
		∂z	2			∂z	2
Γσ	σ						z	Γσ

z = f (x, y).

∂2z + ∂2z = 0. ∂x2 ∂y2

z = 0,

σ

v[z(x, y)] =

σ	. ∂x			+	∂y			+ 2zf (x, y)/ dx dy,
		∂z	2			∂z	2

Γσ z

∂2z + ∂2z = f (x, y), ∂x2 ∂y2

z = f (x, y)

f (x, y)

t

(T − U ) dt,

t0

T U

mi (i = 1, ..., n)			(xi, yi, zi)					Fi
Fix = −	∂U	;	Fiy = −	∂U	;	Fiz = −	∂U	,
	∂xi			∂yi			∂zi
Fix, Fiy, Fiz						¯
						Fi

xi, yi, zi

n

T = 0.5

i

mi(x˙i2 + y˙i2 + z˙i2),

=1

U

t

t0

(T − U ) dt

−

∂U

−

d ∂T

= 0;

−

∂U

−

d ∂T

= 0;

−

∂U

−

d ∂T

= 0,

∂xi

dt

∂x˙i

∂yi

dt

∂y˙i

∂zi

dt

∂z˙i

mix¨i − Fix = 0;

miy¨i − Fiy

= 0; miz¨i − Fiz

= 0

(i = 1, ..., n).

	ϕj (t, x1, x2, . . . , xn, y1, y2, . . . , yn, z1, z2, . . . , zn) = 0,
	j = 1, 2, . . . , m, m < 3n,
3n−m	m	s =
	t
	q1, q2, . . . , qs,
	t	T U
	T − U	L = L(q, q,˙ t)

t1

S = L(q, q,˙ t) dt

t0

t1

δS = δ L(q, q,˙ t) dt = 0.

					t0
					S
∂L	−	d ∂L			= 0,	i = 1, 2, . . . , s.
∂qi		dt		∂q˙i

u = u(x, t) x t U

dx

"

ds = 1 + ux 2dx,

"

(1 + ux 2 − 1)dx.

( 1 + ux 2 − 1) ≈ 12 ux 2 ux