# УСЛОВИЕ ЭКСТРЕМАЛЬНОСТИ ПОВЕРХНОСТИ ВРАЩЕНИЯ ДЛЯ ФУНКЦИОНАЛА ТИПА ПЛОЩАДИ

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• 517.957 + 514.752

.. , ..

, - . - - .

R3, M C2 ,S = . , R3 C2- (x) (x). , [3]

J1(M) =

S

(|x|)dS, J2(M) =

(|x|)dx,

x = (x1, x2, x3), dS . - M ,

J(M) = J1(M) J2(M).

- J1(M) , J2(M) = const. -, [1] - , .

1.

M , x3 -, = (t), P t(. 1). P t . - , t (a, b), 0 2, = (t) .

. 1. . 11. 20072008

.

.

,..,

20072008

39

• r(t, ) = {(t) cos , (t) sin , t},rt = { cos , sin , 1},r = { sin , cos , 0}.

E = (rt, rt) = 2 + 1, F = (rt, r) = 0, G = (r, r) =

2.

dS2 = (2 + 1)dt2 + 2d2.

. 1

J1(M) =

S

(|x|)dS =2

0

d

b

a

EG F 2dt = 2b

a

2 + 1dt,

= ((t)).

J2(M) =

(|x|)dx =b

a

dt

2

0

d

0

d = 2

b

a

h((t))dt,

40 .. , .. .

• = ((t))

h() =

0

(y)ydy.

- . M J2(M), (t),

(a) = A, (b) = B,

, J1(M). , - J(M) = J1(M) J2(M). (t) , ,

J [(t)] =

b

a

(

2 + 1 + h())dt. (1)

t, F = F (, ), [2]

d

dt(F F) = 0.

d

dt

(

2 + 1 h() 2

2 + 1

)= 0. (2)

= u. -

(u)u

u2 + 1 h(u) (u)uu2

u2 + 1

=

(u)uu2 + 1

h(u) = .

= const.

u2 =

((u)u

+ h(u)

)2 1. (3)

dt = du((u)u

+h(u)

)2 1

.

. 1. . 11. 20072008 41

• t =

du((u)u

+h(u)

)2 1

+ C. (4)

, C.

1. M R3 ,

t =

du((u)u

+h(u)

)2 1

+ C

(2) (1).

M .

2.

1. () = 1, () = 0.

J [(t)] =

b

a

2 + 1dt.

(4)

t =

du(u

)2 1

+ C = archu

+ C.

= u = ch

(t C

)

, .

2. () = u32 , () = 0.

J [(t)] =

b

a

1

2 + 1dt.

t =

du1

2u 1

+ C.

42 .. , .. .

• 1/2 = k, u = k, = sin2 2

t = k

1 d + C =k

2

(1 cos )d + C = k

2( sin ) + C.

= u = k sin2

2=

k

2(1 cos )

.

3.

- , -. , () = () = 1. , - , , . , .

2. (t) (2) t0 (a, b) . t = t0 , .

. , , t0 = 0 , , t > 0 (t) , t < 0 .

F (u) =1(

(u)u+h(u)

)2 1

.

(3)

u2 = F2(u).

t1 = , t2 = , > 0. u1 = u(t1),u2 = u(t2), u0 = u(0).

u1

u0

F (z)dz = t1 = ,

u0

u2

F (z)dz = t2 = .

. 1. . 11. 20072008 43

• ,

u2

u1

F (z)dz = 0,

F (u) : u1 = u2. - (2).

Summary

EXTREMALITY CONDITION OF SURFACE OF REVOLUTIONFOR AREA-TYPE FUNCTIONAL

V.A. Klyachin, T.V. Tkacheva

Present article is devoted to investigation of extremal rotation surfaces for squaretype functional. The solutions of differentional Euler-Lagrange equation are obtained.Also, the symmetry property of these surface is proved and demonstrated examplesfunctionals and its corresponding solutions are constructed.

1. . . . .:, 1989. 312 .

2. .., .., .. : . .: , 1986.

3. .. // . . . . 2006. . 70, 4. C. 7790.

4. .. // .. 260. 1981. 2. . 293295.

44 .. , .. .

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