Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://shamolin2.imec.msu.ru/art-110-1.pdf Äàòà èçìåíåíèÿ: Tue Oct 1 16:38:59 2013 Äàòà èíäåêñèðîâàíèÿ: Thu Feb 27 21:38:18 2014 Êîäèðîâêà:

. xx (2011). . 22­39 517.9; 531.01; 531.552; 539.3

-,
c 2011 . . . , . .



1. 2. 3. 4. 5. 6.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rn . . . . . . . . . . . . . . . . . . . . . . . . . Rn . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

......... ......... ......... ......... ......... ......... ......... .........

22 23 25 27 29 30 37 38

1.





, n- . . .-. . . , , [1] n- . , n- . , « , , , ». , R3 , , , , , . , , R3 R2 . , n- , , , , , ( «» «» «» « »). , . , . .
( 0801-00231, 08-01-00251 08-01-00353).
ISSN 1512­1712 c , 2011




23

. , n , . , -, , , . , n . , , , 1 1 Grad u + (Grad u)T , ij = (uj,i + ui,j ) (1.1) 2 2 , . (1.1) , i, j = 1,...,n. Rn . , n. , , , , , . , , n- - i1 ...in , n . i1 ...in , ij :
{2}

=


i1 ...i
n

i1 j

j1 ...j

n

=

. . .

1

in j
1

1

... i1 jn . = .. . . . ... in jn ... .. . ...
i i1 j

i1 ...in j1 ...jn

,

(1.2)

i1 j

i1 ...i

n- 1

l lj1 ...j

n- 1

=
i lm

. . .

. . .

n- 1

=
km

i1 ...in-1 l lj1 ...jn-1

,

(1.3) (1.4) (1.5) (1.6)

n- 1 j 1

n- 1 j n- 1

jki1 ...i

n- 2

i1 ...i
n- 1

n- 2

= (n - 2)! (jl
l

-

jm kl

),

ji1 ...i

i1 ...i

n- 1

= (-1)
i1 ...i
n

n -1

(n - 1)! jl ,

i1 ...i

n

= n! .

( (1.3)­ (1.6)) 1 n. , 1 2 (.. R2 ). , , . 2.


Rn Ox1 ...xn ei , ei · ej = ij . A{k} k ei1 ··· eik : A
{k}

= Ai

1

...i

k

ei1 ··· eik ,

1

k

n.

(2.1)

il im , 1 l < m k , A{k} A{k}s A{k}a : As1 i Aa1 i
...il ...im ...i ...il ...im ...i
k

k

1 = (Ai 2 1 = (Ai 2

1

...il ...im ...i ...il ...im ...i

k

+ Ai - Ai

1

...im ...il ...i ...im ...il ...i

k

), ).

(2.2) (2.3)

1

k

1

k


24

. . , . .

Rn A{k} B{j } k j , k + j n, (. [2]) C{n-k-j } = A{k} â B{j } n - k - j C
i
k+j +1

...i

n

=

i1 ...i

n

Ai

1

...i

k

B

i

k+1

...i

k+j

.

(2.4)

k + j (2.4) , (2.3) A{k} B{j } . (2.4) A{k} B{j } k j 1 . , nk nj j k Cn Cn . C{n-k-j } (2.4) n k Cn -k-j = Cn+j . A{k} a
i
k+1

...i

n

=

i1 ...i

n

Ai

1

...i

k

,

a

{n-k}

=A

{k}

â1=1âA

{k}

(2.5)

, , a{n-k} = k dual A{k} . A{k} a{n-k} Cn . (2.5) ik+1 ...in l1 ...lk ik+1 ,...,in , :
i
k+1

...in l1 ...l

k

a

i

k+1

...i

n

= (-1)
{k}

k (n - k ) {k}

k !(n - k )! A

l1 ...l

k

.

(2.6) /(k !(n - k )!):

, a a
{n-k}

{n-k}

A
{k}

, A
{k}

(-1)

k(n-k) a{n-k} {n-k}

= dual A

=

A

= dual

(-1)k(n-k) a k !(n - k )!

.

n 2 k (-1)k(n-k) k !(n - k )! , A{k} a{n-k} . R3 k = j = 1 (2.4) , (2.4), , C = A â B. R3 . , = rot v/2, v -- , (-) {2} = Grad v - (Grad v)T /2 1 (2.7) ij k ij . 2 (2.7) R3 . (2.4) k = 1 A = ei (/xi ) ei i , j Rn : ij =
ij k

k ,

k =

(Rot B

{j } {n-j -1}

)

(â B

{j } {n-j -1}

)

=

i1 ...i

n

B

i2 ...i

j +1 ,i1

e

i

j +2

··· e

i

n

(2.8)

B Rn : (rot B)
{n-2}

(â B)

{n-2}

=

i1 ...i

n

B

i2 ,i

1

ei3 ··· ein .

(2.9)

, R3 ilk Blk,i , (2.8) n = 3, j = 2, (Rot B
{2} {2}

)

=

ilm

B

lk,i

em ek ,

. , . -- «» « » -- , .
1

.




25

-- n, . Div Grad . , , Div (Rot B
{j } {n-j -1}

Div Grad -- , ( ) Grad Div
{j } {j +1}

)

0

{n-j -2}

,

Rot (Grad B

)

0

{n-j -1}

.

(2.4), , , A = AI eI B = BJ eJ R2 C = IJ AI BJ 1 . -- : rot B = IJ BJ,I = B2,1 - B1,2 . 3.


R

n

A, B C -- Rn , A â (B â C) n - (1 + (n - 1 - 1)) = 1. (1.4) - n - 2 , A â (B â C ) =
mi1 ...i
n- 2

l

(Am

jki1 ...i

n- 2

Bj C k ) el =
n- 2

= (-1) = (-1)

n -1

lmi1 ...i

i1 ...i mk

n- 2

jk

Am Bj Ck el = ) Am Bj Ck el = (n - 2)! [B(A · C) - C(A · B)], (3.1)
n -1

n -1

(n - 2)! (lj

- lk

mj

= (-1) , . R2 R3 C = CK eK ,
Mil

« ». , n = 2 (3.1) A = AI eI , B = BJ eJ (e1 , e2 ),
ML3 JK 3

(AM

JK i

BJ C K ) el = - = - (
MJ LK

AM BJ CK eL = ) AM BJ CK eL = B(A · C) - C(A · B),

-

MK LJ

, , . (3.1), , (2.4). (3.1) A = B = , A = C, B = , Rn : rot rot C = (-1) C â rot C = (-1)
n -1 n -1

(n - 2)! (grad div C - C),

(3.2)

(n - 2)! (Cm k Cm - Cm m Ck )ek = = (-1)
n -1

(n - 2)! C · (Grad C)T - Grad C ,

(3.3)

. . 3.1. -- grad |v|2 grad |v|2 /2 + [(-1)n /(n - 2)!] v â rot , Rn . vi,j vj ei v /2 + (vi,j vj - vj,i vj )ei , (3.3) v. , ­ (3.4)

v 1 (-1)n dv = + grad |v|2 + v â rot v. dt t 2 (n - 2)!
1

, , (A, B), « » .


26

. . , . .

3.2. Rn . n- u ( ) , . 2u , t2 -- , F -- . (3.5) (3.2) ( + )graddiv u + u + F = u = grad div u + : ( +2)graddiv u + 2u (-1)n rot rot u + F = 2 . (n - 2)! t (3.7) (-1)n rot rot u, (n - 2)! (3.5)

(3.6)

div (3.7) , = div u c2 - 1 2 = - div F, t2 c1 = +2 . (3.8)

rot (3.5), n - 2: 2 . (3.9) c2 - 2 (rot u){n-2} = -(rot F){n-2} , c2 = 2 t (3.9) , .. n(n - 1)/2 . c2 . , n- -- -- c1 c2 , . 3.3. Rn . n- u C 2 u = grad + Rot
{n-2}

,

(3.10)

{n-2} . (3.10) (3.11) ul = ,l + ji1 ...in-2 l i1 ...in-2 ,j . {n-2} = dual {2} , . . i1 ...in-2 = kmi1 ...in-2 (1.4) - n - 2
n km

(3.11),
jl,j {2}

ul = ,l +(-1)n · 2(n - 2)!

u = grad +(-1) · 2(n - 2)! Div

(3.12)

(3.10) (3.12) 1 . u = div u. n(n - 1)/2 {2} . rot (3.12):
kli1 ...i
1 n- 2

u

l,k

= (-1)n · 2(n - 2)!

kli1 ...i

n- 2



jl,j k

(3.13)
{ 2}

, .

{ 2}

div Div






27

(3.13)

i1 ...i

n- 2

mp

. (1.4) n(n - 1)/2
jp,j m

4.

-

jm,j p

=

(-1)n (u 2(n - 2)!

p,m

-u

m,p

).

(3.14)



n- -- n- . -- -- [1, 3, 4] (. [5­8]). [9] -- , [10] n - 1 . [11, 12], , n2 /4 (n2 - 1)/4 n. - . [7] so(n) n- . , , . . Rn [13] , n , . n(n - 1)/2 , , . n = 4. 4.1. . n- O. - r = xi ei (|r| = r = xi xi ) r = v = vi ei [2]: v=
{n-2}

( t) â r ,

vk =

i1 ...i

n- 2

lk

i

1

...i

n- 2

( t) x

l

(4.1)

{n-2} -- n - 2 , . (4.1) (. (2.5), (2.6))
{2}

= dual

{n-2}

,



{n-2}

= dual

{2} 2(n - 2)!

(4.2)

n N = n(n - 1)/2 : v = - , (Grad v) (rot v)
{n-2} {2} {2}

( t) · r ,

vk = lk (t)xl .
{2}

(4.3) , (4.4)

=v
i

k,l

el ek = lk el ek = =
lki1 ...i
n- 2

=

lki1 ...i

n- 2

v

k,l

ei1 ··· e

n- 2

=

lk ei1 ··· e

i

n- 2

= 2(n - 2)!

{n-2}

. (4.5)

{n-2} {2} RN . n = 3, N = 3 {1} ; n = 2, N = 1 -- , .


28

. . , . .

¨ = v = w = wi ei , {n-2} = r i1 ...in-2 ei1 ··· ein-2 ( ) {2} = ij ei ej ( ) (4.6) i1 ...in-2 = i1 ...in-2 , ij = ij . (4.1), 1 : wk =
i1 ...i
n- 2

lk



i1 ...i

n- 2

xl -

i1 ...in-2 kl j1 ...jn-2 ml

i

1

...i

n- 2

j

1

...j

n- 2

xm ,

w=

{n-2}

âr+ xm ,

{n-2}

â
{2}

{n-2}

âr ,
{2}

(4.7) (4.8)

wk = lk xl - lk

mk

w = -

·r+

·

{2}

· r.

4.2. Rn . n- , vn . , , xn Rn , {n-2} i
1

...i

n- 2

=

1 n-2

n -2

(-1)
j =1

n -j

i

1

...¯j ...i i

n- 2



ij n {n-3}

(4.9) n - 3.

i1 ...¯j ...in-2 i1 ...ij -1 ij +1 ...in-2 -- i (4.9) (4.1) vk = 1 n-2
n -2

(-1)
j =1

n -j

i1 ...i

n- 2

lk

i

1

...¯j ...i i

n- 2



ij n

xl =

=

1 n-2
n -2

n -2

(-1)
j =1

n -j

i1 ...i

j -1

ni

j +1

...i

n- 2

lk

i

1

...¯j ...i i

n- 2

xl =

=

1 n-2

(-1)
j =1

2(n-j )

i1 ...¯j ...i i

n- 2

lkn

i

1

...¯j ...i i

n- 2

xl =

i1 ...i

n- 3

lkn

i

1

...i

n- 3

x

l

(4.10)

(4.10) , vn = 0 . (4.10) (4.1), Rn-1 , {n-3} -- xn = 0 ( xn Rn ). Rn (4.9):
nl

= 0,

l = 1,...,n - 1.

(4.11)

4.3. Rn . Rn Q = Qk ek , ( ) K{n-2} = Ki1 ...in-2 ei1 ··· ein-2 , k{2} = kij ei ej T , t V : Qk =
V

vk dV =

i1 ...i

n- 2

lk

i

1

...i

n- 2

xl dV =
V i1 ...i lk

=
1

lk V

xl dV = M

n- 2

i

1

...i

n- 2

x

(c ) l

= Mlk x

(c ) l

(4.12)

() .


j1 ...jn-2 ml i1 ...in-2 kl j1 ...jn-2 i1 ...in-2

29

Ki

1

...i

n- 2

=

j

1

...j

n- 2

xk xm dV I
V

j

1

...j

n- 2

,

(4.13) (4.14)

kij = (n - 2)!
V

(mj xi - mi xj )xm dV , 1 ml 2

T=

1 2

|v|2 dV =
V

1 2

j1 ...jn-2 ml i1 ...in-2 kl

j

1

...j

n- 2

i

1

...i

n- 2

xk xm dV =
V {2}

kl V

xk xm dV
...j

1 =- 2

r ·
V

{2}

·

1 · r dV I 2

j1 ...jn-2 i1 ...in-2

j

1

n- 2

i

1

...i

n- 2

. (4.15)

(4.12)­(4.15) (r) -- , ML-n ; M -- ; (c ) j1 ...jn- xl -- ; I{2n-4} = Ii1 ...in-22 ei1 ··· ein-2 ej1 ··· ejn-2 -- 2n - 4 c I K
{n-2} j1 ...jn-2 i1 ...in-2

=

j1 ...jn-2 ml i1 ...in-2 kl V

xk xm dV .

(4.16)



T 1 K{n-2} = I{2n-4} {n-2} , T = {n-2} I{2n-4} {n 2 n - 2 . 5.


-2}

,

(4.17)

R

n

(1.1), u -- {2} , n(n + 1)/2 (5.1) uj,i + ui,j = 2ij , i, j = 1,...,n, n ui . - M - r = xi ei u = ui ei --
{2}

= ij ei ej = [(u

j,i

- ui,j )/2]ei ej ,
i

, u - r = xi ei ui = ui + ji (xj - xj )+
C

M



[ik +(

jk,i

-

ik,j

)(xj - xj )] dxk ,

(5.2)

C -- , M M . (5.2) n- (., , [14, . 6667]). , . . ij 0, (5.2) (4.3) (ij = ij ) (5.3) ui = ui + ji (xj - xj ), M , . (, (5.1) ) [15]. Rn {2n-4} = Ink {2} 2n - 4 [16, 17] (5.4) i1 ...in-2 j1 ...jn-2 = i1 ...in-2 kl j1 ...jn-2 mp lm,kp .


30

. . , . .

i1 ...in-2 j1 ...jn-2 , (5.1). , Nn (5.4) . (5.4) , {2n-4} n - 2 , n - 2 . (5.4) i1 ...in-2 qs j1 ...jn-2 tr 2n - 4 (1.4). 2R
sq tr



st,q r

+

qr,st

-

sr,q t

-

qt,sr

= 0,

(5.5)

.. ­ R{4} , 4 n. Rsq R
sq tr tr

= -Rqstr = -Rsq

rt

= Rtr

sq

(5.6)

+ Rstrq + Rsrqt = 0.

(5.7)

(5.5) , n- n- , Rsqtr 0. Rn (5.5) . 1) s, q , t r ( , ). Nn1 . (5.6) 4 4 (5.7) Nn1 = 3Cn - Cn = n(n - 1)(n - 2)(n - 3)/12. 2) s, q t, r. Nn2 = 3 3Cn = n(n - 1)(n - 2)/2. , n = 3. 3) s, q t, r Nn3 = n(n - 1)/2. Nn = N n2 (n - 1)2 . (5.8) 12 , , . [18] ( )
n1

+N

n2

+N

n3

=

"" . , Rsq
tr,p

+R

sq r p,t

+ Rsq

pt,r

0

(5.9)

, . (5.9) , ij . "" n- [19], -- [20]. 6.
,

() .




31

. . "" ( ( , )) . , , , . , , , () . [21, 22] , , ( ) . , , () . [23, 24] () , . , , () . , -- . . 6.1. . m , . , , : Dx1 x2 x3 x4 (6.1) diag{I1 ,I2 ,I2 ,I2 }, diag{I1 ,I1 ,I3 ,I3 }. (6.2) Dx1 x2 Dx3 x4 . 6.2. so(4). , N S D -- : DN = R() ( -- [20­22]). (6.1) vD D Dx1 . (6.2) . () S S = s()sgn cos · v 2 , |vD | = v

s -- , , [21, 22]. R S , ,


32

. . , . .

[22] ( ..): R = R() = A sin , S = Sv () = Bv 2 cos ; A, B > 0

-- , so(4), , so(4), [25, 26]: + +[, + ] = M, = diag{1 ,2 ,3 ,4 }, 1 = (-I1 + I2 + I3 + I4 )/2, ..., 4 = (I1 + I2 + I3 - I4 )/2, [·, ·] -- so(4); M -- , R4 , so(4). so(4) 0 - 6 5 - 3 6 0 - 4 2 - 5 4 0 - 1 3 - 2 1 0 1 , 2 , 3 , 4 , 5 , 6 -- so(4). , , : i - j = I j - I
i

(6.3)

i, j = 1,..., 4. M R4 â R4 - so(4), R4 so(4). 6.3. R4 . C , : mwC = F wC = wD +2 DC + E DC, wD = vD +vD , E = . (6.4)

F -- , ( F = S); E -- . 6.4. . . , Dx1 ( (6.1) Dx1 x2 ( (6.2)) , C . , .




33

6.5. (6.1). , , , v = const. (6.5)

(6.5) [22]. (0,x2N ,x3N ,x4N ) -- N Dx1 x2 x3 x4 , {-S, 0, 0, 0} -- , 0 x2N -S 0 x
3N

x

4N

0

0

so(4) : {0, 0,x4N S, 0, -x3N S, x2N S } R6 M so(4) = , (v, , 1 ,2 ) -- R4 , x
2N

= R()cos 1 ,

x

3N

= R()sin 1 cos 2 ,

x

4N

= R()sin 1 sin 2 .

(6.3) (4 + 3 ) 1 +(3 - 4 )(3 5 + 2 4 ) = 0, (2 + 4 ) 2 +(2 - 4 )(3 6 - 1 4 ) = 0, (4 + 1 ) 3 +(4 - 1 )(2 6 + 1 5 ) = x
4N

S, S,

(3 + 2 ) 4 +(2 - 3 )(5 6 + 1 2 ) = 0, (1 + 3 ) 5 +(3 - 1 )(4 6 - 1 3 ) = -x (1 + 2 ) 6 +(1 - 2 )(4 5 + 2 3 ) = x
3N 2N

(6.6)

S.

, (6.1) (6.6):
0 1 = 1 , 0 2 = 2 , 0 4 = 4 .

(6.7)

:
0 0 0 1 = 2 = 4 = 0.

so(4) (n2 = AB /(2I2 )): 0 3 = n2 v 2 sin cos sin 1 sin 2 , 0 5 = -n2 v 2 sin cos sin 1 cos 2 , 0 6 = n2 v 2 sin cos cos 1 . 0 z1 = 3 cos 2 + 5 sin 2 , z2 = -3 sin 2 cos 1 + 5 cos 2 cos 1 + 6 sin 1 , z3 = 3 sin 2 sin 1 - 5 cos 2 sin 1 + 6 cos 1 , «» T S3 ( (6.5) (6.7), )


34

. . , . .

( = DC ):

2 = -z3 + n0 z = n2 v 2 sin 3 0 z = z2 z3 ctg 2 z1 = z1 z3 ctg 1 = z2 ctg , 2 = -z1 ctg

v sin ,
2 2 cos - (z1 + z2 )ctg , 2 + z1 ctg ctg 1 ,

(6.8)

- z1 z2 ctg ctg 1 , csc 1 . (6.9)

(6.8), (6.9) (6.8). , , . z1 ,z2 z =
2 2 z1 + z2 ,

z = z2 /z

1

(6.8), (6.9) : 2 = -z3 + n0 v sin , z = n2 v 2 sin cos - z 2 ctg , 0 3 z = zz3 ctg , 2 z = 1+ z z ctg ctg 1 , zz cos csc 1 , 1 = 2 1+ z 2 = -z1 (z, z )ctg csc 1 .

(6.10)

(6.11) (6.12)

, (6.8) : (6.10) -- , (6.11) (, ) -- . , (6.10)­(6.12) (6.10), -- (6.11) , «» (6.12). , (6.10) T S2 . (6.10) [22, 23]. :
2 z 2 + z3 - n2 vz3 sin + n2 v 2 sin2 0 0 = C1 = const, z sin z3 z , , sin = C2 = const. G sin sin (6.11) 2 1+ z = C3 = const sin 1

, , ± cos
2 3 1

C -1

= sin{C3 (2 + C4 )},

C4 = const.

, sin sin 1 , , (v, , 1 ,2 ) , sin = 0 sin 1 = 0 () .




35

6.6. (6.2). () S = {S1 ,S2 , 0, 0} N = (0, 0,x3N ,x4N ) Dx1 x2 x3 x4 : S1 = S sin , x
3N

S2 = -S cos , x
4N

= const,

= R cos 1 ,

= R sin 1 ,

-- , Dx1 x2 , 1 -- , Dx3 x4 . , 0 0 x3N S1 S2 0 x
4N

0

.

CD Dx1 x2 , DC : DC = { sin , - cos , 0, 0}, v
D

D :

vD = {v cos sin 2 , v cos cos 2 ,v sin cos 1 ,v sin sin 1 }, |vD | = v, 2 -- , Dx1 x2 . , (6.3): (4 + 3 ) 1 +(3 - 4 )(3 5 + 2 4 ) = 0, (2 + 4 ) 2 +(2 - 4 )(3 6 - 1 4 ) = x
4N

S2 , S1 , S2 , (6.13)

(4 + 1 ) 3 +(4 - 1 )(2 6 + 1 5 ) = -x (3 + 2 ) 4 +(2 - 3 )(5 6 + 1 2 ) = -x (1 + 3 ) 5 +(3 - 1 )(4 6 - 1 3 ) = x {0,x
4N

4N 3N

3N

S1 ,

(1 + 2 ) 6 +(1 - 2 )(4 5 + 2 3 ) = 0, S2 , -x
4N

S1 , -x

3N

S2 ,x

3N

S1 , 0} R6 M so(4) =

. , (6.13) :
0 1 = 1 = const, 0 6 = 6 = const.

(6.14)

(6.14):
0 0 1 = 6 = 0.

so(4) 2 = -n2 v 2 sin cos sin 1 cos , 0 3 = -n2 v 2 sin cos sin 1 sin , 0 4 = n2 v 2 sin cos cos 1 cos , 0 5 = n2 v 2 sin cos cos 1 sin , 0 n2 = AB /(I1 + I3 ). 0 - , , , ( ): v = const, 2 = const. (6.15)

, (6.4) .


36

. . , . .

, S + T, , (6.15), . . , , (6.4) , , , (6.15). : 5 sin cos 1 - 3 sin sin 1 -4 sin cos 1 + 2 sin sin 1 vD = v -5 cos sin 2 + 4 cos cos 2 , 3 cos sin 2 - 2 cos cos 2 0 0 E DC = - 5 sin - 4 cos , 3 sin + 2 cos 2 2 -5 sin - 4 5 cos - 3 sin - 2 3 cos 2 2 4 5 sin + 4 cos + 2 3 sin + 2 cos , 2 DC = 0 0 (6.4) : v cos cos 1 - v 1 sin sin 1 - 5 v cos sin 2 + + 4 v cos cos 2 - 5 sin - 4 cos = 0, v cos sin 1 + v 1 sin cos 1 + 3 v cos sin 2 - - 2 v cos cos 2 + 3 sin + 2 cos = 0. (6.17) (6.16), (6.17) so(4) . z1 = 3 cos 1 + 5 sin 1 , z3 = 2 cos 1 + 4 sin 1 , w1 = -z1 sin 2 + z3 cos 2 , w3 = z2 sin 2 - z4 cos 2 , z2 = 3 sin 1 - 5 cos 1 , z4 = 2 sin 1 - 4 cos 1 , w2 = z3 sin 2 + z1 cos 2 , w4 = z4 sin 2 + z2 cos 2 , (6.16)

: = -w3 + n2 v sin , 0 1 = w1 ctg , w1 = w3 w1 ctg , w2 = -w4 w1 ctg ,
2 w3 = -n2 v 2 sin cos cos( + 2 ) - w1 ctg , 0

(6.18)

w4 = -n2 v 2 sin cos sin( + 2 )+ w1 w2 ctg . 0




37

, (6.16) = -w3 + n2 v sin , 0
2 w3 = n2 v 2 sin cos cos( + 2 ) - w1 ctg , 0

(6.19)

w1 = w3 w1 ctg w4 = -n2 v 2 sin cos sin( + 2 )+ w1 w2 ctg , 0 w2 = -w4 w1 ctg 1 = w1 ctg . (6.21) , (6.19), (6.21) , , : 1 (w1 ,w3 , sin ) =
2 2 w1 + w3 - n2 vw3 sin + n2 v 2 cos( + 2 )sin2 0 0 = C1 = const, w1 sin

(6.20)

(6.22) (6.23) (6.24)

2 (w1 ,w3 , sin ) = C2 = const, 3 (w1 ,w3 , sin , 1 ) = C3 = const.

( ), . , (6.22)­ (6.24). w = w3 sin( + 2 )+ w4 cos( + 2 ), (6.20) dw = -w , d1
2 w + w 2

w



= w1 sin( + 2 ) - w2 cos( + 2 )

dw = w , d1

: = C4 = const.

, , , . 7.

, M 0 ( ., , . . [27, 28], . . [29, 30], . . [12] ). so(4) â R4 . so(n) â Rn n- ( ).


38

. . , . .

1. . , , . .: , 1996. 2. .., .. Rn // . . 2001. . 380. 1. . 47­50. ¨ 3. Frahm W. von. Uber gewisse Differentialgleichungen //Math. annalen. 1874. B. 8. S. 35­44. 4. Blaschke W. Nicht-Euklidische geometrie und mechanik // Math. Einrelshriften. Hamburg. 1942. B. 34. S. 39. 5. . . .: , 1975. 6. .., .., .. . I // . . . . 4. .: , 1985. . 179­ 285. 7. .., .. . .: , 1995. 8. .. . .: , 1991. 9. Bottema O., Beth H.J.E. Euler' equations for the motion of a rigid body in n-dimensional space // Koninklijke Nederlandse akademie va Wetenschappen. Indagationes Mathematicae Proc. 1951. V. 13. 1. P. 106­108. 10. .., .. Rn // . . 2002. . 383. 5. . 635­637. 11. .. // . . 1970. . 4. 3. . 73­78. 12. .. n- // . . 1976. . 10. 4. . 93­94. 13. .., .. Rn // . . 1. . . 2003. 5. . 37­41. 14. .., .. . .: , 2006. 272 . 15. Kroner E. Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Berlin: Springer, 1958. ¨ 16. .., .. // . 2004. . 68. . 6. . 1043­1048. 17. Pobedrya B.E., Georgievskii D.V. Equivalence of Formulations for Problems in Elasticity Theory in Terms of Stresses // Russian J. Math. Physics. 2006. V. 13. No. 2. P. 203­209. 18. .. // . . . . 2008. 5. . 44­51. 19. .. // .. . .: , 1965. . 373­443. 20. .., .. N - // . . 1. . . 1993. 2. . 78­83. 21. .., .. // . . 1. . . 1989. 3. . 51­54. 22. .. . .: "", 2007. 256 . 23. .. , // . 1999. . 364. 5. . 627­629. 24. .. , // . . 1997. 2. . 65­68. 25. .., .. // . . . . 29. .: , 1986. . 3­80.




39

26. .., . ­ ­ () // . . 1981. . 15. 3. . 54­66. 27. .. // . . 1986. . 287. 5. . 1105 ­ 1108. 28. .. n , // . . 1983. . 272. 6. . 1364­1367. 29. .. ­ // . . 1983. . 270. 5. . 1094­1097. 30. .. so(4) // . . 1983. . 270. 6. . 1298­1300.

. . . . . E-mail: georgiev@mech.math.msu.su . . . . . , E-mail: shamolin@imec.msu.ru