Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://shamolin2.imec.msu.ru/art-173-2.pdf Äàòà èçìåíåíèÿ: Mon Mar 9 13:07:42 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 00:48:56 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: 3

. .
. . . e-mail: shamolin@imec.msu.ru 517+531.01 : , , . , , . . , . , . , , , , , . , . , . Abstract M. V. Shamolin, Integrable cases in the dynamics of a multi-dimensional rigid body in a nonconservative field in the presence of a tracking force, Fundamentalnaya i prikladnaya matematika, vol. 19 (2014), no. 3, pp. 187--222. This paper is a survey of integrable cases in the dynamics of a five-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite topical; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of nontrivial symmetry groups of such systems, we can prove that these systems possess variable dissipation with zero mean,


( 12-01-00020-).

, 2014, 19, 3, . 187--222. c 2014 « »


188

. . which means that on the average for a period with respect to the periodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either the energy pumping or dissipation can occur. Based on the facts obtained, we analyze dynamical systems that appear in the dynamics of a five-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions that can be expressed through a finite combination of elementary functions.


, , . . , , , , , . , « » (. [1, 9, 10, 16, 18]). , , . , - ( ) . , , [26, 35, 52, 56]. : , Rn â so(n), . . ( ): (I2 = I3 = I4 = I5 ). , . , , , , (. [21, 23, 29--33, 52, 56]).




189

, , , , . . . . , « » . . [7] . . . . (. [11--15, 78]).

1.
1.1.
m ( , E5 ). , , : Dx1 x2 x3 x4 x5 diag{I1 ,I2 ,I2 ,I2 ,I2 } diag{I1 ,I1 ,I3 ,I3 ,I3 }. (2) Dx2 x3 x4 x5 . (1)

1.2. so(5) R

5

n- Rn ( ) SO(n) ( ) (3) Rn â SO(n), n+ n(n - 1) n(n +1) = . 2 2


190

. .

, , n(n +1). , -- ( [8--10, 24, 25]), so(5), , so(5), [3, 4, 18, 25]: + +[, + ] = M, = diag{1 ,2 ,3 ,4 }, -I1 + I2 + I3 + I4 + I5 I1 - I2 + I3 + I4 + I5 , 2 = , 1 = 2 2 I1 + I2 - I3 + I4 + I5 I1 + I2 + I3 - I4 + I5 3 = , 4 = , 2 2 I1 + I2 + I3 + I4 - I5 5 = , 2 (5) (4)

M = MF -- F, R5 , so(5), [ ] -- so(5). ( ) so(5) 0 -10 9 -7 4 10 0 -8 6 -3 -9 8 0 -5 2 , (6) 7 -6 5 0 -1 -4 3 -2 1 0 1 ,2 ,...,10 -- so(5). , , : i - j = Ij - Ii (7) i, j = 1,..., 5. , , R5 â R5 so(5), (8) (DN, F) R5 â R5 R5 â R5 so(5), DN = {0,x2N ,x3N ,x4N ,x5N }, F = {F1 ,F2 ,F3 ,F4 ,F5 }, (10) (9)




191

F -- , . 0 F1 x2N F2 x3N F3 x4N F4 x5N F5 . (11)

(4) M = {x4N F5 - x5N F4 ,x5N F3 - x3N F5 ,x2N F5 - x5N F2 ,x5N F1 , x3N F4 - x4N F3 ,x4N F2 - x2N F4 , -x4N F1 ,x2N F3 - x3N F2 ,x3N F1 , -x2N F1 }. (12) , , [52, 56--58, 61, 68]. « » , . -- , R5 : (13) mwC = F, wC -- C , m -- , ( ) : wC = wD +2 DC + E DC, wD = vD +vD , E = , (14)

wD -- D, F -- , ( F = S), E -- ( ). , (4), (13) R5 â so(5) F. (3) .

2.
( (1)) « » ( , « , ») S [17, 19, 20, 51, 63, 64, 66, 71, 74]. (v, , 1 ,2 ,3 ) -- () D (D -- , ),


192 =

. .

0 10 -9 7 -4

-10 0 8 -6 3

9 -8 0 5 -2

-7 6 -5 0 1

4 -3 2 -1 0

--

, Dx1 x2 x3 x4 x5 -- , , CD Dx1 (C -- ), Dx2 , Dx3 , Dx4 , Dx5 , I1 , I2 , I3 = I2 , I4 = I2 , I5 = I2 , m -- - . Dx1 x2 x3 x4 x5 : DC = {-, 0, 0, 0, 0}, vD = {v cos , v sin cos 1 ,v sin sin 1 cos 2 , v sin sin 1 sin 2 cos 3 ,v sin sin 1 sin 2 sin 3 }.

(15)

(1) : S = {-S, 0, 0, 0, 0}, (16) . . F = S. ( [27, 28], . ), R5 , , v cos - v sin - 10 v sin cos 1 + 9 v sin sin 1 cos 2 - - 7 v sin sin 1 sin 2 cos 3 + 4 v sin sin 1 sin 2 sin 3 + S 2 2 2 2 + (10 + 9 + 7 + 4 ) = - , m v sin cos 1 + v cos cos 1 - 1 v sin sin 1 + + 10 v cos - 8 v sin sin 1 cos 2 + 6 v sin sin 1 sin 2 cos 3 - - 3 v sin sin 1 sin 2 sin 3 - (9 8 + 6 7 + 3 4 ) - = 0, 10 v sin sin 1 cos 2 + v cos sin 1 cos 2 + 1 v sin cos 1 cos 2 - - 2 v sin sin 1 sin 2 - 9 v cos + 8 v sin cos 1 - - 5 v sin sin 1 sin 2 cos 3 + 2 v sin sin 1 sin 2 sin 3 - - (8 10 - 5 7 - 2 4 )+ 9 = 0, v sin sin 1 sin 2 cos 3 + v cos sin 1 sin 2 cos 3 + + 1 v sin cos 1 sin 2 cos 3 + 2 v sin sin 1 cos 2 cos 3 - (19) (18)

(17)




193

- 3 v sin sin 1 sin 2 sin 3 + 7 v cos - 6 v sin cos 1 + + 5 v sin sin 1 cos 2 - 1 v sin sin 1 sin 2 sin 3 + + (6 10 + 5 9 - 1 4 ) - 7 = 0, v sin sin 1 sin 2 sin 3 + v cos sin 1 sin 2 sin 3 + + 1 v sin cos 1 sin 2 sin 3 + 2 v sin sin 1 cos 2 sin 3 + + 3 v sin sin 1 sin 2 cos 3 - 4 v cos + 3 v sin cos 1 - - 2 v sin sin 1 cos 2 + 1 v sin sin 1 sin 2 cos 3 - - (3
10

(20)

+ 2 9 + 1 7 )+ 4 = 0, S = s()v 2 , = CD, v > 0.

(21) (22)

(11) 0 x2N x3N x4N x5N . (23) -S 0 0 0 0 , so(5), (4 + 5 ) 1 +(4 - 5 )(4 7 + 3 6 + 2 5 ) = 0, (3 + 5 ) 2 +(5 - 3 )(1 5 - 3 8 - 4 9 ) = 0, (2 + 5 ) 3 +(2 - 5 )(4 10 - 2 8 - 1 6 ) = 0, (1 + 5 ) 4 +(5 - 1 )(3 = -x5
N 10

(24) (25) (26)

+ 2 9 + 1 7 ) = (27) (28) (29)

, 1 ,2 ,3 ,

v

s()v 2 ,

(3 + 4 ) 5 +(3 - 4 )(7 9 + 6 8 + 1 2 ) = 0, (2 + 4 ) 6 +(4 - 2 )(5 8 - 7 10 - 1 3 ) = 0, (1 + 4 ) 7 +(1 - 4 )(1 4 - 6 10 - 5 9 ) = = x4
N

, 1 ,2 ,3 ,

v

s()v 2 ,
10 10

(30) (31)

(2 + 3 ) 8 +(2 - 3 )(9 (1 + 3 ) 9 +(3 - 1 )(8 = -x3
N 10

+ 5 6 + 2 3 ) = 0, - 5 7 - 2 4 ) =

, 1 ,2 ,3 , v v

s()v 2 ,

(32)

(1 + 2 ) = x2
N

+(1 - 2 )(8 9 + 6 7 + 3 4 ) = s()v 2 . (33)

, 1 ,2 ,3 ,


194

. .

, (17)--(21), (24)--(33) so(5): R1 â S4 â so(5). (34) , (17)--(21), (24)--(33) (35) I2 = I3 = I4 = I5
0 1 1 , 0 2 2 , 0 3 3 , 0 5 5 , 0 6 6 , 0 8 8 .

(36)

:
0 0 0 0 0 0 1 = 2 = 3 = 5 = 6 = 8 = 0.

(37)

T, CD = Dx1 v const, (17)--(21), (24)--(33) F1 T - s()v 2 , = DC. (39) (38)

T (38). , T (17)--(21), (24)--(33), cos = 0
2 2 2 2 T = Tv (, 1 ,2 ,3 , ) = m (4 + 7 + 9 + 10 )+

+ s()v
v

2

1-

m sin 3I2 cos v

v

, 1 ,2 ,3 ,

v

, (40)

, 1 ,2 ,3 , + x4 + x3
N

v

= x5

N

, 1 ,2 ,3 ,

sin 1 sin 2 sin 3 +

N

v , 1 ,2 ,3 , v , 1 ,2 ,3 ,

sin 1 sin 2 cos 3 + sin 1 cos 2 + x2
N

, 1 ,2 ,3 ,

v

cos 1 .

(41)

(40) (36)--(38). . -, (), (38). -, , . , (17)--(21), (24)--(33)




195

v cos cos 1 - 1 v sin sin 1 + 10 v cos - = 0, 10 1 v sin cos 1 cos 2 - v cos sin 1 cos 2 + - 2 v sin sin 1 sin 2 - 9 v cos + 9 = 0, v cos sin 1 sin 2 cos 3 + 1 v sin cos 1 sin 2 cos 3 + + 2 v sin sin 1 cos 2 cos 3 - 3 v sin sin 1 sin 2 sin 3 + + 7 v cos - 7 = 0, v cos sin 1 sin 2 sin 3 + 1 v sin cos 1 sin 2 sin 3 + + 2 v sin sin 1 cos 2 sin 3 + 3 v sin sin 1 sin 2 cos 3 - - 4 v cos + 4 = 0, 3I2 4 = -x5 3I2 7 = x4
N N

(42) (43)

(44)

(45) (46) (47) (48) (49)

3I2 9 = -x3 3I2
10

N

= x2

N

, 1 ,2 ,3 , s()v 2 , v , 1 ,2 ,3 , s()v 2 , v , 1 ,2 ,3 , s()v 2 , v , 1 ,2 ,3 , s()v 2 , v

, , v . (42)--(49) v cos + + v cos { cos 1 +[(7 cos 3 - 4 sin 3 )sin 2 - 9 cos 2 ]sin 1 } +

10

+ {- cos 1 +[9 cos 2 - (7 cos 3 - 4 sin 3 )sin 2 ]sin 1 } = 0, (50) 10 1 v sin + + v cos {[(7 cos 3 - 4 sin 3 )sin 2 - 9 cos 2 ]cos 1 - 10 sin 1 } + + {[9 cos 2 - (7 cos 3 - 4 sin 3 )sin 2 ]cos 1 + sin 1 } = 0, (51) 10 2 v sin sin 1 + v cos {[7 cos 3 - 4 sin 3 ]cos 2 + 9 sin 2 } + + {-[7 cos 3 - 4 sin 3 ]cos 2 - 9 sin 2 } = 0, 3 v sin sin 1 sin 2 + v cos {-4 cos 3 - 7 sin 3 } + + {4 cos 3 + 7 sin 3 } = 0, 4 = - 7 = v x5 3I2
N 2 N

(52) (53)

, 1 ,2 ,3 , , 1 ,2 ,3 , v

v

s(), s(),

(54) (55)

v2 x4 3I2


196 9 = -
10

. .

v2 x3N , 1 ,2 ,3 3I2 v2 = x2N , 1 ,2 ,3 , 3I2

,

s(), v s(). v

(56) (57)

. 4 , 7 , 9 , 10 : 4 z1 7 z2 = T3,4 (-1 ) T2,3 (-2 ) T1,2 (-3 ) , 9 z3 z4 10 10 0 0 0 1 0 0 T3,4 ( ) = 0 0 cos - sin , 0 0 sin cos 1 0 0 0 cos - sin 0 0 cos - sin 0 sin cos 0 , T1,2 ( ) = T2,3 ( ) = 0 sin 0 cos 0 0 1 0 0 0 1 0 0 0 , : z1 = 4 cos 3 + 7 sin 3 , z2 = (7 cos 3 - 4 sin 3 )cos 2 + 9 sin 2 , z3 = [(-7 cos 3 + 4 sin 3 )sin 2 + 9 cos 2 ]cos 1 + 10 sin 1 , z4 = [(7 cos 3 - 4 sin 3 )sin 2 - 9 cos 2 ]sin 1 + 10 cos 1 . (50)--(57), O1 = (, 1 ,2 ,3 ,4 ,7 ,9 ,10 ) R8 : =

(58)

0 0 . 0 1

(59)

k, 1 = l1 , 2 = l2 , k , l1 ,l2 Z 2

(60)

, 1 , 2 , 3 . , , (60) . , k l1 , l2 (v, , 1 ,2 ,3 ), k , (50) . , (50)--(57) (60) = -z4 + v s() 3I2 cos
v

, 1 ,2 ,3 ,

v

,

(61)




197

z4 =

v2 2 2 2 cos s()v , 1 ,2 ,3 , - (z1 + z2 + z3 ) 3I2 v sin v s() + -z3 v,1 , 1 ,2 ,3 , + 3I2 sin v - z1 v,3 , 1 ,2 ,3 , + z2 v,2 , 1 ,2 ,3 , v
4

+

v

,

(62)

z3 = z3 z

cos 2 2 cos cos 1 +(z1 + z2 ) + sin sin sin 1 v s() cos 1 + + z4 v,1 , 1 ,2 ,3 , - z2 v,2 , 1 ,2 ,3 , 3I2 sin v v sin 1 cos 1 v2 + z1 v,3 , 1 ,2 ,3 , s()v,1 , 1 ,2 ,3 , - , (63) v sin 1 3I2 v
4

z2 = z2 z +

v 3I2 v + 3I2 v2 + 3I2 z1 = z1 z

1 cos 2 cos cos cos 1 2 cos - z2 z3 - z1 + sin sin sin 1 sin sin 1 sin 2 s() cos 1 v,2 , 1 ,2 ,3 , + -z4 + z3 sin v sin 1 s() 1 cos 2 v,3 , 1 ,2 ,3 , + -z1 sin v sin 1 sin 2 s()v,2 , 1 ,2 ,3 , , v

(64)

cos cos cos 1 cos 1 cos 2 - z1 z3 + z1 z2 + sin sin sin 1 sin sin 1 sin 2 cos 1 1 cos 2 v s() v,3 , 1 ,2 ,3 , + + z2 z4 - z 3 3I2 sin v sin 1 sin 1 sin 2 v2 s()v,3 , 1 ,2 ,3 , - , 3I2 v
4 1

- (65) (66)

v s() cos + v, 1 = z3 sin 3I2 sin 2 = -z 3 = z v,
1 1 2

, 1 ,2 ,3 ,
2

v

, v , v ,

cos s() v + v, sin sin 1 3I2 sin sin 1

, 1 ,2 ,3 ,
3

(67) (68)

cos s() v + v, sin sin 1 sin 2 3I2 sin sin 1 sin 2 v = -x2 v

, 1 ,2 ,3 ,

, 1 ,2 ,3 ,
N

N

, 1 ,2 ,3 ,

v

sin 1 +

+ x3

, 1 ,2 ,3 ,

cos 1 cos 2 +


198 + x4 + x5 v,
2 N

. .

, 1 ,2 ,3 , , 1 ,2

N

, 1 ,2 ,3 ,
N

+ x4 + x5 v,
3

, 1 ,2 , 1 ,2

N

, 1 ,2 ,3 ,
N

+ x5

, 1 ,2

cos 1 sin 2 cos 3 v ,3 , cos 1 sin 2 sin 3 , v = -x3N , 1 ,2 ,3 , v ,3 , cos 2 cos 3 + v ,3 , cos 2 sin 3 , v = -x4N , 1 ,2 ,3 , v ,3 , cos 3 , v

+

v

sin 2 +

v

sin 3 +

v (, 1 ,2 ,3 , /v ) (41). (, 1 ,2 ,3 , /v ) (, 1 ,2 ,3 ,z1 /v , z2 /v , z3 /v , z4 /v ) (59). (50)--(57) (60) k : (60) k (50)--(57), (60) , , . , . . , (38) T cos = 0 (40).
/2

lim

s() cos

v

, 1 ,2 ,3 ,

v

= L 1 ,2 ,3 ,

v

.

(69)

, |L| < + ,
/2

lim





v

, 1 ,2 ,3

v

s()

< +.

(70)

= /2 m Lv 2 2 2 2 2 ,1 ,2 ,3 , = m (4 + 7 + 9 + 10 ) - , (71) 2 2I2 4 , 7 , 9 , 10 . , W , T =T
v




199 (72)

T =T

v

mv 2 ,1 ,2 ,3 , = , 2 R0

R0 -- CW . (71) (72) , , T (60), .

3.
3.1.
[27, 28], s, x2N , x3N , x4N x5N s() = B cos , x2 x3 x4 x5
N

N

N

N

v , 1 ,2 ,3 , v , 1 ,2 ,3 , v , 1 ,2 ,3 , v , 1 ,2 ,3 , v = 0,

= x2 = x3 = x4 = x5

N0

(, 1 ,2 ,3 ) = A sin cos 1 , (, 1 ,2 ,3 ) = A sin sin 1 cos 2 , (73) (, 1 ,2 ,3 ) = A sin sin 1 sin 2 cos 3 , (, 1 ,2 ,3 ) = A sin sin 1 sin 2 sin 3 ,

N0

N0

N0

A, B > 0,

, ( , 1 , 2 , 3 ). v (, 1 ,2 ,3 , /v ), v,s (, 1 ,2 ,3 , /v ), s = 1, 2, 3, (61)--(68), : v , 1 ,2 ,3 , v = A sin , v,
s

, 1 ,2 ,3 ,

v

0, s = 1, 2, 3. (74)

, (38), (60) ( (61)--(68)) = -z4 + z4 = AB v sin , 3I2 (75) (76)

AB v 2 2 2 2 cos , sin cos - (z1 + z2 + z3 ) 3I2 sin


200 z 3 = z3 z
4

. .

cos 2 2 cos cos 1 +(z1 + z2 ) , (77) sin sin sin 1 cos cos cos 1 1 cos 2 2 cos - z2 z3 z2 = z2 z4 - z1 , (78) sin sin sin 1 sin sin 1 sin 2 cos cos cos 1 cos 1 cos 2 - z1 z3 z1 = z1 z4 + z1 z2 , (79) sin sin sin 1 sin sin 1 sin 2 cos 1 = z3 , (80) sin cos 2 = -z2 , (81) sin sin 1 cos 3 = z1 . (82) sin sin 1 sin 2 , : AB zk n0 vzk , k = 1, 2, 3, 4, n2 = , b = n0 , · = n0 v , (83) 0 3I2 (75)--(82) = -z4 + b sin , z4 = sin cos z 3 = z3 z
4 2 2 2 cos , - (z1 + z2 + z3 ) sin 2 2 cos cos 1 +(z1 + z2 ) , sin sin 1 1 cos 2 cos cos 1 2 cos - z2 z3 - z1 , sin sin 1 sin sin 1 sin 2 cos cos 1 cos 1 cos 2 - z1 z3 + z1 z2 , sin sin 1 sin sin 1 sin 2

(84) (85)

cos (86) sin cos z2 = z2 z4 (87) sin cos z1 = z1 z4 (88) sin cos , (89) 1 = z3 sin cos 2 = -z2 , (90) sin sin 1 cos 3 = z1 . (91) sin sin 1 sin 2 , (84)--(91), , , T S4 S4 , (84)--(90) . (84)--(91) , , . w4 z4 z3 w3 , w2 z2 z1 w1



2 2 2 z1 + z2 + z3 ,

201 , (92)

w4 = z4 ,

w3 =

w2 =

z2 , z1

w1 =

z

3 2 2

2 z1 + z

(84)--(91) : = -w4 + b sin , w4 = sin cos - w3 = w3 w4 cos , sin
2 w3

(93) cos , sin (94) (95) (96)

w2 = d2 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 )

2 1+ w2 cos 2 , w2 sin 2 2 = d2 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ), 2 1+ w1 cos 1 , w1 sin 1 1 = d1 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ),

w1 = d1 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 )

(97) (98)

3 = d3 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ), cos , sin cos d2 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ) = -Z2 (w4 ,w3 ,w2 ,w1 ) , sin sin 1 cos d3 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ) = Z1 (w4 ,w3 ,w2 ,w1 ) , sin sin 1 sin 2 zk = Zk (w4 ,w3 ,w2 ,w1 ), k = 1, 2, 3, -- d1 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ) = Z3 (w4 ,w3 ,w2 ,w1 )

(99)

(100)

(92). , : (93)--(95) 3, (96), (97) (, ) -- 2. , (93)--(98) (93)--(95), -- (96), (97) , «» (98). , (93)--(95) T S2 S2 .

3.2.
(93)--(95) , . (93)--(95)


202

. .
2 sin cos - w3 cos / sin dw4 = , d -w4 + b sin dw3 w3 w4 cos / sin = . d -w4 + b sin

(101)

= sin , (101)
2 dw4 - w3 / = , d -w4 + b dw3 w3 w4 / = . d -w4 + b

(102)

w3 = u1 , (102) du2 1 - u2 1 + u2 = , d -u2 + b du1 u1 u2 + u1 = , d -u2 + b du2 1 - u2 + u2 - bu2 1 2 = , d -u2 + b du1 2u1 u2 - bu1 = . d -u2 + b w4 = u2 , (103)

(104)

(105)

(105) du2 1 - u2 + u2 - bu2 1 2 = , (106) du1 2u1 u2 - bu1 : d u2 + u2 - bu2 +1 2 1 u1 = 0. (107)

, (106) u2 + u2 - bu2 +1 2 1 = C1 = const, u1
2 2 w4 + w3 - bw4 sin +sin2 = C1 = const. w3 sin

(108)

(109)




203

1. (93)--(95) [79, 82--87], b = 0: = -w4 ,
2 w4 = sin cos - w3

cos w3 = w3 w4 . sin
2 2 w4 + w3 +sin2 = C1 = const, w3 sin = C2 = const.

cos , sin

(110)

(111) (112)

, (111), (112) (110). b = 0
2 2 w4 + w3 - bw4 sin +sin2

(113)

(112) (93)--(95). (113), (112) (93)--(95) b. (93)--(95). (108) u1 = 0 : u2 - b 2
2

+ u1 -

C1 2

2

=

2 b2 + C1 - 1. 4

(114)

, 2 b2 + C1 - 4 0 (115) (93)--(95) , (114). , (108) (105) U1 (C1 ,u2 ) = 1 2 C1 ±
2 C1 - 4(u2 - bu2 +1) , 2

2(1 - bu2 + u2 ) - C1 U1 (C1 ,u2 ) du2 2 = , d -u2 + b

(116)

(117)

C1 (115). (93)--(95) d = (b - u2 )du 2(1 - bu2 + u ) - C1 C1 ±
2 2 2 2 C1

- 4(u2 - bu2 +1) /2 2

.

(118)


204

. .

( ), , ln | sin |. b 2 = r1 , b2 = b2 + C1 - 4, 1 2 (118) u2 - - 1 4
2 d(b2 - 4r1 ) 1

(119) (120)

(b -

2 1

2 4r1

) ± C1

b-

2 1

2 4r1

-b

dr (b -
2 1 2 4r1

1 2 b2 - 4r1 1

) ± C1

=

1 = - ln 2 I1 = dr
2 1 3 2 b - r3 (r3 ± C1 )

2 b2 - 4r1 b 1 ± 1 ± I1 , (121) C1 2

,

r3 =

2 b2 - 4r1 . 1

(122)

(122) . I. b > 2. 2 b2 - 4+ b2 - r3 1 C1 1 + ± ln I1 = - 2-4 r3 ± C1 2b b2 - 4 2 b2 - 4 - b2 - r3 1 C1 1 + +const. (123) ln 2-4 r3 ± C1 2b b2 - 4 II. b < 2. 1 ±C1 r3 + b2 1 I1 = +const. arcsin b1 (r3 ± C1 ) 4 - b2 I1 = b w4 -, (126) sin 2 I1 . (I) b > 2. b2 - 4 ± 2r1 1 C1 + ± I1 = - ln 2 - 4r 2 ± C 2-4 2b b2 - 4 b1 1 1 1 b2 - 4 2r1 C1 + +const. (127) ln 2 - 4r 2 ± C 2-4 2b b2 - 4 b1 1 1 r1 =
2 b2 - r3 1 +const. C1 (r3 ± C1 )

(124)

III. b = 2.

(125)




205

(II) b < 2. I1 = (III) b = 2. I1 =
2 1 2 ±C1 b2 - 4r1 + b2 1 1 1 arcsin +const. 2 4 - b2 b1 ( b2 - 4r1 ± C1 ) 1

(128)

2r1
2 C1 ( b - 4r1 ± C1 )

+const.

(129)

, (93)--(95); , . 2. C1 (108). ( ): w4 w3 ln | sin | + G2 sin , , = C2 = const. (130) sin sin , (93)--(98) . , , ( ) (96), (97), , «» (98). ( ) (96), (97) : 2 1+ ws cos s dws = , s = 1, 2. (131) ds ws sin s
2 1+ ws = Cs sin s +2

= const,

s = 1, 2.

(132)

, «» (98), (98) (96) dw2 2 = -(1 + w2 )cos 2 . (133) d3 (132)
2 2 C4 cos 2 = ± C4 - 1 - w2 ,

(134) (135)



dw2 1 2 = (1 + w2 ) d3 C4

2 2 C4 - 1 - w2 .


206

. .

, , : (3 + C5 ) = C4 dw2
2 2 2 (1 + w2 ) C4 - 1 - w2

,

C5 = const.

(136)

tg(3 + C5 ) = C4 w2
2 2 C4 - 1 - w2

,

C5 = const.

(137)

, , «» (98), arctg C4 w2
2 C4 2 - 1 - w2

± 3 = C5 ,

C5 = const.

(138)

, (17)--(21), (24)--(33) (73) . (38), (36), (37), (109), , (123)--(130), ( ) , , (132) (138). 1. (17)--(21), (24)--(33) (38), (73), (37) ( ), . .

3.3.
sin ¨ = 0, + b cos +sin cos - [1 2 + 2 2 sin2 1 + 3 2 sin2 1 sin2 2 ] cos 2 1 + cos - (2 2 + 3 2 sin2 2 )sin 1 cos 1 = 0, 1 + b 1 cos + 1 ¨ cos sin (139) 2 2 1 + cos +21 2 cos 1 - 3 2 sin 2 cos 2 = 0, 2 + b 2 cos + ¨ cos sin sin 1 2 1 + cos +21 3 cos 1 +22 3 cos 2 = 0, b > 0, 3 + b 3 cos + 3 ¨ cos sin sin 1 sin 2 , , . . [75--77, 80, 81]. , 8, 3




207

, . (140) T S3 {, 1 , 2 , 3 , ,1 ,2 ,3 } S4 {, 1 ,2 ,3 }, , (139) 3 0, 1 0, 2 0, 3 0 (142) (141)

. , (139) (140) . , . 2. (17)--(21), (24)--(33) (38), (73), (37) (139). , = , 1 = 1 , 2 = 2 , 3 = 3 , b = -b . . [52].

4.
4.1.
. , , . . x = (x1N ,x2N ,x3N ,x4N ,x5N ) -- N ( ) , Q = = (Q1 ,Q2 ,Q3 ,Q4 ,Q5 ) -- , . (x1N ,x2N ,x3N ,x4N ,x5N ) , [5, 6, 22, 36, 41--43]. , : x = Q + R, (143)

R = (R1 ,R2 ,R3 ,R4 ,R5 ) -- -, . R


208

. .

: 0 -10 R1 10 R 2 0 1 8 R = R3 = - -9 v 7 R 4 -6 R5 -4 3

9 -8 0 5 -2

-7 6 -5 0 1

4 -3 2 -1 0

h h h h h

1 2 3 4 5

. (144)

(h1 ,h2 ,h3 ,h4 ,h5 ) -- . , x1N 0, x2
N N

= Q2 - h

x4

10 9 , x3N = Q3 + h1 , v v 7 4 = Q4 - h1 , x5N = Q5 + h1 . v v
1

(145)

4.2.
[27, 28] Q2 = A sin cos 1 , Q3 = A sin sin 1 cos 2 , Q4 = A sin sin 1 sin 2 cos 3 , Q5 = A sin sin 1 sin 2 sin 3 , s, x2N , x3N , x4 s() = B cos , x2 x3 x4 x5
N N

A > 0,

(146)

x5

N



B > 0, = A sin cos 1 - h 10 , v 9 , v

N

N

N

v , 1 ,2 , v , 1 ,2 , v , 1 ,2 , v , 1 ,2 ,

= A sin sin 1 cos 2 + h

7 , v 4 = A sin sin 1 sin 2 sin 3 + h , v = A sin sin 1 sin 2 cos 3 - h

h = h1 > 0, v = 0.

(147) , ( ) (. . ), h2 = h3 = h4 = h5 . v (, 1 ,2 ,3 , /v ), v,s (, 1 ,2 ,3 , /v ), s = 1, 2, 3, (61)--(68), :




209

v , 1 ,2 ,3 , v, v,
1

v

= A sin - =

h z4 , v (148)
3

2

v , 1 ,2 ,3 , v , 1 ,2 ,3 ,

h z3 , v h = - z2 , v, v

, 1 ,2 ,3 ,

v

=

h z1 . v

, (38), (60) ( (61)--(68)) AB v B h sin , (149) = - 1+ z4 + 3I2 3I2 Bhv AB v 2 B h 2 2 2 cos - z4 = sin cos - 1+ z4 cos , (150) (z1 + z2 + z3 ) 3I2 3I2 sin 3I2 cos B h + z3 = 1+ z3 z4 3I2 sin B h Bhv 2 2 cos cos 1 + 1+ - z3 cos , (151) (z1 + z2 ) 3I2 sin sin 1 3I2 B h cos cos cos 1 B h - 1+ z2 z4 z2 z3 z2 = 1+ 3I2 sin 3I2 sin sin 1 1 cos 2 B h Bhv 2 cos - 1+ - z2 cos , (152) z1 3I2 sin sin 1 sin 2 3I2 B h cos cos cos 1 B h - 1+ z1 z4 z1 z3 z1 = 1+ 3I2 sin 3I2 sin sin 1 cos 1 cos 2 B h Bhv + 1+ - z1 cos , (153) z1 z2 3I2 sin sin 1 sin 2 3I2 cos B h , (154) z3 1 = 1+ 3I2 sin cos B h 2 = - 1+ , (155) z2 3I2 sin sin 1 cos B h . (156) z1 3 = 1+ 3I2 sin sin 1 sin 2 , AB Bh zk n0 vzk , k = 1, 2, 3, 4, n2 = , b = n0 , H1 = , · = n0 v , 0 3I2 3I2 n0 (157) (149)--(156) = -(1 + bH1 )z4 + b sin ,
2 2 2 cos - H1 z4 cos , z4 = sin cos - (1 + bH1 )(z1 + z2 + z3 ) sin

(158) (159)


210 z3 = (1 + bH1 )z3 z z
2 4

. .

z

1

1 2 3

cos 2 2 cos cos 1 +(1+ bH1 )(z1 + z2 ) - H1 z3 cos , sin sin sin 1 cos cos cos 1 - (1 + bH1 )z2 z3 = (1 + bH1 )z2 z4 - sin sin sin 1 1 cos 2 2 cos - (1 + bH1 )z1 - H1 z2 cos , sin sin 1 sin 2 cos cos cos 1 - (1 + bH1 )z1 z3 = (1 + bH1 )z1 z4 + sin sin sin 1 cos 1 cos 2 +(1+ bH1 )z1 z2 - H1 z1 cos , sin sin 1 sin 2 cos , = (1 + bH1 )z3 sin cos = -(1 + bH1 )z2 , sin sin 1 cos = (1 + bH1 ) z1 . sin sin 1 sin 2

(160)

(161)

(162) (163) (164) (165)

, (158)--(165), , , T S4 S4 , (158)--(164) . (158)--(165) , , . w4 z4 w3 z3 w2 z2 z1 w1 w4 = z4 , w3 = , z2 , w1 = z1 z
2 1 3 2 2

2 2 2 z1 + z2 + z3 , w2 =

z +z

,

(166)

(158)--(165) : = -(1 + bH1 )w4 + b sin , w4 = sin cos - (1 + bH w3 = (1 + bH1 )w3 w4
2 1 )w3

(167) cos - H1 w4 cos , sin (168) (169) (170)

cos - H1 w3 cos , sin 2 1+ w2 cos 2 w2 = d2 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ) , w2 sin 2 2 = d2 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ),



2 1+ w1 cos 1 , w1 sin 1 1 = d1 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ),

211

w1 = d1 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 )

(171) (172)

3 = d3 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ), d1 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ) = (1 + bH1 )Z3 (w4 ,w3 ,w2 ,w1 ) d2 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ) = = -(1 + bH1 )Z2 (w4 ,w3 ,w2 ,w1 ) d3 (w4 ,w3 ,w2 ,w1 ; , 1 ,2 ,3 ) = = (1 + bH1 )Z1 (w4 ,w3 ,w2 ,w1 ) zk = Zk (w4 ,w3 ,w2 ,w1 ), k = 1, 2, 3, -- cos , sin sin 1 sin 2 cos , sin sin 1 cos , sin

(173)

(174)

(166). , : (167)--(169) 3, (170), (171) (, ) -- 2. , (167)--(172) (167)--(169), -- (170), (171) , «» (172). , (167)--(169) T S2 S2 .

4.3.
(167)--(169) , . (167)--(169)
2 sin cos - (1 + bH1 )w3 cos / sin - H1 w4 cos dw4 = , d -(1 + bH1 )w4 + b sin (175) dw3 (1 + bH1 )w3 w4 cos / sin - H1 w3 cos = . d -(1 + bH1 )w4 + b sin = sin , (175) 2 dw4 - (1 + bH1 )w3 / - H1 w4 = , d -(1 + bH1 )w4 + b dw3 (1 + bH1 )w3 w4 / - H1 w3 = . d -(1 + bH1 )w4 + b

(176)


212

. .

, w3 = u1 , (176) du2 1 - (1 + bH1 )u2 - H1 u2 1 + u2 = , d -(1 + bH1 )u2 + b du1 (1 + bH1 )u1 u2 - H1 u1 + u1 = , d -(1 + bH1 )u2 + b du2 (1 + bH1 )(u2 - u2 ) - (b + H1 )u2 +1 2 1 = , d -(1 + bH1 )u2 + b du1 2(1 + bH1 )u1 u2 - (b + H1 )u1 = . d -(1 + bH1 )u2 + b w4 = u2 , (177)

(178)

(179)

(179) du2 1 - (1 + bH1 )(u2 - u2 ) - (b + H1 )u2 1 2 = , du1 2(1 + bH1 )u1 u2 - (b + H1 )u1 : d (1 + bH1 )(u2 + u2 ) - (b + H1 )u2 +1 2 1 u1 = 0. (181) (180)

, (180) (1 + bH1 )(u2 + u2 ) - (b + H1 )u2 +1 2 1 = C1 = const, u1
2 2 (1 + bH1 )(w4 + w3 ) - (b + H1 )w4 sin +sin2 = C1 = const. w3 sin

(182)

(183)

3. (167)--(169) [2, 34, 48, 62, 65, 69, 70], b = H1 : = -(1 + b2 )w4 + b sin , 2 cos - bw4 cos , w4 = sin cos - (1 + b2 )w3 (184) sin cos - bw3 cos . w3 = (1 + b2 )w3 w4 sin
2 2 (1 + b2 )(w4 + w3 ) - 2bw4 sin +sin2 = C1 = const, w3 sin = C2 = const.

(185) (186)




213

, (185), (186) (184). b = H1
2 2 (1 + bH1 )(w4 + w3 ) - (b + H1 )w4 sin +sin2

(187)

(186) (167)--(169). (187), (186) (167)--(169) b, H1 . (167)--(169). (182) u1 = 0 : u2 - b + H1 2(1 + bH1 )
2

+ u1 -

C1 2(1 + bH1 )

2

=

2 (b - H1 )2 + C1 - 4 . 4(1 + bH1 )2

(188)

, 2 (b - H1 )2 + C1 - 4 0, (189) (167)--(169) , (188). , (182) (179) U1 (C1 ,u2 ) = U2 (C1 ,u2 ) = 1 {C1 ± U2 (C1 ,u2 )}, 2(1 + bH1 )
2 C1

2(1 + bH1 )u2 - 2(b + H1 )u2 +2 - C1 U1 (C1 ,u2 ) du2 2 = , d b - (1 + bH1 )u2

(190)

(191)
2 2

- 4(1 + bH1 )(1 - (b + H1 )u2 +(1+ bH1 )u ),

C1 (189). (167)--(169) (b - (1 + bH1 )u2 )du2 . 2(1 - (b + H1 )u2 +(1 + bH1 )u2 ) - C1 {C1 ± U2 (C1 ,u2 )}/(2(1 + bH1 )) 2 (192) ( ), , d = ln | sin |. u2 - b + H1 = r1 , 2(1 + bH1 )
2 b2 = (b - H1 )2 + C1 - 4, 1

(193)

(194)

(192)


214 - 1 4

. .
2 d(b2 - 4(1 + bH1 )r1 ) 1 2 (b2 - 4(1 + bH1 )r1 ) ± C1 1 2 b2 - 4(1 + bH1 )r1 1 dr1

-
2 b2 - 4(1 + bH1 )r1 1

- (b - H1 )(1 + bH1 ) 1 = - ln 2 I1 =
2 1

2 (b2 - 4(1 + bH1 )r1 ) ± C1 1

= (195)

2 b2 - 4(1 + bH1 )r1 b - H1 1 I1 , ±1 ± C1 2

2 , r3 = b2 - 4(1 + bH1 )r1 . 1 2 b - r3 (r3 ± C1 ) (196) . I. |b - H1 | > 2.

dr

3

(196)

I1 = - +

1 2 (b - H 1
1

)2

-4 ln

ln

(b - H1 )2 - 4+ r3 ± C1 (b - H1 )2 - 4 - r3 ± C1

2 b2 - r3 1

±

C1 (b - H1 )2 - 4 C1 (b - H1 )2 - 4 +const. (197)

2 b2 - r3 1

2 (b - H1 )2 - 4



II. |b - H1 | < 2. I1 = III. |b - H1 | = 2. I1 = b + H1 w3 - , sin 2(1 + bH1 ) I1 . I. |b - H1 | > 2. r1 = I1 = - + 1 2 (b - H 1 2 (b - H
1 1

1 4 - (b - H1 )2

arcsin

±C1 r3 + b2 1 +const. b1 (r3 ± C1 )

(198)

2 b2 - r3 1 +const. C1 (r3 ± C1 )

(199)

(200)

)2

-4 ln

ln

(b - H1 )2 - 4 ± 2(1 + bH1 )r1 b - 4(1 + bH b - 4(1 + bH
2 1 2 1 22 1 ) r1

± C1

±

C1 (b - H1 )2 - 4 C1

+

(b - H1 )2 - 4 2(1 + bH1 )r1
22 1 ) r1

)2

-4

± C1

(b - H1 )2 - 4

+const. (201)

II. |b - H1 | < 2. I1 = 1 4 - (b - H1 )2 arcsin ±C1 b1 (
2 b2 - 4(1 + bH1 )2 r1 ± C1 ) 1 2 b2 - 4(1 + bH1 )2 r1 + b 1 2 1

+const.

(202)




215

III. |b - H1 | = 2. I1 = 2(1 + bH1 )r1
2 C1 ( b - 4(1 + bH1 )2 r1 ± C1 ) 2 1

+const.

(203)

, (167)--(169); , . 4. C1 (182). ( ): w3 w4 , ln | sin | + G2 sin , (204) = C2 = const. sin sin , (167)--(172) . , , ( ) (170), (171), , «» (172). ( ) (170), (171)
2 1+ ws cos s dws = , ds ws sin s

s = 1, 2.

(205)


2 1+ ws = Cs sin s +2

= const,

s = 1, 2.

(206)

, «» (172), (172) (170) dw2 2 = -(1 + w2 )cos 2 . (207) d3 (206)
2 2 C4 cos 2 = ± C4 - 1 - w2 ,

(208) (209)



1 dw2 2 = (1 + w2 ) d3 C4 (3 + C5 ) = C4 dw2 (1 +
2 w2

2 2 C4 - 1 - w2 .

,
2 2 ) C4 - 1 - w2

,

C5 = const.

(210)


216

. .

tg(3 + C5 ) = C4 w2
2 C4 2 - 1 - w2

,

C5 = const.

(211)

, , «» (172), arctg C4 w2
2 C4 2 - 1 - w2

± 3 = C5 ,

C5 = const.

(212)

, (17)--(21), (24)--(33) (147) . (38), (36), (37), (183), , (197)--(204), ( ) , , (206) (212). 3. (17)--(21), (24)--(33) (38), (147), (37) ( ), . .

4.4.
¨ +(b - H1 ) cos +sin cos - - [1 2 + 2 2 sin2 1 + 3 2 sin2 1 sin2 2 ] 1+ 1 +(b - H1 )1 cos + 1 ¨ cos 1+ 2 +(b - H1 )2 cos + 2 ¨ cos 1+ 3 +(b - H1 )3 cos + 3 ¨ cos b > 0, H1 > 0. cos2 sin cos2 sin cos2 sin sin = 0, cos
2

- (2 2 + 3 cos +21 2 sin cos +21 3 sin

sin2 2 )sin 1 cos 1 = 0,

1 - 3 2 sin 2 cos 2 = 0, 1 1 cos 2 +22 3 = 0, 1 sin 2

, ,

(213) , . . [52, 59, 60, 67, 72, 73]. 8, 3




217

, . T S3 {, 1 , 2 , 3 , ,1 ,2 ,3 } (214) S4 {, 1 ,2 ,3 }, , (139) , 3 0 1 0, 2 0, 3 0 (216) (215)

. , (213) (214) . , . 4. (17)--(21), (24)--(33) (38), (147), (37) (213). , = , 1 = 1 , 2 = 2 , 3 = 3 , b = -b , H1 = -H1 . . [52].


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218

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[44] . . // . . -. . 1, , . -- 2001. -- 5. -- . 22--28. [45] . . // . . -- 2001. -- . 37, 6. -- . 74--82. [46] . . // . . -- 2002. -- . 57, 1. -- . 169--170. [47] . . // . . -- 2004. -- . 40, 4. -- . 137--144. [48] . . so(4) â Rn // . . -- 2005. -- . 60, 6. -- . 233--234. [49] . . , , // . . -- 2005. -- . 403, 4. -- . 482--485. [50] . . // . . . -- 2005. -- . 69, . 6. -- . 1003--1010. [51] . . // . . . -- 2006. -- 3. -- . 45--57. [52] . . . -- .: , 2007. [53] . . // . . -- 2007. -- . 43, 10. -- . 49--67. [54] . . // . . . -- 2007. -- 3. -- . 187--192. [55] . . // . . -- 2007. -- . 62, 5. -- . 169--170. [56] . . : , , // . . . -- 2008. -- . 14, . 3. -- . 3--237. [57] . . , , // . . . -- 2008. -- . 72, . 2. -- . 273--287. [58] . . // . . -. . 1, , . -- 2008. -- 3. -- . 43--49. [59] . . , // . . -- 2008. -- . 418, 1. -- . 46--51. [60] . . // . . . . . -- 2009. -- . 65. , . -- . 132--142.




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