Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://shamolin2.imec.msu.ru/art-107.pdf Äàòà èçìåíåíèÿ: Fri Oct 14 09:26:03 2011 Äàòà èíäåêñèðîâàíèÿ: Mon Oct 1 20:33:34 2012 Êîäèðîâêà: IBM-866

. xx (2011). . 84í97 517.9; 531.01

,
c 2011 . . .

. (4D) , . (2D) (3D) , , , (. [1, 14, 15, 18, 23, 28í30, 38í41, 43]).



1. 2. 3. 4.



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84 85 87 91 95

1.





. , ë¨ ( ) . , , , . , , , . [18, 38] , , ( , ) . , , () . [38í40] () ,
( 08-01-00231-).
ISSN 1512í1712 c , 2011




85

. , , () . , , -- . . 2.


2.1. . [38í40]. (v, ) -- , -- , I , m -- - , ( , . , [38]) : v cos - v sin - v sin + 2 = Fx , v sin + v cos +v cos - = 0, I = yN (, ,v )s(), Fx = -

(2.1)

s( )v 2 , > 0. m T, (2.2) VC const (VC -- , . [38]), (2.1) Fx , , . s yN yN (, ,v ) = A sin - h1 , h1 > 0, A, B > 0, v = 0, v , ( ) . (2.2), (2.1) : s() = B cos , W1 = R1 {v }½ S1 { mod 2 }½ R1 { }; + v = v (, ), = - + n2 sin cos2 + 2 sin - h1 B cos2 , 0 I = n2 sin cos - n2 sin2 cos + 3 cos + h1 B 2 sin cos - h1 B cos , 0 0 I I h1 B 2 2 2 (, ) = - cos + n0 sin cos - sin cos , I = v , n2 = AB /I , § = v . 0 (2.3) (2.4)


86

. .

2.2. . (2.3), (2.4) (2.4). 1. (2.3), (2.4) , , -- , . . , (.. ). , , , , . , (2.2) (2.1) Fx 0, , 0 (v, , ) = v 2 + 2 2 - 2 v sin = V
2 C

(2.5)

. (2.3), (2.4) , , 1 (v, , ) = v 2 (1 + 2 2 - 2 sin ) = V
2 C

(2.6)

. (2.6) (2.3), (2.4) , , VC = 0 2 VC . v2 = 1+ 2 2 - 2 sin W1 (2.3), (2.4) , (2.6) ( ) (2.3), (2.4) . () (2.3), (2.4). {(v, , ) W1 : VC = const}. n0 h1 B AB , = n0 , = sin , n2 = H1 = 0 In0 I (2.4) v = v (, ), = - + sin cos2 + 2 sin - H1 cos2 , = sin cos - sin2 cos + 3 cos + H1 2 sin cos - H1 cos , (, ) = - 2 cos + sin2 cos - H1 sin cos . (2.6) (2.3) ( (2.7)). (2.8) - [ 2 - 2 ]+ H1 [ - 1] d = . d - + + [ 2 - 2 ] - H1 [1 - 2 ] (2.7) (2.8)




87

= t , d = td + dt

: d = a 2 ( t) + a 3 ( t) 3 , a 1 ( t) dt a1 (t) = -(1 + H1 )t2 +( + H1 )t - 1, a2 (t) = (1 + H1 )t - , a3 (t) = - H1 t - t2 . p = 1/ 2 , dp = c 1 ( t) p + c 2 ( t) , dt 2t(1 + H1 ) - 2 2 - 2H1 t - 2t2 , c 2 ( t) = . c 1 ( t) = (1 + H1 )t2 - ( + H1 )t +1 (1 + H1 )t2 - ( + H1 )t +1 p1 ( ): 1) D = ( - H1 )2 - 4 > 0: 2(1 + H1 )t - ( - H1 ) - D 2 p1 = k [(1 + H1 )t - ( + H1 )t +1] § 2(1 + H1 )t - ( - H1 )+ D 2) D = ( - H1 )2 - 4 < 0: p1 = k [(1 + H1 )t2 - ( + H1 )t +1] § exp arctg 3) D = ( - H1 )2 - 4 = 0: p1 = k (1 + H1 )t2 - ( + H1 )t +1 § exp 2L1 1+ H1 ‘ 1 , L1 = ‘ 1. 1+ H1 2(1 + H1 )t - ( + H1 ) -D ;
H1 - D

;

, , , k t, . . 3.


3.1. . m . (v, , ) -- , {x , y , z } -- , I1 ,I2 ,I2 -- , , (. [38]) : v cos - v sin +y v sin sin - z v sin cos + (2 +2 ) = Fx , y z sin sin +z v cos - x v sin sin - x y - v sin cos + v cos cos - v v sin sin + v cos sin + v sin cos +x v sin cos - y v cos - x z + x = 0, hy , I2 y +(I1 - I2 )x z = -AB v 2 sin cos sin - v hz , I2 z +(I2 - I1 )x y = AB v 2 sin cos cos - v z = 0, y = 0, (3.1)


88

. .

Bv 2 cos , A, B , , h > 0. m T, , (2.2), (3.1) Fx (T - B cos )v 2 /m; , (2.2), (3.1) . , (3.1) . , , : Fx = - x = x0 = const . , .. , h = 0. : z1 = y cos +z sin , zi = Zi v, i = 1, 2,
x0 x0



(3.2) = 0;

z2 = -y sin +z cos , = v , = v , v = vv . (3.3)

(3.1) (2.2)

= 0 :



v = v (, Z1 ,Z2 ), 2 2 2 2 = -Z2 + n0 sin cos + (Z1 + Z2 )sin , 2 cos , Z2 = n2 sin cos - Z2 (, Z1 ,Z2 ) - Z1 0 sin Z = -Z (, Z ,Z )+ Z Z cos , 1 1 2 12 1 sin cos , = Z1 sin
2 2 (, Z1 ,Z2 ) = - (Z1 + Z2 )cos + n2 sin2 cos , 0

(3.4)

(3.5) AB . I2

n2 = 0

3.2. . ( ) ( ) (3.3)í(3.5) . , (2.2), (3.3)í(3.5) ( (3.2)) . , VC = v cos , v sin cos - z , v sin sin + y . (3.1) (3.2) (x0 = 0) (2.2): 2 2 2 (3.6) v 2 - 2vz2 sin + 2 (z1 + z2 ) = VC 0 . , (3.6), 2, (3.3)í(3.5): 2 2 2 (3.7) v 2 (1 - 2Z2 sin + 2 (Z1 + Z2 )) = VC 0 , (3.7) v : v2 =
2 VC 0 2 2. 1 - 2Z2 sin + 2 (Z1 + Z2 )

(3.8)




89

, (3.8) (3.3))í(3.5) -- . (3.4)
2 2 = -Z2 + b sin cos2 + b(Z1 + Z2 )sin , 2 2 Z2 = sin cos + bZ2 (Z1 + Z2 )cos - bZ2 sin2 cos - Z 2 1

(3.9) cos 2 2 , Z1 = bZ1 (Z1 + Z2 )cos - bZ1 sin2 cos + Z1 Z2 sin b = n0 , : n0 . , = sin , (3.9)) :
2 2 2 dZ2 + bZ2 (Z1 + Z2 ) - bZ2 2 - Z1 / , = 2 )+ b (Z 2 + Z 2 ) d -Z2 + b (1 - 1 2

cos , sin

2 2 dZ1 bZ1 (Z1 + Z2 ) - bZ1 2 + Z1 Z2 / = 2 2. d -Z2 + b (1 - 2 )+ b (Z1 + Z2 )

(3.10)

(3.10) du2 = d du1 = d (3.11)

uk , k = 1, 2,, uk = Zk . 1 - bu2 + u2 - u2 2 1 , -u2 + b 2 (u2 + u2 )+ b(1 - 2 ) 1 2 (3.11) 2u1 u2 - bu1 . -u2 + b 2 (u2 + u2 )+ b(1 - 2 ) 1 2 : (3.12)

1 - bu2 + u2 - u2 du2 2 1 = . du1 2u1 u2 - bu1

, , (3.12): d 1 - bu2 + u u1
2 2

+ du1 = 0,

(, Z1 ,Z2 ) :
2 2 Z1 + Z2 - bZ2 + 2 = const . Z1

(3.13)

(3.13), , (3.4) , :
2 2 Z1 + Z2 - bZ2 sin +sin2 = const . Z1 sin

(3.14)

, (3.14), (3.11) : 2 - 2bu2 +2u2 - C1 U1 (C1 ,u2 ) du2 2 = , d -u2 + b - 2b 2 + b 2 (C1 U1 (C1 ,u2 )+ bu2 ) U1 (C1 ,u2 ) = : (b - u2 ) + b 3 (C1 U1 (C1 ,u2 )+ bu2 - 2) d = . du2 2 - 2bu2 +2u2 - C1 U1 (C1 ,u2 ) 2 (3.16) C1 ‘
2 C1 - 4(u2 - bu2 +1) 2 , 2

(3.15)


90

. .

(3.16) ( (3.15)) : 2(u2 - b)p - 2b(C1 U1 (C1 ,u2 )+ bu2 - 2) dp , = du2 2 - 2bu2 +2u2 - C1 U1 (C1 ,u2 ) 2 p= 1 . 2 (3.17)

, (.. ). (3.17) C2 ; . (3.3)í(3.5) (.. , ) , , Z1 / d , = 2 2 d -Z2 + b (Z1 + Z2 )+ b (1 - 2 ) d u1 = 3 (u2 + u2 )+ b (1 - 2 ) d -u2 + b 1 2 2u1 u2 - bu1 du1 , = 2 (u2 + u2 )+ b(1 - 2 ) d -u2 + b 1 2 (3.19) (3.18)

(3.11). (3.18), (3.19) : du1 = 2u1 - . (3.20) d (3.12) (C1 -- ) (3.20), , du1 =‘ d ‘ du b2
2 1 1

b2 - 4(u2 - C1 u1 +1); 1

(3.21)

, , (3.21), - 4(u - C1 u1 +1) 1 arcsin 2 = + C3 , C3 = const . (3.22)

(3.22) ( ) u1 -
2 C1 +( b2 2 b2 -

4)

.

(3.23)

(3.23) : cos2 [2( + C3 )] =
b u2 - 2 G1 2

u

2 1

,

(3.24)

G1 = [u2 - bu2 ]2 +2[u2 - bu2 ][u2 +1]+[u2 +1]2 + b2 u2 . 2 2 1 1 1 cos2 [2( + C3 )] = (Z2 - sin )Z1 2. (Z2 - sin )2 + Z1

, b = 2, (3.24)

= Z2 - sin Z1

= 1 , 1/2. , .




91

2. (3.3)í(3.5) , , . , , . 4.


4.1. . m , . , ; : Dx1 x2 x3 x4 (4.1) diag{I1 ,I2 ,I2 ,I2 }, diag{I1 ,I1 ,I3 ,I3 }. (4.2) Dx1 x2 Dx3 x4 -- . 4.2. so(4). , N S D -- : DN = R() ( [18, 38í40]). (4.1) vD D Dx1 . (4.2) . () S S = s()sgn cos § v 2 , |vD | = v,

s -- , , (. [18, 42]). R S , , [18] ( . . ): R = R() = A sin , S = Sv () = Bv 2 cos ; A, B > 0. -- , so(4), , so(4), (. [3, 6í9, 11í17, 19, 20, 25]): + +[, + ] = M, = diag{1 ,2 ,3 ,4 }, 1 = (-I1 + I2 + I3 + I4 )/2, ..., 4 = (I1 + I2 + I3 - I4 )/2, M -- , R4 , so(4), [§, §] -- so(4). so(4) 5 - 3 0 - 6 6 0 - 4 2 , - 5 4 0 - 1 3 - 2 1 0 (4.3)


92

. .

i , i = 1,..., 6, -- so(4). , , i, j = 1,..., 4 i - j = I j - I i . R4 ½ R4 - so(4), R4 so(4). 4.3. R4 . C , mwC = F, (4.4) wC = wD +2 DC + E DC, wD = vD +vD , E = , F -- , ( F = S), E -- . 4.4. . (4.1). . , Dx1 ( (4.1)) , C . , . (2.2)), .. . 4.5. (4.1). (2.2) . (0,x2N ,x3N ,x4N ) -- N Dx1 x2 x3 x4 , {-S, 0, 0, 0} -- , 0 x2N x3N x4N , -S 0 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 .

(4.3) (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
0 1 = 1 , 0 2 = 2 , 0 4 = 4 . 3N 2N

(4.5)

S.

, (4.1) (4.5) : (4.6)




93

:
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 , ë¨ T S3 ( (2.2) (4.6), ) ( = DC ):
2 2 3 v = cos n2 v 2 sin2 - (z1 + z2 + z3 ) , 0 2 2 2 2 3 = -z3 + n0 v sin cos + sin (z1 + z2 + z3 )/v , z = n2 v 2 sin cos - (z 2 + z 2 )ctg , 3 0 1 2 2 z = z2 z3 ctg + z1 ctg ctg 1 , 2 z1 = z1 z3 ctg - z1 z2 ctg ctg 1 , 1 = z2 ctg , 2 = -z1 ctg csc 1 .

(4.7)

(4.8)

(4.9)

(4.7)í(4.9) (4.8), (4.9), , , (4.8). , , . z=
2 2 z 1 + z2 ,

z =

z2 , z1

z = n0 vZ,

zk = n0 vZk ,

k = 1, 2, 3,

z = Z ,

n0 v ,

(4.7)í(4.9) (b = n0 , [b] = 1): v Z Z Z = v (, Z, Z3 ),
2 (, Z, Z3 ) = b cos [sin2 - (Z 2 + Z3 )], 2 2 2 3

(4.10) (4.11)

= -Z3 + b sin cos + b sin (Z + Z ),
3

= sin cos - Z 2 ctg - Z3 (, Z, Z3 ), = ZZ3 ctg - Z (, Z, Z3 ), =Z
2 1+ Z ctg ctg 1 ,



1 = 2 = -

ZZ

2

1+ Z Z1

ctg ,
2

(4.12)

1+ Z

ctg csc 1 .

(4.13)


94

. .

, (4.8) : (4.11) -- , (4.12) (, ) -- . , (4.11), -- (4.12) , ë¨ (4.10) (4.13). , (4.11) T S2 . 4.6. . (4.10)í(4.13)
2 2 v 2 (1 - 2bZ3 sin +(Z 2 + Z3 )) = VC ,

(4.14)

(2.2). v . (4.11) , , , ( ) (. [18, 21í43]):
2 Z 2 + Z3 - bZ3 sin +sin2 = C1 = const, Z sin G(Z, Z3 , sin ) = C2 = const .

(4.15) (4.16)

(4.12)
2 1+ Z = C3 = const (4.17) sin 1 , , , 2 , cos 1 (4.18) = sin{C3 (2 + C4 )}, C4 = const . ‘ 2 C3 - 1 , sin sin 1 , , (v, , 1 ,2 ) , sin = 0 sin 1 = 0 () .

3. (4.7)í(4.13) (4.14)í(4.18), , -- ( ). 4.7. . [18, 21í43] , . , , (M 0) (., , . . [2, 3], . . [4, 5], . . [16] ; , ). , , so(4) ½ R4 . , ë ¨ . . . . . . . , . . . (. [10]).




95

1. . ., . ., . . // . . . ë ¨. -- . . . 23, 2007. -- . 34. 2. . . // , 287(5), 1986, 1105í1108. 3. . . n , // , 272(6), 1983, 1364í1367. 4. . . í // , 270(5), 1983, 1094í1097. 5. . . so(4) // , 270(6), 1983, 1298í1300. 6. . ., . . Rn // . . . " ". -- . . . -- 2001. -- . 7. -- . 1. -- . 315. 7. . ., . . Rn // . -- 2001. -- . 380. -- 1. -- . 47í50. 8. . ., . . Rn // . -- 2002. -- . 383. -- 5. -- . 635í637. 9. . ., . . Rn // . . í. . 1. . . -- 2003. -- 5. -- . 37í41. 10. . ., . . // . . . 23, 2007. -- . 5í6. 11. e . ., . . Rn // . . . ë ¨. -- . . . 23, 2007. -- . 24í25. 12. . ., . . Rn // . . . ë ¨. -- . . . 23, 2007. -- . 30. 13. . ., . . ní // . . . ë ¨. -- . . . 23, 2007. -- . 31. 14. . ., . ., . . : , , // . . . " ". -- . . . -- 2001. -- . 7. -- . 1. -- . 301. 15. . ., . ., . . : , , // . . . ë ¨. -- . . . 23, 2007. -- . 16. 16. . . n- // . ., 10(4), 1976, 93í94. 17. . ., . íí () I // . ., 15(3), 1981, 54í66. 18. . ., . . // . . 1. . , 1989, 3, 51í54. 19. . ., . . . í .: - " 1995. í 448 . 20. . ., . . // . . , 29. .: , 1986. 3í80. 21. . . // Book of Abs. Third Int. Conf. "Differential Equations and Applications SaintPetersburg, Russia, June 12í17, 2000; í , 2000, . 198.


96

. .

22. . . // . . . . . . (, 21í 26.08.2000). -- : . . ., 2000. . 196í197. 23. . . , , // . . V . . " "(--2000) (, , 05í13.09.2000). -- , 2000, . 169. 24. . . , // . . . . . . (, 12í14.09.2000). -- : í " 2000. -- . 294í295. 25. . . // . -- 2000. -- . 375. -- 3. -- . 343í346. 26. . . // . . . " ". -- . . . -- 2001. -- . 7. -- . 1. -- . 309. 27. . . , // (Dynamical Systems Modelling and Stability Investigation). . . (22í25.5.2001): Thes. of Conf. Rep. -- Kyiv, 2001. -- C. 344. 28. . . , , // . . VIII . . . . (, 23í29.08.2001). -- : , 2001. -- . 599í600. 29. . . , , // . . . . . . (, 01í06.07.2002). -- : . . ., 2002. . 142í144. 30. . . so(4) ½ R4 // . . . . ë ¨ (í2005), , 27 --2 2005 . : í ë-¨, 2005. í . 97í98. 31. . . so(4) ½ R4 // . . íí . 60. --- . 6, 2005. íí . 233í234. 32. . . // . . . . . . . . , 10í15.07.2006. -- : . . í, 2006. -- . 226í228. 33. . . // . . . ë ¨. -- . . . 23, 2007. -- . 21. 34. . . , // . . . ë ¨. -- . . . 23, 2007. -- . 27. 35. . . --, // . . . ë ¨. -- . . . 23, 2007. -- . 37. 36. . . // " í2007": . . . ., , 4í8 2007 . -- .: -. . í, 2007. -- . 178. 37. . . // . . . . " . 70í . . , 20í24 . 2007 ., . -- .: , 2007, . 110í112. 38. . . . í .: , 2007. 39. . . , // , 364(5), 1999, 627í629. 40. . . , // . , 1997, 2, 65í68.




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41. Shamolin M. V. Methods of Analysis of Dynamics of a 2Dí 3Dí or 4Dírigid Body With a Medium, In: Abst. Short Commun. Post. Sess. Of ICM'2002, Beijing, 2002, August 20í28; Higher Education Press, Beijing, China, p. 268. 42. Shamolin M. V. 4D rigid body and some cases of integrability, In: Abstracts of ICIAM07, Zurich, Switzerland, June 16í20, 2007; ETH Zurich, 2007, p. 311. 43. Shamolin M. V. The cases of integrability in 2Dí, 3Dí and 4Dírigid body, In: Abstr. of Short Commun. and Post. of Int. Conf. "Dynamical Methods and Mathematical Modelling Valladolid, Spane (Sept. 18í22, 2007); ETSII, Valladolid, 2007, p. 31.

. . . . . , E-mail: shamolin@imec.msu.ru