Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ics.org.ru/upload/iblock/389/13-dvizhenie-dvukh-sfer-v-idealnoy-zhidkosti-i-uravneniya-dvizheniya-v-evklidovom-prostranstve-pervye-i_ru.pdf Äàòà èçìåíåíèÿ: Wed Oct 28 19:29:29 2015 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 00:30:18 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 17

. I. .
. . , . . , . .
, ; (. ) , 426034, , . , 1 E-mails: borisov@rcd.ru, mamaev@rcd.ru, ramodanov@mail.ru 12 2007 . . , , . . : . . , . . , . . , . I. . , , 2007, . 3, 4, c. 411­422.

: , , ,

A. V. Borisov, I. S. Mamaev,S. M.Ramodanov Motion of two spheres in ideal fluid. I. Equations o motions in the Euclidean space. First integrals and reduction
The paper deals with the derivation of the equations of motion for two spheres in an unbounded volume of ideal and incompressible fluid in 3D Euclidean space. Reduction of order, based on the use of new variables that form a Lie algebra, is offered. A trivial case of integrability is indicated. Citation: A. V. Borisov, I. S. Mamaev, S. M. Ramodanov, Motion of two spheres in ideal fluid. I. Equations o motions in the Euclidean space. First integrals and reduction, Rus. J. Nonlin. Dynamics, 2007, Vol. 3, No. 4, pp. 411­422.

Keywords: motion of two spheres, ideal fluid, reduction, integrability MSC 2000: 53A05, 15A21

, 2007, . 3, 3, . 411­422


412

. . , . . , . .

1.
, ( . [18]). , . [1, 2] ( [7]), . , , , [3, 9, 10, 12, 13]. (, , ) . [3] , 1843 . « ». ) , ) , ) . (., , [8]). 1863 1875 .. . ( ), . ( Shiotz) , Paris Electrical Exhibition 1881 . . ( . ) [6]. ( ) [14] , , . , , , , (. . , , , ). , . , . . 1871 «Philosophical Magazine» ( ), (, ) , , . , , , , -- , -- . [3] 1879 . -- , , . [3] ( [9]), .

, 2007, . 3, 3, . 411­422


. I

413

, , , . , , , , ms . ms m, . ms < m ( ) : d , , d , . [19], , , . . [6], « » («Kinetic buoyancy principle»), [20], . . , [10] , . ( , , [11]). [10] , , . , , , , . , [12]. 1895 . . . [13] , , , , , . , . . XIX . , . , , , , . [2] . . [4, 5]. , , , , , ( , ), . (, ) , (. [2]). , , , , . , , , , (. [17]). , , . , -

, 2007, . 3, 3, . 411­422


414

. . , . . , . .

, , . .

2.
[14], (, [14] . [2]). , , , « + », . , . . x = = (x1 ,x2 ,x3 ) q = (q1 ,q2 ,q3 ) -- O1 O2 . R1 R2 , -- m1 m2 . U V -- , V = = V + V n U = U + U n , V , U O1 O2 , V n , U n (. 1). « + » [1]:
2 2 2T = a1 Un + b1 U 2 + a2 Vn + b2 V2 +2c1 (U n , V n ) - 2c2 (U , V ).

(2.1)

ai ,bi ,ci , , s ; o(s-12 ) [1, 3]:
33 3R1 R2 a1 = m1 + 1 M1 1+ 2 s6 33 3R1 R2 a2 = m2 + 1 M2 1+ 2 s6 2 4 3 3 3 1 + R2 + 9R2 + R2 (R1 +64R2 ) 4 s2 4s4 16s6 2 4 3 3 3 1 + R1 + 9R1 + R1 (R2 +64R1 ) 4 s2 4s4 16s6

,

,

b1 = m1 + 1 M1 2 b2 = m2 + 1 M2 2 c1 = c2 =
3

1+ 1+

33 3R1 R2

s

6

1+ 1+ + +

2 3R2

s

2

+ +

4 6R2

s

4

+ +

6 11R2

s

6

,

33 3R1 R2

s 1+ 1+

6

2 3R1

s

2

4 6R1

s
8

4

6 11R1

s

6

,

33 R1 R2

33 R1 R2

s

4s

6

33 2 2 R1 R2 (R1 + R2 )

s

,

33 2R1 R2

s

3

33 R1 R2

s

6

33 2 2 3R1 R2 (R1 + R2 )

s

8

.

, 2007, . 3, 3, . 411­422


. I

415

. 1
3 -- ; Mi = 4 Ri .

, T x, q , x, q , ( T ). , , . . [1], ai ,bi ,ci , . « + » T = Ts + Tf , Tf Ts -- . U , V , Un , Vn -- U , V , . Ts = 1 m 2
1 2 2 Un + U + 1 m 2 2 2 Vn + V 2

3

.

Tf [2, 8]: Tf = - 2
S

ds, n

(2.2)

S S1 S2 = S , n -- S , -- , S . , = Un 1 + U 2 + Vn 3 + V 4 , 2 (4 ) -- , , () ()

, 2007, . 3, 3, . 411­422


416

. . , . . , . .

; 1 (3 ) -- , () , () . (2.2),
2 2 2 2Tf = A1 Un + B1 U + A2 Vn + B2 V2 +2C1 Un Vn - 2C2 U V .

, , B1 = -
S

2

2 ds, n

C2 = -

2

4
S

2 =- 2 n

2
2

S1 S

4 ds, n

B2 = -
S

4

4 ds. n

« + »
2 2 2 T = (A1 + m1 )Un +(B1 + m1 )U +(A2 + m2 )Vn +

+(B2 + m2 )V2 +2C1 Un Vn - 2C2 U V .

(2.3)

, Un U , : , Un . B1 ,B2 ,C2 , [2, §98]. Ox, O O1 (. 2). 2 , , 2 n = - cos , 2 n = 0, (2.4)

S

1

S

2

-- O1 x.

. 2

, ,
3 R1 cos

, µ0 = O1 x. (2.4), . µ1 = -µ
03 s
3 R2

3 2R1

2r

2

,(r -- O1 ),

H1 , -

O1 . (2.4), µ2 H2 , H1

, 2007, . 3, 3, . 411­422


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417

. . µ1 ,µ2 ,... f1 ,f2 ,... O1 : R2 f1 = s - s2 ,
3 R2 µ1 =- 3, µ0 s

f2 =

2 R1 ,... f1

3 R1 µ2 = - 3 ,... µ1 f1

s = |O1 O2 |.

( µi , 2 .) 4 µ0 , µ1 , µ2 , µ3 ,... s, s - f1 ,s - - f2 ,s - f3 ,... O1 ,
3 µ0 = 2R2 ,

µ1 = -

3 R1

s

3

µ0 ,

µ2 = -
2 R2

3 R2

f f2 = ,

1

3

µ1 ,

µ3 = -
2 R1

3 R1

(s - f2 )2 .
2

,... .

R2 f1 = s - s1 ,

f

f3 = s -

1

s-f

2 42 = (µ0 + µ2 + µ4 + ...) cos - 2 2 R1 , (2.4), B1 =
S
1

µ1 µ3 + 3 + ... R1 cos + ... 3 f1 f3

(2.5)

2 cos ds,

. B1 =
33 3R1 R2 3 (µ0 +3µ2 + ...) = 2 R1 1+ + ... 3 3 3 sf1 -6

=

33 3R1 R2 + o(s = 1 M1 1+ 2 s6

),

M1 =

3 4R1 3

B2 : B2 = 1 M2 2 1+
33 3R1 R2 6

s

+ o(s

-6

),

M2 =

3 4R2 . 3

, , 4 , (2.5), C2 = - 2 4
s
1

R3 R3 2 ds = (µ1 + µ3 + ...) = 2 1 3 2 + o(s n s

-3

).

B1 ,B2 ,C2 , (2.3) (2.1), s-6 : bi = Bi + mi , c2 = C2 .

, 2007, . 3, 3, . 411­422


418

. . , . . , . .

(2.1) (2.3). A1 ,A2 ,C1 . . , , - -- . [1, 3, 9].

3.
H (x, q , y , p), p = T , y = T , E (3), SO(3) . , e(3): Pi = yi + pi , Mi = p
i+1 qi+2

q

x

-p

i+2 qi+1

+y

i+1 xi+2

-y

i+2 xi+1

, i 1, 3.

(3.1)

, E (3) 4 . . C (., , [17]), , (3.1). . (. [18]) , ( , ). : 1. f1 = s2 = 2. f2 =
i=1 3 i=1

(xi - q1 )2 ;

3

y,

2 i

3

f3 =
i=1

p2 ; i

3.
3 3 3

f4 =
i=1

(xi - qi )yi ,

f5 =
i=1

(xi - qi )pi ,

f6 =
i=1

y i pi .

[15], = dpi dqi pi , qi sp(n). f1 ,... ,f6 sp(n) ( . ). , .

, 2007, . 3, 3, . 411­422


. I

419

, ( , ). , :
2 2 H = a2 (f1 f2 - f4 )+ a1 (f1 f3 - f5 )+ c1 (f5 f4 - f6 f1 )- 2 2 - c2 f5 f4 + b1 f5 + b2 f4 ,

ai = bi = bi
2

ai
1

,

c1 = 2c2
2

2c1
1

,

1

= 2s2 (a1 a2 - c2 ) > 0, 1 i = 1, 3.

,

c2 =

,

2

= 2s2 (b1 b2 - c2 ) > 0, 2

f1 ,... ,f6 {fi ,fj } = - 0 -4f4 4f5 -2f1 2f1 2f5 +2f 4f 0 0 2f 2f 0
4

2 6

4

-4f 0 0 -2f -2f 0

5

6 3

2f1 -2f1 -2f2 -2f6 2f6 2f3 0 -f 5 - f f5 - f4 0 -f 6 + f 2 -f 3 + f

4 6

2f5 - 2f4 0 0 . f6 - f2 f3 - f6 0

(3.2)

4, {x, y }, {q, p} 12; , ( ) 4 . , (3.2). , [17], , ( ). P 2 (M , P )2 : P = f2 + f3 +2f6 ,
2

f D = (M , P ) = f f
2 2

3 6 5

f f f

6 2 4

f5 f4 . f1

(3.3)

[16], (3.2) x = (S1 ,S2 ,S3 ,N1 ,N2 ,N3 ). f1 = 22 (S2 - S3 ), N f2 = - 22 (S3 + S2 ) - 3 N2 + 3 , 8 f4 = N1 +2S1 , 22 = 38 .
1

N f3 = - 22 (S2 + S3 )+ 3 N2 + 3 , 8 f5 = -2S1 + N1 , 22

N f6 = 22 (S2 + S3 )+ 3 , 8

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420

. . , . . , . .

N2 N 0 S3 S2 -1 2 2 N2 N -S3 0 -S1 - -1 2 2 N2 N1 -S2 S1 0 - {xi ,xj } = 2 2 N1 N2 N2 0 -N3 2 2 2 - N2 N1 - N1 N3 0 2 2 2 0 0 0 0 0 P2 =

: 0 0 0 0 0 0

(3.4)

N3 2 2 2 2 2 , D 2 = 2N3 (S3 - S1 - S2 ) - 2N2 S1 N1 + N2 (S3 - S2 )+ N1 (S2 - S3 ). (3.5) 2 N1 ,N2 ,N3 l6 , (3.2). , l6 so(2, 1) .

4.
(l, g, L, G) , [7], [17]. {S1 ,S2 ,S3 }, . [17, 18] ( -- , , , ). - ( , , ) , , , , . so(2, 1) S3 = L. (3.4), {l, L} = 1 {L, S1 } = - S1 = -S2 , l {L, S2 } = - S2 = S1 . l

S1 = A sin l, S2 = A cos l. A , {S1 ,S2 } = = S3 , S1 = L2 - G2 sin l, S2 = L2 - G2 cos l, S3 = L.

2 2 2 G2 = S3 - S1 - S2 so(2, 1) 1 (r â (y - p))2 . 16

{G2 ,N1 } = - {G2 ,N2 } = -

N1 · 2G = S1 N1 + N2 S2 - N2 S3 , g

N2 · 2G = -S1 N2 + N1 S2 + N1 S3 , g

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421

, (3.4), N1 = F (L, G) · g g L2 - G2 - L sin cos l - cos sin l 2 2 2 2 G g g L2 - G2 sin sin l - cos cos l 2 2 2 2 G L2 - G2 ) · (D 2 - 4P 2 G2 ) G ,

N2 = F (L, G) ·

L-

,

F (L, G) =

(L +

.

L, G, l, g , , , . D p . . P 2 = 0 . , (3.3) , D 2 = 0, N1 = N2 = 0, , , 2 2 2 S1 ,S2 ,S3 . D 2 = S3 - S1 - S2 , . . . « » ( -1312.2006.1), ( 05-01-01058), « » ( 07-01-92210) INTAS ( 04-80-7297).


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