Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mce.biophys.msu.ru/archive/doc15532/doc.pdf Дата изменения: Wed Oct 17 18:35:18 2007 Дата индексирования: Mon Oct 1 20:25:12 2012 Кодировка:

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. ., . . -- -- 2005, . 2, . 606 ­ 607 Golo V. L., Sinitsyn D. O. -- MCE -- 2005, vol. 2, p. 606 ­ 607

. . (. [1]), , , . . , ( ) « ». «» (« »), , , , (. 1). , , , , .
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6. Part 6. Mathematical modelling researches and methods

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. , , . 2: , «» , . , , . . .

II.

r r ( x ) = 0 , ( x ) : 3 r ( x ) = xi2 + i xi4 - 1 , i ­ .

(

)

i =1

(. [1]): r && = . x (1) r x rrr L=rв p. (1) . i , : r r r x = cos ( t + ) e1 + sin ( t + ) e2 , rrr e1 , e2 , e3 , r L , 2 =
609

.

r e3 r & x 2 ,


6. Part 6. Mathematical modelling researches and methods

r & i , 2 = x 2 = L2 . , .. , :

&3 L1 = 4 &3 L2 = 4 &3 L2 = 4

L2 L3 L2 L3 L1 L2 LL2 1 L2

2 ( 3 - 2 ) L1 + 3 L2 - 2 L2 2 3 2 - 3 L1 + ( 1 - 3 ) L2 + 1 L2 2 3 2 2 L1 - 1 L2 + ( 2 - 1 ) L2 2 3

(2)

. (. [2], [3])

{Li , L j } =


k =1

3

ijk Lk

( ijk ­ -, (ijk ) , i, j , k ­ 1 3, )

H=

32 L 16


i =1

3

i i - 1 . L

L



2

L2 H , . .

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(2), .. r r L (t ) L0 . : r 1 L0 :

610


. ., . . -- -- 2005, . 2, . 606 ­ 611 Golo V. L., Sinitsyn D. O. -- MCE -- 2005, vol. 2, p. 606 ­ 611

. L10 = 0, L20 = 0, L30 0; . L10 = 0, L20 0, L30 = 0; . L10 0, L20 = 0, L30 = 0;

r 2 L0 :
. L10 = 0, L20 0, L30 0, 3 L2 0 - 2 L2 = 0; 2 30
2 . L20 = 0, L30 0, L10 0, 1 L20 - 3 L10 = 0; 3 2 . L30 = 0, L10 0, L20 0, 2 L10 - 1 L2 = 0; 20

3 L0 :
L10 0, L20 0, L30 0,

r

1 2 - 2 3 + 3

2 L10

=
1

1 2 + 2 3 - 3

L2 20

=
1

-1 2 + 2 3 + 3

L2 30

1

1 i , 2., 2., 2. ­

2 + 2 3 + 31 > 0. , , (2) . , :

2 3 2 1 2 1 - 1

> 0, 31 > 0 2 > 0 , 3 ­ 1
- 2 3 + 31 > 0, + 2 3 - 31 > 0,

1

. 2 > 0; 1 . 2 3 > 0; . 31 > 0;

2

. 2 - 2 3 + 31 > 0; 1 . 2 + 2 3 - 31 > 0; 1 . - 2 + 2 3 + 31 > 0; 1

3

i .

611


6. Part 6. Mathematical modelling researches and methods

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612


. ., . . -- -- 2005, . 2, . 606 ­ 613 Golo V. L., Sinitsyn D. O. -- MCE -- 2005, vol. 2, p. 606 ­ 613

2 - 2 3 + 31 > 0, 1 2 + 2 3 - 31 > 0, 1 - 2 + 2 3 + 31 > 0. 1
II 5 4 ; , (3) . III 3 2 ; i : ) 2 3 > 0 2 0; ) 31 > 0 1

(3) i :

2 3 0; ) 2 > 0 1

31 0.
IV 2 1 ; i : ) 1 = 0 2 3 0; ) 2 = 0 31 0; ) 3 = 0 2 0. 1

. 4: I ­ IV.

ei



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6. Part 6. Mathematical modelling researches and methods

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. ., . . -- -- 2005, . 2, . 606 ­ 615 Golo V. L., Sinitsyn D. O. -- MCE -- 2005, vol. 2, p. 606 ­ 615

.





,





: 1. .. , . 1,7. .: , 2004. 2. ... . .: , 2003. 3. .., ... . , , . I. ­ : « », 1999. 4. .., .., ... : . .: , 2001. VISUALIZATION OF GEODESICS ON A WEEKLY DEFORMED SPHERE Golo V. L., Sinitsyn D. O.
(Russia, Moscow)
We aim at visualizing geodetic lines on a surface, which is close to the standard sphere. We offer a means of the asymptotic representation of geodesics based on the fact that a short segment of the geodesic can be well approximated by an appropriate "great circle" of the sphere. We apply the method to a surface generated by a specific deformation of the sphere. We obtain a graphic description of geodesics with the help of an auxiliary Hamiltonian system. The main point about the auxiliary system is that it is much simpler than the initial one for geodesics, and even admits of the exact solution. The topology of solutions to the system is described by the separatrix graph; the vertices corresponding to geodesics close to great circles and the edges (separatrixes) to geodetic coils joining them. The variation of parameters determining the deformation of the sphere generates several topological types of the graphs.

615