Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mce.biophys.msu.ru/archive/doc15793/doc.pdf Дата изменения: Mon Oct 29 13:22:40 2007 Дата индексирования: Mon Oct 1 20:43:56 2012 Кодировка:

.. , .. , .. , .. . . , . . , , , , . THE EFFECTIVE DIMENSION OF A PHASE TRAJECTORY AND THE PRINCIPAL MODES OF MOTION OF DYNAMICAL SYSTEMS Belega E.D., Rybakov A.A., Trubnikov D.N., Chulichkov A.I. Abstract. A method for describing the behavior of a dynamical system is proposal. The method based on the notions of the effective dimension of a phase trajectory and the principal modes. To visualize trajectory it is suggested displaying its effective-dimensional component. The method is applied to the internal dynamics of triatomic argon clusters. . :
330


.. . -- -10, 2002, .330 - 344

. , , . . - . , . , [1], . , - , , . , - , ( ), . , - . . , , , [2]. , , " ". ,
331


3. (II)

[3]. , ­ , ­ . [4,5]. , , , , . , -- : , , , [6]. ­ ( ). , , - . , Ar3 . , t [ 0, T ] x(t )

Rn n > 3 , . Rk Rn , k < n , , " ", 332


.. . -- -10, 2002, .330 - 344

: t [ 0, T

]



x(t ) x(t ) = xk (t ) + xn - k (t ) , xk (t ) Rk , xn - k (t ) Rk ­ x(t ) Rk t [ 0, T
T -1

Rk , t [ 0, T



]

xn - k (t )

]

"-

xn (t ) .
2



" (k ) = T


0

xn - k (t ) dt -

k - Rk . [7] , Rk ,

(k )
T





k = 1, ..., n , k em Rn S =

1 x(t ) x * (t )dt , T 0

m , m = 1, ..., k . xx* ( Rn Rn ) ­ () x Rn . , t j = 1, ..., N ,

Sii =

1 N


j =1

N

xi (t j ) xi (t j )

i, i = 1, ..., N . ( ) = min {k (k ) } Rk , 333


3. (II)

xk (t ) Rk

x(t ) Rn ,


; ( , ) ­ Rn , ek ­ S, ek = 1 , k=1,...n.

(

x(t ), ek

)

, ek Rn , t [ 0, T ] , k- -



x = ( q, p ) R2 n q Rn p Rn . n- Rn , , :

(

q q q (t ),ek ) ek Rn


T


q



ek ,

1 T

(
0

q q (t ), ek

)

2

dt kq , ekp ­
T

1 k- S = q (t )q (t )dt , T0
q


q

kq , 1q ... nq . q

t [ 0, T ] , en . , n-

, e1 Rn

Rn

(

p (t ), ekp ) ekp Rn -

,

kp


i =1

n

i

p

; ek

p

334


.. . -- -10, 2002, .330 - 344

­ k- S =

q

1 q (t )q (t )dt , T 0

T

kq , 1q ... nq .

, . , q(t ) =

{e } {e } ( e , q(t ) ) e
q k p k

q k

q k

k =1

p (t ) = q (t ) , p (t ) =




k =1

n

{e } {e }
q k p k

q q ek , q (t ) ek ,


-

,

kq =

1 T

T

(
0

q ek , q(t ) ) dt 2

kp =

1 T

T

(
0

ekp , p (t ) ) dt .
2

, , t1 , ..., t N ,

X

i, j

= xi (t ) , i = 1, ..., 2n , j = 1, ..., N
p

. S

S

p i ,i

=


j =1

N

X

i + n, j

X

i+ n , j

,

i, i = 1, ..., n ,

kp ,

k = 1, ..., n , , X i + n , j ,
i = 1, ..., n , j = 1, .., N ekp , k = 1, ..., n ­ . , (n=6), .
335


3. (II)

q(t ) R6 ,
e1 e2 S , ( ). , R6 , . , , q q p e1 e2 S , , p R6 , . , ,
q q
q

1p + 2p


i =1

6

.

i

p

, , ­ , 6- , : k , k=1,...,6, k- p (t ) R6 , t [ 0, T ] . [7]. , .
336


.. . -- -10, 2002, .330 - 344

, p (t ) R6 , () , ­ . . p (t ) R6 t [ 0, T

]



p (t ) =

(
k =1

6

p(t ), ekp ) ekp .

(1)



(

p (t ), ekp

)

(
p

p (t ), ekp ) ekp k. -


p

p6 (t ) ek R6 , ek , k ­ , k- , ( , ). , . , U (q ) = U 0 +

(Qq ,q ) 2

, U 0 = const , Q ­

, [8]. ,
337


3. (II)

q= p p =-Qq q (0) = q0 , p (0) = p0 .




q(t ) =


{b
k =1

n

k

cos k t + ck sin k t}ek ,

p (t ) = q(t ) =


{
k =1

n

-bk k cos k t + ck k sin k t}ek

(2)

k ­ Q, {ek } ­ -

, b1 , ..., bn , c1 , ..., cn : bk = ( ek , q0 ) , ck = ( ek , p0 ) k=1,...,n. q ekp = ek = ek , k=1,...,n, ,
q p

kq kp q

S S . , S

1T T (bk cos k t + ck sin k t )2 dt ek e*k k =1 0 ek , Sq =
n



kq =

1 (bk cos k t + ck sin k t ) 2 dt k=1,...,n. T 0
p
n

T

, S

-

1T 2 T k (-bk cos k t + ck sin k t )2 dt ek e*k k =1 0 ek ,

Sp =

338


.. . -- -10, 2002, .330 - 344



kp =



2T k

T


0

(-bk cos k t + ck sin k t ) 2 dt ,

k=1,...,n.

kq kp , ,

. q0 U (q ) , q0 Q (q ) ,

Qi , j ( q) =

d 2U ( q) , i, j = 1, ..., n dqi dq j

(3)

,

U (q ) = U (q0 ) +
lim
z 0

1 2 ( Q(q0 )(q - q0 ), (q - q0 ) ) + o( q - q0 ) 2

o( z ) = 0 . , z

q , [9], , (4) (2), . , () ,

(

p (t ), ekp ) ekp .

() (4) , . , , 339


3. (II)

, , .
.1

E=-2.5 : M=0 (. . 1, 2) M=0.8Mmax (. . 1, 2). ( , - ). . 1-2, , , M=0.8Mmax.

. 2

340


.. . -- -10, 2002, .330 - 344

, , E = 1.5. . 3, , p (t ) R6 , t [ 0, T ] Ar3 , e1 e2 . , 70% , , . 3 , (28% ) . 3 . , . , 0 T. [10], , . 3 , , . , , (4) . . . . 3 p (t ) R6 ,
q t [ 0, T ] , , e1q e2

q

q

( E=-2.5, ). . 3 , , 84% . [10]. 10% 341


3. (II)

. . 3 . , . . 3 E=-2.5 , . (. . 3 ). , . 3 , , , . , (M = 0.8) . , , E=-2.5 [1]. , , , .

342


.. . -- -10, 2002, .330 - 344

.3

, 343


3. (II)

. , . . , , , . 1. Milne T.A. and Green F.T. // J. Chem. Phys. 1967, v.47, pp.4095-4101. 2. Beck T.L., Leitner D.M., and Berry R.S.// J. Chem. Phys. 1988, v.89, pp.1681-1694. 3. Amitrano C. and Berry R.S. // Phys. Rev. E. 1993, v.47, pp.3158. 4. .., .., .., ..// . . 1997. . 355. 6. . 750. 5. Belega E.D., Trubnikov D.N., Lohr L.L. // Phys. Rev. A. 2001. 63. 043203. 6. Berry R.S. // I.J. of Quant. Chem. 1996, v.58, pp.657-670. 7. .., .., .., ..// . . . . . 2002. . 42. 12. . 1909. 8. .., .. // . .1. . .:. 1988. 9. ..// . ­ .:. 1984. 10. .., .., .., .. // . ­ .:. 1972., . 66.

344