1. (1137).

2. (1139).

3. (1141).

4. (1142).

5. (1145).

6. (1147).

7. (1149).

8. (1152).

9. (1154).

10. -

(1157).

11. (1160).

12.

(1162).

(1166).

1.

. , - . .

- ( ) , (., , [1 4]). , - , . - . , ( ) - ,, (. [5]). - ( , , ..) . , - (., , [6 8]), - (, ..) .

- - - (). - . .

- .

.. , .. . . .. ,117334 , . 2, E-mail: zakharov@itp.ac.ru, kuznetso@itp.ac.ru

5 1997 .

1997 . 167, 11

.. , ..

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

PACS numbers: 52.30.q, 52.35.Ra, 52.55.Fa

# .. , .. 19971 , . 167, 11

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

-, "" . , , , . - ( ), . , - -. , - () . ,, -, .. [9, 10] (. [11]), .. [12]. - [13, 14]. [15], [16].

, (., , [17]). [18]. 1932 .. [19], .. [20] . 1952 . .. [21] - . . (., -, [22]).

, , [23]. (.. ) [25]. 1966 . [24]. . - [26, 27], [27] - . - [28], , [29].

- , - .

[30,31], - ( ) [32]. , , [30]. - , . - 1970 . [33]. - [34], - (., , [35]). - , [31] , . - - . -, (.. -), , , [30], , - .

. - , . , - . [39], [38], , - , - [40], - . , - .

[1] - , 1986 . , . [1] . - . -, , , . ; - (relabeling group) - , . , . [41] 1982 ., 1938 ., 1960 . [44], (1967 .) [46] . , :

1138 .. , .. [ 1997

- [42] (. . 31 [54]), [17] () ( 8 [54]), [56, 57], - . , - .

, - - . [47, 1] - . ,.. , - , .

-. , ,, (. [48]). , - . , . - , .. , .

2.

. . - - qkt, pktk 1; . . . ;N,

_qk qHqpk ; _pk qHqqk

: 2:1

Hp1; . . . ; pN; q1; . . . ; qN -, .

, , pi qi R 2N. , , , 2p. , , , , , . c . - , q1; . . . ; qN -

p1; . . . ; pN. , - N- ( -) M, - RN. G T M - M. "" . , - .

(.,, [55]) .

, , , - . , , . G N, . , G - O. , Oij Oji. xi - . - , Oij

qOijqxk qOjk

qxi qOki

qxj 0 : 2:2

, detOij 6 0. , -

G, , G H

Oij _xj qHqxi : 2:3

, xi xi~x1; . . . ; ~xN, qx1; . . . ; xNq~x1; . . . ; ~xN1 6 0, (2.3) -. O :

~Olm qxiq~xl Oijqxjq~xm

:

, - , . ( detOij 0). - (2.2)

Oij qAiqxj qAjqxi

: 2:4

Aix "" Oij. (2.3) , dS 0,

S Ai _xi H dt : 2:5

1*

. 167, 11] 1139

(2.5) , Oij , .. (2.4) G. , Ai - G, - . , .. (2.1)( ). G , , , (2.4) . Oij , (2.3)

_xi Rij qHqxj : 2:6

Rij Rji , Oij: , (2.2) -

Rimq

qxmRjk Rkm qqxm Rij Rjm

qqxm

Rik 0 : 2:7

R A B, G:

fA;Bg X

RijqAqxi

qBqxj

: 2:8

Rij

fA;Bg fB;Ag ;

(2.7) - fA;Bg;C fB;Cg;A fC;Ag;B 0 : 2:9 Oij, Rij . R . , gij . , (2.7).

O 0 II 0

g I

, I .

R. - .

detRik 0, (2.3) -. xai a 1; . . . ; k - Rij (.e. xiRij 0). (2.6) -

xai _xi 0 ; a 1; . . . ; k : 2:10

, R, (2.10)

f ax1; . . . ; xn const ; a 1; . . . ; k :

xai

xai qf a

qxi:

f a . (2.7) , k f a

. , - . G- (2.6) ( ). - . , - .

, Rij

Rij eij;mxm : 2:11

(2.7) , eij;m

eik;m ejm; l eji;m ekm; l ekj;m eim; l 0 ;

.. L. xi, xj, ,

fxi; xjg Rij eij;m xm : 2:12

, G L.

Rij, , . (2.10) . L, L, - l . L l . - (2.10) l , (2.12) L l . (. [36, 37]) .

, . - , , - - . .

1140 .. , .. [ 1997

, R xi . " R-", , . .

3.

- - . , , .- . - , - . , , -, [19, 20],, . , , , , . -.

qr; t - pr; t, - :

qpqt dH

dq;

qqqt dH

dp: 3:1

H p q.

H X1n0

Xnk0

Gkn r1; . . . ; rk; rk1; . . . ; rn

pr1 . . . prkqrk1 . . . qrn dr1 . . . drn : 3:2

p q . - Gkn (ri rj). , H0

H0 12

Ar r 0 pr pr 0 2Br r 0 prqr 0

Cr r 0qrqr 0 dr dr 0 : 3:3 H0 - .

-- :

pr 12p3=2

pk expikr dk ; pk pk ;

qr 12p3=2

qk expikr dk ; qk qk :

(3.3)

H0 12

Ak pk p

k 2Bk pkqk Ckqkqk

dk :

- :

Ak Ak Ak ; Ck C k Ck ;Bk B1k iB2k B1k iB2k :

(3.1) k-

qpkqt dH

dqk;

qqkqt dH

dpk:

H0. - ,

o1;2 B2k AkCk B21k

q:

-,

AkCk B 21k > 0 ; 3:4

, , . , , , -. - (3.4) . -

Br Br

B2k 0 ; o2k AkCk B21k :

pk qk ak ak:

pk Ukak U k a k Uk Uk ;qk Vkak V k a k Vk Vk : 3:5

H0

H0 oka k ak dk ; 3:6

qakqt i dH

da k: 3:7

ok o1;2.

. 167, 11] 1141

(3.5) (3.3), (3.6) Uk Vk. , :

UkVk U kVk i ;

Uk i B1k io0k2p

Ako0kexpijk ;

Vk iAk2o0k

rexpijk :

o0k sign AkAkCk B 21k1=2, jk - , . ( - ak: ak ! ak expijk.)

ok B2k signAkAkCk D21k1=2

. , 1. : . ( , ..) . , , ( , ..). . , . .

- a a , (3.5) (3.2). , H

H1 Vkk1k2a k ak1ak2 k:s:dkk1k2 dkdk1dk2

13

Ukk1k2a k a k2a k2 k:s:dkk1k2 dk dk1dk2 ; 3:8

U V :

Ukk1k2 Ukk2k1 Uk2k1k ; Vkk1k2 Vkk2k1 : -

H2 12

Tk1k2k3k4a

k1a k2ak3ak4dk1k2k3k3

Yi

dki :

H a a

. (3.7) - - - , a a

. (- H1 , ), ..

, - , - (3.5) , , , - .

, , -H0 oik aik. .

, -, - .

4.

-, pr.

qrqt div rHHj 0 ; 4:1

qjqt HHj

2

2 wr 0 : 4:2

j , wr qe=qr -, er . -

H

rHHj22

erdr : 4:3

, (4.1),(4.2) -:

qjqt dH

dr;

qrqt dH

dj:

, r - , j .

- . , , :

qrqt div rv 0 :

S L dt

(rv2

2 er j

qrqt div rv

)dr dt :

1 , , .

1142 .. , .. [ 1997

v : v HHj, r j (4.1), (4.2). -

H jqrqt

dr L

(4.3). -

.

H0

1

2r0HHj2 c2s

dr2

2r0

dr

jk i

k

ok2r0

1=2ak a k ;

drk k

r02ok

1=2ak a k

; 4:4

ok kcs, dr r r0 r0, cs qp=qr01=2 . - - H1,

H1

1

2drHHj2 c2sg

dr3

2r20

dr ;

Ukk1k2 Vkk1k2 :

Ukk1k2 Vkk1k2 1

16p3r01=23gc2s

kk1k2

okok1ok21=2

okok1ok2

1=2k1kk1kk1

okok24ok1

1=2k1kk2kk2

ok2ok1ok

1=2kk2k1k2k1

: 4:5

, (4.1), (4.2) , . -, Ein . HHr:

Ein

er n2HHr2 . . .

dr : 4:6

; :

qrqt div rHHj 0 ;

qjqt 12HHj2 dEin

dr wr nDr :

- , .. , (4.6), r j .

[17,19, 20]. ca , p - r:

qrqt div rv 0 ; 4:7

qvqt vHHv HHpr

r HHwr : 4:8

, - (), "" -. , mr; t, :

dmdtqqt vHH

m 0 : 4:9

(4.9) .

L

rv2

2 er j

qrqt div rv

lqmqt vHHm

dr :

4:10

L v, r m :

v lrHHm HHj ; 4:11

qjqt vHHj v

2

2 wr 0 ; 4:12

qlqt div lv 0 : 4:13

(4.11) l m (., , [17]). (4.12) - , (4.13) l.

l m v. l, m,j l 0, m 0, j 0, v (4.11). (4.11) dr, :

dj lr

dm dj 0 l0

rdm 0

df dj j 0 lr

dm l0

rdm 0 : 4:14

, j 0 j f -, m m 0. [21]:

l r qfqm

; l 0 r qfqm 0

; 4:15

. 167, 11] 1143

.

v, - l, m j, (4.8),,

lrHHqmqt vHHm

HHm

qqt

lr vHH l

r

HHqjqt vHHj v

2

2 wr

0 :

, j l - (4.12), (4.13). -, - (4.7), (4.9), (4.12), (4.13). . v - l0, m0 j0, (4.9), (4.12), (4.13). - ,

qrqt dH

dj;

qjqt dH

dr;

qlqt dH

dm;

qmqt dH

dl; 4:16

H

rv2

2 er

dr

. - l m 0 r;j.

(4.11) - :

qrqt div rv 0 ;

qqt vHH

pmHHwr 0 ;

p mv1 v

2

c2

1=2:

p

m l

rHHm HHj :

, l; m r;j , (4.12)

H

rmm2c p2c21=2 er

dr :

(4.8) [22], e r, S. (4.9), (4.11)

qqt vHH

S 0

de rTdS wdr : -

v HHj lrHHm b

rHHS : 4:17

j; r, l; m S; b :

qrqt dH

dj div rv ;

qjqt dH

dr v

2

2 vHHj w ;

qlqt dH

dm div lv ;

qmqt dH

dl vHHm ;

qbqt dH

dS div bv rT ;

qSqt dH

db;

H rv2=2 er;S dr. (4.8). , - , - . : -? , . , (4.11) m, . - , (5.4), S.

, a1; a2; a3 a, - , t

r ra; t : 4:18 , a - :

a ra; 0 : ,

a ar; t :

1144 .. , .. [ 1997

( (4.18) -.) a. a

da

dt qa

qt vHHa 0

2, ll; al, l 1; 2; 3, :

v ulHHal : 4:19

ul ll=r, r a

r r0aJ

;

r0a , J det Jij (4.18), Jij qxi=qaj ( . 5, 6). u J:

u JTv ;

T . (4.19) .

, , , . .

a a~a : (4.19)

v ~ulHH~al ;

u

~ul uk qakq~al :

, , u3, 1, (4.19) - (. [49]):

v HHf l1r

HHm1 l2r

HHm2 : 4:20

m2 S, (4.20) (5.4). -, , S

Sr; t const

o , , - a1r; t const. , , , , - , . , , .

[17].

5.

, . , . , -, .o, - -. - [1].( .)

(4.16) :

fF;Gg

dFdr

dGdj dFdj

dGdr

dFdl

dGdm dF

dmdGdl

dr :

5:1

l, m, r j

v lrHHm HHj ;

- F r, j, l m:

dFdr

l dF

dr

v

lHHmr

dFdv

;dFdj div dF

dv;

dFdl HHm

rdFdv

;dFdm div

lrdFdv

: 5:2

(5.2) c l, m, r, j, r v. (5.1) [15]:

fF;Gg

HHdFdr

;dGdv

HH

dGdr

;dFdv

dr

rot v

r;

dFdv dG

dv

dr ; 5:3

(2.9) -.

qrqt div rv fr;Hg ;

qvqt v;HHv HHwr fv;Hg ;

H rv2=2 er dr. (5.3) , v

p rv -. 2 [50].

. 167, 11] 1145

[14]:

fF;Gg r

HHdFdr

;dGdp

HH

dGdr

;dFdp

dr

p;

dGdp

HHdFdpdFdp

HHdGdp

!dr : 5:4

(5.4) p r, ,

pir; pjr 0 pjr 0HH 0i pirHHjdr r 0 ;

pir; rr 0 rHHi dr r 0 : 5:5

(2.12) ; [58, 14].

(5.4), (5.5) . , , [12]. - , [13], , p r - .

, , (5.3). [15]:

fF;Gg

HHdFdr

;dGdv

HH

dGdr

;dFdv

dr

rot v

r;

dFdv dG

dv

dr

HHSr;

dFdv

dGdS dG

dvdFdS

dr : 5:6

: , - . , , , , , .. , , , - -, , , - . , , , , (5.3) (5.6) . () . , , - .

- . j div v 0. , - -

l m,

fF;Gg

dFdl

dGdm dF

dmdGdl

dr :

(5.2)

dFdlHHmr;dFdv HH 1

Ddiv

dFdv

;

dFdm div l

r

dFdv HH 1

Ddiv

dFdv

;

, r 1

fF;Gg

rot v;

dFdv HH 1

Ddiv

dFdv

dGdv HH 1

Ddiv

dGdv

!dr : 5:7

, G - Ar, divA 0. X rot v [18]:

fF;Gg

X ;

rot

dFdX rot dG

dX

: 5:8

X:

qXqt rot vX 5:9

[9, 18]:

qXqt fX;Hg

H v2

2dr :

(5.8) . X , c:

O Dcvx qcqy ; vy

qcqx

:

(5.9) (5.8)

qOqt fO;Hg qO

qxqcqy qO

qyqcqx qO;c

qx; y ; 5:10

fF;Gg O

qdF=dO; dG=dOqx; y dx dy ; 5:11

H 12

HHc2 dx dy :

[1],

1146 .. , .. [ 1997

(5.11) bqc=qy:qqt

Dc b qcqx qDc;c

qx; y : 5:12

, O! O by - (5.11). (5.12) [1]:

fF;Gg O by qdF=dO; dG=dO

qx; y dx dy ; 5:13

H :

H 12

HHc2 dx dy :

, (5.11) (5.13) , - [40]. [38, 39].

, , , . , - , - . -, , . , , , . ,, -, .. , .

6.

, [41, 43], - , , (. 7, 8). , , .

[42] -,

IL XHHSr 6:1

. X rot v ; v - :

qvqt vHHv HHp

r; 6:2

S , , :

qSqt vHHS 0 ; 6:3

r :

qrqt div rv 0 : 6:4

, - - (., , . 31 [54]).

IL (6.1) , IL a; , .

, - : . - . , rjt0 a. , () () -:

r ra; t 6:5 a, . r -

vr; t _ra; 6:6

t. (6.4) (6.3):

rr; t raJ

; Sr; t S0a ; 6:7

J det Jij

Jia qxiqaa (6.5). J 6 0 , - (6.5) .

- . , , . (6.6). dr, - :

dr ra da; t ra; t :

(6.6), - :

ddrdt dr;HHv : 6:8

dr da,

dxi Jikdak ; 6:9

:

d

dtJ UJ : 6:10

. 167, 11] 1147

U

Uij qviqxj ;

B 12UUT

, :

O 12UUT :

J

d

dtJ1 J1U ; 6:11

d

dt

qaaqxi qaa

qxj

qvjqxi

: 6:12

, - - (6.9)

dxi2 gikdaidak

gik Jli Jlk :

IL, , . , IL fa:

Ii ILa f a da : 6:13

, (6.13) , , . , : S , (6.7) (6.1) a. , (6.7) r0 1 3,

rr; t 1J: 6:14

(6.1) (6.13) (6.7) J da dr, :

Ii v; HHf HHS dr : 6:15

r, f S a ar; t. -

a. :

Ii

_xi JEijkqaaqxj

qabqxk

qfaqaa

qS0aqab

da :

JEijkqaaqxj

qabqxk Eabg qxiqag ; 6:16

Ii Aja _xi qxiqaj da : 6:17

Aa HHf HHS0 6:18

, :

diva Aa 0 : 6:19

, , .. (6.17) - . S0 a. Aa - .

- Aa . - s:

a as; as l as : 6:20

,

A l0

dasds

da as ds

: a as . A (6.17), - :

IK C

vr; t; dl : 6:21

C , (6.20).

pr;S. , , - . , , (6.21). - C. - , , - (6.19), - . p pr;S (6.19) (6.18). -3 b ba, Jab r0.

1148 .. , .. [ 1997

Aa Sa const. -, , "" (.. ), (6.21). , - : , S

ar; t const,

. -

. [11]

p dq : 6:22

o-

h p2

2 wr ;

p _r, w . (6.22) , ,

v dr ;

.

, , [52, 51], . -.

7.

, .

.1. Il I1; . . . ; In -

, ,

dIkdt qIk

qt vHHIk 0 :

Il . - , Ieu rIk -

qIeuqt div Ieuv 0 :

(6.2), (6.3) (6.4), , : (6.1) s 4. -

Ii r fIL; s dr ; 7:1

fIL; s .2. (6.2)

, :

xi HHi pr : 7:2

J -,

qxiqak

xi 1rqpr; sqak

: 7:3

(7.2) (7.3) (6.7) (6.14) .

, [17]:

S dt L

dtdr

r

_x2i2 er; s

; 7:4

e , w -

de rTds w dr ; 7:5

T . , dS 0

(7.3). (7.4) r -

a. - :

S dt da

_x2i2 ~er;S

: 7:6

x - a; ~e e=r r s, (6.7) (6.14). - r ~e x (6.14), (7.6).

(6.16)

J 16EijkEabg

qxiqaa

qxjqab

qxkqag

;

4 - s, S.

. 167, 11] 1149

dS dt da

xidxi r2 q~eqr dJ

dt da

xi 1

2

qqaa

r2

q~eqr

EabgEijk

qxjqab

qxkqag

dxi

dt da

xi 1r

qqaa

r2

q~eqr

qaaqxi

dxi 0 7:7

xi 1rqqaa

r2

q~eqr

qaaqxi

:

, - (7.3),

pr; s r2 q~eqr

:

( - (7.5).)

, - .

, ,

P

_x da rvr; t dr ;

,

E

_x2

2 ~er; s

da

rv2

2 er; s

dr ;

- : - .

, [41], - , - . , : , , . , - , o . , - b ba, 1:

J det qbiqaj 1 : 7:8

-, . ( , .) . , . ( ) da a,

b a da ;

:

qaiaqai

0 ; 7:9

(7.8). p pr; s

-, , -, , -, S0a const. , (6.19), aa HHs a 0 : 7:10

(7.9) :

a rot f ;

, div f 0, a (7.9) (7.10) -,

a HHs HHc :

c , a.

( ; .,, [80]), :

1. :

d

dtHHa _xi HHaxi 0 ;

.. a

IL HHa _xi HHaxi : 7:11

: [17] (. [44, 45]).

:

J Tt J J TJt O0 ; 7:12

T , O0

IL :

O0ij EijkILk :

(7.12) [62] - .

(7.11) . - (6.16), (7.11) -

IL JX;HHa r0ar X;HHa : 7:13

1150 .. , .. [ 1997

a r t. a , (7.13) X0a:

IL X0a : (7.13) - B X=r:

Br; t JB0a :

, X=r .

, (7.13) - : .

B, (6.2) (6.4):

d

dtB B;HHv : 7:14

d=dt q=qt vHH. (7.14) (6.8) dr, B, dr . - . , , (7.14) J 1, (6.11) X=r, . - (7.13). ( ) - . B(7.14) -.

2. pr; s - : IL HHs:

IL HHas0; HHa _xi HHaxi

:

-

Eabgqxiqaa

qxjqab

qxkqag EijkJ ;

IL XHHsr :

(6.1). , , , - .

, . : X HHs. . (7.11)

Eabqxiqaa

qxjqab EijJ ;

, :

Or consta :

, .

. . , (7.13)

IL X;HHa : 7:15

- IL

X0a rota u ; u (4.19). , u ( ), . -, , . X0a. - b ba qb1b2b3=qa1a2a3 1, X0a -

~O0ib qbiqaj O0ja : 7:16

(7.16) ; (. [45]) (4.15) 5.

~X0b , z, , 1:

~O01 X0HHab1 0 ; 7:17~O02 X0HHab2 0 ; 7:18~O03 X0HHab3 1 : 7:19

(7.17) (7.19) "-" X0a - ba. , . - (7.17) (7.19):

da

ds X0a :

"" X0a; s "" . ( b)

5 [81].

. 167, 11] 1151

db1ds 0 ; 7:20

db2ds 0 ; 7:21

db3ds 1 : 7:22

b1 b2 . b1 b2 c1 c2 , s. , (7.20) (7.22) , , - , , , - c1 c2. , -, .

, rotb ~u ~X0b, "" ~u:

~u1 qfqb1 ; 7:23

~u2 qfqb2 b1 ; 7:24

~u3 qfqb3 : 7:25

(4.19) - ( . [44]):

v b1HHb2 HHf :

-

Xr; t HHb1 HHb2 qrqb3 b; t : 7:26

, . , (7.26), r b, - (7.17) (7.19). - (7.17) qb1b2b3=qa1a2a3 1.

, - . - .

8.

, . . - - :

v lHHm HHj :

- . , . , . , - . - . M 2 -, l m .

, - , - , , :

dl dm dl 0 dm 0 : ; -, , - , S 2.

, M 2 R3

, . - :

X rot v HHl HHm : 8:1 lr const, mr const. - l m , M 2 - g. , , . - .

(. [56, 57]),

I v; rot v dr : 8:2

. -, [57],

X k1n1d

r l1s1

ds1

k2n2d

r l2s2

ds2 ;

n1;2 , ds1;2 .

r l1s1 r l2s2,

v; dl1 mk2 ;v; dl2 mk1 ;

m . k2, k1 , I:

v; k1 dl1 k2 dl2 v; rot v dr 2mk1k2 :

1152 .. , .. [ 1997

n , . - (8.2) , , .

I, , - , , , - .

, - (8.1) - .

l m S 2. l m y j,

X 2AHH cos y HHj ; A . , z j 2p. O y j n- n2 1 [58]:

Oa eabgn; qbn qgn

: 8:3

, n n0. R3

S 3. - S 3 ! S 2. - p3S 2 Z, .. N . - nr n1 nr n2 n1;2 const. N [59]. , I [60]:

I v;X dr 64p2NA2 :

- .

, [60] A h=2m.

. , M2, - , - . p3 . p3 g (g5 1). - .

- N 1 ( ):n; s qs3q ; 8:4

q 1 irs1 irs1 ;

s .

,

x iy sinhUcoshU cos b expia ; z

sin acoshU cos b

04U

(7.1), Ii r fIL;S dr -

, , :

fIi; . . .g 0 :

, 2, (8.6) -.

, (7.1) - . , , (8.6).

(8.6) , (7.1) . . (4.17) m , (7.1) (8.6) -

fF;Gg

dFdr

dGdj dFdj

dGdr

dFdl

dGdIL dFdIL

dGdl

dFdb

dGds dF

dsdGdb

dr :

, , [53], (7.6) , (9.11) (4.20), -.

9.

- . . . - . , , . . , , .

- g, z. . div v 0, r :

qrqt vHHr 0 :

L , div v 0 :

L

rv2

2Ur; r a

qrqt vHHr

j div v

dr :

9:1

Ur; r g,

Ur; r grr?; zz z 0

zz 0r0z 00 dz 00

: 9:2

(9.2) z z 0, r0z 0 :

r0z 0 rr?; z :

z 0 : z 0r. (9.2) - .

v j - [26]:

rv HHj aHHr 9:3

divr1HHj aHHr 0 ;

. r, a:

qaqt vHHa v

2

2 qU

qr 0 ;

qU=qr gz z 0. (9.3) (4.8)

a r, p:

p r v2

2 rgz z 0

qqt vHH

j const :

, ; :

H

rv2

2Ur; r

dr ;

a r -:

qaqt dH

dr;

qrqt dH

da:

r a - . (9.3), , , - . . -, , 4, -, "" . , - [63].

1154 .. , .. [ 1997

a dr. -, -, :

HB

r0v2

2Ur; z

dr :

- , - .

. , r , ..

r r0yz Zr?; t

:

yz , Zr?; t z 0. Zr? - dsn dr?

1 HHZ21=2, :

n HHZ1 HHZ21=2, -:

U

r0gZ2

2 sh

1 HHZ2q

1i

dr? :

(s -).

,

dZdtqqt vHH

Z vz : 9:4

L dr?

Zh

dz

r0v

2

2 j div v

cqZqt vn

1 HHZ2

q dr? U ; 9:5

vn vz vHHZzZ

1 HHZ2q

, c ar0 , -.

v r0v HHj; j - Dj 0. v ( Z) - :

jzZ c : 9:6

Z.

(9.5), v, , Lv. (9.6)

Lv dr?

Zh

dz

r0v

2

2 vHHj

:

dLv Z . :

dr?

r0v

2

2 vHHj

dZ :

v j, , ,

dj jz dZ jz qjqz

dZ :

dLv dr? vn

1 HHZ2

qqjqz

dZ :

qcqt r0gZ s div

HHZ1 HHZ2

q r0v

2

2 vHHj qj

qzvn

1 HHZ2

q zZ

: 9:7

H, ,

H dr?

Zn

dzr0HHj2

2U ;

[25]:

qZqt dH

dc;

qcqt dH

dZ:

H - . - - Z c; , . - ,

H 12

g sk2jZkj2 dk

1

2

kcjckj2 tanhkh dk

12 2p

Lkk1k2ckck1Zk2dkk1k2 dkdk1 dk2 . . . ; 9:8

Lkk1k2 1

2k2 k21 k22 kk1 tanhkh tanhk1h

r0 1 : (9.8) kZ,

- .

2*

. 167, 11] 1155

kh! 0Lkk1k2 ! kk1 ;

.. H1 - c Z:

H1 12

ZHHc2 dr1 : 9:9

, (.,, [66]) , - (9.9), H0 , h3:

qZqt hDc div ZHHc h

3

3D2c dH

dc;

qcqt gZ sDZ HHc

2

2 dH

dZ;

H 12

gZ2 sHHZ2 hHHc2 h3Dc2 ZHHc2 dr? :

(., [67]):

Lkk1k2 ! kk1 kk1 :

-:

Zk

ok2g sk2

1=2ak a k ;

ck ig sk22ok

1=2ak a k ;

ok kg sk2 tanhkh1=2

. 4,

[1].

qmqt vHHm 0 ;

L dr?

Zh

dz

r0v

2

2 j div v lmt vHHm

cqZqt vn

1 HHZ2

q dr? U : 9:10

v l m:

r0v lHHm HHj ; 9:11

(4.9) (4.13). c :

jzZ c : 9:12

Z. - c (9.7), v (9.11).

, - [64]. [24, 25] - (5.7), :

fF;Gg

rot v;

dFdv dG

dv

dr

S

dFdS

dGdc dG

dSdFdc

ds : 9:13

F G v div v 0 S; ds . (9.13) dF=dv dG=dv . - (, [65]):

v w HHF :

w . F -

DF 0 ; qFqn vn :

c. :

qvqt v;HHv HHp ; qS

qt vn

:

div v 0 ; pS sk ;

k , (9.13) -:

qvqt fv;Hg ; qS

qt fS;Hg :

, (9.13) , - l m, c Z, (9.11).

"" [23].

, (9.11), , - "" l m. - - . (9.1) (9.5).

[69], -

1156 .. , .. [ 1997

. , , , (9.13) - .

: , - , , [68]. () , , .

10.

- (4.1), (4.2). , :

qrqt div rv 0 ;

qvqt vHHv HH

e

mj 3T

mr0dr; 10:1

Dj 4pe drm; dr r r0 :

e, m , T .

Ees 18p

HHj2 dr e

2

2m2

drrdrr 0jr r 0j dr dr

0

ET 32

T

mr0

dr2 dr :

,

e

mj e

2

m2

drr 0jr r 0j dr

dEesdr

;

3T

mr0dr dET

dr: 10:2

(10.2) , (10.1) - (4.1), (4.2) Ein (4.6). H (4.4), o2k o2p 3k2T=m (o2p 4p0e2=m2 ), U V (4.5), g 0.

- , - Te . -

o=k, vTe, . , re r expef=Te, .

qrqt div rv 0 ; qv

qt vHHv e

MHHj ;

Dj 4peM

r r0 exp

ejTe

; 10:3

M . :

H rv2

2dr Ein : 10:4

Ein , - Ees 1=8p

HHj2 dr

ET TeM

r0

ejTe 1exp

ejTe 1dr :

Ein - r,

dEindrjr 0

14p

Ddjr 0drr

e2r0M

expejTe

djr 0drr

dr 0 :

, ,

14p

Ddjr 0drr

e2r0M

expejTe

djr 0drr

e

Mdr r 0 :

,

dEindr e

Mj :

, (10.3) (4.1), (4.2).

, (krd 5 1, rd Te=4pn0e21=2 ) (10.3) . , -

ejTe dr

r0 12

drr0

2 rd 2D dr

r0:

(10.4) (. (4.6)):

Ein r0c

2s

2

"drr0

2 13

drr0

3 rd 2

HH

drr0

2#dr ;

c2s TeM

:

, -, , -

. 167, 11] 1157

:

qrqt div rv 0 ;

qqt vHH

p eE e

cvH 3THH dr

r0;

rotE 1c

qHqt

;

rotH 4pc

erm

v 1c

qEqt

;

divE 4pe drm:

j A, A : divA 0.

, - , p1:

p p1 e

cA :

p1 :

qp1qt HHm2c4 p2c21=2 v rot p1 eHHj 3THH

drr0

:

A

B 14pc

HHj 1

c

qAqt

E

4pc:

4:

p1m l

rHHm HHj :

l; m; r;j; B;A :

qlqt dH

dm;

qmqt dH

dl;

qrqt dH

dj;

qjqt dH

dr;

qAqt dH

dB;

qBqt dH

dA;

H

rmp2c2 m2c41=2 3

2T

dr2

mr0 18protA2

dr

2pc2B2 cBHHj 14p

jDjdr ; 10:5

- , .

, - . [34, 35].

, - () .

, , .

qrqt div rv 0 ;

qvqt vHHv HH de

dr 14prrotHH ; 10:6

qHqt rot vH :

, -, , . , . , v=c. , , - E v=cH .

(10.6) [70], H=r . , H r .

, :

L

rv2

2 er H

2

8p S

qHqt rot vH

jqrqt div rv

cdivH

dr :

L v, r H,

rv H rotS rHHj ; 10:7qjqt vHHj v

2

2 or 0 ; 10:8

qSqt H4p v rotS HHc 0 : 10:9

, -. (10.7), - (10.8), (10.9). c . divS 0

c D1div v rotS c0 ;

c0 Dc0 0. , H0 S , S! 0 r!1.

c0 H0r4p

:

1158 .. , .. [ 1997

(10.6).

, - [33]

qrqt dH

dj;

qjqt dH

dr;

qHqt dH

dS;

qSqt dH

dH;

H

rv2

2 er H

2

8p c divH

dr

.

[71]. , :

v HHf r1mHHlMHHL ;H HHl HHL :

l m, L M, r f . , H v - S (10.7).

- H S j

Dj div 1r0H rotS ;

H

r0v2

2H

2

8p c divH

dr :

r;f H;S :

fF;Gg

dFdr

dGdj dFdj

dGdr

dFdH

dGdS dFdS

dGdH

dr :

10:10

, , : v, - r H. - (5.6) , [15]:

fF;Gg

HHdFdr;dGdv

HH

dGdr

;dFdv

dr

rot v

r;

dFdv dG

dv

dr

H

r;

rot

dFdH dG

dv

rot

dGdH dF

dv

dr:

10:11 (. [15]):

HHSr;

dFdv

dGdS dG

dvdFdS

dr : 10:12

(10.11), (10.12) , H 0, .

, -, [73]:

C r fS;

HHHr

S;

HHHr

2S; . . .

dr : 10:13

, - (10.13),

In HHHr

nS : 10:14

. , - I, B, r p [74, 29]. :

qIqt vHHI 0 ; 10:15

qBqt vHHB BHHv ; 10:16

qpqt vHHp pHHv p rot v 0 : 10:17

- .

, B p (10.16) (10.17) I:

B 0 IB ; p 0 Ip ; 10:18

.. B 0 p 0 , B p. I, p, B

I 0, p 0, B 0, :

p 0 HHI ; B 0 1rrot p 0; I 0 1

rdiv rB : 10:19

, (10.18) (10.19), -

I 0 vB ; B 0 1r p p 0 ; B 0 1

rHHI HHI 0 :

10:20

:

p rB B 0 ; I 1r

p; HHI 0 HHI 00 ;

I 1r

HHI 0HHI 00 HHI 000 : 10:21

- , - (10.18) (10.21)

. 167, 11] 1159

. , , r, p, B ( , (10.18) (10.21)), [29]:

I 00 p B ; I 0ik 1

r

pHHIi HHIk

;

I 0k BHHIk ; I 0 1

r

HHI1HHI2 HHI3

: 10:22

, .

. . [82], , - , , - .

- . - p pr;S I - S, BH=r, p A H rotA, [72]:qAqt v rotA HHvA vHHA AHHv rot v A :

, (10.19) :

A0 HHS ; B 0 Hr; I 0 0 :

, - .

(10.19) B (10.18), ( S) :

I2 HHHSr :

. (10.14):

In HHHr

nS :

(10.20)

I3 AHr :

I3 (6.1)

Ik AH dr ;

[57].

I1 S, I2 I3, H A, (10.22), ,

C r fI1; I2; . . . dr : 10:23

(10.22) . -, S, . , Ii(i 1; 2; 3) , H r, I2 AH=r. [48]:

C r fI2;

HHHr

I2; . . .

dr : 10:24

-,

Ct v;H dr ;

.

, - (10.11), (10.12).

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

11.

- . - f, - ( E HHj) - n0:

qfqt vHH f HHj qf

qv 0 ;

Dj 4p

fdv n0e m 1 : 11:1

. -

1160 .. , .. [ 1997

r; v (11.1) - "", f "". , . , - , x, f ( ) f0x:

fr; v; t f0x ili v Vr; x; t : - :

fr; v; t Fr; x; tdv Vr; x; tdx : 11:2

(11.2) (11.1)

qFqt div FV 0 ; 11:3

qVqt VHHV HHj ; 11:4

Dj 4p

Fx; r; t dx n0: 11:5

(11.3) (11.5) "" F:

Ein 18p

HHj2 dr

12

Fx; r dx n0

Fx 0; r 0 dx 0 n0

jr r 0j dr dr

0 :

4 (11.3) (11.5) (4.1), (4.2).

(11.3) (11.5) . "" V HHF - [1]:

qFqt dH

dF;

qFqt dH

dF;

H Fv2

2dx dr Ein ;

, :

fS;Tg dx dr

dSdF

dTdF dT

dFdSdF

:

f [1]. (11.2), , [16]:

fS;Tg f

qqv

dSdf

qqr

dTdf

qqr

dSdf

qqv

dTdf

dr dv :

- ,

[16], - .

, . , - "", -:

ht divh0

U dz 0 ; 11:6

Ut UHHUW qUqz HHh 0 ; 11:7

qWqz divU 0 : 11:8

h hr; t r x; y, 0 < z < h - , U Ur; z - , W Wr; z - , g 1. , (11.6) (11.8) - .

x 0 < x < l, z. t :

z zr; x; t; hr; t zr; l; t :

, z (9.4):

dz

dt qz

qt UHHz W : 11:9

x l, , (11.9) (11.6). , x z, :

qqt

z

qqt

x zt

Zqqx

; HHz HHx HHzZ

qqx

: 11:10

,

qqz 1

Zqqx

; 11:11

Zr; x; t qzqx

:

(11.9) x (11.10), (11.11),

qZqt div ZU 0 : 11:12

( - x.)

(11.7), -

qUqt UHHU HHh 0 ; 11:13

. 167, 11] 1161

h Z

h l0

Zr; x; t dx : 11:14

(11.3) (11.5) (11.14). ( x; y) - U HHj [30]:

qjqt dH

dZ;

qZqt dH

dj; 11:15

H 12

dx dr ZHHj2 1

2

dr h2 :

x, Z U:

qZqt q

qxdHdU

;qUqt q

qxdHdZ

:

, [32] . , , [32], (11.15).

[76], , , , . [76] , , -. x b, r. , , -, , O, , , , x. (7.6), , . - .

[38, 39], (5.12), , .

12.

, -, , , -. , . - , -

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

, ok ak, (3.7):

qakqt i dH

dak; 12:1

H H0 H1 . . . ;

H0 okjakj2 dk ; 12:2

H1 Vkk1k2a k ak1ak2 k:s:dkk1k2 dkdk1 dk2

13

Ukk1k2a k a k2a k2 k:s:dkk1k2 dk dk1 dk2 :12:3

ak ck :

ak ck Lkk1k2ck1ck2dkk1k2 dk1 dk2

Mkk1k2ckc

k2dk2kk1 dk1 dk2

Nkk1k2c

k ck1dkk1k2 dk1 dk2 . . . 12:4

, . ,

fck; ck 0 g dkk 0 ; fck; ck 0 g fck; ck 0 g 0 :

ak ck

Vkk1k2ck1ck2ok ok1 ok2

dkk1k2 dk1 dk2

2

V k2kk1ckck2

ok2 ok ok1dk2kk1 dk1 dk2

Ukkk2ck ck1

ok ok1 ok2dkk1k2 dk1 dk2 . . . 12:5

- H1 , - , - a a a aaa. ( H1) . - . " -", -

1162 .. , .. [ 1997

ok . . . oki oki1 . . . okn 0 ;k . . . ki ki1 . . . kn 0 ;

n- . - (3.8), (.(12.5)):

ok ok1 ok2 0 ;k k1 k2 0 12:6

ok ok1 ok2 0 ;k k1 k2 0 : 12:7

, , - ok -. , , , , . , H1, aaa aaa, (-).

(12.7) . , ok jkj, -, o0 0 o 00k < 0. ,, - . (12.7) .

(12.6), (12.7) , . -

H3 Tk1k2k3k4a

k1a k2ak3ak4dk1k2k3k4

Ydki ; 12:8

ok1 ok2 ok3 ok4 0 ;k1 k2 k3 k4 0

ok. (12.8) - Tkk1k2k3 (. [83]):

Tkk1k2k3 T 0kk1k2k3 2Uk2k3;k2k3U

kk1;kk1

okk1 ok ok1

2 Vk2k3k2k3Vkk1kk1

okk1 ok ok1 2 Vkk2kk2V

k3k1k3k1

ok3k1 ok1 ok3

2 Vk1k3k1k3Vk2kk2k

ok2k ok ok2 2 Vk1k2k1k2V

k3kk3k

ok3k ok ok3

2 Vkk3kk3Vk2k1k2k1

ok2k ok1 ok2: 12:9

, - :

Hd Vkk1k2a k ak1ak2 k:s:dkk1k2 dkdk1 dk2 ; 12:10

1! 2 2! 1;

Hex 13

U kk1k2a k a k1a k2 k:s:dkk1k2 dk dk1 dk2 ; 12:11

-, - 0! 3; -

Hsc Tkk1k2k3a

k ak1ak2ak3dkk1k2k3

Ydki ; 12:12

2! 2; .. ,

-. , :

Hint Vkk1k2bka k1ak2 k:s:dkk1k2 dk dk1 dk2 : 12:13

(12.13) - , ..

-, , -, , . , - . , , -. , k1, k2 k3. , ,,

ok1 ok2 ok3 ;k1 k2 k3 :

ak :ak a1k a2k a3k ;

a1, a2, a3 . ki jkij. (12.10) Hd :

Hint 2Va 1 k1a2k2a3k3 k:s:dk1k2k3

Ydki :

, H0

ok1 j oki jvi ; vi qoqki

. 167, 11] 1163

: cix aik expiokit

.

H! HX

oijcij2 dj :

-

cix 1

2p3=2cij expijr dj ;

- [77]:

qc1qt v1HHc1

iV

2p3=2c2c3 ; 12:14

qc2qt v2HHc2

iV

2p3=2c1c

3 ; 12:15

qc3qt v3HHc3

iV

2p3=2c2c

3 : 12:16

- . - (12.11):

qc1qt v1HHc1

iU

2p3=2c2c

3 ; 12:17

qc2qt v2HHc2

iU

2p3=2c1c

3 ; 12:18

qc3qt v3HHc3

iU

2p3=2c2c

3 : 12:19

- (12.12) . k0, ,

ak ck expiok0 t ; k k0 k ;

H! H ok0ck2 dk ;

ok ok0 k ok0 kvg 12

q2oqkaqkb

kakb ;

cr:

ict vgHHc oab2

q2cqxaqxb

T2p3 jcj2c 0 ; 12:20

oab q2o

qkaqkb:

(12.20) -- . ,

oab vg2k0dab nanb o00na nb

n k

k

;

ict vgcx vg2k0

D?c o00

2cxx

T

2p3 jcj2c 0 ;

12:21

x . vg ( , , - -, vg); () - () ; , (12.21) .

, c,

ict D?c scxx Zjcj2c 0 ; 12:22

s signo00vg Z signTvg. U Zjcj2, , s. , s Z. Z > 0, , - . Z < 0 - . . sZ 1, sZ 1. - s Z 1, - . - ( . [84]).

, s Z :

ict Dc jcj2c 0 ; 12:23ict D?c cxx jcj2c 0 ; 12:24ict Dc jcj2c 0 ; 12:25ict D?c cxx jcj2c 0 : 12:26

,

ict dHdc

; H jHH?cj2 sjcxj2

Z2jcj4

dr :

12:27

(12.20) , Tk1k2k3k4 -; (12.9) T ki k0. . , , o0 0. (. [78]),

1164 .. , .. [ 1997

Vkk1k2 , k, k1 k2 . , (12.9) T ki k0 .

limk!0jVk0 ; k0k;kj2ok kvg :

,

Vkk0k0 k3=4 ; ok k1=2 ;

. - Vkk0k0 k1=2, ok k, , Tk0k0k0k0 . - . , ok k! 0. , , .. - . - - - - . - , - . (, , (12.13).) , - o - E I:

E

o I :

ck2, H0 ok0

ck2 dk ok0jcj2 dr : -

Hint dojcj2 dr ;

do - dr v:

do qoqr0

dr k0v

( ). v HHj , dr j

(. 4), [66, 79]:

ict vgHHc oab2

q2cqxaqxb

qoqri

dr k0HHjc T2p3 jcj

2c 0 ; 12:28

qqt

dr r0Dj k0HHjcj2 0 ; 12:29

r0qjqt c2sdr

qoqr0jcj2 0 ; 12:30

T ~Tk0k0k0k0 , .

(12.20) (4.3):

Hc ic vgHHc 1

2oab

qcqxa

qc

qxb

qoqr0

dr k0HHjjcj2

12

T

2p3 jcj4 c2s

dr2

2r0 r0

HHj22

dr : 12:31

vg cs, (12.29),(12.30) . vg < cs vgDk4Tjcj2, Dk k , (12.29), (12.30) q=qt vgHH:

vgHHdr r0Dj k0HHjcj2 0 ;

r0vgHHj c2sdrqoqr0jcj2 0 : 12:32

, , - , -:

ict vg2k0

D?c o00

2cxx

qoqr0 k0c

2s

r0vg

drc

T

2p3 k0r0vg

qoqr0

jcj2c 0 ; 12:33

vg

qqx

2dr k0

vgjcj2

D

c2sdr

qoqr0jcj2

: 12:34

- [80].

(12.32) (12.34) dr, j jcj2; - (12.33) . ,

vgHHdr dHdj ; vgHHj dHdr

H Hc (12.31). HDS Hc -

. 167, 11] 1165

:

HDS Hc drvgHHjdr i

c vgHHcdr ;

ict dHDSdc

:

vg > cs, (12.29), (12.30) q=qt vgHH. , (12.29), (12.30) -. ,

kcs kvg ;

(12.7) - . vg 4 cs, , -, , do - dr cs=vg. (12.28) (12.30) :

ict o00

2cxx

vg2k0

D?cqoqr

dr T2p3 jcj2

c 0 ;

qqt vg qqx

2 c2sD?

dr qo

qr0Djcj2 :

. -. , H1 - (4.5) , - a aa aa a , , ok - n : ok kcs

1 nr0k2=2c2s

.

ak ux :

ak ukkp ; u

10

uk expikx u k expikx

dk ;

:

ut csux buux csguxxx 0 ; 12:35

g nr02c2s

; b 12

csr0

1=21 g :

- [85], (4.5) -. , - x, u

:

qqxut csux buux csguxxx cs

2HH2?u : 12:36

(12.36) x -. , csux, . , , .

, , - . , . - , .

. (Y. Pomeau), . (L. Berge ) .. . (..) - ENS (Ecole Normale Super-ieur), , Landau CNRS. INTAS, - " ".

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Hamiltonian formalism for nonlinear waves

V.E. Zakharov, E.A. KuznetsovLandau Institute for Theoretical Physics, Russian Academy of Sciences,

ul. Kosygina 2, 117334 Moscow, Russia

E-mail: zakharov@itp.ac.ru, kuznetso@itp.ac.ru

TheHamiltonian description of hydrodynamic type systems for application to plasmas, hydrodynamics, andmagnetohydrodynamics isreviewed with emphasis on the problem of introducing canonical variables. The relation to other Hamiltonian approaches, in particularnatural-variable Poisson's brackets, is pointed out. It is shown that the degeneracy of noncanonical Poisson's brackets relates to thespecial type of symmetry, the relabeling transformations of uid-particle Lagrangian markers, from which all known vorticityconservation theorems, such as Ertel's, Cauchy's, Kelvin's, as well as the vorticity frozenness and the topological Hopf invariant,derive. The application of canonical variables to collisionless plasma kinetics is described. The Hamiltonian structure of Benney'sequations and of the Rossby wave equation is discussed. Davey Stuartson's equation is given theHamiltonian form. A general methodfor treating weakly nonlinear waves is presented based on classical perturbation theory and the Hamiltonian reduction technique.

PACS numbers: 52.30.q, 52.35.Ra, 52.55.Fa

Bibliography 85 referencesReceived 5 June 1997

. 167, 11] 1167

1. 2. 3. 4. 5. 6. 7. --- 8. 9. 10. 11. 12.