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: [email protected], [email protected]
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: [email protected], [email protected]
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.