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BeCTHHK CIT6fY. Cep. 10, 2006,
Y,ll.K 621.384
BbIII.
4
E. )1,. Komu'Ha
cI»OPMHPOBAHHE 3A,ll;AHHOH ,ll;HHAMHKH IIY"llKA
B MArHHTHOM IIOJIE
BBe.n;eHHe. B · pa60Te paccMaTpHBalOTcH BonpOcbI nOcTpOeHHH MarHHTHbIX nOJIeii, BbI3bIBalOIII.Hx 3a.n;aHHYIO .n;HHaMHKY 3apH)KeHHbIX ~acTHu,.
IIpo6JIeMbI peweHHH pa3JIH~HbIX
06paTHbIX 3a.n;a~, B ~acTHocTH, B ::me'KTpo.n;HHaMHKe Bcer.n;a Haxo.n;HJIHCb B u,eHTpe BHHMaHHH
MHorHX HCCJIe.n;OBaTeJIeii.
u,epa,
A. P.
JIYKaca,
B.
A.
TaK, B pa60Tax r.
n.
3y6oBa
[1-7]
rpm-i6epra,
B. T.
TpoMarHHTHbIX nOJIeii no 3a.n;aHHbIM .n;BH)KeHHHM 3apH)KeHHbIX
HCnOJIb3yeTcH no.n;xo.n;, npe.n;JIO)KeHHblii
B.
OB~apOBa,
B.
MeJIbT­
HCCJIe.n;OBaJIHCb B03MO)KHOCTH nOCTpoeHHH ::meK­
n.
~acTHu,. B
.n;aHHoii CTaTbe
3y6oBbIM, Kor.n;a no 3a.n;aHHOMY nOJIIO CKOpOC­
Teii,onpe.n;eJIHIOIII.eMY .n;BH)KeHHe 3apH)KeHHbIX ~acTHu"
CTPOHTCH ::meKTpOMarHHTHoe nOJIe.
IIycTb HMeeTCH nOJIe CKopocTeii
dr
dt
= 'T}r(r, e, z)
de·
' -d
t
1 (r,e,z )
= -'T}e
r
dz
dt
3.n;eCb
= 'T} z(r, e, z)
U,HJIHH.n;pH~eCKHe Koop.n;HHaTbI,
r,e,z -
3a.n;aHHoii cPYHKU,HH
'T}
(1)
'T}(r,e,z), 'T}e(r,e,z),'T}z(r,e,z) -
B U,HJIHH.n;pH~ecKoii CHCTeMe Koop.n;HHaT,
t -
KOMnOHeHTbI
BpeMH.
,n;BH)KeHHe 3apH)I(eHHOii ~acTHu,bI B cTau,HOHapHOM MarHHTHOM nOJIe onHweM cJIe.n;ylOIII.eii
CHCTeMoii .n;HcPcPepeHu,HaJIbHbIX ypaBHeHHii:
dr
-=Yr ,
dt
de _ ~y;
dt - r e,
dz
dt
=
(2)
Yz ,
dY
m·- =eY xB
dt
'
B KOTOPOH
Y
= (Yr , Ye, Yz ) -
.n;HHaT, m - Macca ~acTHu,bI,
BeKTOp CKOPOCTH ~acTHu,bI B U,HJIHH.n;pH~ecKoii CHCTeMe KOOp­
e-
3apH.n; ~acTHu,bI,
B
= (Br, Be, B z )
- BeKTopHaH cPYHKU,HH,
onpe.n;eJIHIOIII.aH MarHHTHYIO HH.n;YKU,HIO.
B. n. 3y6oBa nOKa3aHO, ~TO cTau,HOHapHoe MarHHTHoe nOJIe (T. e. BeKTopHaH
B, y.n;OBJIeTBOpHIOIII.aH ypaBHeHHIO MaKCBeJIJIa div B = 0), .n;BH)KeHHe 3apH)KeHHbIX
~acTHu, BKOTOPOM npOHcxo.n;HT B COOTBeTCTBHH C nOJIeM CKopocTeii (1), MO)KHO npe.n;CTaBHTb
m
B BH.n;e B = -- rot 'T}. 9TO 03Ha~aeT, ~TO B KOHcPHrypau,HoHHoM npOCTpaHCTBe TpaeKTO­
e
B
pa60Tax
cPYHKU,HH
pHH 3apH)KeHHbIX ~acTHu" onpe.n;eJIHeMble CHCTeMoii
CHCTeMbI
©
(1)
(2),
6y.n;yT COBna.n;aTb C TpaeKTopHHMH
npH o.n;HHaKOBbIX Ha~aJIbHbIX YCJIOBHHX.
E. ,ll.. KOTHHa, 2006
77
O,II,HaKO ,II,aHHOe rrpe,II,CTaBJIeHlfe He lfCqeprrbIBaeT Bcex B03MQ)KHOCTeH, If rrplf rrpaKTHqeC­
KOH peaJIlf3aIJ;lflf OKa3aJIOCb 60JIee eCTeCTBeHHbIM lfCKaTb MarHlfTHoe rrOJIe B Blf,II,e
[8]
m
B = - - rot 7] + h7],
e
r,II,e
h
B
= her, 0, z) -
(3)
rrpOlf3BOJIbHa5I <PYHKIJ;H5I, y,II,OBJIeTBOp5IlOIII,a5I YCJIOBlfIO
,II,aHHOH pa60Te rrpe,II,CTaBJIeHlfe
(3)
div(h7])
= 0.
lfCrrOJIb3yeTC5I ,II,JI5I <pOpMlfpOBaHlf5I ,II,lfHaMlfKlf 3a­
p5I)KeHHbIX qaCTlfIJ; B aKClfaJIbHO-ClfMMeTplfqHOM CTaIJ;lfOHapHOM MarHlfTHOM rrOJIe.
IIocTaHoBKa 3a,n;aQH. PelIleHlfe MHorlfX 3a,II,aq <POKYClfPOBKlf
If TpaHcrropTlfpOBKlf 3a­
p5I)KeHHbIX qaCTlfIJ; B aKClfaJIbHO-ClfMMeTplfqHOM CTaIJ;lfOHapHOM MarHHTHOM rrOJIe CBO,II,lfTC5I
K lfCCJIe,II,OBaHlflO ,II,lfHaMlfKlf qaCTlfIJ; B rrJIOCKOCTlf
(r, z).
PaccMoTplfM CJIe.n.yIOIII,y1O 3a,II,aqy. ,lI,JI5I aKClfaJIbHO-ClfMMeTplfqHOrO rrOJI5I lfMeeM
B r·
= Br(r, z),
Bz
= BzCr, z).
Be
= 0,
IIycTb 3a,II,aHO IIOJIe CKopocTeH
dr dz = fer, z),
r,II,e
fer, z) -
(4) HeKOTOpa5I 3a,II,aHHa5I BeKTOpHa5I <PYHKIJ;lf5I, 06eCrreqlfBalOIII,a5I Heo6xo,II,lfMOe,II,Blf­
)KeHlfe 3ap5I)KeHHbIX qaCTlfIJ; B rrJIOCKOCTlf
(r, z) .
Tpe6yeTC5I HaHTlf CTaIJ;lfOHapHoe aKClfaJIbHO-ClfMMeTplfqHOe MarHlfTHoe rrOJIe, B KOTOPOM
,II,Blf)KeHlfe 3ap5I)KeHHbIX qaCTlfIJ;, orrlfChIBaeMoe ClfCTeMOH
(4)
orrpe,II,eJI5IeMbIMlf ypaBHeHlfeM
B rrJIOCKOCTlf
(2),
COBrra,II,aJIO 6hI C ,II,Blf)KeHlf5IMlf,
(r, z) .
PaccMoTplfM CJIe.n.yIOIII,y1O ClfCTeMY ,II,lf<p<pepeHIJ;lfaJIbHbIX ypaBHeHlfH :
at
oL
- fl oz
oh
- q 0z
3,II,eCh
F
q =
2
C1
= (~~ + ~~ f)
L2
r
2'
/
9 =
hL
-2'
r
(1 + f2),
+ l or
= d1 ,
oL
oh
+ 9 0 z - f q or
+
oL
f 9 or
(5)
= d 2.
m L
m L2
m
l = - 2' d 1 = - 3 - -Fq
er
er
e
d2 = h
f
i + q (~~ -
(c
f F ) ),
C1 -
2
+ rg (q(l + f))
1/2
,
rrOJIHa5I CKOPOCTb qac-
TlfIJ;hI.
.
IIo,II, pelIleHlfeM ClfCTeMhI
<PYHKIJ;lflO
L
(5) 6Y,II,eM rrOHlfMaTb ,II,Ba)K,II,hI HerrpepbIBHO- ,II,lf<p<pepeHIJ;lfpyeMYIO
= L(r, z) If HerrpephIBHO-,II,lf<p<pepeHIJ;lfpyeMYIO <PYHKIJ;lflO h = her, z), o6paIII,alO­
III,lfe 3Tlf ypaBHeHlf5I B TO)K,II,eCTBa.
I1CIIOJIb3Y5I rrpe,II,CTaBJIeHlfe
(3)
If yqlfThIBa5I, qTO
Be
= 0, a 7]2 = const, MO)KHO rrOKa3aTh,
qTO lfMeeT MeCTO CJIe.n.ylOIII,a5I TeopeMa.
TeopeMa. IIycm'b f - .I/,'I06a.R, 'Henpep'bU3'HO-auifjifjepe'H'll,upye.Ma.R, ifjy'H'K:'ll,U.R" npu 'K:omopou
cucme.Ma (5) UMeem pew,e'Hue. T02aa cYUl;ecmeyem a'K:cua.l/,'b'HO-CUMMempu,,{'Hoe Ma2'HUm'HOe
nO.l/,e B, e 'K:OmOpOM aeu:JICe'Hue 3ap.R,:JICe'H'H'b£X "{acmu'll" OnpeaM.R,e.MOe Cucme.MOU (2), 6yJem
coenaaam'b e n.l/,OC'K:ocmu (r, z) c aeU:JICe'HU.R,MU ypae'He'HU.R, (4) npu OaU'Ha'K:oe'b£x 'Ha"{a.l/,'b'H'b£X
yC.l/,oeu.R,x. IIpu"{e.M 'K:OMnO'He'Hm'b£ Ma2'HUm'H020 no.l/,.R, onpeae.l/,.R,'IOmc.R, no ifjopMY.l/,aM
(6)
78
TaKMM 06pa30M, B C.lIY'lae aKCMa.lIbHO-CMMMeTpM'-IHOrO MarHMTHOrO nO.lIH np06.lIeMa ¢Op­
(5).
MMpOBaHM51 3a.n.aHHOll .n.lmaMMKM 3apmKeHHbIX 'laCTMlJ, CBO.n.MTCH K peIlIeHMIO CMCTeMbI
~:lMC.lIeHHOrO HaXO)K.n.eHMH KOMnOHeHT MarHMTHoro nO.lIH
PaccMoTpMM .n.a.TIee a.lIrOpMTM
YpaBHeHlul xapaKTepHCTHK.
PaccMoTpMM CMCTeMY
PH.n.Ka, OHa 3aBMCl1T OT np0113BO.lIbHOll <PYHKlJ,Ml1
f (r, Z),
(5).
(6).
9TO CMCTeMa nepBoro no­
onpe.n.e.lIHIOII1,eti nO.lIe CKopOCTell B
rr.lIOCKOCTM
(r, z).
JIW-reCKoll.
rIpMMeHl1M .n..lIH ee peIlIeHMH MeTO.n. xapaKTepMCTMK
rIPM .lII060M BbI60pe noti <PYHKlJ,MM CMCTeMa 6y.n.eT HB.lIHTbCH rMnep6o­
[9].
BbIIIMllieM .n..lIH .n.aHHoli
CMCTeMbI ypaBHeHMH HarrpaB.lIeHMll xapaKTepMcTMK M .n.M<p<pepeHu.Ma.TIbHble COOTHOIlIeHMH Ha
HMX.
IIepBoe ceMeticTBo: .
dr = fdz,
qdh+ gdL - d2 dz
(7)
= O.
BTopoe ceMei1cTBO:
dz = fdr,
ldL - d1dr
3aMeTMM,
'ITO
.n.M<p<pepeHlJ,Ma.TIbHble
TO.lIbKO OT <PYHKlJ,MM
f,
(8)
= O.
ypaBHeHMH
HarrpaB.lIeHMll
L
a He OT MCKOMbIX <PYHKlJ,Mll
M
xapaKTepMCTMK
3aBMCHT
h.
(5).
3a,n;aqa rypca. PaccMoTpMM 3a.n.a'lY HaXO)K.n.eHMH 'lMC.lIeHHOrO peIlIeHMH CMCTeMbI
,n.lIH 3Toro . Heo6xo.n.MMO 3a.n.aTb .n.onO.lIHMTe.lIbHO HeKoTopble YC.lIOBMH.
(7), (8)
peHlJ,Ma.TIbHble COOTHOIlIeHMH
6Y.n.eM MCKaTb peIlIeHMH
L
M
h
.n..lIH CMCTeMbI
CMCTeMbI
(5)
(5),
I1cnO.lIb3YH .n.M<p<pe­
paccMoTpMM 3a.n.a'lY fypca, T. e.
npM YC.lIOBMM, 'ITO Ha .n.BYX xapaKTepMCTMKax,
BbIXO.n.HII1,MX M3 O.n.HOll TO'lKM, 3a,n:aHbI 3Ha'leHMH
L
M
h,
a COOTBeTCTBYIOII1,Me <PYHKlJ,HM COBna­
.n.aIOT B o6II1,ei1 TO'lKe. IIo.n.pa3YMeBaeTCH TaK)Ke, 'ITO 3a.n.aHHble <PYHKlJ,MM
L
M
h
Ha Ka)K.n.oll
xapaKTepMCTMKe TaKOBbI, 'ITO .n.M<p<pepeHu.Ma.TIbHble ypaBHeHMH xapaKTepMCTMK y.n.OB.lIeTBO­
pHIOTCH.
By.n.eM npe.n.nO.lIaraTb .n.a.TIee, 'ITO <PYHKlJ,MH
none CKopocTei1, TaKOBa, 'ITO
rrpHMaH r
=0
f(O, z)
= f(r, 0) = o.
f(r, z),
3a.n.alOII1,aH COOTBeTCTBYIOII1,ee
HeTpy,n:HO 3aMeTMTb, 'ITO B 3TOM C.lIy'lae
HB.lIHeTCH xapaKTepMcTMKoti ,n:.lIH nepBoro ceMellCTBa, npHMaH
z
=0-
.n..lIH
BToporo.
3a.n.a'lY fypca Tenepb MO)KHO C<POPMY.lIMpOBaTb C.lIe,n:yIOII1,MM 06Pa30M. HatiTl1 peIlIeHMe
CMCTeMbI
r
(5),
T. e.
<PYHKlJ,Ml1
L
M
h
npM YC.lIOBMM, 'ITO OHM 3a.n.aHbI Ha xapaKTepMCTMKax
= 0 M Z = 0:
L(O, z) = Lro(z), h(O, z) = hro(z), Br(i, j), L(r,O) = Lzo(r), h(r,O) = hzo(r), 0 ~ r ~ rm. J
3.n.eCb
Lro(z), Lzo(r), hro(z), hzo(r) -
3a.n.aHHble <PYHKlJ,MM, Zn,
rm -
3a.n.aHHble Be.lIM'lMHbI.
IIPM 3TOM 3Ha'leHMH COOTBeTCTBYIOII1,MX <PYHKU.Mll, 3a.n.aHHblX Ha xapaKTepMCTMKax, COBna­
.n.alOT B 06II1,ell TO'lKe: .
Lro(O)
= Lzo(O),
hro(O)
= hzo(O).
ECTeCTBeHHO npe,n:nO.lIaraeTCH, 'ITO 3a.n.aHHble <PYHKU.MM
Lro(z), Lzo(r), hro(z), hzo(r)
Ha
Ka)K.n.oll xapaKTepMCTMKe TaKOBbI, 'ITO .n.M<p<pepeHlJ,Ma.TIbHble COOTHOIlIeHMH Ha xapaKTepMC­
TMKax BbIIIO.lIHHIOTCH.
OTMeTMM, 'ITO 3a.n.aHMe <PYHKlJ,Mll
L
M
h
Ha xapaKTepMCTMKax COCTaB.lIHeT HeKOTOpyIO
np06.lIeMY, nOCKO.lIbKY 3TOT BbI60p onpe.n.e.lIMT TO M.lIM MHoe MarHMTHoe nO.lIe.
11
XOTH no.lIY­
'leHHOe MarHMTHoe nO.lIe 06eCne'lMT BbIIIO.lIHeHMe OCHOBHOti 3a.n.aqM, T. e. 06eCne'lMT 3a.n.aH­
HYIO .n.MHaMMKY 3apH)KeHHbIX qaCTMlJ" OHO MO)KeT He y.n.OB.lIeTBOpHTb TeM M.lIM MHbIM npaKTM­
'leCKMM Tpe6oBaHMHM, CBH3aHHbIM C pea.TIM3alJ,Meti ,n:aHHoro nO.lIH.
C
3TMX n03Mu.Mti YKa3aH­
Hble <PYHKU.MM MorYT 6bITb paCCMOTpeHbI KaK ynpaB.lIHIOII1,Me <PYHKU.MM, M MO)KHO CTaBMTb
79
3a,LI,a'IY HaXQ)K,LI,emUI MarHMTHbIX rrOJIeH, 06eCrre'IMBaIOIIIMX He TOJIbKO 3a,LI,aHHYIO ,LI,MHaMMKY
3ap5l)KeHHbIX 'IaCTMlI" HO M Tpe6yeMble xapaKTepMCTMKM MarHMTHbIX rrOJIeH.
PaCCMOTpMM O,LI,MH H3 B03MQ)KHbIX BapHaHTOB 3a,LI,aHH5I <PYHKlI,HH
BYIOIIIHX xapaKTepMCTMKax.
CT05IHHa5I.
=
L ro
I1YCTb
Tor,LI,a 3Ha'IeHM5I <PYHKlI,HH
0, Lzo(r)
hro(z), hzo(r)
=
-kr2,
L
3,LI,eCb
M
h
k -
Ha COOTBeTCT­
HeKOTOpa5I rro­
orrpe,LI,eJI5IeM M3 COOTBeTCTBYIOIIIHX
,LI,H<p<pepeHIl,HaJIbHbIX COOTHOIIIeHHH Ha xapaKTepHCTHKax. I10JIY'IaeM
hzO ()
r -- - mk(
C2
e
1
k22)-1/2
m -8 f (r, 0)( C2
r
- -ekr
8z
1
-
H pe1IIa5I rrpM KOHKpeTHOM 3Ha'IeHHH <PYHKIJ,HH
f
dhro ( ;-(0,
z)
f
-
k 22
r )1/2
,
,LI,H<p<pepeHIJ,HaJIbHOe ypaBHeHHe
fz)) dz =
+8
8r (0,
°
C Ha'IaJIbHbIM YCJIOBHeM
h ro
HaxO,LI,HM <PYHKIJ,HIO
= - mk
_
eCl
mCl (
ek
8f
8z
l
r)
(0,0),
hro(z).
MeTo,n; Maceo.
,II;JI5I 'IMCJIeHHOrO pe1IIeHH5I 3a,LI,a'IH fypca 6Y,LI,eM MCrrOJIb30BaTb Me­
TO,LI, Macco, B OCHOBe KOToporo JIe)KMT 3aMeHa ,LI,M<p<pepeHIJ,MaJIbHbIX ypaBHeHMH xapaKTe­
PMCTMK COOTBeTCTBYIOIIIMMM KOHe'IHO-pa3HOCTHbIMM ypaBHeHM5IMM. PaccMoTpMM ypaBHeHM5I
HarrpaBJIeHMH xapaKTepMCTMK, rrOJIY'IeHHble BblIIIe:
dr = fdz,
dz = -fdr.
(9)
(10)
OHM orrpe,LI,eJI5IIOT ,LI,Ba opTOrOHaJIbHbIX ceMeHCTBa xapaKTepMCTMK.
TaK KaK ,LI,M<p<pepeH­
IJ,MaJIbHble ypaBHeHM5I HarrpaBJIeHMH xapaKTepMCTMK 3aBMC5IT TOJIbKO OT <PYHKIJ,MM
OT MCKOMbIX <PYHKlI,MH
L
M
ceMeHCTB xapaKTepMCTMK.
JIY'IeHHOH ceTKM.
f,
I10CJIe 3Toro BbI'IHCJIHM 3Ha'IeHM5I <PYHKIJ,MH
(i,j): r(i,j)
H
z(i,j), i
L
M
h
B Y3JIaX rro­
= 1, ... ,n, j = 1, ... ,m, rrpH rrOMOIIIH CMCTeMbI
= (r(i -1,j) + flhr(i,j -1) + h(z(i,j -1) -
= (z(i,j -1) + hhz(i -1,j) + h(r(i,j -1) -
fl = f(r(i -1), z(i -1,j)) H h = f(r(i,j-l)' z(i,j -1)) (i,j -1) M TO'IKe (i-l,j) COOTBeTCTBeHHO.
r,LI,e
z(i -1,j)))/(1 + hh),
rei -1,j)))/(1 + hh),
3Ha'IeHMe <PYHKIJ,MM
f
B TO'IKe
,II;aJIee, 3aMeHH5I ,LI,H<p<pepeHIJ,MaJIbI, BXO,LI,5IIIIMe B ,LI,M<p<pepeHIJ,MaJIbHble COOTHOIIIeHM5I
(8)
(7),
Ha COOTBeTCTBYIOIIIHX xapaKTepHCTMKax, KOHe'IHbIMM Pa3HOCT5IMH, ,LI,JI5I orrpe,LI,eJIeHM5I
3Ha'IeHMH
L
H
hB
TO'IKaX
L(i,j)
h(i, j)
80
a He
TO rrOCTpOHM CHa'IaJIa ceTKY, 06pa30BaHHYIO rrepeCe'IeHMHMH
I10CTPOHM ceTKY, 06pa30BaHHYIO KpHBbIMM 3TMX ceMeHCTB, T. e. HaH,LI,eM
KOOp,LI,MHaTbI TO'IeK
r(i,j)
z(i,j)
h,
(i - l,j)
H
(i,j -1)
rrOJIY'IMM cHcTeMY ypaBHeHHH
= L(i,j -1) + ~l (i,j -1)(z(i,j) -
= h(i -1, j) + ~q (i, j)(L(i, j) -
r(i,j -1)),
L(i -1, j)) - d2 (i -1, j)(z(i, j) - z(i -1, j)).
q
BbPIHCJIeHHe KOMnOHeHT MarHHTHoro nOJIS!. PaCCMOTpHM <P0PMYJIY )I.JI5I orrpe)I.eJIe­
HH5I KOMrrOHeHT MarHHTHoro rrOJI5I
(6).
ITPH qHCJIeHHOM rrOCTpoeHHH peIlIeHH5I CHCTeMhI
KOMrrOHeHThI MarHHTHoro rrOJI5I TaK)Ke 6Y)I.eM HCKaTh qHCJIeHHO.
h(i,j)
B Y3JIaX ceTKH HaH)I.eM B 3THX)Ke TOqKaX BeJIHqMHhI
3HaHHe 3HaqeHHH
~
-
dz
L(i,j)
ITo 3HaqeHH5IM
Br(i,j), Bz(i,j).
H
OTMeTHM, qTO
.
B Y3JIaX ceTKH rr03BOJI5IeT BhlqHCJIHTh rrOJIHhle rrpOH3BO)I.Hhle
.
,
(5)
L(i, j)
dL
-d '
r
IT
B)I.OJIh COOTBeTCTBYIOIII;HX xapaKTepHCTHK. nCrrOJIh3yeM 3TO )I.JI5I HaXO)K)I.eHH5I qaCTHhlX
rrpOH3BO)I.HhIX
.
aL
or
ITpoH3BO)I.Hhle
.
aL
oz·
dL
dr' B CHJIY
H
CHCTeMhI
(10),
H
dL dz' B
dL
dz
CHJIY CHCTeMhI
(9),
HMeIOT BH)I. = aL + laL.
az
or
OTCIO)I.a MO)KHO HaHTH qaCTHhIe rrpOH3BO)I.Hhle
aL
or
aL
az
,l1,JI5I rrOJIyqeHH5I 3HaqeHHH
= ~dL
. dr + IdL~
dz
dL
= dL _ I
aL
or
. dz
H
aL
az
dr
/(1 + ,
/(1
12)
(11)
+ 12 ).
B Y3JIaX ceTKH Heo6xo)I.HMO 3aMeHHTh B
(11)
)I.H<P­
¢epeHlJ;HaJIhI COOTBeTcTByIOIII;HMH KOHeqHhIMH Pa3HOCT5IMH:
~~(i,j) = (L(i,j + 1) -
L(i,j))/(r(i,j
~~ (i,j) = (L(i + l,j) -
L(i,j))/(z(i
IIpH 3TOM 3HaqeHHe <PYHKlJ;HH
1 6epeTC5I B
+ 1) -
+ l,j)
r(i,j)),
- z(i,j)).
(i,j).
Br(i,j), Bz(i,j) B Y3JIaX ceTKH orrpe­
aL aL
or' az' L, h, I, r B rrpaBhIX qaCT5IX 6epYTc5I B
TOqKe
TaKHM 06Pa30M, KOMrrOHeHThI M a rHHTHoro rrOJI5I
,AeJI5IIOTC5I <p0pMYJIaMH
(6),
COOTBeTCTBYIOIII;HX TOqKaX
3aKJIIOQeHHe.
B
r)I.e <PYHKlJ;HH
(i, j).
pa60Te paCCMOTpeH MeTO)I. rrOCTpOeHH5I yrrpaBJI5IIOIII;HX MarHHTHhIX
rrOJIeH B lJ;HJIHH,ApHqeCKOH CHCTeMe KOOp)I.HHaT Ha OCHOBe rrO)I.XO)I.a
B. 11.
3y6oBa.
B
cJIyqae
aKCHaJIhHO-CHMMeTpHqHOrO MarHHTHorO rrOJI5I 3a)I.aqa rrocTpOeHH5I MarHHTHhIX rrOJIeH CBO­
,AHTC5I K peIlIeHHIO CHCTeMhI KBa3HJIHHeHHhlx )I.H<p<pepeHlJ;HaJIhHhIX ypaBHeHHM C qaCTHhIMH
IIPOH3BO)I.HhIMH rrepBoro rrOp5l)I.Ka.
,l1,JI5I rrOJIyqeHHOM CHCTeMhI CTaBHTC5I 3a,n:aqa rypca H
rrpe)I.JIaraeTC5I aJIrOpHTM ee peIlIeHH5I Ha OCHOBe MeTO)I.a Macco.
TaKHM 06Pa30M, HCrrOJIh­
3y51 rrpe)I.JIO)KeHHhIH MeTO)I. HaXO)K)I.eHH5I MarHHTHhIX rrOJIeH, MO)KHO pelliaTh pa3Hoo6pa3Hhle
3a)I.aqH <POKYCHPOBKH 3ap5l)KeHHhIX qaCTHlJ;.
Summary
Kotina E. D. The formation of beam dynamics of charched particles in the magnetic field.
The methods of construction of magnetic fields forming given beam dynamics are developed.
We consider this problem as the inverse problem of determination of magnetic field by preassigned
motion of charged particles.
81
1. rpu1t6epa r. A. Ih6paHHbIe BonpOCbI MaTeMaTll'leCKOH Teopllll ::meKTpllqeCKllX II MarHllTHbIX
HBJIeHllH. M .: Ih,ll,-BO AH CCCP, 1948. 536 C.
2. Meltzer B. Single-component stationary electron flow under space-charge conditions/ / J . of
Electronics. 1956. Vol. 7, N 1. P. 118-127.
.
3. Lucas A. R., Meltzer B. A general theorem for dense electron beams// J. of Electronics and
Control. 1957. · Vol. 4, N 2. P. 160-164.
4. 06'4.ap06 B. T. AKCllaJIbHO-CllMMeTpllqHbIe nyqKll 3a,n;aIlHOH <pOPMbI/ / ,ll;OKJI. AH CCOP.
1956. T . 107, N! 1. C.47-50.
f
'
.
5. 06'4.ap06 B. T. TeopllH <pOPMllPOBaHllH ,meKTpOHHbIX nyqKoB/ / Pa,n;lloTexHllKa II 3JIeKTpo­
HllKa. 1957. T. 2, N! 6. C . 696-704.
6. 3y606 B. H. ,ll;llHaMllKa ynpaBJIHeMbIX CHCTeM. M. : BbIcmaH mKOJIa, 1982. 285 c .
7. 3y606 B. H. KOJIe6aHHH H BOJIHbI. JI.: Ih,n;-Bo JIeHllHrp. YH-Ta, 1989. 415 c.
8. Kotina E. D. Beam dynamics formation in magnetic field/ / Proc. of EPAC-2002 . Paris,
France, 2002. P. 1264-1266.
9. Bepe3U1t H. C., )l(ua~o6 H. n. MeTo,n;bI BbIqHCJIeHllH:B 2 T. M.: <I>u3MaTrll3, 1962. T. 2.
639 c.
CTaTbH rrpe,lJ,CTaBJleHa K rry6mIKa~l:!l:! 'lJleHOM pe,lJ,KOJlJlerl:!l:!
CTaTbH rrocTyrrl:!Jla B pe,lJ,aK~l:!IO
7
l:!IOHH
2006
r.
O. M .
.n;Pl:!BOTl:!HbIM.
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