Pll-2005-121 tlN R-Pi3-Ji)S-IJI

E B EeCI1aJIbKO c A MHxeeB B TI UBeTKOB AH UHpyJIeB HB TIy3bIHHH

BbfllHCllEHHE HblOTOHOBCKoro TIOTEHUHAJIA

fPABHTHPYIOIUEH KOHltlgtHIYP AUHH

C TIOBEPXHOCTblO EJIH3KOH K CltlgtEPOHUY

C TIOMOlllblO CYlMBOJIbHbIX

H lJHCJIEHHbIX METOnOB

HanpaBJleHO B IKYPHaJI laquoBeCTHHK PoccHiicKOro YHHBepCHTeTa

IlPyJK6bI Hap0llOBraquo cepHSl laquollPHKJIa)lHaSI H KOMnblOTepHaSI MaTeMaTHKaraquo

TBepCKOH rOCYIlapCTBeHHblH YHHBepCHTeT

2005

BBEuEHHE

tl3BecTHo qTO 3araqa 0 nOTeHllHaJIax OllHOpOllHOrO 3JUlHnCOHlla B03HHKJIa

nepBOHaqaJIbHO KaK npellMeT TeOpHH nITOTeHHS Be peIlleHHe n03llHee HaIlIJ10

npHMeHeHHe B pa3JlHqHblX qH3HqeCKHX npHnO)l(eHHSx HanpHMep qoPMYnbl llJ1S

HblOTOHOBCKoro rpaBHTallHOHHoro nOTeHIlHaJIa qgt Hcnonb3YIOTCS B mnPOllHHaMHKe

B 3araqaX 0 nOTeHIlHaJIbHOM TeqeHHH HeC)I(HMaeMOH HlleaJIbHOH )l(HllKOCTH BOKpyr

3nnHnCOHlla 0 CHne conpoTHsneHHs lleHcTBYIOIIleH Ha MellJ1eHHO llBH)I(YIIlHHCs B

BS3KOH )l(HllKOCTH 3nnHnCOHll H Tll IIpHqeM B 3araqax 0 KOHqHrypaIlHSX rpashy

BHTHPYIOIIlHX CHCTeM He06xollHMO aHaJIHTHqeCKOe npellCTasneHHe BHYTpeHHero

nOTeHIlHaJIa a B 3araqaX CBS3aHHbIX C onHcaHHeM llBH)I(eHHs OllHOH CHCTeMbl

OTHOCHTenbHO llpyroH - BHeIllHero nOTeHIlHaJIa TaK KaK nOTeHIlHaJI BXOllHT B

ypaBHeHHs onpellensIOIIlHe KOHqHrypaIlHIO H B ypaBHeHHs onHCblBalOIIlHe llHHashy

MHKy rpaBHTHpYIOlIlHX Macc

B TeopHH HblOTOHOBCKoro rpaBHTallHOHHoro nOTeHIlHaJIa B03HHKaeT TpH THna

3araq no TpeM THnaM cHMMeTpHH HaH60nee npocToH cnyqaH KOrlla KOHqHrypashy

1lHs He BpamaeTCS TOrlla OHa 6YlleT HMeTb cqepHqeCKH-CHMMeTpHqHYIO q0PMY

3TOT cnyqaH HaH60nee XOPOIllO H3yqeH [1] Ho qaCTO He06xollHMO yqHTbIshy

BaTb BpaIIleHHe TOrlla KOHqHrypallHs 6YlleT HMeTb cqepHqeCKH-HecHMMeTpHqHYIO

qoPMY [2] B 3TOM cnyqae MbI npHxOllHM K He06xollHMOCTH Hcnonb30BaTb 60nee

cnmKHbIe nOBepXHOCTH B qaCTHOCTH 3nnHnCOHllaJIbHbIe rpaHHlla KOTOPbIX npellshy

cTasnSeT 3MHncOHll [34] l-l3yqeHHeM 3nnHnCOHllaJIbHbIX qHryp paBHoBecHS

3aHHMaJIHCb MaKnOpeH HK06H JISnYHoB H MHorHe llpyrne I1MH 6bInH nonyqeHbI

TOqHbIe aHaJIHTHqeCKHe npellCTasneHHSI nOTeHIlHaJIa qgt B cnyqae KOflla nnOTHOCTb

npellCTasneHa B BHlle CTeneHHbIX qYHKIlHH KOOpllHHaT BHYTpeHHHH nOTeHIlHaJI

SIBnSleTCS TOrlla MHorotmeHoM no CTeneHSM KOOpllHHaT K03qqHIlHeHTbI KOTOPOro

npellCTaBnSlIOT C060H 3nnHnTHqeCKHe HHTerpaJIbI

OllHaKO B peaJIbHblX cnyqasx llnSl HeOllHOp0llHbIX KOHqmrypallHH B03HHKaeT

3araqa B KOTOPOH nOBepXHOCTb KOHqmrypallHH npellCTasnseT H3 ce6S1 60nee

cnO)l(HYIO CTPYKTYPY B pa60Te [5] npellJ10)l(eH MeTOll annpOKCHMallHH nOBepxHoshy

CTH nCeBllOnoBepxHocTblO a HMeHHO B03MYIIleHHoH 3nnHnCOHllaJIbHOH nOBepXHOshy

CTblO napaMeTpbI KOTOPOH onpelleflSlIOTcS H3 ycnoBHS MHHHMYMa KBallpaTa nnOTshy

HOCTH Ha 3TOH nOBepxHocTHbull llns 3THX lleneH 3qqeKTHBHo MO)l(eT 6b1Tb Hcnonbshy

30BaH MeTOll pa3JIO)l(eHHs B pSll EypMaHa-JIarpaH)I(a no MaJIOMY napaMeTpy

3allaqa BblQHCneHHS HblOTOHOBCKoro nOTeHIlHaJIa B03MYIIleHHbix 3nnHnCOHshy

llaJIbHbIX KOHqHrypaIlHH aKTYaJIbHa B CBSI3H C H3yqeHHeM HenHHeHHbIx 3qqeKToB

B aCTpoq1I3HKe reoqH3HKe II Tll OHa npellCTasnseT C060H TpyllHylO MaTeMaTHshy

QecKYIO 3araqy [7]

1

BBEuEHHE

tl3BecTHo qTO 3araqa 0 nOTeHllHaJIax OllHOpOllHOrO 3JUlHnCOHlla B03HHKJIa

nepBOHaqaJIbHO KaK npellMeT TeOpHH nITOTeHHS Be peIlleHHe n03llHee HaIlIJ10

npHMeHeHHe B pa3JlHqHblX qH3HqeCKHX npHnO)l(eHHSx HanpHMep qoPMYnbl llJ1S

HblOTOHOBCKoro rpaBHTallHOHHoro nOTeHIlHaJIa qgt Hcnonb3YIOTCS B mnPOllHHaMHKe

B 3araqaX 0 nOTeHIlHaJIbHOM TeqeHHH HeC)I(HMaeMOH HlleaJIbHOH )l(HllKOCTH BOKpyr

3nnHnCOHlla 0 CHne conpoTHsneHHs lleHcTBYIOIIleH Ha MellJ1eHHO llBH)I(YIIlHHCs B

BS3KOH )l(HllKOCTH 3nnHnCOHll H Tll IIpHqeM B 3araqax 0 KOHqHrypaIlHSX rpashy

BHTHPYIOIIlHX CHCTeM He06xollHMO aHaJIHTHqeCKOe npellCTasneHHe BHYTpeHHero

nOTeHIlHaJIa a B 3araqaX CBS3aHHbIX C onHcaHHeM llBH)I(eHHs OllHOH CHCTeMbl

OTHOCHTenbHO llpyroH - BHeIllHero nOTeHIlHaJIa TaK KaK nOTeHIlHaJI BXOllHT B

ypaBHeHHs onpellensIOIIlHe KOHqHrypaIlHIO H B ypaBHeHHs onHCblBalOIIlHe llHHashy

MHKy rpaBHTHpYIOlIlHX Macc

B TeopHH HblOTOHOBCKoro rpaBHTallHOHHoro nOTeHIlHaJIa B03HHKaeT TpH THna

3araq no TpeM THnaM cHMMeTpHH HaH60nee npocToH cnyqaH KOrlla KOHqHrypashy

1lHs He BpamaeTCS TOrlla OHa 6YlleT HMeTb cqepHqeCKH-CHMMeTpHqHYIO q0PMY

3TOT cnyqaH HaH60nee XOPOIllO H3yqeH [1] Ho qaCTO He06xollHMO yqHTbIshy

BaTb BpaIIleHHe TOrlla KOHqHrypallHs 6YlleT HMeTb cqepHqeCKH-HecHMMeTpHqHYIO

qoPMY [2] B 3TOM cnyqae MbI npHxOllHM K He06xollHMOCTH Hcnonb30BaTb 60nee

cnmKHbIe nOBepXHOCTH B qaCTHOCTH 3nnHnCOHllaJIbHbIe rpaHHlla KOTOPbIX npellshy

cTasnSeT 3MHncOHll [34] l-l3yqeHHeM 3nnHnCOHllaJIbHbIX qHryp paBHoBecHS

3aHHMaJIHCb MaKnOpeH HK06H JISnYHoB H MHorHe llpyrne I1MH 6bInH nonyqeHbI

TOqHbIe aHaJIHTHqeCKHe npellCTasneHHSI nOTeHIlHaJIa qgt B cnyqae KOflla nnOTHOCTb

npellCTasneHa B BHlle CTeneHHbIX qYHKIlHH KOOpllHHaT BHYTpeHHHH nOTeHIlHaJI

SIBnSleTCS TOrlla MHorotmeHoM no CTeneHSM KOOpllHHaT K03qqHIlHeHTbI KOTOPOro

npellCTaBnSlIOT C060H 3nnHnTHqeCKHe HHTerpaJIbI

OllHaKO B peaJIbHblX cnyqasx llnSl HeOllHOp0llHbIX KOHqmrypallHH B03HHKaeT

3araqa B KOTOPOH nOBepXHOCTb KOHqmrypallHH npellCTasnseT H3 ce6S1 60nee

cnO)l(HYIO CTPYKTYPY B pa60Te [5] npellJ10)l(eH MeTOll annpOKCHMallHH nOBepxHoshy

CTH nCeBllOnoBepxHocTblO a HMeHHO B03MYIIleHHoH 3nnHnCOHllaJIbHOH nOBepXHOshy

CTblO napaMeTpbI KOTOPOH onpelleflSlIOTcS H3 ycnoBHS MHHHMYMa KBallpaTa nnOTshy

HOCTH Ha 3TOH nOBepxHocTHbull llns 3THX lleneH 3qqeKTHBHo MO)l(eT 6b1Tb Hcnonbshy

30BaH MeTOll pa3JIO)l(eHHs B pSll EypMaHa-JIarpaH)I(a no MaJIOMY napaMeTpy

3allaqa BblQHCneHHS HblOTOHOBCKoro nOTeHIlHaJIa B03MYIIleHHbix 3nnHnCOHshy

llaJIbHbIX KOHqHrypaIlHH aKTYaJIbHa B CBSI3H C H3yqeHHeM HenHHeHHbIx 3qqeKToB

B aCTpoq1I3HKe reoqH3HKe II Tll OHa npellCTasnseT C060H TpyllHylO MaTeMaTHshy

QecKYIO 3araqy [7]

1

BBHIl) qpe3BbIqaHHOH CnO)KHOCTH KaK aHaJIHTHqeC1CHX TaK H qHCneHHbIX

paCqeTOB B03HHKaeT Heo6xonHMOCTb HCnOnb30BaHIDI KOMnblOTepHbIX MeTOnOB C nOH uenblO HaMH 6bIJ1 BbI6paH naKeT CHMBOnbHOH H qHCneHHOH MaTeMaTHKH

MAPLE B qa~THOCTH onpo6oBaHHbIH B pa60Te [1 OJ B HacToslIueH pa60Te pa3BHBaeTcSI MeTOn TOqHOro npencTaBJIeHHSI HbIOTOHOBshy

CKoro nOTeHUHaJIa KOHcpHIypauHH C nOBepxHocTblO 6nH3KoH K 3nnHnCOHl(aJIbshy

HOH B BHl(e a6comoTHo CXODSImHXCSI PSlnOB Ha OCHOBe CHMBonbHbIX H qHCneHHbIX

MeTonOB BblqHCneHIDI Ha KOMnblOTepe Bce aHaJIHTHqeCKHe npeo6pa30BaHIDI npHshy

BODSITCSI C MaKCHMaJIbHO B03MO)KHOH CTeneHblO nonpo6HOCTH B OTnHqHe OT paHee

ony6nHKoBaHHblx HaMH pa60T [10] 3anaqy BblqHCneHHSI HblOTOHOBCKOro rpaBHTaUHoHHoro nOTeHUHaJIa tP MO)KHO

pa36HTb Ha nBe 3anaqH OTbICKaHHe ero Ha BHemHlO1O TOqKY tPout H 3anaqy OTbICshy

KaHHSI nOTeHUHaJIa Ha BHYTpeHHlO1O TOqKY tPin [lJ HblOTOHOBCKHH rpaBHTauHoHHbIH nOTHUHaJI tP npH 3TOM ynOBJIeTBOpSieT ypaBshy

HeHHSlM

PemeHHeM 3THX ypaBHeHHH SlBJISleTCSI

tP = -Gf p(T )dV (1) ~r - ~IrI

D

rne D 06naCTb 3aHHMaeMaSI KOHcpHrypaUHeH

1 BbI60P nAPAMETPOB B03MYIIJEHHOH 3JIJIHnCOHnAJIbHOH nOBEPXHOCTH H METOJ( PAAOB JiYPMAHA-JIArPAHIKA

rpaHHua KOHqmrypaUHH E HaxonHTCSI H3 ycnoBHSI paBeHcTBa UJ10THOCTH Ha

rpaHHue HYnlO

p(x y z) = 0 (x y z) E E (2)

rpaBHTaUHOHHbIH nOTeHUHaJI tP SlBHO 3aBHCHT OT CPOPMbI rpaHHUbl (2) B SlBshy

HOM BHl(e aHaJIHTHqeCKH ero ynaeTCSI BbIqHCnHTb TonbKO )l1UI npOCTeHmHX CPOPM

nOBepXHOCTH (map 3J1J1HnCOHl() TI03TOMY B 06meM cnyqae TOqHYIO rpaHHUY KOHshy

cpHrypaUHH E MbI 3aMeHHM nceBnorpaHHueH 8D cjJopMa KOTOPOH 3aBHCHT OT

HeH3BeCTHblX nOKa napaMeTPoB Zijk

TIoTeHUHaJI tP 6yneT ynOBneTBOpSlTb Ha rpaHHue ycnoBIDIM tPoutI = tPinI H

(V4-out h (V4-inh~middot

2

CnellyS pa60TaM [6] Bhl6epeM nOBepXHOCTh ~ B BHIe B03MymeHHoH 3JU1Hshy

nco~anhHoH nOBepXHOCTH

(3)

x y z rne Xl X2 = - X3 aI a3 - nonYOCH 3JUIHnCO~a Bpameshy

al al a3 HHS KOTophle HapsIly C Zijk napaMeTpH3YIOT 8D L - MaKCHManhHU CTeneHh

a3 MHoroqneHa no KoopnHHaTaM XtX2X3 napaMeTp cnmOCHYTOCTH e = -

al

YcnOBHS 6nH30cTH (2) H (3) MO)fHO ccpoPMYnHpOBaTh BBeneHHeM CPYHKUHO-

Hana A

(4)

OqeB~HO napaMeTp TJ~D = A12 6yneT npenCTaWUlTh Mepy norpelllHOCTH B

HalllHX ypaBHeHIDIX npH 3aMeHe TOqHOH nOBepXHOCTH KOHcpHrypaUHH E Ha 8D non nceBnonOBepXHOCThlO 8D 6yneM nOHHMaTh nOBepXHOCTh Ha KOTOPOH cpenshy

Hee 3HaqeHlIe KBanpaTa IUIOTHOCTH He npeBocxonHT nOCTaTOqHO ManOH BenHqHHhl

no cpaBHeHHIO C enHHHueH 3Ta BenHqHHa onpenenseT TOqHOCTh pellleHHsI nOCTashy

BJIeHHOH 3anaqH

Y cnOBHe MHHHMMa A npHBonHT K CHCTeMe anre6paHqeCKHX ypaBHeHHH OTshy

HOCHTenhHO at a3 H Zijk

(5)

a l12 = a3 -aA(Zijk = 0) = O

a3

npenCTaBHM nnOTHOCTh KOHcpHrypaUHH p B BHne nonHHOMa CTeneHH P

p

p(P) = L PabcXIX~X3 (6) abc

EcnH Bhl6paTh P nOCTaTOqHO 60nhlllHM TO C BhlCOKOH CTeneHhlO TOqHOCTH

BhIpa)feHHe (6) 6yneT annpoKCHMHpOBaTh nnOTHOCTh peanhHoH KOHcpHrypaUHH

3

CnellyS pa60TaM [6] Bhl6epeM nOBepXHOCTh ~ B BHIe B03MymeHHoH 3JU1Hshy

nco~anhHoH nOBepXHOCTH

(3)

x y z rne Xl X2 = - X3 aI a3 - nonYOCH 3JUIHnCO~a Bpameshy

al al a3 HHS KOTophle HapsIly C Zijk napaMeTpH3YIOT 8D L - MaKCHManhHU CTeneHh

a3 MHoroqneHa no KoopnHHaTaM XtX2X3 napaMeTp cnmOCHYTOCTH e = -

al

YcnOBHS 6nH30cTH (2) H (3) MO)fHO ccpoPMYnHpOBaTh BBeneHHeM CPYHKUHO-

Hana A

(4)

OqeB~HO napaMeTp TJ~D = A12 6yneT npenCTaWUlTh Mepy norpelllHOCTH B

HalllHX ypaBHeHIDIX npH 3aMeHe TOqHOH nOBepXHOCTH KOHcpHrypaUHH E Ha 8D non nceBnonOBepXHOCThlO 8D 6yneM nOHHMaTh nOBepXHOCTh Ha KOTOPOH cpenshy

Hee 3HaqeHlIe KBanpaTa IUIOTHOCTH He npeBocxonHT nOCTaTOqHO ManOH BenHqHHhl

no cpaBHeHHIO C enHHHueH 3Ta BenHqHHa onpenenseT TOqHOCTh pellleHHsI nOCTashy

BJIeHHOH 3anaqH

Y cnOBHe MHHHMMa A npHBonHT K CHCTeMe anre6paHqeCKHX ypaBHeHHH OTshy

HOCHTenhHO at a3 H Zijk

(5)

a l12 = a3 -aA(Zijk = 0) = O

a3

npenCTaBHM nnOTHOCTh KOHcpHrypaUHH p B BHne nonHHOMa CTeneHH P

p

p(P) = L PabcXIX~X3 (6) abc

EcnH Bhl6paTh P nOCTaTOqHO 60nhlllHM TO C BhlCOKOH CTeneHhlO TOqHOCTH

BhIpa)feHHe (6) 6yneT annpoKCHMHpOBaTh nnOTHOCTh peanhHoH KOHcpHrypaUHH

3

IIocJIe nepexo)la K CltPepHqecKHM KOOp)lHHaTaM R 6 ltgt Xk = Rak al sin (J cos ltgt a2 sin (J sin ltgt a3 = cos (J BLlpaJKeHHe llJISI B03MllleHHOH 3JIJIHshy

nCOHJlaJILHOH nOBepxHocTH (3) npHMeT BHJl

R 1 (7)

IlaJILHeHmHe BLIqHCJIeHIDI 6yuyr OCHOBaHLI Ha HCnOJIL30BaHHH BapHaHTa Teoshy

peMLI JIarpaH)Ka [7] TeopeM3 [7] IIyCTL J(z) H w(z) - aHaJIHTHqeCKHe ltPYHKIJHH Z Ha KOHTYPe

C oKp~aIOllleM TOqKY a H BHYTpH Hero )]1)]1 Bcex Z Ha C BLIllOJIIDIeTCSI HepaBeHshy

CTBO KW(Z)C laquo Iz - ale Tor)la ypaBHeHHe Z = a + KW(Z) HMeeT O)lHH KopeHL

Z ~ BHYTPH C H J(~) pa3JIaraeTCsJ B CTeneHHOH a6COJIIOTHO CXOJlsIlllHHCs psJ)l

00 8 ds - I

J(~) = J(a) + L das- I [f(a)w(a)S] (8) s=1

psJ)l (8) nOJIyqHJI B JIHTepaTYPe Ha3BaHHe pjfJla EypMaHa-JIarpaH)Ka

B HameM cJIyqae ~ = R J(~) = ~h+2 a = 1 w(a) = w(a Zijk ak) Tor)la nOJIyqHM

DOJIee )leTaJILHO (9a) 6Y)leT BLlrJIjfJleTL CJIenyIOlllHM 06Pa30M

s=oo sL ( l)S ds-I i+j+k+h+1 h+2 (h 2) L L - Z () -i -1 -k a (9b)a + + --- ijk S 01 U2a 3 -dI ( )

S as - a + 1 s s=1 ijk

r)le Zijk (s) - MHorOqJIeH OT Zijk CTeneHH He npeBOCXOJlsJllleH S

ds- I ai+j+k+h+1 PaCCMOTpHM 60JIee nO)lp06HO -dI ( 1) IIocne 3aMeHLI a = y + 1 as - a + s

1 ds- I (1 + y)i+j+k+h+1 1 (y 6JIH3KO K HYJIIO) nOJIyqaeM --dI Y y=o= -middotltPs

2s 2syS- (1 + _)S 2

IlnsJ onpe)leneHIDI sJBHOro BHJla ltPs nOHazto6HTCsJ COOTHomeHHe [9]

a(l~y)b ] lta) =rr (b+1-2r)ltPs S(a b) [ (1 + - )a+1 r=l

2 y=O

r)le a S 1 b = i + j + k + h + 1

4

H3 Hero It (9b) cnelleT

8=00 sL (_1)Sh8 (s) Rh = L 2 I Zijk(8)amp1~amp~ps(i + j + k +hI) (9c)

8=0 ijk 8

me 8(8) = 0 nplt 8 = 0 U 8(8) = 1 npu 8 gt O ps 1 npu 8 = 0 U 8 = 1 KBanpaT nnOTHOCTU 6y)]eT BblrJUI)]eTb cnelllOmuM 06pa30M

p p

2(p) ~ ~ a+a b+b c+cP 6 6 PabcPalbcxl x 2 x3 (10) abc abc

TaK KaK Mbl nepeIilllU K cltPepUqeCKUM KOOp)]UHaTaM TO C yqeToM (9c) BbIshy

pIDKeHlte J1JUI KBanpaTa llJlOTHOCTU npUMeT BU)1

00 P P 8L ( 1)Sh8(s)

p2 (P) = L L 2 L - I I X

8=0 abc abc ijk 8

() -a+a +i -b+b +j -c+c+k m (h k 1)X PabcPalbc Z ijk 8 a l a 2 a 3 X 1s I + ~ + J + - (11)

r)]e hI = a + a + b + b + c + c YqUTblBWI (4) U (11) A MOXHO npe)]cTaBuTb B 51BHOM BU)1e y)]epxuBWI B

pa3JlOXeHUU A qneHbl Zijk CTeneHU He Bblme 8 m bull r)]e 8 m - MaKCUMaJIbHO yqushy

TblBaeMWI CTeneHb no 8

X (hI + i + j + k - 1)QAtA 2A3 (12)

- -AI -A2 -A3d(QAlA2A3 = al a 2 a 3 u (13)

r)]e Al =a+a+i A2 =b+b+j Al =c+c+k HHTerpaJIbI (13) MOryr 6blTb ynpomeHbl

- 2n(a + a + i l)(b + b + j l)(c + c + k - I)Qa+a +ib+b+jc+c+k ~-------(-hl-+-i-+-J-+-k---l)------- shy

(14) TpeXMepHble MaCCUBbl HeU3BeCTHblX B CUCTeMe (5) Zijk 0603HaqUM qepe3

Ym (m 12 N 1)

TIpu P = 6 L 2 ItMeeM NI 5 CB5I3b MeJKJ1j nepeMeHHbIMu 6y)]eT

BblrJl5l)]eTb cneJ1YlOmuM 06pa30M

al YI = TY2 = eY3 = ZOO2Y4 = Z020Y5 = Z200

5

rlle l - xapaKTepHLIH MaCllITa6 paCIIpelleJIeHHSI Macc rpaBHTHpyromeH CHCTeMLI

B 3TOM cnyqae CHcTeMY ypaBHeHHH (5) YIl06HO 3aIIHcaTL B BeKTopHoM BHIle

f(y e) 0 (Yb Y2 middotmiddotmiddotYs) (15)

)lruI IHCJIeHHoro pellIeHHSI CHCTeMLI (15) HCIIOJIL30BaH peryJISlpH30BaHHLIH

aHaJIOr MeTOlla HLIOToHa [8] C IIapaMeTPoM peryJISlpH3aUHH a = 10-3

Y(n+l)(e) y(n)(e) - Tn [af2(y(n) (e) e) + T (y(n) (e) e) (y(n)(e) e)]-l x

x T(y(n)(e)e)f(y(n)(e)e) (16)

me n - HOMep HTepauHH Tn laquo(O ~ Tn ~ 1) - OIITHMaJILHLIH llIar Ha n-H HTeshy

paUHH pHIeM H3 coo6paxeHHH CXOllHMOCTH (o ~ 01 (y(n) (e) e) - Mashy

TPHUa sIKo6H l(y(n) (e) e) - TpaHcIIoHHpoBaHHasI MaTPHua sIKo6H BeJIHIHHa

f2(y(n) (e) e) = 8n(T) IIpellCTaBJISleT C060H KBanpaT HeBSl3KH OIITHMaJILHLIH llIar

8n(0) )Tn = max ( () 8 (0) + 8 (1) () ~ 01n n

HaMH BLI6paH HMeHHO peryJISlpH30BaHHLIH aHaJIOr MeTOlla HLIOTOHa TaK KaK

HeYIlaIHLIH BLI60p HaIaJILHoro IIpH6JIHJKeHHSI MOJKeT pHBeCTH K TOMY ITO Mashy

TpHua sIKo6H 6YIleT IIJIOXO 06YCJIOBJIeHHOH a B 3TOM CJIyqae MeTOll HLIOTOHa pacshy

XOllHTCSI HaIaJILHOe IIpH6JIHJKeHHe BLI6HPaJIOCL cOOTBeTcTBYlOmHM cqepHIeCKHshy

cHMMeTPHIHOMY cnyqalO

PaCIIpelleJIeHHe IIJIOTHOCTH CIHTaeM 3anaHHLIM T e BLI6HpaeM KOHKpeTHLle

(a+b) 2 a+b+c c

Pabc = (17)G) (DtO+bCYI Y2

al a3 M ~ me Yl = -l Y2 e = bl B3SnH aKCHaJIbHO-CHMMeTpHIHbIH cnyqaH

al a2 al K03qqHUHeHTLI Pa+bc IlJISI P = 6 pHBelleHLI B Ta6nl H B3S1TLI

H3 [10] IlJISI KOHKpeTHoH KOHqmypaUHH COOTBeTCTBYlOmeH MOlleJIH 6LIcTpoBpashy

malOmeHCg HeiiTpoHHoH 3Be31lLI C ypaBHeHHeM COCTOSlHHSI EeTe-llJKoHcoHa

Ta6JmQa 1

0 0 0 2 2 2 4 4a+b 0 6 4 0 40 2 6 2 0c 2 0

0634 -0912 -0690 12941 -0729 -2765 0683 -2845 -0996Pa+bc

BLI6paHHasI HLIOTOHOBCKasI cxeMa 6LIna peaJIH30BaHa B paMKaX IIneTa

MAPLE OHa OKa3aJIaCL IlOCTaTOIHO 3qqeKTHBHOH Tn KaK IHCJIO HTepaUHH

IlJISI IlOCTHJKeHHSI HeBSl3KH 10-30 He IIpeBLICHJIO 10

6

B pe3ynbTaTe nonyqeHbIqHCneHHbIe 3HaqeHIDI Zijk H Yl Y2 OHH npHBelleHbI

B Ta6n 2

DJUI Pa+be npHBelleHHbIx B Ta6n 1 8D 6YlleT nOpjIllKa 10-6 bull

Z020

646815middot10 646815middot10

2 BHYTPEHHHH rPABHTAUHOHHbIH nOTEHUHAJI ltPin(PL)

B pa60Te [9] C Hcnonb30BaHHeM PjIlJOB EypMaHa-narpaHxa llOKa3aHa Teoshy

peMa comaCHO KOTOPOH BHyrpeHHHH rpaBHTaUHOHHbIH nOTeHUHan ltPin MoxeT

6bITb npellCTaBneH a6COnIOTHO H paBHOMepHO CXOlJjllI(HMCjI PjlllOM no K03CPCPHUHshy

eHTaM Zijk npH HeKOTopbIX orpaHHqeHHjlX Ha Zijk H K03cpcpHUHeHTbI npH HHX

KOTopble jlBnjlIOTCjI nOnHHOMaMH CTeneHH P + s(L - 2) + 2 KOOpllHHaT Xk 3lleCb

s - HOMep qneHa pjllJa EypMaHa-narpaHxa

nOcne nepeXOlla K 06061I(eHHbIM ccpepHqeCKHM KOOpllHHaTaM R () cp BnepBble npennOxeHHblM B pa60Te [6] co CMelI(eHHeM ueHTPa KOOpllHHaT B TOqKY

Ha6nIOlleHHjI Xk HMeeM

- 1 (aD) 1 (aD) () Xk = Xk + akR al = - - sin()coscp a2 = - - SIn SIncp a al a al

O~R~R

B 3TOM cnyqae BblpaxeHHe ~ B03M)1I(eHHOH 3nnHnco~anbHoH nOBepXHOshy

CTH (3) nepenHWeTCjI B cnenyIOlI(eM B~e

(19)

me 3

Q=l-Lx~ k=l

7

PeWHB KBaIlpaTHOe ypaBHeHHe OTHOCHTeJIbHO R B JIeBOH lJaCTH (19) nOJIyqHM

ypaBHeHHe l1)1S1 HaXO)J(lJeHIDI R

3JIeMeHT 06beMa HHTerpHpoBaHH51 B HOBbIX KoopnHHaTax 6yneT paBeH

dV

nanbHeHWHe BbIlJHCJIeHmI 6yDYT OCHOBaHbI KaK H B cJIyqae C A Ha HCnOJIbshy

30BaHHH TeopeMbI JIarpaHJKa

ComaCHO (8) B HaweM CJIyqae e= R f(() = eh+2 a = Ro l1(a) =

l1(R01 Ok Xk Zijk)

Torna nOJIyqHM

llJUI COKpallleHIDI 3anHCH 3neCb H nanee HCnOJIb3yeTcSI BBeneHHbIH B [6] oneshy

nl n2 nm 1 ~ paTOp CyMMHpOBaHHSI kl k2 k neHCTBHe KOToporo 3aKJIIOlJaeTCSI B [ m CyMMHpOBaHHH no Ka)J(lJOMj HHJKHeMY HuneKCY OT HYJISI no 3HalJeHHSI cToSIlllero

HaIl HHM HuneKca C onHOBpeMeHHbIM yMHOJKeHHeM Ha COOTBeTCTBYIOIllHe 6HHOMHshy

anbHbIe K03CPCPHIJHeHTbI

1 ds - [n+m+n+l+l]JlJISI onpeneJIeHHSI SIBHoro BH)(a dR~-l (Ro +T + U)s npOH3BeneM 3ashy

MeHY Ro U - T + yU dRo = U dy H BOCnOJIb3yeMcSI paHee npHBeneHHoH

8

JIeMMoii

ds- 1 [~+m+n+l+1 ] = d~-l (Ro + T + U)s

Ro=U-T+yU s 1

= [ h + n + m + 1+ Ix ] (-I)JJ u h+n+mH-2(s-1)-JJTJJ d - X

XJl 2s dys-l

X [(1 + y)h+n+m+I-JJ+l] = (1 + ~)s

= [ h + n + m + 1+ Ix ] (-l)JJ uh+n+m+I-2(s-1)-JJTJJ~ X XJl 2s s

x(h+m+n+l Jl+I)

C yqeTOM nOJIyqeHHoro pe3YJIhTaTa R npHMeT BHlI

00 sL ( 1)8+JJ Rh+2 = ~+2 + (h + 2) 2 2 - S Zijk(S) X

8=1 ijk

j k h+n+m+l+1X] X [ xn m 1 Jl X

X x~-n~-mx~-lo~o2o~uh+n+m+I-2(s-I)-JJTJJ~s (22)

BbIpIDKeHHeIJlUl pacrrpeneneHIUI IIJIOTHOCTH (6) B HOBO CHCTeMe KOOpnHHaT

npHMeT BHlI

(P) - [a bel xa-dxb-Ixc-9R-d+I+9rvdJ _0 (23)P = LJ Pabc d f 9 1 2 3 A 1 A2 USmiddot

abc

BhIpIDKeHHe IJlUI HhlOTOHOBCKOro rpaBHTaIlHOHHOro rrOTeHIlHana (1) B HOBhIX

KOOpnHHaTax C yqeToM (23) rrpHMeT BHlI

9

TrumM o6pa30M (24) MO)ICHO rrpellCTaBHTb B BHIle

00

~(PL) = - Ga6(Fa + LF8 ) (25) 8=1

00

rlle Fa H L F8 COOTBeTCTBeHHO rrepBblH H BTOPOH lJJIeHbl B (24) Fa - lJJIeH 8=1

aHaJIHTHQecKoro npellCTaBJIeHHsI BHyrpeHHero nOTeHUHaJIa COOTBeTCTByromHH Heshy

B03MYllleHHOMY aMHrrcoHllYbull

ILruI sIBHOro rrpellCTaBJIeHHsI Fa Bocnonb3yeMCsI ~OpMYnOH 6HHoMa HbIOTOHa

IlnsI Rh+2

h + 2 - 2v A v T e] x

ApT e X

10

rlle l - xapaKTepHLIH MaCllITa6 paCIIpelleJIeHHSI Macc rpaBHTHpyromeH CHCTeMLI

B 3TOM cnyqae CHcTeMY ypaBHeHHH (5) YIl06HO 3aIIHcaTL B BeKTopHoM BHIle

f(y e) 0 (Yb Y2 middotmiddotmiddotYs) (15)

)lruI IHCJIeHHoro pellIeHHSI CHCTeMLI (15) HCIIOJIL30BaH peryJISlpH30BaHHLIH

aHaJIOr MeTOlla HLIOToHa [8] C IIapaMeTPoM peryJISlpH3aUHH a = 10-3

Y(n+l)(e) y(n)(e) - Tn [af2(y(n) (e) e) + T (y(n) (e) e) (y(n)(e) e)]-l x

x T(y(n)(e)e)f(y(n)(e)e) (16)

me n - HOMep HTepauHH Tn laquo(O ~ Tn ~ 1) - OIITHMaJILHLIH llIar Ha n-H HTeshy

paUHH pHIeM H3 coo6paxeHHH CXOllHMOCTH (o ~ 01 (y(n) (e) e) - Mashy

TPHUa sIKo6H l(y(n) (e) e) - TpaHcIIoHHpoBaHHasI MaTPHua sIKo6H BeJIHIHHa

f2(y(n) (e) e) = 8n(T) IIpellCTaBJISleT C060H KBanpaT HeBSl3KH OIITHMaJILHLIH llIar

8n(0) )Tn = max ( () 8 (0) + 8 (1) () ~ 01n n

HaMH BLI6paH HMeHHO peryJISlpH30BaHHLIH aHaJIOr MeTOlla HLIOTOHa TaK KaK

HeYIlaIHLIH BLI60p HaIaJILHoro IIpH6JIHJKeHHSI MOJKeT pHBeCTH K TOMY ITO Mashy

TpHua sIKo6H 6YIleT IIJIOXO 06YCJIOBJIeHHOH a B 3TOM CJIyqae MeTOll HLIOTOHa pacshy

XOllHTCSI HaIaJILHOe IIpH6JIHJKeHHe BLI6HPaJIOCL cOOTBeTcTBYlOmHM cqepHIeCKHshy

cHMMeTPHIHOMY cnyqalO

PaCIIpelleJIeHHe IIJIOTHOCTH CIHTaeM 3anaHHLIM T e BLI6HpaeM KOHKpeTHLle

(a+b) 2 a+b+c c

Pabc = (17)G) (DtO+bCYI Y2

al a3 M ~ me Yl = -l Y2 e = bl B3SnH aKCHaJIbHO-CHMMeTpHIHbIH cnyqaH

al a2 al K03qqHUHeHTLI Pa+bc IlJISI P = 6 pHBelleHLI B Ta6nl H B3S1TLI

H3 [10] IlJISI KOHKpeTHoH KOHqmypaUHH COOTBeTCTBYlOmeH MOlleJIH 6LIcTpoBpashy

malOmeHCg HeiiTpoHHoH 3Be31lLI C ypaBHeHHeM COCTOSlHHSI EeTe-llJKoHcoHa

Ta6JmQa 1

0 0 0 2 2 2 4 4a+b 0 6 4 0 40 2 6 2 0c 2 0

0634 -0912 -0690 12941 -0729 -2765 0683 -2845 -0996Pa+bc

BLI6paHHasI HLIOTOHOBCKasI cxeMa 6LIna peaJIH30BaHa B paMKaX IIneTa

MAPLE OHa OKa3aJIaCL IlOCTaTOIHO 3qqeKTHBHOH Tn KaK IHCJIO HTepaUHH

IlJISI IlOCTHJKeHHSI HeBSl3KH 10-30 He IIpeBLICHJIO 10

6

B pe3ynbTaTe nonyqeHbIqHCneHHbIe 3HaqeHIDI Zijk H Yl Y2 OHH npHBelleHbI

B Ta6n 2

DJUI Pa+be npHBelleHHbIx B Ta6n 1 8D 6YlleT nOpjIllKa 10-6 bull

Z020

646815middot10 646815middot10

2 BHYTPEHHHH rPABHTAUHOHHbIH nOTEHUHAJI ltPin(PL)

B pa60Te [9] C Hcnonb30BaHHeM PjIlJOB EypMaHa-narpaHxa llOKa3aHa Teoshy

peMa comaCHO KOTOPOH BHyrpeHHHH rpaBHTaUHOHHbIH nOTeHUHan ltPin MoxeT

6bITb npellCTaBneH a6COnIOTHO H paBHOMepHO CXOlJjllI(HMCjI PjlllOM no K03CPCPHUHshy

eHTaM Zijk npH HeKOTopbIX orpaHHqeHHjlX Ha Zijk H K03cpcpHUHeHTbI npH HHX

KOTopble jlBnjlIOTCjI nOnHHOMaMH CTeneHH P + s(L - 2) + 2 KOOpllHHaT Xk 3lleCb

s - HOMep qneHa pjllJa EypMaHa-narpaHxa

nOcne nepeXOlla K 06061I(eHHbIM ccpepHqeCKHM KOOpllHHaTaM R () cp BnepBble npennOxeHHblM B pa60Te [6] co CMelI(eHHeM ueHTPa KOOpllHHaT B TOqKY

Ha6nIOlleHHjI Xk HMeeM

- 1 (aD) 1 (aD) () Xk = Xk + akR al = - - sin()coscp a2 = - - SIn SIncp a al a al

O~R~R

B 3TOM cnyqae BblpaxeHHe ~ B03M)1I(eHHOH 3nnHnco~anbHoH nOBepXHOshy

CTH (3) nepenHWeTCjI B cnenyIOlI(eM B~e

(19)

me 3

Q=l-Lx~ k=l

7

PeWHB KBaIlpaTHOe ypaBHeHHe OTHOCHTeJIbHO R B JIeBOH lJaCTH (19) nOJIyqHM

ypaBHeHHe l1)1S1 HaXO)J(lJeHIDI R

3JIeMeHT 06beMa HHTerpHpoBaHH51 B HOBbIX KoopnHHaTax 6yneT paBeH

dV

nanbHeHWHe BbIlJHCJIeHmI 6yDYT OCHOBaHbI KaK H B cJIyqae C A Ha HCnOJIbshy

30BaHHH TeopeMbI JIarpaHJKa

ComaCHO (8) B HaweM CJIyqae e= R f(() = eh+2 a = Ro l1(a) =

l1(R01 Ok Xk Zijk)

Torna nOJIyqHM

llJUI COKpallleHIDI 3anHCH 3neCb H nanee HCnOJIb3yeTcSI BBeneHHbIH B [6] oneshy

nl n2 nm 1 ~ paTOp CyMMHpOBaHHSI kl k2 k neHCTBHe KOToporo 3aKJIIOlJaeTCSI B [ m CyMMHpOBaHHH no Ka)J(lJOMj HHJKHeMY HuneKCY OT HYJISI no 3HalJeHHSI cToSIlllero

HaIl HHM HuneKca C onHOBpeMeHHbIM yMHOJKeHHeM Ha COOTBeTCTBYIOIllHe 6HHOMHshy

anbHbIe K03CPCPHIJHeHTbI

1 ds - [n+m+n+l+l]JlJISI onpeneJIeHHSI SIBHoro BH)(a dR~-l (Ro +T + U)s npOH3BeneM 3ashy

MeHY Ro U - T + yU dRo = U dy H BOCnOJIb3yeMcSI paHee npHBeneHHoH

8

JIeMMoii

ds- 1 [~+m+n+l+1 ] = d~-l (Ro + T + U)s

Ro=U-T+yU s 1

= [ h + n + m + 1+ Ix ] (-I)JJ u h+n+mH-2(s-1)-JJTJJ d - X

XJl 2s dys-l

X [(1 + y)h+n+m+I-JJ+l] = (1 + ~)s

= [ h + n + m + 1+ Ix ] (-l)JJ uh+n+m+I-2(s-1)-JJTJJ~ X XJl 2s s

x(h+m+n+l Jl+I)

C yqeTOM nOJIyqeHHoro pe3YJIhTaTa R npHMeT BHlI

00 sL ( 1)8+JJ Rh+2 = ~+2 + (h + 2) 2 2 - S Zijk(S) X

8=1 ijk

j k h+n+m+l+1X] X [ xn m 1 Jl X

X x~-n~-mx~-lo~o2o~uh+n+m+I-2(s-I)-JJTJJ~s (22)

BbIpIDKeHHeIJlUl pacrrpeneneHIUI IIJIOTHOCTH (6) B HOBO CHCTeMe KOOpnHHaT

npHMeT BHlI

(P) - [a bel xa-dxb-Ixc-9R-d+I+9rvdJ _0 (23)P = LJ Pabc d f 9 1 2 3 A 1 A2 USmiddot

abc

BhIpIDKeHHe IJlUI HhlOTOHOBCKOro rpaBHTaIlHOHHOro rrOTeHIlHana (1) B HOBhIX

KOOpnHHaTax C yqeToM (23) rrpHMeT BHlI

9

TrumM o6pa30M (24) MO)ICHO rrpellCTaBHTb B BHIle

00

~(PL) = - Ga6(Fa + LF8 ) (25) 8=1

00

rlle Fa H L F8 COOTBeTCTBeHHO rrepBblH H BTOPOH lJJIeHbl B (24) Fa - lJJIeH 8=1

aHaJIHTHQecKoro npellCTaBJIeHHsI BHyrpeHHero nOTeHUHaJIa COOTBeTCTByromHH Heshy

B03MYllleHHOMY aMHrrcoHllYbull

ILruI sIBHOro rrpellCTaBJIeHHsI Fa Bocnonb3yeMCsI ~OpMYnOH 6HHoMa HbIOTOHa

IlnsI Rh+2

h + 2 - 2v A v T e] x

ApT e X

10

B pe3ynbTaTe nonyqeHbIqHCneHHbIe 3HaqeHIDI Zijk H Yl Y2 OHH npHBelleHbI

B Ta6n 2

DJUI Pa+be npHBelleHHbIx B Ta6n 1 8D 6YlleT nOpjIllKa 10-6 bull

Z020

646815middot10 646815middot10

2 BHYTPEHHHH rPABHTAUHOHHbIH nOTEHUHAJI ltPin(PL)

B pa60Te [9] C Hcnonb30BaHHeM PjIlJOB EypMaHa-narpaHxa llOKa3aHa Teoshy

peMa comaCHO KOTOPOH BHyrpeHHHH rpaBHTaUHOHHbIH nOTeHUHan ltPin MoxeT

6bITb npellCTaBneH a6COnIOTHO H paBHOMepHO CXOlJjllI(HMCjI PjlllOM no K03CPCPHUHshy

eHTaM Zijk npH HeKOTopbIX orpaHHqeHHjlX Ha Zijk H K03cpcpHUHeHTbI npH HHX

KOTopble jlBnjlIOTCjI nOnHHOMaMH CTeneHH P + s(L - 2) + 2 KOOpllHHaT Xk 3lleCb

s - HOMep qneHa pjllJa EypMaHa-narpaHxa

nOcne nepeXOlla K 06061I(eHHbIM ccpepHqeCKHM KOOpllHHaTaM R () cp BnepBble npennOxeHHblM B pa60Te [6] co CMelI(eHHeM ueHTPa KOOpllHHaT B TOqKY

Ha6nIOlleHHjI Xk HMeeM

- 1 (aD) 1 (aD) () Xk = Xk + akR al = - - sin()coscp a2 = - - SIn SIncp a al a al

O~R~R

B 3TOM cnyqae BblpaxeHHe ~ B03M)1I(eHHOH 3nnHnco~anbHoH nOBepXHOshy

CTH (3) nepenHWeTCjI B cnenyIOlI(eM B~e

(19)

me 3

Q=l-Lx~ k=l

7

PeWHB KBaIlpaTHOe ypaBHeHHe OTHOCHTeJIbHO R B JIeBOH lJaCTH (19) nOJIyqHM

ypaBHeHHe l1)1S1 HaXO)J(lJeHIDI R

3JIeMeHT 06beMa HHTerpHpoBaHH51 B HOBbIX KoopnHHaTax 6yneT paBeH

dV

nanbHeHWHe BbIlJHCJIeHmI 6yDYT OCHOBaHbI KaK H B cJIyqae C A Ha HCnOJIbshy

30BaHHH TeopeMbI JIarpaHJKa

ComaCHO (8) B HaweM CJIyqae e= R f(() = eh+2 a = Ro l1(a) =

l1(R01 Ok Xk Zijk)

Torna nOJIyqHM

llJUI COKpallleHIDI 3anHCH 3neCb H nanee HCnOJIb3yeTcSI BBeneHHbIH B [6] oneshy

nl n2 nm 1 ~ paTOp CyMMHpOBaHHSI kl k2 k neHCTBHe KOToporo 3aKJIIOlJaeTCSI B [ m CyMMHpOBaHHH no Ka)J(lJOMj HHJKHeMY HuneKCY OT HYJISI no 3HalJeHHSI cToSIlllero

HaIl HHM HuneKca C onHOBpeMeHHbIM yMHOJKeHHeM Ha COOTBeTCTBYIOIllHe 6HHOMHshy

anbHbIe K03CPCPHIJHeHTbI

1 ds - [n+m+n+l+l]JlJISI onpeneJIeHHSI SIBHoro BH)(a dR~-l (Ro +T + U)s npOH3BeneM 3ashy

MeHY Ro U - T + yU dRo = U dy H BOCnOJIb3yeMcSI paHee npHBeneHHoH

8

JIeMMoii

ds- 1 [~+m+n+l+1 ] = d~-l (Ro + T + U)s

Ro=U-T+yU s 1

= [ h + n + m + 1+ Ix ] (-I)JJ u h+n+mH-2(s-1)-JJTJJ d - X

XJl 2s dys-l

X [(1 + y)h+n+m+I-JJ+l] = (1 + ~)s

= [ h + n + m + 1+ Ix ] (-l)JJ uh+n+m+I-2(s-1)-JJTJJ~ X XJl 2s s

x(h+m+n+l Jl+I)

C yqeTOM nOJIyqeHHoro pe3YJIhTaTa R npHMeT BHlI

00 sL ( 1)8+JJ Rh+2 = ~+2 + (h + 2) 2 2 - S Zijk(S) X

8=1 ijk

j k h+n+m+l+1X] X [ xn m 1 Jl X

X x~-n~-mx~-lo~o2o~uh+n+m+I-2(s-I)-JJTJJ~s (22)

BbIpIDKeHHeIJlUl pacrrpeneneHIUI IIJIOTHOCTH (6) B HOBO CHCTeMe KOOpnHHaT

npHMeT BHlI

(P) - [a bel xa-dxb-Ixc-9R-d+I+9rvdJ _0 (23)P = LJ Pabc d f 9 1 2 3 A 1 A2 USmiddot

abc

BhIpIDKeHHe IJlUI HhlOTOHOBCKOro rpaBHTaIlHOHHOro rrOTeHIlHana (1) B HOBhIX

KOOpnHHaTax C yqeToM (23) rrpHMeT BHlI

9

TrumM o6pa30M (24) MO)ICHO rrpellCTaBHTb B BHIle

00

~(PL) = - Ga6(Fa + LF8 ) (25) 8=1

00

rlle Fa H L F8 COOTBeTCTBeHHO rrepBblH H BTOPOH lJJIeHbl B (24) Fa - lJJIeH 8=1

aHaJIHTHQecKoro npellCTaBJIeHHsI BHyrpeHHero nOTeHUHaJIa COOTBeTCTByromHH Heshy

B03MYllleHHOMY aMHrrcoHllYbull

ILruI sIBHOro rrpellCTaBJIeHHsI Fa Bocnonb3yeMCsI ~OpMYnOH 6HHoMa HbIOTOHa

IlnsI Rh+2

h + 2 - 2v A v T e] x

ApT e X

10

PeWHB KBaIlpaTHOe ypaBHeHHe OTHOCHTeJIbHO R B JIeBOH lJaCTH (19) nOJIyqHM

ypaBHeHHe l1)1S1 HaXO)J(lJeHIDI R

3JIeMeHT 06beMa HHTerpHpoBaHH51 B HOBbIX KoopnHHaTax 6yneT paBeH

dV

nanbHeHWHe BbIlJHCJIeHmI 6yDYT OCHOBaHbI KaK H B cJIyqae C A Ha HCnOJIbshy

30BaHHH TeopeMbI JIarpaHJKa

ComaCHO (8) B HaweM CJIyqae e= R f(() = eh+2 a = Ro l1(a) =

l1(R01 Ok Xk Zijk)

Torna nOJIyqHM

llJUI COKpallleHIDI 3anHCH 3neCb H nanee HCnOJIb3yeTcSI BBeneHHbIH B [6] oneshy

nl n2 nm 1 ~ paTOp CyMMHpOBaHHSI kl k2 k neHCTBHe KOToporo 3aKJIIOlJaeTCSI B [ m CyMMHpOBaHHH no Ka)J(lJOMj HHJKHeMY HuneKCY OT HYJISI no 3HalJeHHSI cToSIlllero

HaIl HHM HuneKca C onHOBpeMeHHbIM yMHOJKeHHeM Ha COOTBeTCTBYIOIllHe 6HHOMHshy

anbHbIe K03CPCPHIJHeHTbI

1 ds - [n+m+n+l+l]JlJISI onpeneJIeHHSI SIBHoro BH)(a dR~-l (Ro +T + U)s npOH3BeneM 3ashy

MeHY Ro U - T + yU dRo = U dy H BOCnOJIb3yeMcSI paHee npHBeneHHoH

8

JIeMMoii

ds- 1 [~+m+n+l+1 ] = d~-l (Ro + T + U)s

Ro=U-T+yU s 1

= [ h + n + m + 1+ Ix ] (-I)JJ u h+n+mH-2(s-1)-JJTJJ d - X

XJl 2s dys-l

X [(1 + y)h+n+m+I-JJ+l] = (1 + ~)s

= [ h + n + m + 1+ Ix ] (-l)JJ uh+n+m+I-2(s-1)-JJTJJ~ X XJl 2s s

x(h+m+n+l Jl+I)

C yqeTOM nOJIyqeHHoro pe3YJIhTaTa R npHMeT BHlI

00 sL ( 1)8+JJ Rh+2 = ~+2 + (h + 2) 2 2 - S Zijk(S) X

8=1 ijk

j k h+n+m+l+1X] X [ xn m 1 Jl X

X x~-n~-mx~-lo~o2o~uh+n+m+I-2(s-I)-JJTJJ~s (22)

BbIpIDKeHHeIJlUl pacrrpeneneHIUI IIJIOTHOCTH (6) B HOBO CHCTeMe KOOpnHHaT

npHMeT BHlI

(P) - [a bel xa-dxb-Ixc-9R-d+I+9rvdJ _0 (23)P = LJ Pabc d f 9 1 2 3 A 1 A2 USmiddot

abc

BhIpIDKeHHe IJlUI HhlOTOHOBCKOro rpaBHTaIlHOHHOro rrOTeHIlHana (1) B HOBhIX

KOOpnHHaTax C yqeToM (23) rrpHMeT BHlI

9

TrumM o6pa30M (24) MO)ICHO rrpellCTaBHTb B BHIle

00

~(PL) = - Ga6(Fa + LF8 ) (25) 8=1

00

rlle Fa H L F8 COOTBeTCTBeHHO rrepBblH H BTOPOH lJJIeHbl B (24) Fa - lJJIeH 8=1

aHaJIHTHQecKoro npellCTaBJIeHHsI BHyrpeHHero nOTeHUHaJIa COOTBeTCTByromHH Heshy

B03MYllleHHOMY aMHrrcoHllYbull

ILruI sIBHOro rrpellCTaBJIeHHsI Fa Bocnonb3yeMCsI ~OpMYnOH 6HHoMa HbIOTOHa

IlnsI Rh+2

h + 2 - 2v A v T e] x

ApT e X

10

JIeMMoii

ds- 1 [~+m+n+l+1 ] = d~-l (Ro + T + U)s

Ro=U-T+yU s 1

= [ h + n + m + 1+ Ix ] (-I)JJ u h+n+mH-2(s-1)-JJTJJ d - X

XJl 2s dys-l

X [(1 + y)h+n+m+I-JJ+l] = (1 + ~)s

= [ h + n + m + 1+ Ix ] (-l)JJ uh+n+m+I-2(s-1)-JJTJJ~ X XJl 2s s

x(h+m+n+l Jl+I)

C yqeTOM nOJIyqeHHoro pe3YJIhTaTa R npHMeT BHlI

00 sL ( 1)8+JJ Rh+2 = ~+2 + (h + 2) 2 2 - S Zijk(S) X

8=1 ijk

j k h+n+m+l+1X] X [ xn m 1 Jl X

X x~-n~-mx~-lo~o2o~uh+n+m+I-2(s-I)-JJTJJ~s (22)

BbIpIDKeHHeIJlUl pacrrpeneneHIUI IIJIOTHOCTH (6) B HOBO CHCTeMe KOOpnHHaT

npHMeT BHlI

(P) - [a bel xa-dxb-Ixc-9R-d+I+9rvdJ _0 (23)P = LJ Pabc d f 9 1 2 3 A 1 A2 USmiddot

abc

BhIpIDKeHHe IJlUI HhlOTOHOBCKOro rpaBHTaIlHOHHOro rrOTeHIlHana (1) B HOBhIX

KOOpnHHaTax C yqeToM (23) rrpHMeT BHlI

9

TrumM o6pa30M (24) MO)ICHO rrpellCTaBHTb B BHIle

00

~(PL) = - Ga6(Fa + LF8 ) (25) 8=1

00

rlle Fa H L F8 COOTBeTCTBeHHO rrepBblH H BTOPOH lJJIeHbl B (24) Fa - lJJIeH 8=1

aHaJIHTHQecKoro npellCTaBJIeHHsI BHyrpeHHero nOTeHUHaJIa COOTBeTCTByromHH Heshy

B03MYllleHHOMY aMHrrcoHllYbull

ILruI sIBHOro rrpellCTaBJIeHHsI Fa Bocnonb3yeMCsI ~OpMYnOH 6HHoMa HbIOTOHa

IlnsI Rh+2

h + 2 - 2v A v T e] x

ApT e X

10

TrumM o6pa30M (24) MO)ICHO rrpellCTaBHTb B BHIle

00

~(PL) = - Ga6(Fa + LF8 ) (25) 8=1

00

rlle Fa H L F8 COOTBeTCTBeHHO rrepBblH H BTOPOH lJJIeHbl B (24) Fa - lJJIeH 8=1

aHaJIHTHQecKoro npellCTaBJIeHHsI BHyrpeHHero nOTeHUHaJIa COOTBeTCTByromHH Heshy

B03MYllleHHOMY aMHrrcoHllYbull

ILruI sIBHOro rrpellCTaBJIeHHsI Fa Bocnonb3yeMCsI ~OpMYnOH 6HHoMa HbIOTOHa

IlnsI Rh+2

h + 2 - 2v A v T e] x

ApT e X

10

C yqeToM (26) Fo npHMeT BH)l

p

Fo = L Pabc( -l)I+Tx abc

x [a b C h + 2 (hl) + 1 h + 2 - 211 A 1I r ~] x d f 9 Jl 1I Apr ~ X

h+2-2v-gt+a-d+2(T-~) b-f+gt-p+2(~-x) c-g+p+2x QX Xl bull x 2 x3 AIA2A31 (27)

Al = d + h + 2 - 211 - A A2 = f + A - P A3 = 9 + p

J Al A2 A3 dO3)eCb H )aJIee QA 1 A2A3 = 01 02 02 02 bull

BocnoJlb3yeMcB KaK H paHee CPOPMYJlOH 6HHOMa HblOTOHa nOJlyqHM

uh+n+m+l-2(s-1)-1 TI =

= [ N Jl _ (8 - 1) 1I r ~ N+ 2 - 28 - 211 A] (-1r x

1I r ~ X A P N-2(s-1)-2v-gt+2(T-~) gt-p+2U-x) p+2x N-2(s-1)-2v gt-p p (28)

X 2X Xl X3 01 02 03 bull

3)eCb H )aJIee N = h + l + m + n YlIHTbIBaB (28) HaXO)HM aHaJIHTHlIeCKOe npe)lcTaBJIeHHe)JIB Fs

(-1 )I+T+S P sL

Fs = 2s LLPabc X

8 abc ijk

X [a b C i j kN + 1 (N Jl) - (8 - 1) N + 2 - 28 - 211 A 1I r ~] x dfgnml Jl 1I A pr~x

r Z () a-d+j-n+N+2-2v-2s-gt+2T-2~ b-f+j-m+gt-p+2~-2xX Js ijk 8 bull Xl X2 X

X x~-g+k-l+P+2X bull QA IA2A3 (29)

Al = d + n + N - 28 + 2 - 211 - A A2 = f + m + A - P A3 = 9 + l + P s-l

~l = 1 ~sgtl = n(N + 1 - Jl + 1 - 2r) r=l

H3 (27) H (29) CJle)yeT npe)CTaBIIeHHe ~ co)epxamee Bce CTeneHH KOOP)Hshy

HaT)O P BKJllOlIHTeJIbHO

P

~(P L) = -21rGpoaI L ~abcX~X~X3 (30) abc

11

(2a - 1)(2b - 1) (31)Q2a2b2c = 211 2a+ b(a + b) 12(a+b)2c

1 (1 - X2 )aX 2cdx2c

12(a+b)2c = e- f ( ( 1 ) 2) a+2c+1 -1 1 + - -1 x

e2

B pe3YJlbTaTe HaMH nOJlyqeHo aHaJIHTHqeCKOe npe)lCTaBJIeHHe 4gt JlillI peaJIHshy

3allHH B CHCTeMe CHMBOJlbHbIX BbIqHCJleHHH MAPLE BblqHCJleHHe 4gtabc JlillI 3HaqeHHH P = 468 H L = 246 HepeaJIbHO 6e3

HCnOJlb30BaHHsI MeTO)la CHMBOJlbHbIX BbIqHCJleHHH Ha KOMnbIOTepe KaK CJle)lCTBHe

HaMH 6bIJIa COCTaBJIeHa npoIpaMMa B paMKaX TOro xe naKeTa MAPLE JlillI aHashy

JlHTHqeCKOrO npe)lCTaBJIeHHsI BH~HHero nOTeHIlHaJIa B BHJle pa3JIOXeHHsI no

CTeneHsIM KOOp)lHHaT

3a)laqa CHJlbHO ynp0lllaeTCsI ecJlH 3a)laHO pacnpe)leJleHHsI nJIOTHOCTH T e

H3BeCTHbl Ko~qxpHIlHeHTbI Pabc MbI B3sIJlH Pabc npHBe)leHHble B Ta6Jl3

Ta6mu(a 3

a 0 0 0 0 0 0 0 b 0 0 0 0 2 2 2 c 0 2 4 6 0 2 4

Pabc 1 -0729 0634 -0912 -0690 1294 -2765

a 0 0 0 2 2 2 2 b 4 4 6 0 0 0 2 c 0 2 0 0 2 4 0

Pabc 0683 -2845 -0996 -0690 1294 -2765 1367 a 2 2 4 4 4 6 b 2 4 0 0 2 0 c 2 0 0 2 0 0

Pabc -5690 -2989 0683 -2845 -2989 -09964

ilO)lCTaBHB KO~ltpqlHIlHeHTbl Zijk MbI nOJIyqHJlH nOTeHIlHaJI KaK ltPYHKIlHIO

KOOp)lHHaT 0603HaqHB r2 = xi + x~

4gtin(P = 6 L = 2) = -Gpoai(2 7216 -16191r2 + 0 3540r4 - 0 1776r6 shy

- 1 068437x + 0 20631x - 0 079436xg - 0 3852xr4shy

- 0 2954xr2 + 0 5279xr2) (32)

12

(2a - 1)(2b - 1) (31)Q2a2b2c = 211 2a+ b(a + b) 12(a+b)2c

1 (1 - X2 )aX 2cdx2c

12(a+b)2c = e- f ( ( 1 ) 2) a+2c+1 -1 1 + - -1 x

e2

B pe3YJlbTaTe HaMH nOJlyqeHo aHaJIHTHqeCKOe npe)lCTaBJIeHHe 4gt JlillI peaJIHshy

3allHH B CHCTeMe CHMBOJlbHbIX BbIqHCJleHHH MAPLE BblqHCJleHHe 4gtabc JlillI 3HaqeHHH P = 468 H L = 246 HepeaJIbHO 6e3

HCnOJlb30BaHHsI MeTO)la CHMBOJlbHbIX BbIqHCJleHHH Ha KOMnbIOTepe KaK CJle)lCTBHe

HaMH 6bIJIa COCTaBJIeHa npoIpaMMa B paMKaX TOro xe naKeTa MAPLE JlillI aHashy

JlHTHqeCKOrO npe)lCTaBJIeHHsI BH~HHero nOTeHIlHaJIa B BHJle pa3JIOXeHHsI no

CTeneHsIM KOOp)lHHaT

3a)laqa CHJlbHO ynp0lllaeTCsI ecJlH 3a)laHO pacnpe)leJleHHsI nJIOTHOCTH T e

H3BeCTHbl Ko~qxpHIlHeHTbI Pabc MbI B3sIJlH Pabc npHBe)leHHble B Ta6Jl3

Ta6mu(a 3

a 0 0 0 0 0 0 0 b 0 0 0 0 2 2 2 c 0 2 4 6 0 2 4

Pabc 1 -0729 0634 -0912 -0690 1294 -2765

a 0 0 0 2 2 2 2 b 4 4 6 0 0 0 2 c 0 2 0 0 2 4 0

Pabc 0683 -2845 -0996 -0690 1294 -2765 1367 a 2 2 4 4 4 6 b 2 4 0 0 2 0 c 2 0 0 2 0 0

Pabc -5690 -2989 0683 -2845 -2989 -09964

ilO)lCTaBHB KO~ltpqlHIlHeHTbl Zijk MbI nOJIyqHJlH nOTeHIlHaJI KaK ltPYHKIlHIO

KOOp)lHHaT 0603HaqHB r2 = xi + x~

4gtin(P = 6 L = 2) = -Gpoai(2 7216 -16191r2 + 0 3540r4 - 0 1776r6 shy

- 1 068437x + 0 20631x - 0 079436xg - 0 3852xr4shy

- 0 2954xr2 + 0 5279xr2) (32)

12

COOTBeTCTBYIOll(HH rpaqmK npHBOnHTCSI Ha pHC 1 X3 = X3 bull e

-14

-16

-18

-2

-22

-24

-26

06

01

05 04

03 02 deg

DOCKOJIbKY nporpaMMa OKa3aJIaCb 60JIbWOH H CJIOJKHOH B03HHKJIa Heo6xoshy

nHMOCTb ee TecTHpOBaHHSI HaMH 6blJlH BbIllOJIHeHbl nBa TeCTa DPOBOnHJICSI npeshy

neJIbHblH nepexon npH e ~ 1 K ccpepHqeCKH-CHMMeTpHqHOMy cJIyqalO KOTOPblH

JIefKO paCCqHTblBaeTCSI no He3aBHcHMoH nporpaMMe KpoMe Toro ~ltPin(P = 6 L = 2 4) = 41rGp(P = 6) H neHcTBYSI Ha ltPin OnepaTOpOM JIanJIaCa Mbl

nOJIyqaeM pacnpeneJIeHHe nJIOTHOCTH YMHOJKeHHOe Ha 41rGp K03cpcpHllHeHTbI B

KOTOPOH COBnanaIOT C 3anaHHblMH C TOqHOCTblO 10-24 bull

3 nOTEHI(HAJI HA BHEmHIOIO TOqKY ltpout(N P L)

HaH60JIbwHH HHTepec npenCTaBJISIeT nOTeHllHaJI Ha BHeWHlO1O TOqKY

ltpout(N P L) Korna r gt aI rne r - paCCTOSIHHe no npHTSIrHBaeMOH TOqKH

HaH60JIee 3cpcpeKTHBHblH MeTOn HaXO)KJleHHSI BHewHero nOTeHllHaJIa - MeTOn

pa3JIOJKeHHSI no MyJIbTHnOJIbHblM MOMeHTaM BHeWHHH nOTeHllHaJI HMeeT BHn

-3 ltP = -GJ p(fjdr (33)out 1_ -Ir r

rne dr-3 = dx dy dz

13

Pa3JIo)KHM ltPYHKUHIO ~ B psm TeiUJopaIT - rl

1 1 -r~==~~~====~~====~=IT- ril J(x - X)2 + (y - y)2 + (z - zl)2

00 (_l)o+p+Y ( (o+P+Y 1 ) -~ ~~~ ~ - ~ o8y 8x0 8yp8zY Jx2 + y2 + z2

HCnOJIL3yg (35) nonyqHM BLlpIDKeHHe llJUI BHewHero nOTeHUHaJIa

(36)

3aMeHHB B (36) HHTerpaJI Ha DoPY nonyqaeM

(37)

y ZC yqeToM TOro qTO XI -X3 = - DopY 6YlleT BLlrJIsmeTL

al a3 CJIeJ)IOntHM o6pa30M

DopY ar+p+2a~+1 f p(r)xlx~x~dxldx2dx3 (38)

KaK H B cnyqae BH~HHero nOTeHUHaJIa npellCTaBHM flJIOTHOCTL KOHqmIyshy

paUHH p B BHlle nOJIHHOMa (6) CTeneHH P TIepeitlleM K ccpepHqeCKHM KOOpllHHaTaM Xk = rak 01 = sin 9 cos ltp 02 =

sin 9sin ltp 03 = cos 9 me 0 lt r lt Rmax H KaK B cnyqae C BLIqHCJIeHHeM A BOCnOJIL3yeMcSI TeopeMoH JIarpaH)Ka TIoCJIe STOro BLlpIDKeHHe llJUI B03MymeHHoH

SMHnCOHllaJILHOH nOBepXHOCTH (3) B llaHHOM cnyqae 6YlleT HMeTL BHll

Rh l+h~~ (-1)8 Z A (h++ +k l)-i=-j-k max = L- L- s28 ijk8 t J - 01U203

8=1 ijk

14

ilonmKHB Qabc = JiiY ii~ ii~dn nonyqaeM

N P

~out(NPL) = (-PoG) L LPabcMafJX afJ abc

1 _ 00 ijk (-1)8 x ( ho + 3 Qa+afJ+b+c + ~~ 28 8 Zijk(8)~8(h + l)x

x Q+a+iIl+b+i~+C+k) (39)

8-1 rlle ho = o+8++a+a+c h = ho+i+j+k a ~8(h+1) IT (h+2-2r)

r=1 B cnyqae 8 = 1 ~ 8 = 1

HeTpYllHO llOKa3aTb liTO

- 4 (a - l)(b l)(c - 1)1 (40)Qabc = 1f (a + b + c + I)

YlIHTblBasI (40) BbIpIDKeHHe llnsl BHenIHero nOTeHI(Hana MO)KHO npellCTaBHTb

B BHI1e pa3nO)KeHIDI no MYnbTHnonbHbIM MOMeHTaM cnellYIOII(HM 06pa30M

N P

~out(N P L) = -41fpoG 2 2 PabcMafJX afJ abc

1 (0 + a - 1)(8 + b - 1)( + c - I) x(ho +3 (ho+1) +

00 ijk (_1)8 + L L 28 81 Zijk(8)~8(h + l)X

8=1 8L

(0 + a + i-I) (8 + b +j I)( + c + k - I) (41) x (h + I) )

me N - nOpslllOK MYnbTHnonsr Ha OCHOBe (41) 6bma cocTasneHa nporpaMMa B CHCTeMe CHMBonbHblX BbIshy

lIHcneHHH MAPLE KaK CKa3aHO Bblwe pacnpelleneHHe nnOTHOCTH ClIHTaeM 3ashy

llaHHbIM T e H3BeCTHbI KOslaquoplaquopHI(HeHTbI Pa+bc 3HasI KOTopble H Hcnonb3YsI (10)

HaxOllHM Pabc KOslaquoplaquopHI(HeHTbI Pa+bc npHBelleHbI B Ta6n4 Ha OCHOBaHHH [10] B KOTOPOH

HaMH paccMoTpeHbI TpH ypaBHeHIDI COCTOslHIDI sIllepHOH MaTepHH lieTe-ll)fltOHcoHa

(BJ) OnneHreHMepa-BonKoBa (OV) H Peii)la (R) C Hcnonb30BaHHeM llaHHblX TaGn4 H Bblpa)KeHHsI llnsl BHewHero nOTeHI(Hshy

ana (41) B PaMKax TOro )Ke naKeTa MAPLE nonyqeHo BbIpIDKeHHe )InsI BHeWHero

15

ilonmKHB Qabc = JiiY ii~ ii~dn nonyqaeM

N P

~out(NPL) = (-PoG) L LPabcMafJX afJ abc

1 _ 00 ijk (-1)8 x ( ho + 3 Qa+afJ+b+c + ~~ 28 8 Zijk(8)~8(h + l)x

x Q+a+iIl+b+i~+C+k) (39)

8-1 rlle ho = o+8++a+a+c h = ho+i+j+k a ~8(h+1) IT (h+2-2r)

r=1 B cnyqae 8 = 1 ~ 8 = 1

HeTpYllHO llOKa3aTb liTO

- 4 (a - l)(b l)(c - 1)1 (40)Qabc = 1f (a + b + c + I)

YlIHTblBasI (40) BbIpIDKeHHe llnsl BHenIHero nOTeHI(Hana MO)KHO npellCTaBHTb

B BHI1e pa3nO)KeHIDI no MYnbTHnonbHbIM MOMeHTaM cnellYIOII(HM 06pa30M

N P

~out(N P L) = -41fpoG 2 2 PabcMafJX afJ abc

1 (0 + a - 1)(8 + b - 1)( + c - I) x(ho +3 (ho+1) +

00 ijk (_1)8 + L L 28 81 Zijk(8)~8(h + l)X

8=1 8L

(0 + a + i-I) (8 + b +j I)( + c + k - I) (41) x (h + I) )

me N - nOpslllOK MYnbTHnonsr Ha OCHOBe (41) 6bma cocTasneHa nporpaMMa B CHCTeMe CHMBonbHblX BbIshy

lIHcneHHH MAPLE KaK CKa3aHO Bblwe pacnpelleneHHe nnOTHOCTH ClIHTaeM 3ashy

llaHHbIM T e H3BeCTHbI KOslaquoplaquopHI(HeHTbI Pa+bc 3HasI KOTopble H Hcnonb3YsI (10)

HaxOllHM Pabc KOslaquoplaquopHI(HeHTbI Pa+bc npHBelleHbI B Ta6n4 Ha OCHOBaHHH [10] B KOTOPOH

HaMH paccMoTpeHbI TpH ypaBHeHIDI COCTOslHIDI sIllepHOH MaTepHH lieTe-ll)fltOHcoHa

(BJ) OnneHreHMepa-BonKoBa (OV) H Peii)la (R) C Hcnonb30BaHHeM llaHHblX TaGn4 H Bblpa)KeHHsI llnsl BHewHero nOTeHI(Hshy

ana (41) B PaMKax TOro )Ke naKeTa MAPLE nonyqeHo BbIpIDKeHHe )InsI BHeWHero

15

Ta6Jnn(a 4

OV BJ R Rov BJ

a+bc Pa+bc Pa+bc Pa+bc Pa+bcPa+bc Pa+bc 1 1 11 10 0 1

-046375 -072181 -072895 -068902 -0695472 -0461170 063397 062306 0621654 051814 051818 0635800

-093429 -093256-091425 -0911690 6 -105713 -105863 o -045871 -044935 -071544 -069125 -068320 -0661102

129401 1265432 103862 104169 127984 1253312 -317357 -275605 -276435 -2814622 4 -317536 -282108

o 0520818 053272 064485 068239 063099 0664164 2 -318038 -319872 -277129 -284334 -282819 -2891334 o -106216 -108540 -100615-092951 -099437 -0947886

nOTeH1Jl-laJIa CJIO)KHOH 3JIJIHnCOH)]aJIbHOH KOHqgtHrypauHH B KB3)lpynOJIbHOM npHshy

6JIH)KeHHH AJls N 2 B cqgtepHlIecKHx KOOpllHHaTax

Gm (D(e)at(p) 2 )ltpout(N = 2P = 6L = 2) = ---- 1 + r2 (1- 3 cos ()) +

(42) rlle m - Macca rpaBHTHpYlOmeH KOHqgtHrypaUHH

B pa60Te [10] nOKa3aHo liTO at = 2 G Po

2 ( ) rlle Po - nJIOTHOCTb B

11 Poy e ueH~ KOHqgtHrypaUHH Po - llaBJIeHHe B ueHTPe KOHqgtHrypaUHH y(e) - pacshy

ClIHTaHHbIe qgtYHKUHH napaMeTpa CllJIlOCHYTOCTH e H yqHTbIBalOmHe 6bICTPOTY Bpashy

meHHjJ KOHqgtHrypaUHH TIPH H3MeHeHHH yrnoBoH CKOPOCTH BpameHHjJ qgtYHKUHH

Po H Po TaK)Ke 6yllYT 3aBHCeTb OT napaMeTpa CruIlOCHYTOCTH e

Po = Poo(e) Po = poox(e) (43)

C yqeToM (43) HMeeM

2 (e) Poo(poo)D(e)al = D(e) () ( ) 2 G 2 = Dl(e)K(poo) (44)

x eye 11 PO~

rlle (e) K( ) Poo(poo)

Dl(e) = D(e) x(e)y(e) POO=2G2 11 Poo

TIoCJIe npOBelleHHbIX npeo6pa30BaHHH BbIpa)KeHHe AJIjJ BHellIHero nOTeHUHaJIa

B03M)llleHHOH 3JIJIHnCOH)]aJIbHOH KOHqgtHrypaUHH (42) npHMeT BH)]

Gm ( (1 - 3cos2 (J))ltpout(N = 2P = 6L = 2) = ---- 1 + Dl(e)K(poo) r2 +

(45)

16

fpaqlHK CPYHKUHH Dl (e) npenCTaBJIeH Ha pHC 2 a 3HaqeHIDI K (poo) npHBoshy

lUITCSI B Ta6JI 5 KaK BHIlHO H3 pHC 2 npH e = 1 Dl 06pamaeTCSI B HYJIb a C )MeHbUleshy

HHeM napaMeTpa e Dl YBeJIHlJHBaeTCSI T e K03cpcpHUHeHT pa3JI0XeHIDI B KBashy

npynOJIbHOM npH6JIHXeHHH Dl MOHOTOHHO 3aBHCHT OT napaMeTpa CnJIlOcHYToCTH

KOHcpHrypaUHH TaKHM 06pa30M lJeM 60JIbUle CDJIIOCHYTOCTb KOHcpHrypaUHH reM

CYllleCTBeHHeH BKJIaIl KBaIlPynOJIbHoro qneHa

025

02

015

01

005

O~ltT-r~~-~~-~~-r-~-r~

06 07 08 09 e

PHC 2 3aBHcHMoCTb K03qxpHUHeHToB Dl OT napaMeTpa e KpHBrui 1 COOTBeTCTBYeT

ypaBHeHHIO COCTOSlHIDI OV 2 - BJ 3 - R

Ta~a5

poo rcM1 K(OV) CM K(BJ) CM K(R) CM

41014 1418middot IOu 1911 IOu 1200 IOu 61014 12841011 23791011 14541011

81014 11601011 26361011 16701011

3AKJIIOQEHHE

B pe3YJIbTaTe HaMH nOJIyqeHo aHaJIHTHlJeCKOe npencTaBJIeHHe HblOTOHOBCKOro

rpaBHTaUHOHHoro nOTeHUHaJIa B03MYIIJeHHbIX 3J1JIHnCOHIlaJIbHbIX KOHcpHrypaUHH

Ha BHYTpeHHlO1O TOlJKY npH CPHKcHpoBaHHblx 3HalJeHIDIX P H L B BHIle a6COJIIOTHO

CXOlUlIIJHXCSI PSlIlOB no napaMeTpy B03MYIIJeHIDI (25) B paMKaX naKeTa MAPLE 6bIJIa COCTaBJIeHa nporpaMMa IlJISI npencTaBJIeHHSI nOTeHUHaJIa 4in(P L) B BHIle

CPYHKUHH KOOpnHHaT (32) IlJISI npOH3BOJIbHbIX 3HalJeHHH P L H peaJIH30BaHa IlJISI

cnyqaSl P = 6 L = 2 TIoTeHUHaJI B03MYIIJeHHOH 3J1JIHnCOHIlaJIbHOH KOHCPHryshy

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1 CpemuHcKuu JI H TeopIDI HbIOTOHOBCKOro nOTeHUHaJ1a M fOCTeXH3IlaT 1946 C 316

2 TaccYlrb)J( JI TeopIDI BpaIllaIOIIlHXCJI 3Be3ll M MHP 1982

3 MypamoB P3 lloreHUHaJ1b1 3JlJlHnCOFIQa M ATOMH3IlaT 1976

4 AHmoHoB B A TUMOlUKUHa B H XOmueBHuKoB K B BBeJleHHe BTeopHIO HbIOTOHOBshyCKOro nOTeHUHaJ1a M Ha)Ka 1988

5 Masjukov V v Tsvetkov V P Nonlinear Effect in Theory of Equilibrium Gravitatshying Rapidly Rotating Magnetized Barotropic Configurations and the Gravitational Radiation from Pulsars I Astron and Astrophys Transactions 1993 V4 P41-42

6 ItupylIeB A H ItBemKoB B fl BpamaIOIIlHecJI nOCTHbIOTOHOBCKHe KOHcI)HrypauHH OJlshyHOPOJlHOH HaMarHlflIeHHOH )KHlU(OCTH 6JIHlKHe K 3JlJlHnCOFIQaM I II I AcrpOHOM ~H 1982 T59 C476-482 666-675

7 JIaBpeHmbeB M A llla6am 5 B MeTOJl TeopHH qYHKUHH KOMnJIeKcHoro nepeMeHshyHOro M Ha)Ka 1987 C 688

8 EPMaKOB B B KanumKuH H H OnTHMaJ1bHblH mar H peryJIJlPH33UHJ1 MeTOJla HbIOshyTOHa I )KYPH BbflHCJI MaTeM H MaT cpH3 1981 T 21 ~ 2 C419-497

9 ItBemKoB B fl MaclOKOB B B MeTOJl p~OB EypMaHa-J1arpaIDKa B 3auaQe 06 aHaJ1HshyTlflIecKOM npeJlCTaB1IeHHH HbIOTOHOBCKOro nOTeHUHaJ1a B03MymeHHbIx 3JlJlHnCOFIQaJ1bshyHbIX KOHCPHrypauHH I JlAH CCCP 1990 T 313 ~ 5 C 1099-1102

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E14-2004-148

TpyllbI XII MeiKllyHap0llHOH KOHipepeHUHH no H36paHHbIM np06JIeMaM cOBpeMeHHoH qH3HKH )1y6Ha 2003 379 c (Ha PYCCKOM H aHrJI H3)

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npenpHHT 06bellHHeHHoro HHCTH1)Ta guepHbIX HCCJIellOBaHHH Uy6Ha 2005

Bespalko E V et a1 PI 1-2005-121 Calculation of Newton Potential of the Gravitating Configurations with the Surface Close to the Spheroid by Symbolic and Numerical Methods

An exact analytical representation of the heterogeneous perturbed ellipsoidal configuration of the Newton gravitational potential to the inner point pin is obtained An expression for the potential pin is obtained in the form of a polynomial of the coordinates by the program prepared in the system of symbolic mathematics MAPLE and by numerical calculations based on the regularized analogue of the Newton method A formula for the potential to the outer point pout is found by the quadrupole approximation

The investigation has been performed at the Laboratory of Information Techshynologies JINR

Preprint of the Joint Institute for Nuclear Research Dubna 2005