Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I
By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville-Arnol’d integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytic method for the investigation of the corresponding mapping of imbedding of an inva...
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| Date: | 1999 |
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| Main Authors: | , , , |
| Format: | Article |
| Language: | Ukrainian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
1999
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4737 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510900722597888 |
|---|---|
| author | Prykarpatsky, Ya. A. Samoilenko, A. M. Прикарпатський, Я. А. Самойленко, А. М. |
| author_facet | Prykarpatsky, Ya. A. Samoilenko, A. M. Прикарпатський, Я. А. Самойленко, А. М. |
| author_sort | Prykarpatsky, Ya. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville-Arnol’d integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytic method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration. |
| first_indexed | 2026-03-24T03:04:21Z |
| format | Article |
| fulltext |
YIIK 517.9
A. M. CaMOi~.rleHKO, J]. A. HpHKapHaTCbKHi~ (In-'r MaTeMaluKa HAH Yxpai)m. KHIn)
~oc.rH~DKEHH$I IHBAPIAHTHHX ~ E |
IHTEFPAYIbHHX MHOFOBH~[IB A~[IABATHtIHO 3BYPEHHX
I ! I J IKOM I H T E r P O B H H X UAMIJII~TOHOBHX CHCTEM. I
By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely
Liouville-Arnold integrable Hamiltonian systems on the cotangent phase space, we consider an
algebraic-analytical method for investigating the corresponding mapping of imbedding of an invariant
torus into the phase space. This method enables one to perform the analytical description of the structure
of quasiperiodic solutions to the Hamiltonian system under consideration.
BaayloqvlCb Iia/trlqbepelntiadtl, llo-reoMe'lpnqlli~ "l~eopii KapTalia ilrret'paJlbm, lX ni/tMllOl'Onl, I]til~ (inaapi-
aIITIIHX TopiB) IIOBIliCTIO ilrrei'ponHHx 3a d"liyBi./IJleM--ApilOJll,ltOM I'aMi.IIhTOIIOBHX CHC'I'eM Ila KO/IO-
THqlIOMy qbaaonoMy npocTopi, po3l'3t:,lllyTo a~we6paiqllo-allaJlil~qttrlfl MeTOII ltOCJliloKellHa Bi/.tnonilt-
Hero Bi,,to6pa~',euH~i nKJmltcml:4 iHImpimrmoro ropa 13 ~aaonml npocrip. I.[e ltar MomJlrn~ic'rt~ on~ca-
TI4 aHa./liTl, lqllo ~tpyKTypy KBaainepioltrtqllnx poaB'aagin ]toCJfi]tx<ynaHoi raMiJlb'ronoBoi CHCTeMI, L
1. IIonepeAHi ni~ao~toc'ri. Po3r~aHCMO ~rmaMi,tay crtcrcMy aK BVKTOpHe noJtc K :
M 2n --> T ( M 2n) Ha r:ia/moMy 2n-BriMipHo~y, d imM = 2n e Z+, MHOrOBH~i 3 Hy-
H ~M 2n" Z) = 0, aKait HaarmaeMo qba3oBrtM IIpOCTO- JabOBO~O rpynoIO roMoJaoriil 2 ~ ,
pOM [I--3], i BBa)Kae~dO, mo Ha HbOMy 3aztaHa CHMnJ1eKa~aqHa cxpyKTypa ~(2} e
e A(M2a), TO6TO 3aMKHCHa HeBHpo~ameHa ~qbepeHtda~bHa 2-qbopMa a aJare6pH
rpacMaHa A(M2~). B U~o~y B~nazmy BeKTOpHe no~e K: M2"--* T(M 2#) 6y~c ra-
MiJIbTOHOBHM, aKIr40 icHye TaKa dpyHKtda H e D ( M 2n) : = C**(M2n; R) , ttlo 3a/~o-
BOJIbHSiE HaCTyIIay yMOBy:
iK~"2 (2) = - d H , (1.1)
~e iK: A (M 2 n) _.> A ( M 2 n ) ~ Taz 3BaHe SHyTpituar lqbCpeHLiilosaaH~t [4] Ba/~Oa:~
BeKropHoro rtoaa K: M 2 n _._> T ( M 2 n).
O3HaqeHHJ~ 1.1. Fa~ti.abmonoae aemnopne none
du
- - = K ( u ) (L2)
dt
na cumn~erraut~no~ty ~tnoeoau~i M 2n po3~dpnocmi 2n e Z+ 3 )'~oeo~o (1.1), c3e
t ~ R ~ eaomotci~nu~ napa~temp, nazuaaernbca ~imco~t inmeepoano~o 3a Jliyai.~ne~t
Ouna~tinno~o cucme~to~o, arur icnye piano n e Z+ e.~aOrux ct~ynmli~ H1 =
= H, H 2 . . . . . H n e D (M 2n) marux, u4o eiOnoaiOni aermopni nona Kj : M 2n --->
�9 (2) - d H j , j I, n ym~p~otom~, crin~ennoau~dpny posa'~z- --> T(M2n), 0e tK j~ = = ~ ,
ny [5] a m e @ y JIi naO R , moSmo icny~omb mari ~uc~a c)k e R , i , j , k = l,--n,
U40 [ K i, Kj] = = l cij Ki Ona acix i , j = ~ ,n , i na niS~mozoauOi M~ = { u e
M 2 " ; Hj = hj ~ R , j = ~ } poz~dpnicmb a m e @ u .fli K aermopnux noaio
piona dim K = n.
k Y s a n a ~ y , KOJIH aare6pa J-li K e iHBOJIIOTHBHOiO Ha M 2n, TOOTO sci Cij ----- O)
i, j , k = 1,--'~, cnpa~e]X~mBe TBep~ameHna ApHoab~aa [6] npo re , mo inTerpa~rmnH~
~moroBH/~ M~h ' C M 2n aa yMOBH KOMrlaKTHOCTi r ~[HCI~C0M0p~HI4M n-sHMipHOMy T0-
py T a, Ha gX0My op~iTH ]IHHaMiqH01 CHCTeMH (1.2) eKaiaa~CaTHi X~aainvpio~h~Hn~
�9 A. M. CAMO~.HEHKO, SL A. I'IPHKAPFIATCbKH~, 1999
ISSN 004t-6053. Yrp. ~*am. ~.'vpn., 1999, m. 51, bl ~ 10 1379
1380 A.M. CAMO~JIEHKO, ft. A. FIPHKAPFIATCbKHfl
.rIiHihHHM O6MOTKaM. CaMe uefl BHna~oK MH 6y~IeMO poar:ia~aT~. ]Ia~i 6yaeMo KO-
pHCTyBaTHCb HacrI~I1HHMH ~BOMa TBcp~)KeHH~IMH.
TeopeMa 1.1 (raJficco-Pi6). Hexa~ na OeaKomy ~iO~pumomy oKoni U ~ M 2n
za@ani za~w,~eni ninit~no nezane~ni I -dpopmu r j ~(~) e A l (M2n), j = 1--,-~, u~o sa-
~060Abl~lOmb ~ M 2 n y3tosy ineomomuenocmi
o
~,~a ecix i , j = 1,"-'n, 3e, za ~uzna~ennam, ~emnopni no,~a Kj : M 2n ---.> T ( M 2n) ma-
rd, ur i K ~ (2) = - I]~ 1) , j = 1"~. ToOi ic~vomb dpop.~tu Ildpanqbqba (nepuaozo cmy-
n e ~ ) fj(1)e A I(U), j = 1-~, z marumu enacmueocmamu:
j--l
Oe I (~) - - i ~ e a ~ 3udpepemlia,n~nux dpopm a ameSpi Fpacmana A (M2n), nopo-
a~enua I -dpopmamu ~1) e A 1 (M2,,).
Hcxai~ l-qbop~ fl(~) r j �9 A I(M2n), j = 1, n, 3a./~OBO:ibH~qlOTb yMOBH Teope-
MH I.I. OcKi:ibZl;I BiKIIOBi]xHHI~ ilxea~ 1([3) Ha "U ~ M 2" ~ae K:iac n e Z+ (aaS.
[7]) i e aa~xtmHa~, TO Ha OCHOBi Teope~n ~po6eHiyca-KapTaHa icHyr i~oro iHTe-
rpaJi~Hni~ ~HOrOBHn (M~;n~), ~e ~ 13: M~ --~ M 2n -- i~oro Bi~noBi~He BZ:ia~eH-
Ha, TaKHK mo BezTopni no:IS Kj, j = l,--n, ~0TH~Hi ~O o6paay x[~(M~ ~) ~ M 2",
T~ cavm~ iHayKymT~ si~'lOSi~ai BeZTOpHi noaa Zj : M~ --> T( M~) , j = I, n, Ha
Mnor0sH/Ii M~. BH3HaqHMO 1-cl)opMH X;j~ (1) fi A(n~I(U)), j = 1-~. CripaBelI-
.rIHBa TaKa TeOpCMa.
TeopeMa 1.2 (KapTa.--I~ocT). ~ u c ~ e p e m c i a n b n i l - q b o p m u x[f jO)
e A 1 (~1 (U)), j = 1, n , matomb roam e,aacmuaocmi:
1) l-cl)opmu ~;fjO), j = l,---n, eneza.aeacnumu.a (~c~t(U)) = M~;
2) I-dpop~m rc[f~ l) , j = 1,-'-n, e ~a.a,x~e~umu ~a (TC~I(U)) C M~ i 3a-
3oeo~bn.a~omb )'moau
= o,
(] .4)
b,aa acix j , k = 1, n .
2. I~Terpa~mHi mtoroeH/lH iHTerpOBHHX aHre6paTmao-noaiHoMiaasHHX ra-
MiJ-mTOHOBHX cHeTe~ Ha T * ( R n ) . HHmqe 6y/IeMo poar:iaaaTa cneuiaJmHHi~ BHna-
~OK iHTCFpOBHHX FaMiJIbTOHOBHX CHCTCM Ha CHMII:ICKTHqHOMy MHOFOBH~i g 2 n =
= T*(Rn) , n e Z+, aKHi~ ~onycKae eqbcKTHBHHfl onHc Si]Xnosiaaoi CHMHJICKTHqHOI
CTpyKTypH ~(2) e A 2 ( T * ( R n ) ) . A CaMe, ax ai~OMO a TeOpeMH ~ap6y [6 - 8], ~JL~
ncBuoro oxoay U 6ylI~-aKOi TOmCH U e M 2n icHyr :iOKanbaa CHCTCMa KoopAHHaT
Ha aTsmci ~moroaK~y M 2n, B TcpMiHaX JIKOi MaC Mictle TaKe KaHOHiqHC 306pa.~CHHSI
CHmamKTaWaOi crpyxTypH ~ (2) r A 2 ( T * ( R n) ) :
ISSN 0041-6053. Ytcp. ~lam. ~'vpu.. 1999. m. 51, IV'-' 10
J2OCd~I/X)KEHHfl IHBAPIAHTHHX LIEOOPMAI.Ufl IHTEFPAJIbHHX ... 1381
n
= d p j ^ d q j , (2.1)
j=l
l:Ie (pj, qj): U----> R 2, j = 1--. n , - - Bi/InoBiRHi si~o6paa<eHHa, aKi He 3aBzKJ~H I'IpOnOB-
:>KyIOTbCH F~o6a,nbHO 3 KapTH U c g 2" Ha Becb MHOFOBHB[ M 2n. C•Tyauia Kapa~-
Ha.nbHO 3MimocTbCH, KO.nH M 2 n = T* (Rn). B UhOMy Bmta~Ky M~, oqem4AHO, Mome-
MO 3aBx~m 3am4caam #.nil 6y~b-HKOi 1 -qbopMH a e A l (R n) = T*(R n) po3K.na~
BiLIHOCHO 6a3Hcimx dliHil~HO He3aJICXHHX 1 -qbopM dqj e A l (R"), j = ~ :
n
~( t ) (q ) = ~ pj(q)dqj (2.2)
j= l
s 6yzb-zKifl TOqlAi q r R n, l~e pj : R n_._> R , j = 1---,n, ~ •eaKi r~ta~Ki qbyHKtdi a
D (Rn). 3 (2.2) srlano, mo t~i qbyaKuii pje D (Rn), j = 1,"~, SHKOpHC'roByIoaa, CH aK
r.no6anr~Hi KOOpaj~naTr~ KO/IOTHqHOrO npocTopy T*(Rn). 3acTocyBaBmn Tenep Z~o
1 -dpopMH (2.2) onepattito 3omdmm, oro aHdpepeHtfi~oBanHa HK ~0 1 -~OpMH Ha MHO-
rOBHZ~i T* (Rn), BK~a~aeHoi S T* (R") npHpoZmnM qHHOM, 3 (2.2) OTpHMyeMo, m o r m
T*(R n) icHye rao6aaI , Ho 3a~aHa 2-qbopMa ~(2) ~ A2(T*(Rn)) n Kaaoniunia
qbopMi 22ap6y:
I2 (2) := d a ~t) = ~ dpj^dqj . (2.3)
j=l
Bpaxosy~o~rt anpiopHy 3aMKHeHiCT~, 2-~OpMa (2.3) Ta aesnpoaxeHicrr~ Ha
T*(R'~), ~n6!4paeMo ii aK Kanoniaay cnMn~eKrn'~ay cTpyKTypy ~(2)
e Ae(T* (R")) Ha KOZ~OT~aHO~y MHOrOm~ai M 2~ = T* (Rn).
I'IoKJla/~eMO TaKO:~K, mo B KaHOHiqHHX 3MiHHHX CHMIIJIeKTHqHOi cTpyKTypH
~2 (2) e A2(T* (Rn)) 3a~aHa raMi~TOHOSa CrmTeMa
du
- - = K ( u ) , (2.4)
dt
/~e u : = (q, p) ~ T* (R"), t e R - - eSOJllOIIiflHHIt napa~eTp, HKa Mac crmreMy TOq-
HUX n e Z+ 1 -qbopM ~1) ~ A I ( T * (Rn)), j = 1"~, B iHBOZ~uii, npn,~oMy, aa oaHa-
HeHHHM,
�9 ~ ( 2 ) (1)
-aHj = tK SZ = ~j ,
j = 1,-"~, K = K 1 i seKropHinoaa Kj: T* (R n) ---> v ( r * ( ~ n ) ) , j = 1-~, KOMyraTrl-
~ni Ha T*(R n) "ra aiHiltaO Heaa-~emHi Ha inBapiaHraoMy niaMHOrOSH~i M~ =
= { u r T* ( R ~) : I'~ = hj ~ R , j = 1-~ }. B Tep~iHax qbyn~aii ra~iz~roHa n : =
: = H t r D( T* (Rn)) ~HaMi~Ha CHCreMa (2.4) r, tae a'amn~ KanoHiumat aarmc:
aqj
at '
(2.5)
dpj = _ O H
dt qj'
ISSN 0041.6053. Yrp. ~tam. ~3'pu., 1999, m. 51, bl e 10
1382 A.M. CAMOflJIEHKO, ,,q. A. HPHKAPHATCBKH~
Re j = 1, n, npHqoHy ~a~i Bna~KaTHMCMO, tUO aci aHaHaqeHi BHIIIe dpynKRii Hj r
e D (T* (Rn)), j = 1"~, e aYlre6paiqHO-rlOJliHObliaYlbHHMH Ha T* (Rn).
Ha ni~crani TcopeMH 1.1, IIOKJIaBIIIH B (1) I ,_j := dHr A T*(Rn), j = 1-~, MH MO-
xeM0 cqbop~yJnonaTH ~0no~i~Hy ~eMy.
dleHa 2.1. l / ~ ~inro~l inmeeposnoi" a:lee@ai'~t~o-noelino~danbnoi" ea~dAbmoHoooi"
cucmeam (2_5) ictffe cucme~la pa~iona,abnUX 1 -dpop.~l f~l) ~ A I (T* (Rn)) , j =
= 1,'~, .srJ e mmmu~m ma ninitlno neza/w~nu~tu na inmeepanbno~ty ~moeoeuDi M~,
ornaOeno~o' o T* (R n) 3a Oono~toeoto aiOo6pa,~enn,~ nh: M~ ---> T* (Rn), h r R n.
0CKiJIbKH cniasizInomeHna
dHj = 0 (2.6)
Ha inTerpa~bHOMy IIi/JHHOFOBHRi r YliHiitHHM Bi/~HOCH0 KaHOHiqHHX 3MiHHHX dpj Ta
dqj E A I(T*(Rn)) , j = l,--n, Ha ni~cTaBi neMH 2.1 MO~KHa cqbopMyn~aTH
HaCTynHy nelly.
.JIe~a 2.2. Ha inmeepanbno~ty niO,~moeo~uOi M~ ~ t a ~ e cKpi3b ou3nat~eni ni-
niano nesaneacni 1-dpop~m ~j(i) e T* (Rn), j = ~,n, y ouzn~Oi
t l
~j(l) = Z "ffj.k (q, P) dqk, (2.7)
k=l
Oe (q, p) e gh(M~) c T*(Rn), mari, u~o ~uanaqatomb ~taa~e crpia~, ,ao~a,abnO
~iOo6pa~enna orna~enn~ g : M~ --> R n ~ T*(Rn), nezaae~ne oiO napa~tempia
sr.~aOenn.~ h ~ R n.
3ayoa.~xennst 2.1. 3ritmo a onncaHom BHRIe KOHCTpyKRir xapaKTepHcTHqHa
1-qbopMa (2.7) ~naHa~aeT~CS O~n0ana~no, SZmO CHCTeMa ~aqbepemfianbnnx 1-
dpop~ (2.7) r poa~'s3nom si~nocno na6opy tmdpepemxianis dpj ~ T* ( T* (R n) ), j =
= 1,-~, ~afl~e cKpia~ Ha iHTerpaYlbHOHy HHOrOBH/Ii M~.
TardaM qHHOM, CKOprtcTaBtUHC~ Teope~om 1.3, ~a Mo~KeMO po3rnaHyTH IIHTaHHH
npo rno6an~al t KoopRI~ItaTHH~ onHc [1, 6] ~a~Ke cKpia~ iHTet'paJlhHOr0 HHorosarty
M~. A came, cnpaBez~nnae aacTynne Taep/I~Kenn~.
Tnep~t~ennx 2.1. Ha inmezpanbno~ty niO~moeoeuOi M~ icny~omb n ~ Z+ ne-
zane,w~ux, ~to,w.nuoo w~onoei,mo neoOno3naqnux eno6an~,nux roopOunam tj : M~ --->
--> R, j = l,"'n, marux, v4o ~ ta~e cKpi3~, na M~ t I = t ~ R i
](l)(q,p) = ~*f t [ l ) := dtj, (2.8)
O e j - - 1,n, ( q , p ) r gh(M~), npu,~o~q oermop p ~ T~q(R") e R n ~ q-zane~-
nuiq napa~tempo~L
3. BKna;lenHx inTerpaa~noro MHoronn~y: aare6paiqno-aHaniTnqnnfl
onHc. Bae~eHi aHme rno6an~Hi ~a~ge cKpi3~ goop~HuaTa tj : M~, ---> R , j = 1, n,
Ha inTerpammoso/~Horos.~i M~ aa/ia~OTh, OqCaRRH0. B He~auifl qbop~i si~o6pa-
xeam~ sKnatlCHrta r~h : M~ --4 R n ~ T* (Rn). 3 Mer0m no6yROSH noro ana~irH~-
HOrO ormcy, ari~mo a (2.7) noKna/IeHo, ttl0 Ha ~tcaKitt KalYri U h C M~ aa~taHa KOOp-
~HHarHa r ~tj: U h -4 R, j = I, n, TaKa. mo Bi~to6paA~enrl2 BXJIa/IeHna Xh:
M~---> T*(R n) aa~aeT~ca aHa~irHano y qbopMi 2n r Z+ h-napaMerpaqHHX
Bi~to6pamens
ISSN 0041.6053. Y~p. ~am. ~.'vpu.. 1999, m. $ !. N ~ ] 0
LIOCJIILDKEHH~I IHBAPIAHTHHX J1EtDOPMALII~ IHTEFPA.rlbHHX ... 1383
qj = xj(~),
(3.1)
pj = Hj,h(p. ; h),
~e j = 1,-~ i It = (~t 1, ~2 . . . . . ~n) r R n. 11106 poaauHyrH na.rti MCTO]I 3HaXO]I~KeH-
HA aiao6pameHh (3.1), KOKaTKOBO IIpHHyCTHMO, mo iHTerpaYlbHHfl MHOFOBI4~ M~
KonycKar B Koop~HHaTHHX 3minnnx IX : U h "> R n anaaiTHqHe Hpo~IOB~KeHHa B 06-
a~aCTb KOMHJIeKCHHX rlapaMeTpia IJ. : U h ---> C n TaKHM ~HOM, mo iHTerpaY~HHll MHO-
rOBH~ M~ 6y~e Ko~naKTHnM KOMn~eKCHO-aHaaiTHqnHM ~HOrOBH~OM. OCTaHHa
yMoaa, a orJ~a]~y Ha pa~ioHa~HiCT~ i -qbop~ (2.7) Ha MHOrOBH]Ii M~, He r CyTTe-
BHM O~MeZKCHHSIM Ha K.rlaC po3r..rl~yBaHnx n a M a anre6paiqao-no~iaoMia~nnx CHC-
TelV[ Ha T* (R n).
JXna n o ] l a ~ m o r o onHcy Bi~ofpaxCeHHJl BKJIas (3.1) CKOpHCTaeMOCb MeTO-
~OM KanoniqHHX neperBopeH~ I lyaHKape- KapTaHa [6, 7], axaia ~ar ~O~K~naicaa, in-
BapianTnoro onncy roop~rmaTaoi cTpyKTypu BK~a~enHa (3.1) 3a UOnOMOrO~ Ta~
3BaHOrO MeTOay nopo~oKytoqrtx qbyHKI~i~t, mo aa~oBo~bnarOT~ neani ~aqbepentt ia~-
Hi piBHJmna FaMi~TOHa-- J;IKo6i [6] a qacTnaHa~n noxizmHMn.
3a~a~o Tenep Ha KOaOT~aHOMy npOCTOpi T* (Uh) 1~0 iwrerpa~bHoro MnoroBn~y
M~ zo~a~nO-KaHOHiqni~o0pznHaT~ ~t: Uh-.-->R" Ta ~ : T~(Uh) -->R", a Tep-
MiHax JIKHX 3a/Ia~TbCJt Mafl:hKe cKpiab JIOKaYlbHHfl ~mqbeoMopqbia~ ~* (T* (Uh)) --.>
---> T* ( Rn), ~ o 3a~OBOnbHJ~e yMony KartoHiqriOCTi
/Ie, 3a BH3HaHCHHYIM,
~h `0-(2) = f~(2), (3.2)
n
) = dwj^dl.tj.
j= l
Ha ni~CTaBi 3o6pa~enHa (2.7), ari~ano 3 ( 3 . 1 ) , OTpnMyeMo, mO Biao6pa~enna
BKJIa~CHHa 7C " U h ---> RnC T*(R n) He 3aJIeYKHTI~ eC1DeKTHBHO Bi]~ n a p a M e T p a BKJla -
h ~ R n. ~c o3Haqae, n~o piBHiCTb (3.2) MO~KHa 3aHHCaTH B CKBiBaYleHTHi~ ~eHH,q
~opMi
/le
n
j= l
n
= pj dqj E r*(R"),
j= l
TOSTOILrI~BCiX j = 1,"-'n BH3HaqeHi e.rICMeHTH ~j " Rn"'> R TaKi, RIo Ha U h C M S
a( l ) [u h = H * a (1), a s i a n
n a ~ k (11)
% h) = Pk (3.3)
t = l ~ l l j
BHxo]~qrl 3 JIOKaJIbHOi ]~H~beOMOpt~DHOCTi Ha CBiR o6paa Ma~mc cKpi3b BKYlaI~eH-
HYl H : M~ ~ Rn C T* (Rn), CIIInBi]IHOIIICHH~I (3.3) MO:KHa 3arIHCaTH B eKBiBaJICHT-
Hill qbopMi ~X cIIiBBi/IHOIIIOHH~I B H F J I ~ [9]
ISSN 0041-6053. Yrp. Hwm. acy. pn., 1999. m. 51,1~ 10
1384 A.M. CAMOi;13IEHKO, ft. A. HPHKAPHATChKHI~
( 3~ (ix) 1 -I
PJ = k=l~" ~k~, ~IX Jkj (3.4)
~aaa Bcix j = 1,--n Ha npocTopi. T* (M~).
3 Toro, mo Bi~ao6pa)KeHHa (q, p) : T* (R") ~ R e" za~ae rno6a~bHi Koop~armaTH
Ha KOROTHqHOMy MHO.r0Brlai T*(R") , Ta 3 (3.4). BrlrulrlBar mo Bi/Io6pa2genH~l
(IX, w----) : T* (R") --> R 2 n aa~ar M~XXe cKpiab rno6a,abHi Koop~aHHaTH Ha T* (M~).
~ n a 6i0~btu ~eTanbHOrO OnHCy ttboro Bi~to6pa~KeHHa B TepMiHax BKna~CHHa re:
M~ --> R " c T* (R") CKOpHCTACMOCb MeTOaOM KaHOnianaX nepeTBopeHb HyaHKa-
pe-KapTaaa [6], aa~a~qH riopo~t)Ky~Oqy qbyHKt~ia) S : R n• R n--~ R TaK, mo
cnpaBe~0mBa Ha M~ piBHiCTb
n
~., ~ j ( ix ;h)d l . t j = - ~ , ?j( ix;h)dhj + dg(~;h), (3.5)
j=l j=l
~e tj : M~ --> R, j = 1,-'n, - - ~aeaxi xaHoHiqHo cnpJUgeH! Be2IHqtlHH .~O 3MiHHHX na-
pa~eTpiB BK2Ia~eHHJ~ h e R n. 3 yMOBH (3.5), OHeBH/~HO, BHII21HBaE, mOB KOOp/~HHa-
TaXBKna~eHHa~HOrOBKay M~ B T* ( R") Bizo6pa)KeHHa (?,h): T* ( M~ ) .---> R 2",
3a~arle nopo~gywaow qbyaKttir S : R n• R " - ~ R, reHepye 0~0Ka~HHta ~nqbeo-
Mopqbi3n MHorona/Iy T*(M~) ~ ce6e z HOBHMa KoopaHHaTa~H (~, h) : T* (M~) -->
-.-> R 2". IIpH L~bO~y, O~eBH~HO, mo nopol~x~yw~a dpyHKttia S : Rnx R n .---> R 6y~e
icHyBaTa ZK O~HOaHaaHe Bi~o6pa)KeHHa ~ O~tHOaB'aaHifl O60~acTi U h MHOrOBH~y
M~ zmue Toai, KOnn iHTerpa~
~ a O) = 0 (3.6)
i~o2 ~ Uh
6y~e TOTOA~HO piBHHM HyJIIO B3ROB~K 6y]Xb-RKOFO 3SMKHeHOFO Ky~KOBO-r'~a]~KOFO
tunnxy ~30 (2) c U h. BpaxoBy~OqH, mo iHTOr'paJIbHH~ MHOFOBH/~ nopo/DKeHH~ op6i-
TaMH KOMyTaTHBHHX BCKTOpHHX rlO.rlis Kj : T* (R n) __> T(T* (R n)), j = 1"~, He3a-
ae~KHHX Ha M~, 3a qbopMy~o~o CTOKCa [1, 7, 10] a (3.6) 3Haxolmno
d ( j ' ~ a ( ' ) : ]~t:(d~(l)) : ] ~Z~(2) _ O, (3.7)
o 2 o 2 bo 2
OCKiJToKI4
~'~(2)(Ki, gj) = ~-'~(2)(7~h.Z i, Kh.Zj)= K~'~(2)(Zi, Zj),
flC2)czi, Zj) ~ 0
~ scix i , j = 1,-'~. lke, 30xpeMa, oaHaqar [6], mo iHTerpaYlbHHfl MHOrOBH~ M~ r
npmcnanoM Talc 3BaHOF0 ~arpaHxCBOrO MHOFOBH~[y, Ha ~K0~j' CHMH~eKTHqHa cTpy-
KTypa TOTOZC.HO pinHa Hy.rllO. BpaXosyio,.Irl (3.6), 3 (3.5) MO~Krla BH3HaHI, fI'H qbyHxRiIO
: R n X R " -"> R TamtM qHHOM:
S ~ ; h ) := J~= i ~j(ix(s);h)dix,(s), (3.8)
ISSH 0041-6053. Y~p. ~uzm. ~'ypn.. 1999, m. 51, N e 10
/~OC.J'II/])KEHHJt IHBAPIAHTHHX ~EOOPMA/_H~ IHTEFPAJIbHHX ... 1385
/~e It: [0, 1]-->Rn~TaKaTpaeKTOpi~IB R n mo
It l ,fo=~t~ I~l,=~=~t~R ", It-~[0, t ]aUh.
J;IK npocTaia BaCHOBOK a BHaHa~eHHZ (3.5) aHaxo/m~o, tUo x0op/mHaTni qbyHrafii
~ j : T~ (M~) --4 R Ta t'j: T~ (M~) --4 R, j = 1"~, O~qHCdlIOIo'rbcJl TaKHM qHHOM:
~j (it; h) = ~-~ (It; h)
~t j '
(3.9)
O~ (~; h)
nza Ma~a<e Bcix (g , h) e R 2n.
Ba6epeMo renep KaHOHi~my Koop~aHHaTHy CHCTeMy g : U h ---> R n, ari~Ho a npan-
tmnoM ai~oKpeM~eHHZ 3MiHHHX FaMi~STOHa--HKO6i [1], ~ a qbyHKtIii S : R nx
• Rn---> R , TO6TO~ataMafl;~eBcix (~t ;h)~ R 2n S = S : R n x Rn---> R , l~e, aa BH-
3HaqeHH~IM,
n
s(it, h) : = ~ Sj (~j, h), (3.10)
j = l
npHqoMy Sj : R x R n..~ R, j = 1 ~ , - - aeaKi Mafl~xe cKpiab raaz~i ai/xo6pa~eHnz.
3 (3.9) Ta (3.10) Biapa3y aHaxoaaMO, mo ~a~a Mafl~e Bcix aHaqerm (It ; h) ~ R 2n
wj(it;h) := wj(~tj;h) " (3.11)
a~z KO~HOrO j = 1, n. B ttboMy BHnaI~cy 3 (3.8) Ta (3.1 1) BHrhrmBae
S(~, h ) : = j~=l J. wj ( i t j ' , h )d i t j , (3.12)
z~e Ito E R~6y/xI,-J~Ka qbiKcoBaHa TOqKa Z o6pazy si~o6pa~ceHaa It : T*(M~) -->
-._> R n.
POaI'.rI~IHeMo renep 6izstu/~eTaa-lbH0 BrI3HaqeHH.q (3.5) nopo~a~cy~oi qbyttKllii S :
R n x R n ~ R. 3 (3.5), OqeBH~O, aHn~Hsar tUo MHOrOBHa T* (M~), ZK noKa~mHO
~HqbeoMopqbHrt~ MHOFOBHa T* ( R " ) , r CHMIIneKTHqHHM, CHMIIMeKTHqHa cTpyKTypa
m<oro noKa~HO aa~aeT~Ca Bnpa3oM
/I
/~; ~,-~(2) = ~ dhj Ad?j. (3.13)
j=l
BpaxoBy~oqr~ Tenep KaHOHi'a'ai pinHaHH~ FaMi~mTOHa (2.5) ~aaa aMiHrlHX hj o r~ h
Ta ?jo gh : T* (Mg) --> R, j = l ~ , ' Ha MHOrOBH/~i M~, ~lerKo OTpHMaTH
dt"~ = 5j'k
a~a Bcix j = 1, n . OcTaaHe. OqeB~HO, o3Haqae, tuo ~o~a~a r~o6a.rmHo OTOTO~KmtT~
~j -- ~: M~ --> R, j = 1, n, i aan~caTa Tenep (3.5), nHKopHCTaBmH (3.11); mr
?SSN 0041-6053. YKp. ~ m , ~Tcpn.. 1999. m. 51. N e 10
1386 A.M. CAMOflJ'IEHKO. ft. A. I'IPHKAPnATCbKI41rl
n n
wj ( ~ j ; h ) dg j = - ~ tj (g; h) dhj + dsc~t; h). (3.14)
jffil j=l
Tenep M~ ~o~e~io cKOpHCTaTHC~ panime ncraHoa~eHnMn cniaai~HomeHH~
(2.8) Ta (3.4) i cclaop~y~noaaTH OCnOaHy xapaxTepHc~4q~y TeopeMy.
T e o p e ~ a 3.1. Ha inraeepa.~b~o~ty niS~moeo~u~i icnytom~, n 2 ~ Z+ pat4iona~bttUX
~iOnocno z~dnnux (~t, w) ~ R x R qbyt~rt~i~ fkj : R x R --> R , j , k = 1, n, marux,
uto cnpa~eOnu~i cy~dcni 8udpepeh~ianbni cni~aiOnomennn
Owj ($.tj ; h) (3.15)
f kj([tj ; wj ) -- Oh k ,
posa'~.~ro~t ~rux e nagip n ~ Z+ anzeSpai'~nux cnio~i~nomenb na napa~tempu
wj(~ ), 3. ~ R , j = 1, n , marozo auen,~Oy:
mj
mj -k
W~.~ (~.) + ~ cj~(7~;h)wj = 0, (3.16)
k=l
Oe ~ucna m j ~ Z+, j = 1, n , a aci aenu~unu c j t ( ~ . ; h ) , j = 1, n , e ninianuz~u
adpinnu~tu ~),nrt4ionana~tu sa napa~tempo~t h ~ R n s pat~iona.abnu~tu no 9~ ~ R
roedpit~ienma~m. ~
]l{o~eOenn~. Ha ni~cTaai crIissi/~u0menb (2.8) Ta 3aMiHH (3.4) y a a r z ~ i
p~ = ~
kj '
j = 1,-"n, ia ao6pa~enHa (3.12) Ta oa'oTo~rteHh tj - tj : M~ --~ R, j = 1, n , ai~paay
aHaxo/xrr~o, mo ~ t a K0acaoro j = 1, n cnpaae~a~aai pianocri
::l,(q,p)l,,, = -- =
k=l
:,,<,;w,)
= <-> = d i N = . S(g;w) =
n s Ow k (It k ; h)
= O"~jj ~ k~l= Wk(ktk;h)d~tk = k=l ~hj dktk (3.17)
~IL~ ~XeaKaX n 2 ~ Z+ qbynKmlt fj~ : R n x R n -~ R, j , k = 1"~, r a acix auaneab na-
paMeTpa ~t ~ R n 3 KapTa U h. 3 (3.17) aHax0~aMO, tU0 na U h ~ M~ c n p a s e ~ n a a
piaa icrs
t = z k-~ l Ohj d}'tk" (3.18)
IIpHpinmoloqa n (3.18) KOedpittieHTH npH ~iniiIHO ueaa.ne~KHnX ~adpepemtia.aax dgj,
j = l,'-'n, auaxo~HMo, laO ~aa j, k = 1,"-n Ha Uh aHronyeTsCa piBaiC'l~
aw~ ~k; h)
fj~ (l.t; w) = Ohj " (3.19)
Tenep poarJI2HeMO sHpaa n (3.19) c~pasa ~ qbiKcOSaHOrO k e 1, n aK qbyHK-
t~iW Ymule O/IHOrO rlapabteTpa ~k E R. ~e , aBHqal~lto, ozHaqae, IIio/I~12 UbOrO cl3iK-
ISSN 0041-6053. YKp. ~!am. ~ry. pn., 1999, m. 51, N ~ 10
~[OCJ'IILI~EHH$I IHBAPIAHTHHX LiE~PMAI.iffl IHTEFPAJ'IBHHX ... 1387
coaaHoro k e 1, n srrpaa a~iaa TeTK rlOBHHeH 6yTH ~ynKttiem ~Hme napaMeTpa
~L k E R. 3 npyroro 6oKy, snpaa cnpaBa aamxHTb ~Hme sin qbynKttionanhnoro na-
paMeTpa w k e R, ne k e 1,---n 3aqbircoBane ramie. A tie oaHa~ae, mo i anpa3 3niaa
npH UbOMy 3HaqeHHi k e 1, n Ta Bcix j = 1, n HOBHHeH 6yTI, I qbyHxltieio TiJIIaKH
qbyHKUiOHanbHOrO napaMeTpa w k ~ R . OCTa~He MO~Kna, OqeBH~nO, 3anHc~aTH eqbcK-
T~mHO a K Ha6ip TaKnX cniBBinHomenb n ~ Bcix j = 1, n Ta i ~ k e l , n :
- o . ( 3 . 2 0 )
Ow i
Ha6ip yMOB (3.20) e Ha6OpOM a21re6paiqHrlX crliaBi/~HOUleHb Ha pal~ioHa.rlbHi c16yn-
KuiI i j~, J , k = 1,"-n, aa~exnrtx edpeK'rrlaHO Bin Bino6pax~eHrlJi BKJIa~eHH~q ~ :
U h ---> R n c T * ( R n ) , onHcaHoro paHime. 3a6eaneqrmmri yMOBy (3.20) Brt6opoM
Bino6paXeHHZ BK.na~eHHZ "~ : U h --~ R " c T* ( R n) , Mrl TrIM cablnM, oqeammo, Bi~t-
paay m aa6eaneqHMO Heo6xinHy yMosy (3.19), TO6TO OTpnMyeMo neJtKl4i~ Ha6ip n 2 e
e Z+ qbyHKui~ fj~(I.t k ; h ), j , k = 1, n , paLdOHa~bHO aa.ne~Km4X ~in napaMe'rpiB
I.tk e R, k = 1, n, aKi 3anoBo.nbnmort,, arinHo 3 (3.19), cniBsinHomeHHZ
f jk (~tk ; wk) = ~wk (~tk ; h)
Ohj
nna Bcix j , k = I, n, mo cniBnaaae no qbopni a (3.15). HpnqoMy icHyaaHHa Maitme
cKpiab r~a~zo ~ a ~ e H o r o iHBapiaH'rHoro niffMHoroaHny M~ rapanTye cyMicaica~
CI4CTeM~t nHqbepem~ia:mmtx cniBBinHomeab (3.15) Ha KapTi U h ~ M~ tt.n~ Mal~)Ke
acix 3Haqem, napaMeTpa h e R n. I'lprt LU, OMy yaronmemtM 3 (3.1) i (3.4) poaB'aaKOra
CHCTeMH (3.15) 6yne Ha6ip ame6pai,~HrIX cniBBinHomeH~, (3.16) a y~osamt rla roeqbi-
tdeHTa, SKi cqbopMy.m, osaai B TeOpeMi. Oca~aHHe i ~oaonwrb TeopeMy.
J:IK nac.ni~oK reopeMH 3.1, arinno a ~opMynaMH (3.9) Ta (3.12), n.nz aI~Xi~tHOi ra-
Mi.~bTOHOBOi CHc'reMH (2.5) Ha iHTerpa.m, HoMy MHOrOSr~ni M~ cnpatae~mma aaasti-
Trlqaa qbopMy.rla n.rla eBo.rtlotdi aeKTopnoro HO.rl~l K : T* (R n) _._> T* ( T* (R n)) :
t = (3.21)
j = t . ~h~ d2L,
ne a.rlre6pai,..iHi qbyHKtfii wj : R x R n ---> R, j = 1 ~ , mtaHa,-talo'I'~C.a .aRHO .aK PO3B'.a3-
KH piBH.aHr:, (3.16). OCTaHHi ~orlycKalo'rb, oqeSHnHO, aHa.niTrtqHe npo,ao~xeHHZ B
KOMn.neKcHy n.nOmHHy 3a napaMeTpoM X ~ C. FpyHTy~OqHCb Tenep Ha Bi/~o~o~y
qbaKTi [5], mo KOX_r.-lil~ a.are6pai'-IHila ~bynKt~ii (3.16) Bi~moBiztae OnHOaHatiHa KoMna-
KTHa piMaHOBa rloaepxH$1 Fh j a~re6paitmoro pony nj e Z+, j = 1-~, $1KHi~i BHaHa-
qaeTl, C~l onHoarlaqHo aa/xono~oro~ Ta~ 3BaHOi qbopMyzH PiMa~a--Fypsitta [5, 11] :
k i
2nj - 2 = ~ (V~,k -- 1), (3.22)
k=l
ne ~za z o x a o r o j = 1, n qnc~ao v j , i e Z+, k = 1, kj ~ zpaTnocri kj e Z+ OC06~n-
BHX TOqOK raaymerlHg piMarlosoi noBepXHi Fh j . 3 inmoro 6OKy, BLIIOMa TeOpeMa
Rgo6i [1 I] npo Te, mo a6e.neBrIfl vmoroBa~a 6ynb-azoi piMaaoBoi nosepxrri .F~ pony
g e Z+ r nrlClOCOMOpCl)HHbl KOMn~eKCHObIy Topy T~ ---- Cgl(l, B), 1~e (I, B)~
ISSN 0041.6053. Yrp. ~tam. ~. pn.. 1999, m. 51. N'-' 10
1388 A.M. CAMOfl~IEHKO. $I. A. HPHKAPHATCbKHI;t
KoNnJ1excHa rpaTza si~anoeiRrmx nepio~is Ha C g [12]. Tenep, 3riRHO a TeOpeMo~
ApHo~b~a [6] nix) nHqbeo~opqbiaN Topa T n KOMIIaKTHOF0 iHTerpaJmHOrO MHOFOBH-
~y t~iJmoN irrrerposHoi aa JIiyBi~CN raMi~TOHOBOi cMCTeNH, BCTaHOn~meMO cnpa-
I~RY[HBic'rb HacTyIIHOFO TBep/DKeHH~I.
TBep~xenn~ 3,1. Ko~marmnua inmezpa.abnu~ ~mozoeua M~ t4i,arO,~ inme-
zpoonoi" no,aino~da,abnoi" za~ti,abmonoooi" cucme~tu (2.5) 3uqbeo.~topdpnua 3i~cnia ,lac-
muni ro,~m,aercnozo mopa T~ poOy n e Z+, npu~tozty ame6pai'~ni poOu ocix ame-
6pai'~uux rpuoux (3.16) pioni, mo6mo nj = n, j = 1, n.
Tar, aM qHHOM, qbopMy~a (3.21) a~z eBomottii ~nHaMiqnoi CHCTeNH (2.5) na iHTe-
rpa.waHOMy MHOFOBHRi M~, BKYla/IeHOMy B T* (Rn), ~Iae c13aKTHqHO IIOBHH~ OrltlC
B#'IaCTI4BOCTei~I ii op6iT B 3a.rIe)KHOCTi Bi~ napa~eTpa BKJmKeHH~ h e R n. ]la~ 6ara-
T~OX 3aCTOCyBaHb qbopMym! Trmy (3.21) qaCTO 3aCTOCOSy~)T~, y qbopMi aMiH~X
,d~i~-KyT" [3, 6], ~IKi BBOJ~ffTbC3I TaKHM qHHOM. Hcxa~ Tj : R n__.> R , j = l,--n,
r~a~Ki Na~mc czpiab si/~o6pamenH~, SHaHaqCHi 3a ~IorioMOrOlO qbopMy~I
~ 1 ~ wk(~,;h)d~,, (3.23)
~ J : = 2-x~ oi k=l )
_(k) n~ ~ Re Kp~ai Oj ~ F~'~ , j = 1, n, yTBOp~aTb ~a~a Koxnoro k = 1, n 6aa~ca o/moan-
NipHoi rpynH roMoJloriil H I (F/~ ; Z) piMarIOaOi rlOBCpXHi F~ t , BBCRCHOi paHime
npH aHa~iai a~re6paiqHnx piaH~H~ (3.16). 3Ninn~M ~ : Rn-~ R , j = 1 ~ , BizmoBi-
~a~T~, anHqai~HO cnp~xeHi KanoHiqHO, TaX 3BaHi KyTOBi aMiHai 9 j : M~ -~
--~ R t / 2~Z, j = 1 ~ , ~ma mc~x ~erKo aHatrm Bi~anoBi/my nopo~tmy~qy qbyHKtfi~
KaHOUiqHoro nepeTaopeaHa S' : Rnx R n --> R, ~xe
n /l
wj(~tj;h)dlxj = - ~ q~j dTj + dS'(~t;7) (3.24)
j f t j=~
~zm Mal~xC acix (p; 7) e Rnx Rn" 3 (3.24) amino, mo
S'(~t;y) = S(p.; h(T))
JX2IJt NaRme Bcix (p; T) e RnXRn, lXe Bi/1o6pax<eHHa h : Rn• R n - - o6epHeHe ao
ai~ao6paxeanz (3.23). TaN CaMHN, 3 (3.12) i (3.24) aer~o aHaxo~aMO
~S'(I.t; 7) ~ ( ' ~ '
% = ~'~ = O~'J k, k -- t V!,_ wk (~; h (~')) dX =
7 " = ~ _ O.Wk(~ ' ;h ( ' f ) )ah~ ' (Y)dX = ~ tst%j(T), (3.25)
s.k ffi 1 . 0 Ohs ~'~ j s = 1
/~e, 3a ariaHaqeHrl~lM,/IID! Ma.flxe scix "[ e R n
ah, (7)
(%j (~) : =
j , s = 1,'-'n, npHqorxy NaTprms o)(y) : - {~oq (Y);. J, s = 1-~} Ha3HBa~l'bCJl [1] ,,qac-
rOrHO~)" ~aTpHUe~O ~aMiaHo i cHere~a (2.5). rlpH eBoJ~mttii arma~iaHoi CHCTC~H
(2.5) sZ/Ioaar op6iza, ~taqbeo~opqbnoi aH6paaiia ~aHZHeHiia Kpaai~t O s, S = 1, n , 3
(3.25) va (3.24) nnnmmar
I$$N 0041.6053. Yrp. ~wn. :,,~. pn., 1999 , m. 5 ] , N e I0
~OC.JIIJ~KEHH~I IHBAPIAHTHHX L~EOOPMAI2II;t IHTEFPAJIbHHX ... 1389
A (pjlo" = 2~:~j,,,
m
~ae j = 1, n. OCTaHHr o3Haqar mo eao~mtda/IHHaMiqHOi CHCTeMH Ha KOMnaKTHOMy
MHOrOBa/~i M~ r KBaainepio~qHOm TpaexTopiem Ha Topi T n 3 IvtHOXCHHOIO Ksaai-
nepiogiB {Tj(s) = 27tr l (y): j = 1"~}, s = 1,'-'n.
3aysa~enna 3.1. 3anponoHoBaHa BHme a~re6paiqHo-aHa~iTnmta MeTOZtHKa
no6ygOB~ Bigo6paxeHb BKJIa~eHH~I iHTerpanbHHX MHOEOBrl/~iB lli.rIKOM iHTgFpoBItHX
rlO~iHozia~IbHHX FaMDII~TOHOBHX CrlCTeM B 6araTl, OX BHnazlKax MOmO 6yTH,
OqeBHgrtO, 3aCToconaHa i /~O Ri.rlKOM iHTerpOBHHX patdorm~HO aagamlx
raMi~bTOHOBHX CHCTeM Ha T* (Rn). 5IK npnKaa~t MOmHa poar~x~myTM raMDmTOHOSy
CHCTeMy Ha u~iomnHi R n, aKa onHcye pyx MaTepia~baoi TOUKH B no~i gsox
nprtTarymqrlX aa H~mTOHO~ t~eHTpi~, poazimeHn x Ha qbi~coBanifl Bi]l~a~i [I ].
Fa~ia~ToHia~ Ta~Oi CHCTeMa B e~inTaqH~x ~oop~mHaTax (~, r I ) ~ R 2 aa~aeT~Ca B
TaKiit pal~ioHa~H0-a~re6pai~Hiti qbopMi:
"~ ~2 -- C 2 ~2 -- C 2 4k~
H := H t = 2p~ ~_-~ '~ + 2p2 ~--~--_~2 ~2_r l2 , (3.26)
�9 2 ~e (p~, pn)e T(~,~) ( R ) - - yaara~HeHi iMnya~c)i MaTepia.m, HOi TOqKH, .k ~ R+
rpaBiTattiflHHl~ napaMeTp i c e R+ ~ qbiKcOBaHa BiRrta#m Mira np~TarymmiMH tteH-
TpaMH. KaHoHiqHa cHMn~eKTtiqHa cTpyKTypa y Bi~oKpeMaeHHx aMiHmIX FaMiJI~TO-
Ha--SIKO6i Ha T*(R 2) a~ae'r~ca a~
~,~(2) = dpq'Ad'q + dp~Ad~ ~ AX(T*(R2)).
BiaHOCHO uiei cwpyKvypH icHyr me oaHa pauioHa~bHo-ame6painHa qbyH~ui~ H 2 :
T* (R 2) _~ R ~ a iHaoa~uiL VO6VO ainnoniaHa a y x ~ a HyaccoHa { H~, H 2 } = 0 n a
T*(R2), /~e
l p ~ ( ~ 2 _ C 2) + 1 2 hi H E = ~ ~ p~ (112 - c 2) - k~ - ~ (~2 + ~12). (3.27)
BHKOpHCTOBylOqH pOaBHHeHHfl BHme a~lropaT~ JIoc~i/~eHn~I ih'Terpaa~,noro
MHOrOBH~y M~ ~az ai~noai~HOi iaTerpOBHOi raMi~TOHOBOi CHCTeMH (2.5), aerKo
oTpn~yeMo ~mHm~ Bnpaa ~JIa nopo/I~Kymqoi qbynKuji KaHOHiqmiX nepeTaopem, y Ba-
r ~ i
n
S(~, n ; h) = f dX co~ (X; h) + ~ dX coq (X; h), (3.28)
~e (~0, H 0) ~ R 2 ~ qbiKcOBaHi qncaa, KartoniqHo crlpstxeHi no aMiIm~ix (~, TI ) r
g 2, qbyHKILi~ (W~, W~l)~ T(~,~I)(R2 ) aH3HaqeHi ~IK poaa'~laKI~ azire6pai~IHI4X
= 0 ,
h2 + h A~2 +2k~.
2
- - c 2
= o
piBHmlb BHrna~y (3.16):
(3.29)
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Ozlep~xano 19.02.99
ISSN 0041-6053. Yrp. ~ra. ,x'ypn., 1999, ra. M , l~ IO
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| id | umjimathkievua-article-4737 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:04:21Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f3/b6e7ba007ae530ddeafa80e2b49bdff3.pdf |
| spelling | umjimathkievua-article-47372020-03-18T21:12:54Z Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I Дослідження інваріантних деформацій інтегральних многовидів адіабатично збурених цілком інтегровних гамільтоновнх систем. I Prykarpatsky, Ya. A. Samoilenko, A. M. Прикарпатський, Я. А. Самойленко, А. М. By using the Cartan differential-geometric theory of integral submanifolds (invariant tori) of completely Liouville-Arnol’d integrable Hamiltonian systems on the cotangent phase space, we consider an algebraic-analytic method for the investigation of the corresponding mapping of imbedding of an invariant torus into the phase space. This enables one to describe analytically the structure of quasiperiodic solutions of the Hamiltonian system under consideration. Базуючись на дифереінціально-геометричній теорії Картана інтегральних підмноговидів (інваріантних торій) повністю інтегровних за Ліувіллем-Арнольдом гамільтонових систем на кодотичпому фазовому просторі, розглянуто алгебраїчпо-апалітичний метод дослідження відповідного відображення вкладення інваріантного гора в фазовий простір. Це дає можливість описати аналітично структуру квазіперіодичних розв'язків досліджуваної гамільтонової системи. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4737 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1379–1390 Український математичний журнал; Том 51 № 10 (1999); 1379–1390 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4737/6175 https://umj.imath.kiev.ua/index.php/umj/article/view/4737/6176 Copyright (c) 1999 Prykarpatsky Ya. A.; Samoilenko A. M. |
| spellingShingle | Prykarpatsky, Ya. A. Samoilenko, A. M. Прикарпатський, Я. А. Самойленко, А. М. Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I |
| title | Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I |
| title_alt | Дослідження інваріантних деформацій інтегральних многовидів адіабатично збурених цілком інтегровних гамільтоновнх систем. I |
| title_full | Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I |
| title_fullStr | Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I |
| title_full_unstemmed | Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I |
| title_short | Investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. I |
| title_sort | investigation of invariant deformations of integral manifolds of adiabatically perturbed completely integrable hamiltonian systems. i |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4737 |
| work_keys_str_mv | AT prykarpatskyyaa investigationofinvariantdeformationsofintegralmanifoldsofadiabaticallyperturbedcompletelyintegrablehamiltoniansystemsi AT samoilenkoam investigationofinvariantdeformationsofintegralmanifoldsofadiabaticallyperturbedcompletelyintegrablehamiltoniansystemsi AT prikarpatsʹkijâa investigationofinvariantdeformationsofintegralmanifoldsofadiabaticallyperturbedcompletelyintegrablehamiltoniansystemsi AT samojlenkoam investigationofinvariantdeformationsofintegralmanifoldsofadiabaticallyperturbedcompletelyintegrablehamiltoniansystemsi AT prykarpatskyyaa doslídžennâínvaríantnihdeformacíjíntegralʹnihmnogovidívadíabatičnozburenihcílkomíntegrovnihgamílʹtonovnhsistemi AT samoilenkoam doslídžennâínvaríantnihdeformacíjíntegralʹnihmnogovidívadíabatičnozburenihcílkomíntegrovnihgamílʹtonovnhsistemi AT prikarpatsʹkijâa doslídžennâínvaríantnihdeformacíjíntegralʹnihmnogovidívadíabatičnozburenihcílkomíntegrovnihgamílʹtonovnhsistemi AT samojlenkoam doslídžennâínvaríantnihdeformacíjíntegralʹnihmnogovidívadíabatičnozburenihcílkomíntegrovnihgamílʹtonovnhsistemi |