Certain aspects of theoretical-group approach in problems of nonlinear mechanics
Two aspects of the application of the asymptotic decomposition method to the decomposition and investigation of nonlinear systems are considered. The first one is the extension of the class of functions in terms of which to form the asymptotic decomposition by applying special functions of mathemati...
Збережено в:
| Дата: | 1992 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1992
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/2317 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508242675761152 |
|---|---|
| author | Lopatin , A. K. Лопатин , А. К. |
| author_facet | Lopatin , A. K. Лопатин , А. К. |
| author_sort | Lopatin , A. K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-09-14T12:03:56Z |
| description | Two aspects of the application of the asymptotic decomposition method to the decomposition and investigation of nonlinear systems are considered. The first one is the extension of the class of functions in terms of which to form the asymptotic decomposition by applying special functions of mathematical physics. The other aspect is the extension of the asymptotic decomposition method to a new class of differential systems, namely, Pfaffian systems in involution. |
| first_indexed | 2026-03-24T02:22:06Z |
| format | Article |
| fulltext |
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EeHHH Ha c cj)epe B ccpepHtJeCl<Oll CHCTeMe l<O0J) ll ll 113T f!Mee1 BH.ll. [6, c. 616]
dq!ldv = - I + w cos cp ct g 0 - ebr 2 cos2 fl cosi cp,
d0/dv = w sin cp - ebr2 s in 0 cos O sin <p cos <p,
drldv = 0,
r)le w = const - ynpaaJJmomee ycKop e 1-me ; cocTaBJJ51JOLUH e npH e _ yqHThlBaKn
:ueueHTp aJJ bl-lOCTb rlOJIH npHT51)KeHH51 3e.MJ1H .
nm:jxpepeHUH3JlbHble onepaTOpbl
u 1 = cos cp ctg 0 a~ + sin (jJ :0 .
~ A. K. JlOl1ATl1H, 199'1
22 ISSN 0041-6053. Y,cp_ MQT. ~ypH., 1992, T. 44 . .M I
BXOMU!He B KatieCTBe COCTaBJIHIOW.HX B onepaTop
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V 6n -•<I> 0 = r ~ e cos ' HMelOT BH.U
L = _ V 6n e-i<1> [- _i _ _!_ + cos 0 _j_ + sin 0 _!__]
1 3 r sin 0 ocp r ae or ,
L V 6n ill) l" i iJ cos 0 a . O a j
3 = - 3- e . 0 ;;- + -- ,:,0 + Sll1 . ~ , r srn · ucp r u ur
L = V 12n l- sin 0 _j_ + 0 _!__]
2 3 r ae cos or .
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L = colon II Li, L 2 , L3 II,
B I b s .
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0 0 0
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ISSN 0041-6053. YKp. Mar. :Jteyptt., 1992, r. 44, JI l 23
0 0 0
()
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x (5cos2 8 - 3)), V~n e-q,i ( V ; 2 sin 0 (5cos2 0 - 1)),
1 - 2cpi ( V 10s . 2 8 e) V2n e - 4- sin cos , _1 -3q,i ( V'l~ . 3 e) //
V2n e 8 sm .
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ISSN 004 f -6053. ,YKp. AtaT. ,icypn., 1992, 1. 44, M 1
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ypaBHeHHH
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B COOTBeTCTBHH C on11ca1-1HOH MeTOJl,HKOH HaXOJJ.HM
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0
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0 ,034428
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0
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4
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0,020494
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0,065947
0,0256 12
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a onepaTopbI N 11, N 13 - no cpopMyJJaM
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N 1s = f' 3) p.r (Bia) L.
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d<pldv = - 1 + cos <p ct g 0 - .,,b, 2 {- 2
1
0 (cos <p ctg 0 - 1) +
+ (0,059253 cos 0 sin 0) cos3 cp + [0,059253 (5cos2 0 - l )-
- 0,058525 sin2 0] cos 2<p + 10,071097 sin 0 cos 0 (5 cos2 0 - 3) -
-0,17758 cos 0 sin 0] cos <p + 0,002371 (5 cos2 0 - I)},
d0/dv = sin <p - ebr2 {- 2
1
0 sin <p + 0,059253 sin 2 0 sin .1cp +
+ [0,08294 cos 0 sin3 0 + 0,05992 sin 0 cos 0 (5 cos2 0 - l)J sin 2cp +
+ [O, 17776 cos 2 e sin2 0 + 0,071104 sin2 e cos 2 e (5 cos 2 e - 3) +
+ 0,03555 sin 2 0 (5 cos2 0 - l)] sin <p},
lSSN 0041-6053. YK.p. Mar. 11tCyp11., 1992, r. 44, I-& l
drldv = - ebr3 {r0,059254 s in4 0 + 0 ,05992 sin2 0 (5 cos2 0 - 1) - 0,02370 1 X
x (sin 2 0 cos2 0)1 sin 2(p + [0, 17776 cos 0 sin'3 0 + 0 .071104 sin2 0 cos 0 X
x (5 cos2 H- 3) - 0 ,03555 s in 8 cos 8 (5 cos2 0 - l)] sin (p}.
2. A C HM TITO TH q e CI{ a 51 .II. e I{ 0 Mn O 3 H I.l H 51 II cp a q) cp O •
B hl X C H C T e M B 11 H B O JI 10 i.l H H C M a JI bl M TI a p a M e T p O l\f,
B pa60Te [7] npoae.11.eHO 06oc 110Ba 1-me MeTO,ll,a aCHM l1TOTHlleCl{OH ,ll,eKOMTI03HIUHf
)],MI BTIOJ!He HI-ITerpHpyeMblX m pac\xpOBblX CHCTeM. PacnpocTpaHHM 3TOT p e3y.'lb
TaT Ha TiqlarllcpOBbI CHCTer,11,1 B 1urnomouHH. P accMoTJ)HM ml)ac\xjloay c1-1cTeMy
dx1 = w( 1> (t, x, u) dl1 + ... + w\ml (t, x, u) dtm
(3)
dx,, = w'./l (t, x, u) dt 1 + ... + w;,ml (t, x, u) dtm,
r.11.e x = II x 1, . . . , Xn Ii - Bel{TOp 3aBHCHMbIX nepeMeHHbIX, t = II t1 , . . . , tm li-
Be1nop 11e3aBHCHMbIX nepeMeI-IHbIX, u = II u 1, ... , u,. II - Bel{Top napaMeTpHYec
KHX nepeMeHI-JbIX , 1{03C!Jcj)HUHeHTI,I w;il (t, X, u) - Bel..l..{eCTBeI-IHO-aHaJmTHlJecrme
¢y1-rnmm no sceM apryMeI-ITaM B 06J1acTH G = Rm x R" x Rh.
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,cpyHI<UHH
X;=(P;(t1 , . .. • tm). U; = ,p;(ti,·•·,tm), t=l,n, j=l,h,
npOH3BOJI BbI6opa KOTOporo 011pe.uem1eTCH noc.1e.11.0BaTeJi bHOCTbJO xaparnepoB
S0 , S 1 , S2 , . .. , S,n · (4)
flpe,11.TTOJIO,KHM , l!TO ncjJacjxpoBa CHCTeMa B 111-JBOJIIOUHH (3) IIO,!l.BeprnyTa Ma
.Jlb!M B03Myw,eHH5JM . BBO,!l.51 HOBble nepeMeHHbl e
x' = colon II x;, ... , < II , u' = colon II u'1, . •• , u~ II ,
sarrmneM ee B Bl-1)1.e 3BTOHOMHoiI CHCTeMbl
d ' (I) ' ~(I)' - (m)' ~(m)' -
Xn =(Wn +ewri )dt 1 + ... +(w11 +ew,, )dtm,
dt 1 = div ... , dtm = dim .
(5)
3)1.eCb cpyHKUHH B03Myl.IJ,eHH5l wzi)' (f, x', u')- Bel.IJ,eCTBeH HO-aHaJIHTHl!eCKHe n o
aceM apryMeHTaM B 06J1acr1-1 G = R"' x Rn x Rh.
HaXO)l{,11.eHHe HHTerpaJibHOro M[!Oroo6pa3HH CHCTeMbl (5) B HHBOJl!Oll.I-Ui
CBO,!l.HTCH K HHTero1-1posamuo pacw1-1pe1rno11 ncpac!Jcj)OBOH CHCTeMbl
d , ( (I)' ~(I)' dt- \m)' -(m) ' d-
X1 = W1 + ew 1 ) 1 + ., . + (w 1 + ew1 ) tm , . . . . . . . . . . . . . . . . . . . . . . . .
(6)
. . . . . . . . . . . . . . .
ISSN 0041-6053. Y1w Ata r . ,icypn., 1992, r , 44 . M 1
r,ne Ha napaMeTpHL1ec1rne np0H3B0llHbie HaJiaraIOTC51 ,!l0TT0JIHHTeJJhHbie CB513H H3
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r,D.e
a (ffi~}J)' + effi~)')
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i)t p
i , p = I, 111, a = I,n.
3anmueM onepaT0pbl, aCC0UHHpoBaHHbJe c CHCTeMOH (6), s BH,!le
,* ou; 0 iJu~ 0
U; =-0---;:i-, + ... + -8---8 ,, i= 1,m.
Xj ou l Xj uh
(7)
(8)
(9)
OlleBHJIHO, onepaT0 pb! (8) 51BJI5110TC51 pacwttpe HHblMH onepaTopaMH B03My
meHHOH CHCTeMbl (5), a onepaT0pbl (9) - pacwnpeHHhie onepaT0pbl CHCTeMhl 11y
JieBoro npH6JIH)KeHH5l, TTOJIYll 3l'MO ti H3 B03My1ue~rnoh CHCTeMbl np11 f'. = 0.
Jler1<0 BHJieTh, llT0 YCJ!0BH,I HJ-ITerpHpyeM0CTH (7) 3KBHBaJJeHTHbl K0MMy
TaTHBH0CTH pacw11p e HHblX onepaT0()0B B03MY1UeI-IHOfI CHCTeMbl
(10)
6yneM npennoJJaraTh, 'ITO CHCTeMa HyJieBoro npH6JIHX<eI-llHl B03MYLileHHOH
CJ ICTeMbl (5) HaX011HTC5l B HHB0m0UHH H CTPYKTypa ee HHTerpaJibH0f0 Ml!Of006-
pa3H51 onpe.ueJrneTcH pHll0M xapaKTepoB (4).
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MymaJOII~He cpyHKUH H. nJI51 3T0ro HCTT0JJh3yeM YCJI0BH51 mrrerpH pyeM0CTH ( J 0):
ru,, D~1 + e nu1, D;1 + 1D1, u;n + e2 ru;, u:1 = o.
TTpH e = 0 .TOM.!leCTBO npeBpawa eTC51 B ycJI0BHe HHTerpupyeM0CTH CHCTeMbl
HyJJeB0ro npH6JIH)KeHH51
(11)
3TH ycJI0BH51 npHB0)l51T K orpaHHlleHHHM Ha napaMeTpHlleCKHe npoH3B0JIHbJe.
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np11Mepe.
r l p 11 M e p 2. PaCCMOT()HM 0JJI-I0MepHoe B0JII-I0B0e ypaBHeHHe
(12)
ISSN 0041-6053. Y,cp. MllT. ,ic,ypn., 1992, T. 44. M 1 27,
00)1.BeprnyTOe MaJibIM BO3Mylll.eHHHM
iPu iJ2u -r OU
at2 = ox2 + 1:,f lt, X, u, a[ '
TiycTb mnerpaJibHOe MHOroo6p a 3 He y p aBHeHHH (12) 3a,ZJ.aeTCH cpyHKUHeH
u = cp(x,t),
(13)
( 14)
noJiyqeHHOH B pe3yJibTaTe peweHHH HeKOTOpOH HaqaJibHOH HJIH Kpae~OH 3a.uaq1-1_
HccJie,ZJ.yeM rrOBe)l.e1-rne HHTerpaJibHOro Mlloroo6pa3HH (14) TI O,Zl. BO3)].eHCTBHeM He
JIHHettHbI X B03Myu~eHHH B rrpaBofi qacTH ( 12). C 3TOH ueJihIO n o MeTo,uy acuMITTO
TH'IeCKOH ,!I,eKOMTIO3HUHH Hatt,ZJ.eM peweHHe ypaBHeI-IHH (1 3) B BH)J.e
u = (p(x, t) + ~ e"q\,(x,t),
V= l
r.ue (j)v (x0 , f0) = 0.
BBO,Zl.51 HOBble rrepeMeHHb!e
Z1 = u, Y1 = X, Y2 = t, Z2 = oz/oyl = oulox,
23 = 02/0Y2 = ou!ot, U1 = 02/oyl = a2u/ox 2 , ll2 = 022loy2 = 021.doxot .
3arruweM ypaBHeI-rne (13) B BH,11,e 3KB J1BaJieHTHOH CHCTeMhl ypaBHeHHH rrepsoro
nop5I.UKa
OZ/OY1 = U1,
02/0yl = 22,
a221ay2 = u2,
az11ay2 = z3 ,
02/0ljl = U2, iJz/oy2 = U1 + e[ (!J , Z, U),
YpaBHeHHH ( 15) MO)KHO 3aMeHHTb mpaqxpoBOH CHCTeMOH
dz 1 = 22dy1 + 23dy2 ,
d2 2 = utdy1 + u2dy2 ,
d23 = u.2dY1 + (u1 + Ef) dY2·
(15)
(1 6)
H a p5I,uy C B03MyIUeHHOH CHCTeMOH (16) 6y,11,eM p aCCMaTpHBa Tb CHCTeMy HyJieBo
ro npH6JIH)KeHJ:rn
d21 = 22dy1 + z3dy2 ,
d2 2 = u1dy 1 + u 2dy2 ,
d23 = u2dy1 + u1dy2,
( 17)
Bbl5!CHHM orpaHHqeHHH , IrnJia r aeMhle Ha rrpOH 3BO,Zl.Hble OT rrapaMeTpHqec,rnx
n ep eMeHHblX ycJIOBH5IMH HHTerpup yeMOCTH CHCTeMbl (17). TTe perrHWeM ypaBHe
HHH (17) B BH,11,e CHCTeMhl ypaBHeHHH B qacTI-lblX npO113BO,!I,HbIX
02/0!J 1 = Z2,
OZ)O!J1 = U1,
oz/oy1 = U2 ,
oz/oy2 = 23,
02zlO!J2 = U2,
02/oy2 = u1 .
(1 8)
TipttpaBHJ.rna 5I BTOpb!e rrpoH3BO.UHbi e , TIOJi y4eHHbl e H3 COOTHOWeHHH (1 8). rroJiy
qaeM y c JIOBH5I HHTerp1-1pyeMOCTH
02/oy2 = oz/oy1 , ou/oy2 = ou/oy1, ou/oy2 = ou/oy1 •
TiepBoe 1-13 3THX COOTI-IOWe HHH B CHJIY (18) rrpHBOllHT K TO)K,Zl.eCTBY ll2 = ll2 - CTie
,11,OBaTeJibHO, CHCTeMa (18) HaXO,Zl.HTC5I B HHBOJIIOUHH H rrpoH3BOJI B TIOCTpoeHHH
ee HHTerpaJibHOro MHOroo6pa3H5I orrpe,ueJI5l eTC5I 11BYM5I cpyHK UH5IMH OT O,Zl.HOH
He3 aBHCHMOH rrepeMeHHOH, TaK KaK P1 = 0, P2 = 2 B CHJIY COOTHOWeHHH
ou/oy2 = ou/oy1, ou2/oy2 = OU1loy1. (19)
28 ISSN 0041-6053. YKp. MaT. 3/CypH.., 1992, r, 44, M 1
ITepeii,11.eM OT CHCTeMbl (16) K pacnrnpe1u10?'1 3BTOHOMHOH CHCTeMe
dz 1 = z 2dy1 + z3dy 2 ,
dz2 = u1dY1 + t-12dY2,
dz 3 = u 2dy 1 + (u 1 + ef) dy2 ,
dt1 = d!Ji, dt2 = dy2 •
CooTBeTCTBem10 ,11.JUI CHCTeMbl ( 17) 110JIY'IHM
dz1 = z2dY1 + Z3dy2,
dz 2 = 11 1dy1 + u 2dy2 ,
dz 3 = u 2dy1 + u 1dy2,
d aul d aul
u1 = ar;- Y1 + at2 dy2;
O nepaTOpbl, accoum1poBaHI-1bie C CHCTeMOH (21), 3311HCb!B3JOTC51 B mrne
" a a a a + au 1 a au. iJ
ul = atl + 22 azl + UI OZ2 ---j- Uz OZ3 at; OU1 + at: OUz I
" a a a a au1 a au. a U--+z-+u-+u- ~ - ----+--~--
2 - at2 3 az] 2 az2 1 823 ' dt2 aul at2 au2 .
(20)
(21)
C ooTBeTCTBeI-JHO pacnrnpeIJH bi e onepa-ropbl, accou1111poBaI-1Hbie C CHCTeMOH (20),
3 311 HlllYTC51 Talc
(22)
~ ~ a
r n e U2 = f -8-.
Z3
Oner aTOpbl D 1• f; 2 KOJl,IMYTaTHBHbl B n!Jly ycJIOBHH (19) HHBOJIIOTHBHOCTB
CllCTeMbl (21)
(23)
no cneJiaHH0My BbIWe npe,unoJJo:>KeHHIO B03Myw,eHH35l CllCTeMa (20)
,ll0Jl:>KHa H3X0,UllTbC5l B llHB0Jl!OllHH B CBJJY TeX ,Ke ycJJOBHH, T. e . onepaT0 pbl
u 01• f; 02 D,0Jl:>KHbl 6bITb J<OMMyTaTHBHbl npll JII060M 3H3'IeHHH napaMeTpa C B
CHJJY (23), T. e. [U 01 , U 02 ] = 0. oTKy,ua cJJenyeT BHD. cpy1-IKUHH B03Mylll.eHB51
(24)
TaKHM o6pa30M, npH Bbll10JJHeHHH pase J-ICTl3 (23), (24) np1-1xonHM K sa_ua,,e 06
8CHMTITOTHtleCKOH .ue K0MI103 HUHH BTI0JIJ-Je HHTerpHpyeMOH mpacpcpOBOH CHCTeMbl
(20), HyJJesoe npH6JJHJKem-1e KOTOpoii TaKJKe BD0J!He HHTerpupyeMO.
3anaeMC51 K0HKpeTHb!M BH,11.0M peweHH51 ypaBHeHH51 (12) B BHlle ll JIOCKOH'.
B0J!Hbl u = cos (Y1 - Y2)- TeM C3Mb!M onpe,11.eJJeHo HHTerpaJibH0e MH0roo6pa3 He
ISSN 0041-6053. Y,w Mar. 31eypn., 1992, r . 44, M 1 29
mpaqxpoBOli CHCTeMhl HyJieBoro npH6JIH)KeI-I!Ul
2 1 = cos(t1 - t2), 2 2 = - sin (t1 - t2), 2 3 = sin (t1 - t2), (25)
U1 = - cos U1 - t2 ), U2 = cos U1 - t2)-
Coon10111e11H5l (25) noJIHOCTb!O onpe,ueJ1 5l!OT Ha<rnJ1bHYIO T04KY M Cto1 , fo2,
201 , 2 02 , 2 03 , 11 01' u02). TToJIO)I<ttM t,11 = t02 = 0. T or,ua 2 01 --:: 1, 2o2 = 0, 2 03 =
= 0, U.0 1 = 0 , llo2 = I H onpe).l.eJieHbl TalOKe 31-!34eHH5l CBO0O)lHbl X 11apaMeTpH
qec1rnx nepe\1e11 llblX
au2liJt, = - sin (tl - t 2), iJu/iJtl = sin U1 - t2)
I11nerpaJi bl ncpacj)(j.>OBOH CHCTeMbl HYJl eBoro n p116JIH)1<eH1rn
<Pi = l/.2 - cos (t , - t2),
(jl2 = ll1 + cos U1 - t2),
qi3 = 2 3 - sin (t 1 - t2) + (t 1 - t2) cos (t 1 - t 2 ) - t1u2 - t2u1,
{jl4 = 22 + sin (tl - t 2) - (tl - t2) cos (/ I. - t2 ) - tlul - t 2ll2 ,
(26)
1 2 I 2
{jl5 = 21 - cos U1 - t2) + 2 t1U1 + 2 t1 cos (tl - t2 ) - t122 - f 2Z3 - (tl -
onpe,uemieM H3 Bl"lOJIHe HHTeqrnpyeMOH CHC"fe\1161
,,.._ ,,.._
r.ne B K03cj:xjmuHeHrnx onepa-ropoB U1, U2 rno6011 11 1,1e napaMe-rpnqecKHe npoH3·
BO)lHble iJU1f iJf1, iJu2fiJf1 33MelleHbl 11!)3Bb!MH 43CT511,1H p aBeI-ICTB (26) .
.[LJIH cpyHKUHH f B03Myll(eHH5l npHMeM BblpmKe1-me
t = (1 + <p5)2.
TTpoH3Be.neM B onepa-ropax (22) 3aMetty nepeMeHHhIX t1 = t;, t2 = t;,, x; = <p 1,
x; = <P2, x; = %, x; = {jl4, x; = cp5 , B H0BblX nepeMe HHblX onepaTOpbl t22 >
npHMYT BHJ1
CJienoBaTeJibHO, B H0BbIX nepeMeHHbIX B03Myr.ue1-rnan CHCTeMa (20) npHHHMaeT
BH,!l,
dx; = 0,
dx~ = 0,
dx~ = e (I + x;)2dy2 ,
dx; = 0,
dx; = - et; (l + x~)2dy2 ,
dt; = dy1 , dt; = dy2 •
(27)
TTpHMeHHM K CHCTeMe (27) anropHTM aCHM TITOTWJeCKOH .UeK0MTI03HIJ.HH, COBeplllHB
aaMeHy nepeMeHHhIX
30
xj = exp(eS)xj , i = T:5,
t; = exp (eS) t 1, i = U,
(28)
ISSN 0041-6053. Y,cp. Mar. :HCypH.., 1992, r. 44, M l
r.n:e
2 5
S = S1 + eS2 + ... , S,,, = ~ "r,,,; ii + ,L "r,,,; 0~i .
i=l i= I
BbrmmteM B H0Bb!X rrepeMeHHb!X orrepaT0pbl Lfo1, [} 02, accou1mpoBaHHb!e C
CHCTeMor1 (27) C T0lJH0CTb!O no BeJIHLIHH BT0poro rropHnKa M3JI0CTH:
r.ne
" i)
U 01 = 7ft + e (- [U 1 , S1 ] + F 11] + e2 (- [U1 , S2 ] + F 12 ),
1
F 11 = u1 = o,
F 21 =V2 =(1+x5) 2 0~ -t2 (1+ x5)2 0~ ,
3 6
~ l
F12 = - [Uv S1 ] + 2 [[U1 , S1], S1],
~ l
F 22 = - [U 2 , S1 ] + 2 f[U 1 , S1], S1].
OrrepaTop S1 HaxonHM H3 CHCTeMbI ypaaHeHHH
[U 1 , S 1] = 0, [U 2 , S 1] = F 21 - pr F 210
r.ne pr F 21 = N 21 = (l + X5)2a!ax3.
OocJie HeCJIQ)l(Hb!X Bb!KJl3)l0K IT0JiylJaeM
S t~ 2 a
1 = - 2 - (l + X5) 7fx •
6
OrrepaTOp S2 HaxonHM H3 CHCTeMbI ypaaHeHHH
[U 1 , S 2 1 = 0, [U2 , S2 ] = F 2 2 - p r F 22 ;
r.nt
nocJie HeCJIO}KI-!blX BbllJHCJieHHH orrpe)l.eJI HeM
t~ c)
S2=-3(l+ xs)3 OX3.
(29)
TaKHM o6pa30M, orrepaTOpbI (29) rrocJie npeo6pa3oBaHHH (28) rrpHMY,T BRA
A a
Uo1 = 7ft,
1
U" a (I + , 2 a
02 =75t-/- e X5 , -0-,
2 X3
a ueHTpaJIH30Ba HH35l CHCTeMa, C00TBeTCTBYIOIUaH B03MyrueI-IHOll CHCTeMe (271.
Bll)l.
dx 1 = 0,
dx2 = 0,
dx..._ = e ( I + x 5)2dy1 ,
ISSN 0041-6053. Y,cp. Mar. "'YPH., 1992, r. 44, M 1
(30)
aa
dx4 = 0,
dx5 = 0,
dt1 = dy 1 , dt 2 = dy2 •
IlpomnerprrpyeM 3TY CHCTeMy npH Hal!aJibl-Ib!X ycJIOBH51X X1 = ·. · = Xs =
= 0 npH t1 = t2 = 0. B p e3yJihTaTe n oJiyllHM x1 = x 2 = x 4 = Xs = 0, X3 =
= et2 • I1e p eMeHHb1e x', t' Bhipa:>KaIOTC5l q ep e3 n epeMeHHhie x, t no cpopMyJiaM
{28). me cJie.nyeT noJI0:>KHTh S = S1 + c:S2 .
BbIITHilleM 3 Hal!eHH51 x' C T0lJH0CTb!O .no BeJIH'lHH BT0poro n op H,LI.Ka M3JI0CTH
x~ = 0,
x; = 0,
3
• ·t 2 t2
X3 = e 2-e 3'
x; = 0,
t~ e2 4
x~ = - e 2 + 4 t2.
OKOHll3TeJibHO .l(JIH <PYHKUHH u, z noJiyllttM BbipmKeHHH
U 2 = COS (t1 - f 2 ),
U 1 = - COS (t1 - f2 ),
2 4
2 1 = cos(t1 - t2 ) + t2x; + x~ = cos (t1 - t2 ) + e ~ - e2 :; , (3 1)
z2 = - sin (t1 - t2 ),
3
z 3 = sin (t1 - t3) + x; = sin (t1 - t2 ) + et2 - e2 i ,
Y6e.Lll-IMC5l , l!TO Hafi.neHHbie cjJyHKUHH C TOlll-IOCTb!O .no e2 y.noBJieTBopmoT
-CHCTeMe ypaRHeHHH (I 5), a CJieJlOBaTeJJb HO, H HCXOllHOMY B03Myw,eHHOMY ypaB
HeHHIO (1 3). no.ncTaBJIHH (31) B ypaBHeHH5l (1 5) , noJiy•rneM
az21at1 = - cos u1 - t2 ) = u1 ,
iJz /iJt, = - sin (t , - f 2 ) = 22,
iJzaf iJt 1 = COS (t 1 - f 2 ) = U 2 ,
iJziiJt 2 = cos (t 1 - t2 ) = u2 ,
iJz/iJt2 = sin (t1 - t2 ) + ( et 2 - c: 2 t; ) = z3•
4T00bl nponep ttTb nocJieJIHee ypaBHe Htte, CJJenyeT oueHHTb BeJIHlJHHY (I + cp5)11•
nonCTaBJIHH 4.JY HKUHH (31) B HtITerpaJJ cp5, 1-JaXOJIHM
t~ e2 4
(j)5 ~ - e 2 + 4 t2,
c Jienosa TeJihI-10,
( 1 + ())5)2 ~ ( I - eti),
iJz 3/iJt 2 = - cos(t1 - t2 ) + (e- e2t~) = u 1 + e (I + cp5)2 = - cos (t, - t~) +
+ e( l - et~). .
.32 1/SSN 0041-6053. YKp. MaT. :J1Cyp11, ,, 1992, T. 44, Ni 1
Oepexo.n: K ue~npa.ntt30Bam1011 rncTeMe (30) oT B03My111,em1or1 (27) 06.ner qa
e-r HHTerpHpOB3HHe, T3K KaK u e1-npa.nH30B3 1·11-1351 CHCTeMa aBTOHOMl·ta H nepeMeH
Hbie B HeH pa3.UeJTeI-l bl Ha 6b!CT)Jbte H Me.l]_JTellllbie .
I. Kpbl..100 H. M., Boc0A106oe 11. 11. BBeJ\e ,rnc B ncmrneii11y10 Mexa111my.- K11 en : 1-btt-no AH
YCCP, 1937.-- 363 C.
2. EocOAIOUOO H. ff ., M umpon0.1bCKUll 10. /\. AcHM IITOTil'lec,rne MeT0l(bl n -reopHH HeJIHHeHHb/X
Ko.1e6a 1rni-i. - M. : Hay1<a, 1974. - 504 c.
4. J\1umpo!IOAbCKUii 10 . A . I1poG.1eM1.,1 aCHMfffOTH'ICCKOll Teop1rn 1-1 ecTau11011ap!Ib!X KOJie6a H-
1111ii.- M.: Hay1rn, 1964.- 43 1 c.
4. A·/ umpono.1bcK11u JO. A., JlonamuH A. K. T eopen11<0-rpy1rnono11 IJ0J\X0,1 n acHMITT0TH'lecKHX
MCT0)(aX HeJ11rnei-i11oii MeXal!HKH. - KHCB : Hay1<. J\YMKa , 1988.- 272 C.
5 . Jlonamu!-l A. K. Ac1ntnTOTH•IeCI{351 neK0MIT0'.oHUH51 JUHj1<(1cpemmaJ1bHblX CHCTeM C M3,1b1M napa
MeTp0M n npoc-rpaHCTBe npe.l(cTan.1e1m1'.i K011e•11-10~1ep11oi-i rpynnLI J l11 // ~r"K p. MaT. )Kyp11.-
1987. - 39, N2 1.- C. 56-64 .
6. fpoauoamuii r. JI., /1()0/-lOB 10. /-! . , ToKupee B. fj_ M exaHHK3 K0CMH'leCK0ro no.1eTa C M3JIOH
nuoii .- M.: H ayKa, 1966.-- 679 c.
7. MurnponOAbCKUt'i /0. A., Jlonamtll-l A. K. A c HMllTOHP!eCKa H .l(eK0Ml103HUllH nnomi e llHTerpn
pyeMblX mjJacjxpon1.,1x cncTeM // YKp. MaT. )KypH.- 1988.- 40, N2 3.- C. 349-361.
flOJIY'leJ-JO 22 . 10.91
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| id | umjimathkievua-article-2317 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T02:22:06Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d6/d762e70be120682e263df458b01858d6.pdf |
| spelling | umjimathkievua-article-23172023-09-14T12:03:56Z Certain aspects of theoretical-group approach in problems of nonlinear mechanics Некоторые аспекты теоретико-группового подхода в задачах нелинейной механики Lopatin , A. K. Лопатин , А. К. Mathematical Physic Nonlinear System Special Function Decomposition Method Differential System Математична фізика Нелінійна система Спеціальна функція Диференціальна система Two aspects of the application of the asymptotic decomposition method to the decomposition and investigation of nonlinear systems are considered. The first one is the extension of the class of functions in terms of which to form the asymptotic decomposition by applying special functions of mathematical physics. The other aspect is the extension of the asymptotic decomposition method to a new class of differential systems, namely, Pfaffian systems in involution. Рассматриваются два аспекта применения метода асимптотической декомпозиции к исследованию нелинейных систем. Первый — расширение класса функций, по которым проводится асимптотическое разложение за счет применения специальных функций математической физики. Второй — распространение метода асимптотической декомпозиции па новый класс дифференциальных систем — пфаффовы системы в инволюции. Розглянуто два аспекти застосування методу асимптотичної декомпозиції до нелінійних систем. Перший розширення класу функцій, за якими проводиться асимптотичний розклад за рахунок застосування спеціальних функцій математичної фізики. Другий — поширення методу асимптотичної декомпозиції на новий клас диференціальних систем — пфаффові системи у інволюції. Institute of Mathematics, NAS of Ukraine 1992-02-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2317 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 1 (1992); 22-33 Український математичний журнал; Том 44 № 1 (1992); 22-33 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/2317/1538 Copyright (c) 1992 A. K. Lopatin |
| spellingShingle | Lopatin , A. K. Лопатин , А. К. Certain aspects of theoretical-group approach in problems of nonlinear mechanics |
| title | Certain aspects of theoretical-group approach in problems of nonlinear mechanics |
| title_alt | Некоторые аспекты теоретико-группового подхода в задачах нелинейной механики |
| title_full | Certain aspects of theoretical-group approach in problems of nonlinear mechanics |
| title_fullStr | Certain aspects of theoretical-group approach in problems of nonlinear mechanics |
| title_full_unstemmed | Certain aspects of theoretical-group approach in problems of nonlinear mechanics |
| title_short | Certain aspects of theoretical-group approach in problems of nonlinear mechanics |
| title_sort | certain aspects of theoretical-group approach in problems of nonlinear mechanics |
| topic_facet | Mathematical Physic Nonlinear System Special Function Decomposition Method Differential System Математична фізика Нелінійна система Спеціальна функція Диференціальна система |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2317 |
| work_keys_str_mv | AT lopatinak certainaspectsoftheoreticalgroupapproachinproblemsofnonlinearmechanics AT lopatinak certainaspectsoftheoreticalgroupapproachinproblemsofnonlinearmechanics AT lopatinak nekotoryeaspektyteoretikogruppovogopodhodavzadačahnelinejnojmehaniki AT lopatinak nekotoryeaspektyteoretikogruppovogopodhodavzadačahnelinejnojmehaniki |