On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability
For nonholonomic systems, we introduce the notion of the function of Hamiltonian action, with the use of which we investigate the stability of nonholonomic systems in the case where the equilibrium state under consideration is a critical point of the corresponding Lagrangian (Whittaker system).
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| Datum: | 1999 |
|---|---|
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1999
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4740 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510908228304896 |
|---|---|
| author | Sosnitskii, S. P. Сосницький, С. П. |
| author_facet | Sosnitskii, S. P. Сосницький, С. П. |
| author_sort | Sosnitskii, S. P. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | For nonholonomic systems, we introduce the notion of the function of Hamiltonian action, with the use of which we investigate the stability of nonholonomic systems in the case where the equilibrium state under consideration is a critical point of the corresponding Lagrangian (Whittaker system). |
| first_indexed | 2026-03-24T03:04:28Z |
| format | Article |
| fulltext |
YI2K 531.36
C. II, COCHHHBKI41~ (hI-T MaTeMaTHKrt HAH YKpahlrl, KHIa)
I I P O < I ) Y H K I I I I O ,/~II 3 A F A M I J " I b T O H O M ooo.
~Id-I~I H E F O J I O H O M H I , IX C H C T E M TA II 3 A C T O C F B A H H J I
I I P H ~ O C , r I I ~ K E H H I C T I ~ K O C T I
For nonholonomic systems, we introduce a notion of a function of Hamiltonian action. By using this
function, we study the stability of nonholonomic systems in the case where equilibrium under
consideration is a critical point of the corresponding Lagrangian (of the Whittaker system).
~[Jl~l lleI'OJIOIIOMI|HX CHCTeM I~BOII;HThCJI nolla'lUr~l qbyngltii ltii ~ FaMiJib'rono~, a ]l~OnOMOl'OIO agoi
/~oc~li/~ye'rbca C'l'i~lKieTI, neroJIoIIOMHnX CHc'reM y armaltgy, gOJm noJio~emla pinnolmvH, uto poa-
rJla/tae-n,ca, e Kpn-rl, iquolo TOqgOIO nilmoBilmovo Jlarpau~iana (CHC'reMH YiTrexepa).
~IK BiaOMO [1], no~o~eHHa piBHoBarn nerozoHoMnax CaCTeM, Ha a i ~ i n y Bi~t ro~o-
HOMHHX, He O~OB'~I3KOBO 36ira1OTbCJ~ 3 KpHTHqHHMH TOqKaMH ai/Irloai/~noro .narpari-
atiaHa, tUo MO~e 6yTH ~epe:~oM ~O~aTKOBnX Tpy~notuiB nprl Z~oc~i~t:~eHni CTi~t-
XOCTi CHCTeMH, OC06JIHBO, KOJIH nHTaHHJt He pO3B'~aycTbCJ~ a paMKaX .niHil~noro
Ha6J-IH~f.eHH.,q. CTi~KiCTb IIOJ'IO~eHb piBHOBaFH HeFOJIOHOMHHX CHCTCM, aKi C Kp~I-
THqHH~H TO'aKaMn zarpaHa<iaHa, Ztocz i~ynan YiTTeKep [2]. I.[efl 6i~,tu By3b~n~
K~ac HeFOJIOHOMHHX CHCTeM, J~Krli~ B rlo/Ia.rlbllIOMy Ha3HBaTHMeMO CHCTeMaMrl YiT-
TeKepa, nepm 3a BCe tIiKaBriit TaM, ~O 3a CBOIMH B.rlaCTHBOCT.~IMH Bill ,/~OCa'l'b CXO~C,.HI~I
Ha rOJIOHOMHi CHCTeMH. [~i~]bme TOFO, B NeBHOMy CeHCi CHCTeMH YiT-reKepa MO~KHa
poar~IJl]laTH ~qK piaHOBrl~ 36ypeHHX rO~OHOMrlIIX CrlCTeM, Ma~qa Ha yaaai
MO)KJIHBiCTb 3aCTOCyBaHHJI 1~0 HHX MeTO~iB /IOC~Ii/~KeHH~ CTiflKOCTi Fo.rIOHOMHHX
CHCTeM [3--6]~ BI4$1BJI~IeTr~CJI TaKO~K, tlAO /l~J~ /IaHHX CHCTebl, 3a aaa~noriem a rono-
HOMHHMI4, KOHCTpyKTHBHI4M e po3r.rI,q]l ~yHKttii ~tii 3a FaMi.rlbTOHOM S. 3OKpeMa, B
~e~ax ni~xd]xy [7], mo rpyHTyeT~,Ca Ha BrmOpaCTaHHi qbyHKuii ~ii ~t:~a ]~OC~Ii~t-
:~eHHJ~ CTit~toc'ri ro.rtOHOMrlrlX CHCTeM, ,tl,3"I$l CHCT~M YiTTeKepa B~aen,ca oTpmaaTr~
yMOBH HeCTiI~IKOCTi piBHoaarn.
I. PO3F~J~HeMO HCFO~OHOMHy cricTeMy
d ~L 3L = Br (q)X, X = (X l . . . . . Xt) T (1)
dt O(l Oq
B(q) ~t = 0, (2)
ae B (q) = ( bij (q)) ~ MaTprmz poaMipHOCTi I X n, i = 1 . . . . . l, j = 1 . . . . . n, l < n ;
~ , - -~nO~HnKr t B'aaelt, L(q,(1), B (q ) ~ C2(Dq • R ; ) , a a a r p a n m i a a L ariarla-
q aC"l'b C.q BHpa3oM
1 . T
L(q,(l) = T ( q , ( t ) ' H ( q ) = ~q A ( q ) ( l - H ( q ) . (3)
TyT qbym<t~ii T i FI ~ ~i~no~i~Ho ziaeTH~na i noTenraia~bHa enepr i i caCTeMn.
BBamaT~eMO, mo KBa~apaTr~na qbopMa T(0, q) ~o~aTno BaaHaqena, I I ( 0 ) = 0;
~I-I/3q(O) -- 0 i TO'~Ka q = q = 0 Tn~ cat, m ~ e CTa~OM piBnoRara CaCTeMrt (1)--(3).
HeinTerpoBai cniBBi/XHOmeHrm (2), aKi 06Mea<y~OTt, y~ara.m,neni mBn~KOCTi Cr~C-
Teblrl ( rankB(q) = l ) , e Hero:~ot~OMHm, m ~'aaJaMrl.
~IK i y suna/IKy roJaono~nax CaCTeM, ]ZJm CPICTeMH (I)- (3) Mac Mictte iHTerpaJ~
enepri i
T(q,(l) + Fl(q) = h = const. (4)
BiaoMo [8, 9], tUo piBnaHHa pyxy nero~oHo~aoi cHereMH MO~Xy'r. 6y'm oxpn~aHi
Ha n~crani np~HtJ~ny ra~ia~TOHa y qbopMi FezmJacpa
�9 C. I'l. COCHHL[bKHI~. 1999
ISSN 0041-6053. Yrp. autm. ~.'vpn., 1999, m. 51, N'- I0 1411
1412 C.n. COCHHI.IbKHI~I
Ii
8 L(q, (7) d'c = O. (5)
o
Ha BillMiHy si~ FOJ'IOHOMHHX CHCTeM, npHHu~ln l"at,ti.rtbTOHa y cl~opMi (5) B ~ e He e
npm-mmms~ c'ra~ioHapHOi Aii, Kon~ c n p a ~ e ~ B a piBHiC'~
tl
5 ~ L(q, q) dx = O. (6)
o
Fept~ [10], Ma6yTb, nepmHM BKa3aB Ha Te, mo rlpHHttrlrl r'aMialbTOHa y qbopMi (6) ~ a
Hero.rIOHOMHHX CHCTeM B~Ke He e crlpaBe/~UIHBHlVl. OLIHaK BaT~J'IHBHM e c13aKT, mo i
/lYla BHIZa~zy Hero.rlOHOMHHX CrlCTeM narpamKian L(q, (t) 3aJ-IHItlae~l'bCJl KJIIOqOBOIO
xapaKTeprlCTHKOIO CrlCTeMrl.
BHXO/~qH 3 AaHOFO ~aKTy, pO3FJ-I}IHeMO t~yHKtli~o
t
S ~ L(q, (I) dx, (7)
0
Ae Be$IHqHHH
q=q(t , qo,(lO), q = q(t, q0, q0),
(8)
qo = q(t = o), qo = , / ( t = o),
JtKi BXOAZ'I~ y IIi/~iHTCFpaJIbHrtlt Bl4pa3 piBHOCTi (7), e 3aFBJlbHHM pO3B'$13KOM piBH~lHh
(I), (2). AHaJIoriqHO BI4IIa~Ky rOJIOHOMHHX CrlCTeM, Ha3BeMO qby~ro.tim S clJyHKuieIO
~ii 3a Fa~di.rmTOHOM.
l'IprlrlyCTHMO, mO pO3B'Z3C)K (8) e npo]~OB~KyaaI-IH/r Ha BCtO BiCb t ~ R i, OTme,
Bi]~noBi]lar BH3HaqCHHIO n0TOKy [11]. 11~ 06CTaBHHa He o6rdc:~tye 3ara.nbHOCTi poa-
rnaAy, ocKim~KH HH~'~C MOBa RTrtMe npo HeCTilaziCTb piBHOBarH. 3aMiHHBtUrl none-
pe/mbo B (8) t Ha X i 3/fiRcmmmrt B piBH0CTi (7) iHTcrpyBam~a, OTpm~acMo
s = g(x, qo, Oo)[& ~ c~;,~'? (gxss), (9)
ae BeKrop (q0, #0) HaJ'le~KHTb OKOSIy s8 = {(q0, q0) ~ Dq x Rq, Ilqo �9 qo II < 5} TO"-
KH q = q = 0. BpaxOBym.a, mo cniBBiAnomermzMH (8) BHzHa,~aen, ca nor iK i, Ta-
KHM "HHOM,
qo =q(- t ,q , (1) (lO =(1(-t ,q,(l) , (10)
Ha ni~cTaBi (9) Mae'~o
: C(I'I't)(RxDqxR~). (11) S S*(~, q(x), q(x))]~ E -tr
B noAay~moMy 3o6paauMo cHcreMy (I)- (3) y BHrn~Ai
BH
~s
i~ = - ~H + B T(q) 2,,, (12)
aq
B(q)~p H = 0,
1 H(q,p) = =prA(q)-tp + H ( q ) = h = const.
Z
(13)
~c
ISSN 0041-6053. Ys47. ~.am. ~. pn., 1999, m. 51,1~ 10
FIPO OYHKIdlIO iIII 3A FAMIIIBTOHOM IlJt~l HEFOIIOHOMHHX CIdCTEM... 1413
PO3F.qRHCMO B R n MHO~LtlHy ~, mO BH3Haqa~rbc$1 piBHJIHH$1MH
B (q) q = 0. (14)
OCKi.rlbKH rankB(q) = l, TO piBH.qHH~t (14) 3aa~.,lIa MOTKrta poaa'aaaTH Bi/IHOCHO
~KHX-He6y~b I KOMHOHeHT BeKTOpa yaara~HeHHx Koop/II4HaT q. I'I03HaqHMO 06Me-
~KCHHa ~oain~HOi qbyHKUii ~ ( q ) Ha ~ qcpe3 ~ ( q ) . l'lop$l/~ 3 MHO~HHOIO ~ BH3Ha-
qI4MO TaKO~ MHO~HHH
f2 = { ( q , p ) e s e : H = h = 0},
[ 2 - = {(q,p) ~ se : H = h < 0},
~2 + = {(q,p) ~ s E : H = h > 0},
OH
~2] = {(q,p) e se : H = h < O , H - q - ~ q + ( B ( q ) q ) X > 0},
OH
~2] = {(q,p) r s e " H = h > 0, H - q ~ q q +(B(q) q)X > 0}.
"7
TeopeMa 1. l-[punycmu.~w, u~o icnye mare qucno s > 0 ( D e D s~ = { q e R n,
II q II < e } ) , npu ~rozty ouKonytombc~ yztosu:
1) to*= { q e s ~ :f-I(q)<O}.r
2) -OH(q------~) + B T ( q ) X ~ O V q ~ co = { q ~ s ~ : H ( q ) < O } , O~ 200.
Oq
Toai no.no~eunn piot-tosaeu q = (I = 0 cucme~tu ( 1 ) - (3) necmi~re.
Cno~aaxy/~OBe~eMo TaKy neMy.
JIeMa. Hpu ourona.ni ),~toa meope~lu 1 ~-t ~ 0 .
Aro~e~eun~. Cnepmy lIoKa:~KeMo, Illo HeIIopO~HbOIO r MHO~KHHa
OH
n o = ( ( q , p ) e s~: H = O , - q ~ q > O , B ( q ) q = O } .
Ha ni/IcTaai TeopeMrl npo cepe~He [12, c.301 ] MaeMo
OH
H(q,p)-n(qo, O) = 0 = (q - q o ) ~ q (qO + O ( q - q o ) , O p ) +
OH
+ p-.~p (qo + O ( q - q o ) , O p ) , Oe ] 0 , 1 [ , (15)
ae qo e Oto*, q e to* ( q , p ) e ~ .
Bn6epeMo TO~Zri q0 e 0t0*, q ~ CO* (q0, q e f2) Ha npoMeni, mo saxo~trrr~ 3 TOq-
Ka q = 0, TaK trio ae~Top (q - q0) C tO*. 3o rpe~a , aKmO pa~iyc-seKTop q npH
CTaryBaaHi ~oro a TOqKy q = 0 He BHXOanT~ 3a Mead Co*, TO apyano sH6paTa
q0 = 0. TaKHia mt6ip TOqOK q0 i q aa6eane~ye KOniHeapHiCT~ aeKTopiB q i q - q0.
BpaxoB)nOqa p i a u i c ~
1 r A - I OH �9 L =pq-H= ~p p-l-I = P-~U V(q,p)et~,qeto,
pO6HMO BHCHOBOK, tUO/~o/~allOK (q- qo)OH/Oq B IIpaBill qacrHHi (15) Bi/~'r
A OCKi.rI~KH BeKTOpH q --q0 i q0 + 0 (q - - q0) KOniseapHi, TO Ha ni~c'ra~i (15), 6epy-
qH/IO yBarH BH3Ha'-IeHHJt to, HepeKoHyCMOC}I B TOMy, I/IO s ~ .
ISSN 0041-6053. YKp. ztam. ~.'vpn. 1999, m. 51. N e 10
1414 C.H. COCHHI_[bKH~I
3aqbiKcye~o wenep TOqKy (qO, pO) e s OCKiflbKH ['2 r Mez~em Asia Mnomxann
. a~il~cnn~o ~aae a6ypeHna TOqKg (q 0, p 0) :
[ l ( q * - q 0 ) @ ( P * pO)l[ < ~1, ~ =cons t ,
TaKe, mo TO,~a (q*, p*) sine r eae~eH'ro~ ~HOXHnH [l -. Toni ~nacsxi~oK nene-
pepsnocTi/Xo6ymy q ~H/3q i ~nKona~Ha pi~Hoeri (14) V(q ,p ) e ~0 ~InCaaO ~1 (a
OTZ~Ce, i Bi~rlOBi/IHr 36ypeHH~l) MOZ~na BI, I6pavn HaCTia'IbKn Ma.rlnM, too6 BHKOHyBa-
~ncb nepisHoc'ri
H ( q * , p * ) - q - ~ q q=q.p= �9 +(B(q*)q*)X > O, H(q*,p*) < 0.
3ni~c~ po6n~o sncnoao~ npo cnpa~e~a~nni~r~ neun.
Hac~iOor,. B y~m~ax ne~tu ~ ~ (3.
fl[ose~enn~ meope~u 1. l'IpnnyCXaMO, mo no~omeHna piBHOBarrt q = q = 0
BI4XiRHoi CI4CTeMI4 (1)--(3) cvii~ze.
Y Bi/mosi/mocTi a [7] poars~neMo qbynxtlim
V = q P
$ 2 + 1 '
all) * Dq • R:,) Sl = S*( t ,q , -~p = S! ( t ,q ,p) e
][i noxi~na na~onx ~eKwopHoro no~a, mo BnaHaqaeTbca pi~nanHa~H (12), Mac
BHF J'l~l/l~
d___V.V = L (I - ~) § ( H - qOH (B(q)q)X) ~. = 2qp S - ~ " (t6)
at
Y Bi~nosi~HOCTi a npnnymeHHa~l npo c'riflKiCTh pisHosarH 3a~x~n icnye/lo~aTHa
nisTpaexTopia ~/]" C s e, aKa npoxo/ma~ ,~epea "ro,~Ky (q*, p* ) e ~ . BpaxoBy~oyn
mo 06ea'annny, npoinverpyeMo pinnica~ (16) ssaosm ai/xpi3Ka ni~'rpaeKTopii ~t~,
aKn~ ai~mosi~ae npoMimKy [q, t2], Re qncam q , tZ "taxi, mo
t l t2 ~ (17)
l [ t t C .
Ilpn m,o~y aaysa~n~o, mo oc~iamcn ?~ - - ~o~na~ama ~no~chHa, TO rao~ym,
m s ~ o e x i ai/xnositmoi ao6pamym,~oi To,~n (q (t, q*, p* ), p (t, q*, p* )) T npn ii pyci
s.~aosm ~,] pisnouipHo o6~exennfi, i, oxxe , ~o ~n a sKaaaaaa "raKe '~Hc~o a > 0, mo
pia~mtta t2 - t I > a , He~LIIeZKHO BiR TOF0, HaCKi~lbK~ Bea-IHKHMH r 3Ha~ICHH~I t 1, t 2 ~
r R. B peaysmTaTi iwrerpy~anna pisnocTi (16) ~ar
q p [r I~z ( Its)
Sl---'~+l[ t, = aretgSl t, + o arctgS 1 r, +
t2
+ ~ ( t4 -q~H/bq + (a(q)q)Z) dr. (18)
t, S l + 1
(l)yitKui~ aretgSt, mo qbirypye y npa~i~i ,~acTmd (18), e ~noroana,41tom qbyn~-
ttieao a to,rearm po~ra~ayxenna St = + ~. PasoM ~ vrr~ nepe~an ~noata~nn pisna ra-
I$SN 0041-6053. Ytcp. ~qam. ~ p n , . ] 999. m. 5 l . IV e ! 0
rlPO OYHKI_IIIO ~II 3A FAMIYlbTOHOM JMIJ:I HEFOJ1OHOMHHX CHCTEM... 1415
MDIbTOHiaHa H = h < 0 i Ma:loro OKO~ly rlOJIO~KeHH$1 piBHOBaFH r MHOFOBH~OM, B
6yllb-.qKii~ TOqI~i aKoro di)yHKiIi~l S 1 He nCpCTaOplOeTl, CJl B HeCKiHqeHHiCTb. Bea
06MeZ~eHH~I 3araYa, HOC'li poarsxa/ly 6y~eMO BBaT~a'I'H Ha~ta.rli, mo SII t =t~ > 1.
BH6epeMo npoMiz~oK [t t, t2 ] S Mez<ax BHKouamta yMOSH (17) ~toeraa~r~o Mas~ma,
tao6 Mowma 6y.no o6Me~7"acb o6~ac'rio rOSIOBHHX aHaqeHb qbyHralii arctg S~, BH-
KopHcTo~y~OqH, Hanprm~a~, ao6pa~eHHZ ocranm, oi y BHr.az~i
arctgS~ = rt 1 + 1 _
2 - S - ~ 3St 3 . . . . .
To/~i a piBHOCTi (18) rrpH ~OCTaTHbO Ma~o~l e > 0 oTpHMyeMo
l i t 2 + O( 1_.2__]t21 o i l ] t 2 1 = ! ( H - q ~ H l ~ q + ( B ( q ) q ) ~ , )
~l ,, ( 3 s : l , , ) + ~,Sll , , ) --~l+i d r . (19)
3ayBa~ytoqi~, mo y ai~noBi~HOCTi 3 (17 )npaaa qaCTnHa piSHOCTi ( I9 ) ~ao~XaTHa,
npHxo~rXMO ~aO cynepe'~HocTi, ocKi~mKH, ari~Ho 3 cTpyKTypoIo s tarpaK~iana
dS! ~ _+It,,
L = dt = piI-H = pTA-Ip-I'I > 0 V(q,p)~ 711t ~,
BHpaa B i'i .niBii~ qaCTHHi Bi~'CMHH~. OT:~e, rlpHIIyl.HeHH~l l'IpO CTiflKiCTb ~oc.ni~DKyna-
HOrO HO~O:~eHH$t pinHOBarH HOMHJIKOBC. TcopeMy 1 ~OBe/ICHO.
B TOMy qaCrHHHOMy BHna/lKy, KOnH HcroJIOHOMHa cHercMa ( 1 ) - (3) mrpo/~a<yea~-
ca B rO~OHOMHy: B(q) = 0, yMOSH 1, 2 TcopeMH Bi~anoBi/~o Hcpexo/~aa~ a TaKi:
1") r = {qes~ :Il(q)<O} c f~ , O~ ~r
2*) ~H(q) /~q~O V q e co.
3ayoa~ennst. ~ a nepeBipKH yMOBH 2 TeOpCMrl 1 3py~HO CKOpHCTaTrlC~ ao6pa-
~enn.qM/~ocaTi/DKyBaaoi HerostOHOMHOi cricTer, irl y ~opMi Bopomt~ [1, 13].
TeopeMa 2. . f l ru~o o m o ~ i q = 0 ctbynrtli.~ FI ( q) ~tae .aolca.~bltUfl ~taKcu.~o,~t
(.eoaoo' ,,o o mpo u ), = {q < 0} mo
pi~no~azu q = q = 0 cucme,~tu ( I ) - ( 3 ) n e c m i a r e .
~ a a iaoaelaeHH~ TeopeMH 2 Z~OCTaTH~O cKopHcTaTnC~ Hac:fi~OM :torah i cxeMom
~OBe/~eHH~t TeOpeMH 1, 3ayBa~HBttIH npH tmoMy, mo al-taa-lOrOM MHO~HHH ~'~/ 6
MHOZC.~Ha ~'~.
Hacni~o~. s o moqtfi q = 0 dpynr~i,~ H (q) ~tae cmpoeut~ noxanbnu~
,~tarcu~ty~, mo no,ao~ennn pionooazu q = (I = 0 cucme,~tu ( I ) - (3) necmiare.
~IK sma~HBar 3 TeopeM 1, 2, HeFO21OHOMHiCTb CHCTeMH 3Haxo/Irrrb CSOe Bi/Io6pa-
aCeHHa S xapaxTepi yMOB HecTiiaKOCTi, npoTe He sapTo aa6yBaT~, mo oe ramt i r smme
IIOCTaTHiMn. To~y Bi/InOBi/I~, Ha IIHTaHH~I npo BnJIHB HeFOJIOHOM~-IHX B'$13ePl Ha CTiI~I-
~:iCT~ pisHoaara, Ha ~a2I~, nOKH U~O aa.rmtuaer~c~ HenoBHOIO.
2. ~IK ai~o~o [13], naTauua npo cTilt~ier~ Hero~ouo~,.uax CaCTe~ ~ar CBOi cne-
ttrldpiqHi p~4CH, ~O e si~ao6pa~eHHa~, aoepeMa, Ha~SHOCri S HerOaOHOmmX c~ereMax
MHOrOnH~ty noaoa<eH~ piaHoaara, poaMipHica~ mcoro He ~eHtue ~Hcna HeroaoHo~-
HHX s 'aael l , A~e tie He oaHaqae, mo noa6amaeHO CeHCy/Xoc~aia~ermx criltKOCri qbiK-
cosarloro HO~O~KeHHJI pisHoBara4, oco6J1rlBo, ZOMBI BOHO ~ae a~o ry apo6rrm BHCHOBOK
npo CTiflKiCTb BCbOrO MHOrOBrI~y I~O~O:~r piBHOBaFH.
Teope~a 3, ~/~u~o o mo~t~i q = 0 r H (q) ;~tae cmpoeu~ .aoxa,abnua
~arcu~t)'~t, mo ~momaui9 nonoacenb piano~aeu cucme~tu ( 1 ) - ( 3 ) , u4o auunattaembc,~
piot.tnnn,~t~tu
ISSN O041-6053. YKp. ;~lam. ~. pn., 1999.m: 51. N ~ !0
1416 C. IT COCHH[ABKHI;I
_ 3rl(q_.._)) + B r ( q ) ~ = O, (7 = O,
necmi~xu~.
glo~eOeunsL 3ri~aHO 3 TeopeMolo 2, r lo3io~erlr lH piaHOBarH q = q = 0 CHCTeMrl
( 1 ) - - ( 3 ) a e e r i l t K e . I IoKameMo, LRO nec'ri t tKiCTb Mac Micae i a i z ~ o c a o 3~iHHI,IX q .
J2~H m, o r o n p H n y e r H ~ o CyllpOTHBHe, RIO Hecr i~Kic ' rb a a n o r o no.no~KeaHa piBHOBara
c y n p o B o z ~ y e T ~ C H ~,Hme aecTi l tKic 'no ~O~O 3MiHm~X q. To~ti, s e 3 a . n e ~ H o si~t TOrO,
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O]tepz<aHo 11.03.98
ISSN 0041-60.$3, YKp. ~am. ;ay. pu., 1999, m. .$1, N ~ ! 0
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| id | umjimathkievua-article-4740 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:04:28Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/04/93be7bd39245c711d42d1e595fd24e04.pdf |
| spelling | umjimathkievua-article-47402020-03-18T21:12:54Z On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability Про функцію дії за гамільтоном для неголономних систем та її застосування при дослідженні стійкості Sosnitskii, S. P. Сосницький, С. П. For nonholonomic systems, we introduce the notion of the function of Hamiltonian action, with the use of which we investigate the stability of nonholonomic systems in the case where the equilibrium state under consideration is a critical point of the corresponding Lagrangian (Whittaker system). Для неголопомних систем вводиться поняття функції дії за Гамільтопом, за допомогою якої досліджується стійкість неголопомпих систем у випадку, коли положення рівноваги, що розглядається, є критичною точкою відповідного лагранжіана (системи Уіттекера). Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4740 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1411–1416 Український математичний журнал; Том 51 № 10 (1999); 1411–1416 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4740/6181 https://umj.imath.kiev.ua/index.php/umj/article/view/4740/6182 Copyright (c) 1999 Sosnitskii S. P. |
| spellingShingle | Sosnitskii, S. P. Сосницький, С. П. On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability |
| title | On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability |
| title_alt | Про функцію дії за гамільтоном для неголономних систем та її застосування при дослідженні стійкості |
| title_full | On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability |
| title_fullStr | On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability |
| title_full_unstemmed | On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability |
| title_short | On the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability |
| title_sort | on the function of hamiltonian action for nonholonomic systems and its application to the investigation of stability |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4740 |
| work_keys_str_mv | AT sosnitskiisp onthefunctionofhamiltonianactionfornonholonomicsystemsanditsapplicationtotheinvestigationofstability AT sosnicʹkijsp onthefunctionofhamiltonianactionfornonholonomicsystemsanditsapplicationtotheinvestigationofstability AT sosnitskiisp profunkcíûdíízagamílʹtonomdlânegolonomnihsistemtaíízastosuvannâpridoslídžennístíjkostí AT sosnicʹkijsp profunkcíûdíízagamílʹtonomdlânegolonomnihsistemtaíízastosuvannâpridoslídžennístíjkostí |