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Lie Symmetry and Generalized Mei Conserved Quantity for Nonconservative Dynamical System

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Lie Symmetry and Generalized Mei Conserved Quantity for Nonconservative Dynamical SystemMei,for,Lie,and,LIE,AND,MEI,FOR

    Lie Symmetry and Generalized Mei

    Conserved Quantity for Nonconservative

    Dynamical System

    Commun.Theor.Phys.(Beijing,China)49(2008)PP.11481150

    ?ChinesePhysicalSocietyVo1.49,No.5,May15,2008

    LieSymmetryandGeneralizedMeiConservedQuantityforNonconservative DynamicalSystem

    JINGHong?-XingandLIYuan?-Cheng

    CollegeofPhysicalScienceandTechnology,ChinaUniversityofPetroleum(EastChina),Dongying257061,China

    (ReceivedJune8,2007)

    AbstractBasedoilthetotaltimederivativealongthetrajectoryofthesystem,forIloilcoilservativedynamicalsysteml,

    thegeneralizedMeiconservedquailtityindirectlydeducedfromtheLiesymmetryofthesystemisstudied.Firstly,the

    Liesymmetryofthesystem

    given.Then.thenecessaryandsuIticientconditionunderwhichtheLiesymmetryjsa Meisymmetry

    presentedandthegeneralizedMeiconservedquantityindirectlydeducedfromtheLiesymmetryofthe

    systemisobtained.Lastly,anexamplejsgiveiltoillustratetheapplicationoftheresult. PACSnumbers:03.20.+i.02.20.Sv.11.30.-j

    Keywords:Liesymmetry,Meisymmetry,generalizedMeiconservedquantity,nonconservativedynamical

    system

    1Introduction

    Itiswellknownthattherearethreemainkindsof symmetriestofindconservedquantitiesofmechanicalsys

    terns.i.e..Noethersymmetry.[Liesymmetry. [2]andMei

    symmetryforthefclrminvariance[3J1.Noethersymme

    tryisaninvarianceofaHamiltonactionunderthein

    finitesimaltransformationofgroupandMeisymmetry meansthatthedynamicalfunctionsintheequationsof motionstillsatisfytheequations'primaryformafterin

    finitesimaltransformations.Theabovetwosymmetries canleadtoNoetherconservedquantityt4JandMeicon

    servedauantity[respectively.Liesymmetryisaninvari

    anceofdifierentialequationsundertheinfinitesimaltrans

    formations.UndersomeconditionsLiesymmetrycan leadtoNoetherconservedquantity.JHomanconserved quantity.e,generalizedHojmanconservedquantity.[.] Lutzkyconservedquantity,[10,11]generalizedLutzkycon

    servedquantitv[12,13]andMeiconservedquantity. [14,l5]

    Recently,Fangeta1.gaveanewtypeofconserved quantity[16](orgeneralizedMeiconservedquantity)di

    rectlydeducedfromMeisymmetryofthesystems.On thebasisofthetotaltimederivativealongthetrajectory ofthesystem,thispapergeneralizesFang'sresultstonon

    conservativedynamicalsystemandobtainedthegeneral

    izedMeiconservedquantityindirectlydeducedfromthe Liesymmetryofthissystem.

    2LieSymmetryforNonconservative

    DynamicalSystem

    ConsideranonconservativedynamicalsystemofN

particles.anditsconfigurationisdeterminedbyngen

    eralizedcoordinatesqs(s=1,...,n).Themotionofthe systemissubectedtotheidealbiolateralholonomiccon

    straintsandthentheequationsofmotionforthesystem a.re

    (L)=Q,(s1

    whereL=L(t,q,)istheLagrangianofthesystem

    Qs=Qs(t,q,)arethenonpotentialgeneralizedforces and

    d0

    dt04

    a

    Oq

    aretheEuleroperators.

    Supposethatthesystemisnonsingular,i.e

    det(a00404k

    )?0

    fromEqs.(1),wecanfindallthegeneralizedaccelerations as

    =

    (t,q,),(s=1,,n).(4)

    Choosetheinfinitesimaltransformationsofgroupwith respecttotimeandcoordinatesas

    tt+cr(t,q,),

    qs(t)=q(t)+E?(t,q,),(s=1.,n),(5)

    whereEisaninfinitesimalparameter,and?sarein

    finitesimalgenerators.

    TheinvarianceofEqs.(1)undertheinfinitesimal transformations(5)leadstothedeterminingequationof Liesymmetry

(.'{E())('(Q),(s1.,n)

    Theirequivalentformsare

    dT

    dtdt

    1),(as)gssJ

    (s1,...,n),

    where

    )=00

    +dG)Oos

    (2)

    d

    dt

    +(

    0.

    00

    +q+

    .ddr

    qs

    (1)Sowehavethefollowingproposition 2)OG

No.5LieSymmetryandGeneralizedMeiConservedQuantityforNonconservativeDynami

    calSystem1149

    Proposition1Fornonconservativedynamicalsystemisfythedeterminingequations(6)or(7)

    ,theyareLie

    (1),ifinfinitesimalgeneratorsr(t,q,)and?(t,q,)sat.symmetrica1. 3NecessaryandSutticientConditionunderWhichLieSymmetryIsaMeiSymmetryof

    NonconservativeDynamicalSystem Makingthecalculation,wehave

Es{2?({(=()OrOL+d\(O~aOL)O~kOL+[()]

    一旦

    Oqs

    ()一『()ti]+0dr/l.OLddrOLdt04kdtLO4dtO4kOqdtdtdt,\,/\/"J'\/aati

    (8=1,,n).(9)

    BasedontheMeitheoryofthenonconservativedynamicalsystem,thecriterionequationsofthesystemare[

    {()())=()(Q),(s=1,...,).(10)

    Thereforewehave

    Proposition2Fornonconservativedynamicalsystem(1),thenecessaryandsufficientconditionunderwhichtheLie

    symmetryisaMeisymmetryisthattherighthandsideofformula(9)vanishes,i.e. d,OrOL,OrOLd,akOL,akOLd0,dk,a]0,dk,OL

    dt\,atiOt/OqOt'dtO4Oqk/OqOqk'dtLo4\,dt/O4kJOq\,dt/04k-4---t-I一一l…一I

    l一一IIl一一Il

    

    d

    [0i'dr/I"I.OL]+0/dr/I"I.OLddrOL=.,(s=1).(11)

    PmofSubstitutingEqs.(6)andrelation(11)intoformula(9),weobtainEs{X(1)())=()(Q).Accordingto

    criterionequations(10),weknowthattheLiesymmetryisaMeisymmetryofthesystem. 4GeneralizedMeiConservedQuantityDeducedfromLieSymmetryofNonconservative DynamicalSystem

    UsingtheLiesymmetryofthenonconservativedynamicalsystem,wecanindirectlyfindthegeneralizedMei

    conservedquantity.Wehavethefollowingproposition.

    Proposition3Fornonconservativedynamicalsystem(1),undertheinfinitesimaltransformations(5),ifthein-

    finitesimalgeneratorsr(t,q,)ands(t,q,

    )satisfyEqs.(6)andrelation(11),andthereexistsagaugefunction GM=GM(t,q,)suchthat

    (1)()+(1)IX(1)(+(.厂一丁)+()+(1)(Qs)(已一)+=.,(12)

    thentheLiesymmetryofthesystemwillindirectlyleadtothegeneralizedMeiconservedquan

    tity

    ,M:?(),+(Gf)+GM:c.nst.,(13)

    wheref=f(t,q,)isacoordinationfunctionmakingGMexisteasily. PmIftheinfinitesimalgeneratorsr(t,q,)and(,q,

    )satisfyEqs.(6)andrelation(11),usingProposition2, weknowthattheyareMeisymmetrical,sothecriterionequations(10)ofMeisymmetryhold.

    Basedonthetotaltime

    derivativealongthetrajectoryofthesystem,takingthederivativeof/Mtot,andusingEqs.(10)

    and(12),weobtain

    aIX?()If1)()+[,)+(警:)

    1)(df1),+

    

    ()(Q)(ti.)

    =

    [()一一c1(Q)](已:,)=..(4)

1150JINGHong-XingandLIYuanChengVl01.49

    5IllustrativeExample

    TheLagrangianofanonconservativedynamicalsys

    temis1

    L1q

    l2++)q3,(15)

    thenonpotentialgeneralizedforcesare

    Q1=,Q2=1,Q3:0.(16)

    TrytostudytheLiesymmetryandthegeneralizedMei conservedquantityofthesystem.

Thedifferentialequationsofmotionofthesystemcan

    beexpressedas

    l一一,q21,q3一一1.(17)

    FromEqs.(7)and(17),weobtainthedeterminingequa-

    tionsoftheLiesymmetry dd6

    dtdt

    dd?2

    dtdt

    dd(3

    dtdt

    .

    ddT

    ql..d..——t...d...t

    .

    ddT

    q2-at..d...t

    GM:2+q2.

    Equation(13)yieldsaconservedquantity

    :一;+2+g2一;香2一如:c.nst. Selectf=43,then

    GM:q3一;2.

    Equation(13)yieldsaconservedquantity

    :一;一;2+q3一如一;:c.nst. When,=T=0,weobtain 2(一旌)=(一口z)(3)j.nhGM 2.

    dT

    

    0.d

t

    ~

    ddT

    q3+2

    Takinginfinitesimalgeneratorsas

    7-0,?l:?2=0,=03+t, weget

    ,,()(L)03t.

    (19)

    (20)

    Itiseasytoverifythattheinfinitesimalgenerators(19)

    satisfyEqs.(18)andrelation(11),accordingtoProposi tion2,theyareLiesymmetricalandalsoMeisymmetrica1.

    SubstitutingEqs.(19)and(20)intoEq.(12),wehave

    References

    [1]

    [2]

    [3

    [4]

    dGM

    =

    .

    (21)

    6

    (22)

    (23)

    (24)

    (25)

    (26)

/M03t=const.(27)

    Conclusion

    Inthispaper,anewtypeofconservedquantityofnon

    conservativedynamicalsystem.i.e.thegeneralizedMei conservedquantity,isintroduced.Itcannotonlybedi

    rectlydeducedfromMeisymmetrybutalsobeinduced indirectlybyLiesymmetryofnonconservativedynami

    calsystem.TheMeiconservedquantityofthesystem deducedindirectlyfromtheLiesymmetryl5Jisaspecial caseofourresultfwhenf=T1.Thecoordinationfunc

    tionfcanbeselectedproperlytofindthegaugefunction GMeasily,somoreconservedquantitiesofMeisymmetry fornonconservativedynamicalsystemcanbeobtained. A.E.Noether,Math.Phys.KIII(1918)235.

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    strainedMechanicalSystems,BeijingInstituteofTech

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    [6]Y.C.Li,Y.Zhang,andJ.H.Liang,ActaPhys.Sin.51 (2002)2186(inChinese).

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    H.B.Zhang,L.Q.Chen,R.W.Liu,andS.L.Gu,Acta Phys.Sin.54(2005)2489(inChinese).

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