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Structural Equation and Mei Conserved Quantity of Mei Symmetry for Appell Equations in Holonomic Systems with Unilateral Constraints

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Structural Equation and Mei Conserved Quantity of Mei Symmetry for Appell Equations in Holonomic Systems with Unilateral Constraintsof,and,Mei,for

    Structural Equation and Mei Conserved

    Quantity of Mei Symmetry for Appell

    Equations in Holonomic Systems with

    Unilateral Constraints

    Commun.Theor.Phys.(Beijing,China)52(2009)PP.572576

    ?ChinesePhysicalSocietyandIOPPublishingLtdVo1.52,No.4,October15,2009 StructuralEquationandMeiConservedQuantityofMeiSymmetryforAppell EquationsinHolonomicSystemswithUnilateralConstraints

    JIALiQun,1,2,tZHANGYao-Yu,2CUIJinChao,andLUOShao-Kai3

    1SchoolofScience,JiangnanUniversity,Wi214122,China

    ElectricandInformationEngineeringCollege,PingdingshanUniversity,Pingdingshan467002,China

    InstituteofMathematicalMechanicsandMathematicalPhysics,ZhejiangSci-TechUniversityHangzhou31001,China

    (ReceivedOctober23,2008)

    AbstractStructuralequationandMeiconservedquantityofMeisymmetryforAppellequationsinholonomic

    systemswithunilateralconstraintsareinvestigated.Appellequationsanddifferentialequationsofmotionforholonomic

    mechanicsystemswithunilateralconstraintsareestablished.Tj1edeftnitionandthecriterionofMeisymmetryfor

    Appellequationsinholonomicsystemswithunilateralconstraintsundertheinfinitesimaltransformationsofgroupsare

    alsogiven.TheexpressionsofthestructuralequationandMeiconservedquanti

    ofMeisymmetryforAppellequations

    j

holonomicsystemswithunilatermconstraintsexpressedbyAppellfunctionsareobtained.A

    nexampleisgivento

    illustratetheapplicationoftheresults.

    PACSnumbers:02.20.Sv,11.30.-j,45.20.Jj

    Keywords:holonomicsystemwithunilateralconstraints,Appellequation,structuralequati

    onofMeisym-

    metry,Meiconservedquantity

    1Introduction

    Thetheoryofanalyticalmechanicsisstillevolving.At thesametime,Appellequationshavebecomeoneofthree famousmechanicalsystemsinanalyticaJsystemsandplay animportantroleinanalytica1theories.lJThealgebrare

    ductionandthegeometryreductionarethemainmethods asmodernmethodsofreductionindynamicalsystem.It isanimportantalgebrareductionmethodtoseekforcon- servedquantitybyuseofLieGroupsandLieAlgebras researchsymmetryofthesystem.

    In1918.Noetherrevealedthepotentialrelat,ionbe- tweenthesymmetryandtheconservedquantity.How-

    ever,thescientificsignificanceoftheNoethertheorywas notreallyrealizedinthefieldofAnalyticalmechanicsnil

    tilthe1970's.Fromthenon,thestudyonbothsymmetry andconservedquantityareflourishedandthereforeplen

    tifu1andsubstantialoutcomesareachieved.3_14]Inre

    centyears,someresearchresultshavebeenmadeforMei symmetry.[t520IHowever.

    foralongterm.toofewre

    searchresultshavebeengiveninAppellequations.2In

    ordertoseekthemethodsofsolvingAppellequations,Mei firstlygaveNoetherconservedquantity[21]deducedindi-

rectlyfromNoethersymmetryaccordingtoforminvari

    ance.Inthesameway,theconservedquantitiesofvariable massholonomicsystemswereobtainedbyLieta1.[221Asa resultofNoethersymmetryandLiesymmetryfromform invariancerespectively,theconservedquantityofAppell equationsfortherotationalrelativisticholonomicsystem wasachievedbyLuo.23InRef.

    [24],Meisymmetryand

    MeiconservedquantityofAppellequationforthemechan- icalsystemsoftheChetaev'stypeconstraintwerestudied bymeansoftherelationsbetweenLagrangefunctionand Afunction.

    Aboveresultsofthestudyprovidedanewideatolook forMeiconservedquantityofAppellequation.Butthere isnostructureequationandtheexpressionofMeicon

    servedquantitywhichcanbedenotedwithAppellfunc

    ..

    ii,tionThereforetsnotdissolvedtothecoreaboutthe problemofMeiconservedquantityofAppellequation. Thepracticalapplicationinthenaturalandengi

    neeringtechnology,themajorityofconstraintsarebe

    longtotheunilateralconstraintsnotthebilateralcon. straints.Therefore,tostudyingunilateralconstraints hasanimportantpracticalsignificance.Inrecentyears, someresearchresultshavebeenmadeforsymmetryand conservedquantityofunilateralconstraintsystems.In Ref.251,Zhangstudiedforminvarianceforunilateral holonomicconstrainedsystems,inBf.261hepresented

    symmetryandconservedquantityofunilateralholonomic systemsinthephasespace,andinRef.f27]hestudiednon-

    Noetherconservedquantityofnon-holonomicconstrained systemswithunilateralnon-Chetaev'stype.LiandFang studiedMeisymmetryofvariablemasssystemswithuni

    lateralholonomicconstraints.28

    ThispaperpresentsastructureequationofMeisym

    metryandtheMeiconservedquantityinAppellfunction. SupportedbytheNationalNaturalScienceFoundationofChinaunderGrantNo.10572021a

    ndthePreparatoryResearchFoundation

    ofJiangnanUniversityunderGrantNo.2008LYY011 7E-mail:jlq0000@163.corn

    N0.4StructuralEquationandMeiConservedQuantityofMeiSymmetryforAppellEquation

    sin573

    2AppellEquationandDifferentialEquations ofMotionforaHolonomicSystemwith

    UnilateralConstrains

    Supposethattheholonomicsystemwithunilateral constraintsofparticleswithmassesanditspositionvector respectively,andletthepositionofthesystembedeter

    minedbygeneralizedcoordinates,itissubjectedtothe idealunilateralholonomicconstraints

    (~,q)0(=l,2.,g),(1)

    Qs=Qs(~,q,d)arethegeneralizedforces,theenergyof accelerationforthesystemisexpressedas

    s=7q1=m

    andAppellequationsarewrittenintheform

    -Qs-Qs+As():

    0(=1,2.,g)(when=0)

    Os

    a=Q(s=1.,),

    >0(=1,...,g)(when>0).

(3)

    (4)

    (5)

    (6)

    Foreach,0,0,0(=1,,g)isex

    isted.TheuniformsubscriptsdenoteEinsteinsummation conventioninEqs.(2)and(3)aswellasinthefollowing text.InEq.(3),arethemultipliers.Where

    As_A,q,(7)

    Asaregeneralizedconstrainforcecorrespondingtothe generalizedcoordinateqs.ByvirtueofEqs.(3)and(5), wecansolveallgenerizedaccelerationsas

    =A(t,q,d)(s=1.,n)(when=0),

    =

    B.(t,q,d)(s=1.,n)(when>0).(8)

    3MeiSymmetryofAppellEquation

    Introducetheinfinitesimaltransformationsoftimeand generalizedcoordinatesas

    t=t+At,qs(t)=q8(~)+Aq(8=1.,),(9)

    or

    t=t+~(~,q,d),)=q8(t)+e~8(t,q,)

    (s=l.,n),(10)

    where~isaninfinitesimalparameterand,&rein

    finitesimalgenerators.Similarly,weintroducevector(0) ofinfinitesimalgeneratorsagain

    (0)=+~80----

    ,

    aswellasitsfirstandsecondextended

    eratorsare

:+(象一仉鲁),

    (11)

    infinitesimalgen

    (12)

    =+[()一蟊].(3)

    Thetotalderivativesalongthetrajectoryofthesystem

    withrespecttotin(13)are =+++0dtOt一十口+A+A

    (when=0),(14)

    d

    =

    0

    ++B

    0+0+

    q++

    (when>0).(15)

    FromEq.(10),wehave dq*

    d2

    0

    dq+E

    d+0=+E(?一仉)+0()

    =+(?一以?0).一百]+O(e.) Undertheiniinitesimaltransformations(10),letdynamic

    functionsS,Q,AsofsystembeS,Q:,A:,takingthe

    TaylorexpansionofS,Q;,Awithrespectto(t,q,d,ei),

    wecaneasilyobtain s=s(q,dq*,)=,q)

    +{OS++(

+0s一仉)?一?s],}

    J

    =

    Qs(q,dq*)

    +0(E)

    =

    Q(t,q,d)

    +[等针()]

    +0(E),

    A=A(q,dq*)=A(,)

    ++(??.)]+0(E)

    =

    ,q'q)+E(+)+o(

    thatis

    S=s(t,q,q,q)+(.)()+O(.),(17) Q=Q(t,q,d)+~()(Q.)+O(e.)(s:1,,n),(18) A=A.(,q,)+E()(A.)+O(~.)(s:l,,n),(19) =

    (~,q)+eX(.)()+O(E.)=1.,9).(20)

    De~nition1IfAppellequation(3)keepsitsformin

    variantwhendynamicfunctionsS,Qs,Asaresubstituted

    bythefunctionsS,Q:,A:undertheinfiniteaimaltrans formations(i0),namely 8S

    a=Q+A:(8=l.,n)(21)

    thentheinvarianeeiscalledMeisymmetryofAppellequa-

    tion(3).

    坠札

    574JIALiQan,ZHANGYao-Yu,CUIJin-Chao,andLUOShao-KaiVb1.52

    Definition2Ifconstraintequations(4)keepsitsform

    inv~iantwhenfunction|8issubstitutedbythefimction undertheinfinitesimaltransformations(10),namely =

    (,q)=0(=l.,g)(when,>0).(22)

    Definition

    metry,then

    thesystems

    isunderthe

    3IfEqs.(5)and(6)satisfiedtheMeisym

    theinvarianceiscalledtheMeisymmetryof

    withunilateralholonomicconstraints,which constraints.

    DefinitionIfAppellequation(5)keepsitsformin- variantwhendynamicfunctionsS,Q8aresubstitutedby thefunctionsS,Qundertheinfinitesimaltransforma- tions(10),namely

    aS

    a=Q(8=1.,)

    thentheinvarianceiscalledthe

    temswithunilateralholonomic

    fromtheconstraints.Then,we

    (23)

    Meisymmetryofthesys

    constraints,whichisfree

    have

    =

    (t,q,d)>0(=1.,g)(when>0).(24)

    Definition5IfEqs.(3),(4),andEq.(5)satisfiedthe Meisymmetry,thentheinvarianceiscalledtheMeisym- metryofAppellequationforthesystemwithunilateral

holonomicconstraints.

    4CriteriaofMeiSymmetry

    SubstitutingEqs.(17)(20)intoEqs.(21)(24)and

    neglectingtermsthatare0(E.)orhigher.Takingnotice Eqs.(3)(6),weobtain

    )(s)=)(Q8+A8)(s-1,,(25)

    (.)()=o(=1.,g)(when>0),(26)

    )=)(Q.)

    (t,q)+eX(.)()>0

    (when>0),

    (8=1.,n)

    =

    1.,g)

    (27)

    (28)

    Eqs.(25)(28)arecalledthecriterionequationsoftheMei symmetryforaholonomicsystemofAppellwithunilat

    eralconstraints.Then,wehave

    Criterion1Iftheinfinitesimalgenerators,admit

    Eqs.(25),(26),thentheinvarianceofEq.(3)andEq.(4) undertheinfinitesimaltransformations(10)iscalledMei symmetryofAppellequationforthesystemswithunilat

    eralholonomicconstraints,whichisundertheconstraints. Criterion2Iftheinfinitesimalgenerators,6admit Eq.(27),thentheinvarianceofEq.(5)underthein

    finitesimaltransformations(10)iscalledMeisymmetryof Appellequationforthesystemwithunilateralholonomic constraints,whichisfreefromtheconstraints. Criterion3Iftheinfinitesimalgenerators,admit Eqs.(25),(26),and(27),thentheinvarianceofEqs.(3),

    (4),and(5)undertheinfinitesimaltransformations(10) iscalledMeisymmetryofAppellequationforthesystems withunilateralholonomicconstraints. 5StructuralEquationandMeiConserved

    Quantity

    PropositionIfthelnfinitesimalgeneratorsxz0,xi8and thegaugefunctionGM=GM(t,q,)ofMeisymmetry satisfies

    (2)(s)+(1[(2)(s)]+(?s一口.)[(2)(s)]

    +.)(Qs+A8)]dAs+dGM=0

    (when=0),

    '(s)+'[''(s)]+(?一口)['2()]

    +[(1(Q.)]dBs+dGM=0

    (when>0),(29)

    thentheMeiconservedquantitiesdeducedbytheMei symmetriesareexpresseda8follows

    =(2)(s)+(6一仉?.)+GM

    =const.(30)

    InEq.(29),

    =d00

    iscalledgeneralizedEulerarithmeticoperators. Pm0|Firstofall}whenthesystemisunderthecon- straints,weprovidetheMeiconservedquantitiesdeduced bytheMeisymmetriesofAppellequation(3)satisfy Eq.(3o).

    UsingEq.(14),wehave

    diM

    =

    [048

    +]

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