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Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Relativistic Birkhoffian Systems

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Lie Symmetrical Perturbation and Adiabatic Invariants of Generalized Hojman Type for Relativistic Birkhoffian Systemsof,Lie,and

    Lie Symmetrical Perturbation and

    Adiabatic Invariants of Generalized Hojman Type for Relativistic Birkhoffian

    Systems

    Commun.Theor.Phys.(Beijing,China)47(2007)PP.2530

    @InternationalAcademicPublishersVo1.47,No.1,January15,2007

    LieSymmetricalPerturbationandAdiabaticInvariantsofGeneralizedHojmanType forRelativisticBirkhoffianSystems

    LUOShao-Kai1,tandGUOYong-Xin2

    Institute0fMathematicalMechanicsandMathematicalPhysics,ZhejiangSci-TechUniversity,Hangzhou310018,China

    2DepartmentofPhysics,LiaoningUniversity,Shenyang110036,China

    (ReceivedJune29,2006)

    AbstractForarelativisticBirkho

    ansystem.theLiesymmetricalperturbationandadiabaticinvariansofgeneral-

    jzedHojmantypea/'estudiedundergeneralinfinitesimaltransformations.Onthebasisoftheinvarianceofrelativistic

    Birkhoflianequationsundergenerajinfinitesimaltransformations,Liesymmetricaltransformationsofthesystemare

    constructed.whichoniydependontheBirkhoffianvariables.TheexactjnvariantsjntheformofgeneralizedHojman

    conservedquantitiesledbytheLiesymmetriesofrelativisticBirkhoffiansystemwithoutperturbationsaregiyen.Based

    onthedefinitionofhigher-orderadiabaticinvariantsofamechanicaJsystem,theperturbationofLiesymmetriesfor

    relativisticBjrkhofliansystemwiththeactionofsmalldisturbanceisinvestigated,andanewt

ypeofadiabaticinvariants

    ofthesystemisobtained.Intheendofthepaper,anexampleisgjyentoillustratetheapplication

    oftheresults.

    PACSnumbers:03.20.+i,03.30.+p,02.20.SV,l1.30.-j Keywords:relativity,Birkhoffiansystem,Liesymmetricalperturbation,exactinvariant,adi

    abaticinvariant

    ofgeneralizedHojmantype

    1Introduction

    In1917,Burgersfirstproposedadiabaticinvariants, whichreferredtoaspecialkindofHamiltonsystem.lJ

    Aclassicaladiabaticinvariantisacertainphysicalquan- titythatchangesmoreslowlythansomeparametersof thesystemwhichvaryveryslowly.l一驯Fbramechanical

    system.thereexitsintimaterelationbetweentheintegra- bilityofthesystemandthevariationsofitssymmetries andinvariantsundertheactionofsmalldisturbance.and ?

    thereforetheresearchesonsymmetricalperturbationand adiabaticinvariantsareofgreatsignificance.Moreand moreattentionhasbeenpaidtoresearchonsymmetrical perturbationandadiabaticinvariantsofmechanicalsys. tems,andsomeimportantresultshavebeenreportedin recentyears_【一12]Buttheadiabaticinvariantsobtained byallthesestudiesbelongonformofNoether.Recently, Y.Zhanggaveanewadiabaticinvariantsofgeneralized Hojmantype.I3JButthestudieswereconfinedtotheclas- sicalmechanicssystem.

    Relativisticanalyticalmechanicsisanimportantas- pectwiththemoderndevelopmentinthefieldofthee- reticalphysics.Since1987,weobtainedthetheoryof

analyticalmechanicsfortherelativisticsystem.1423In

    1927,G.D.BirkhoffmadeprimaryresearchesonBirkhof- fiandynamics.104.In1983.R.M.Santillistudiedthetrans. formationtheoryofBirkhofIianequationsandgeneraliza- tionofGalileirelativity,andsummarizedcomprehensively theoriginofBirkhofIianequationsandthelaterstudies onthem.125JLater,F.X.Meiconstructedthetheoretical flameofBirkhofliandynamics.[2839TheBirkhoffiandy-

    namicsismoregeneralthantheHamiltoniandynamics. TheHamiltoniandynamicshasbeenextensivelyapplied inthefieldofmodernphysics,sotheBirkhoffiandynam- iasshouldplayanimportantroleinthefieldofmodern physics.Recently,weconstructedthebasictheoryofrel- ativisticBirkhoffiandynamics,andgaveitsbasictheoret. icalflame.[4o46

    Thestudyofadiabaticinvariantshasbecomeapop- ularsubjectinmechanics-113,47atomicandmolecular

    physicsI48,49etc.Inthispaper,wefurtherstudythe

    Liesymmetricalperturbationandadiabaticinvariantsof generalizedHojmantypeforarelativisticBirkhoffiansys tern.Firstly,wegivetheexactinvariantsintheformof generalizedHojmanconservedquantitiesforrelativistic Birkhoffiansystemwithoutperturbations.Then,based onthedefinitionofhigher.orderadiabaticinvariantsofa mechanicalsystem,westudytheLiesymmetricalpertur- bationforrelativisticBirkhoffiansystemwiththeactionof smalldisturbance.andobtainanewtypeofadiabaticin- variantsofthesystem.Intheendofthepaper,wepresent anexampletoillustratetheapplicationoftheresults. 2LieSymmetriesandExactInvariantsof

RelativisticBirkhofflanSystems

    HerewewillconsideramechanicalsystemofNparti

    cles.Attimet,thethparticle'svelocityist,thelimit

    ingvelocityisc,theclassicalmassm.',anditsrelativistic TheprojectsupportedbyNationalNaturalScienceFoundationofChinaunderGrantNos.103

    72053and10472040,theNaturalScience

    FoundationofHunanProvinceunderGrantNo.03JJY3005,theScientificResearchFoundat

    ionofEductionDepartmentofHunanProvince

    underGrantNo.02C033andtheOutstandingYoungTalentsTrainingFundofLiaoningProvi

    nceunderGrantNo.309005

    tE-mail:mmmplsk~163.

    tom

LUOShao-KaiandGUOYong-XinVl0I_47

    maSS1S

    milo{(i=l,...,?)(1)

    WecanconstructtheBirkhoffianBandtheBirkhoff's functions(=1.,2)ofarelativisticsystemas

    B=B(mi(t,ap),t,a),

    =

    (m(t,ap),t,a),

    (,=1.,2n),(2)

    andlet

    B=B(t,a)=B(m(t,ap),t,a),

    R==(t,a)=R;(mdt,ap),t,a).(3)

    Foranidealholonomicorfreerelativisticsystem,the Birkhoflianequationsofthesystemcanbewritteninthe form[43]

    (ORr/一警=.

(,=1.,2),(4)

    wheretherepeatedsubscriptsrepresentthesummation. Ingeneral,itissupposedthatthesystem(4)isnonsingu- 1ar,ie

    aet()?0,:(亲一).

    FromEqs.(4),wecanobtain

    (等一百OR;)(-1】…)

    where

    :_r=f.

    (5)

    (6)

    (7)

    Equations(6)canbeexpressedasfollows: ahr(t,a)(,=1,,2n).(8)

    Weintroduceinfinitesimaltransformationfortandar t=t+At,

    ar*(t)=ar()+Aa(=1.,2)

    andtheirexpandingformsaxe

    t=t+.(t,0),

    0=0+?(t,0)(,=1,,2n),

    (9)

    (1o)

    whereEisaninfinitesimalparameter,and.andare calledtheinfinitesimalgenerators.Undertheinfinitesimal transformations(1o),weintroduceavectorofgenerator ..+

    TheinvarianceofEq.(8)undertheinfinitesimal mations(10)leadstothe'satisfactionoftheLie ricaldeterminingequations

一面

    d

    -,

    0=.(),

    where

    d0.,

    0

    瓦十

    (11)

    transfor.

    symmet-

    (12)

    (13)

    Definition1FortherelativisticBirkhofliansystemf41, ifthegeneratorso,?oftheinfinitesimaltransforma- tions(10)satisfythedeterminingequations(12),thenthe correspondingtransformationsaxecalledLiesymmetrical transformations,andthecorrespondingsymmetriesare calledtheLiesymmetriesofthesystem.

    TheLiesymmetriesdonotalwaysimplyexactinvari. ants.Now,wegivetheconditionsandtheformsunder whichaLiesymmetrycanleadtoHojmanexactinvaxi

    antsofarelativisticBirkhofliansystem.

    Theorem1FortherelativisticBirkhofliansystem(4) withoutperturbations,ifthegenerators.,ofthein

    finitesimaltransformations(10)satisfytheLiesymmetri

    caldeterminingequations(/2),andthereexistsafunction A0=0(t,ar)sarisfyingthefollowingcondition: O+ln.:oar.dt'(14)

    thenthesystem(4)possessesthefollowinggeneralized

Hojmanconservedquantity,i.e.theexactinvaxiants

    =

    1

    ..

    0(A07"0)+10(.)d.. (15)

    PofDifferentiating10withrespecttotimet.andcon

    sideringtheLi~symmetricaldeterminingequations(12)

    andthecondition(14),wehave

    

    d/0

    =

    筹一dt()+.(h)面苏一一J+m =

    0f

    dtr

    do

    _

    x(

    0

    .]

    +X(

    o

    01

    Oh.

    +

    d1

    n

    +(+.).

    =o

    Therefore,therelativisticBirkliofliansystem(4)hasthe exactinvariants(15).

    Theconservedquantity(151isanexactinvaxiantof generalizedHojmantype,whichrevealstheintrinsicrela- tionbetweentheinvariantandtheLiesymmetryofundis

    turbedrelativisticBirkhofliansystem.

    3LieSymmetricalPerturbationandAdia-

    baticInvariantsofGeneralizedHojman

    TypeforRelativisticBirkhofllanSystems

    Firstlywegivetheconceptofhigherordera~iiabatic

    invariants.

    Definition2IfL=(t,a,)isaphysicalquantityin- cludinginwhichthehighestpowerisinarelativistic Birkhofliansystem,anditsderivativewithrespecttisin directproportiontogz+,thenLiscalledathorder

    adiabaticinvariantsofarelativisticBirkhofllansystem.

NO.1LieSymmetricalPerturbationandAdiabaticInvariantsofGeneralizedHojmanTypef

    or

    SupposetherelativisticBirkhoffiansystemcorre- sp0ndingtoEqs?(4)isperfurbedbysmallquantitiessQ,wllere thentheequationsofmotionofthesystembecome……

    :OB*+OR;=Q(

    ,=1.,2n).(16)

    ExpandingEqs.(16),weget

    a=(,a)+Q(,=1,,2n)(17)

    DuetotheactionofQ",theprimarysymmetriesand invariantsofthesystemmayvary.Supposethevariation isasmallperturbationbasedonthesymmetricaltrans

    formationsofthesystemwithoutperturbation,and7_and

denotethegeneratorsoftimeandspacerespectively

    afterbeingperturbed,then

    7_=7_0+7_1+e27_2+.,

    =++.+

    ,

    andtheinfinitesimalgeneratorvectoris )'丁晏+.

    SubstitutingEqs.(18)intovector(19),wehave

    (.)='(m=l,2.),

    where

    =丁瓦0+m0.

    (18)

    (19)

    (20)

    (21)

    Aftertheactionofsmallforcesofperturbation,theinvari

    anceofEqs.(17)undertheinfinitesimaltransformations

    (10)leadstothefollowingLiesymmetricaldetermining

    equations,

    

    d

    

    d

    T--gOd*"VQd

    =(.)()+eX(.)(Q)

    

    0+(+0

    SubstitutingEqs.(18)intoEq. ficientsofEonthetwosides

    other,wehave

27

    (22)

    (23)

    (22),andlettingthecoef-

    ofequalitybeequaleach

    

    r一十Qd丁一1

    =

    ()+1(Q)(24)

    whereweappointthat7-=0and=0when=0.

    Theorem2FortherelativisticBirkhoffiansystemf161

    whichisdisturbedbysmallforcesofperturbationQ,

    iftheinfinitesimalgeneratorsrand?satisfythLie symmetricaldeterminingequations(24)ofdisturbedsys

    tem,andthereexistssomefunctionA=(,0)satisfying thefollowingcondition:

    +s(+dln

    =.(25)

    thentherelativisticBirkhoffiansystem(16)hasath-

    orderadiabaticinvariantsofgeneralizedHojmantype.as

    f0lloWS:

    L=10m)+10m)_])(26)

    whereweappointthat=A0when=0

    Proo/Differentiatingwithrespecttotimet,andconsideringEqs.(18),(20),(22),(25),and(26),wehave

    +丁一未丁+a珊】

    =

    {[dTrnW*#vQT~--I_x(O)oc

    +

    Oh.(t

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