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[doc] Two New Types of Conserved Quantities of Mei Symmetry for Holonomic Mechanical System

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[doc] Two New Types of Conserved Quantities of Mei Symmetry for Holonomic Mechanical System

    Two New Types of Conserved Quantities of Mei Symmetry for Holonomic Mechanical

    System

Commun.Theor.Phys.(Beijing,China)49(2008)PP.5356

    ;@ChinesePhysicalSocietyVo1.49,No.1,January15,2008

    ;TwoNewTypesofConservedQuantitiesofMeiSymmetryforHolonomicMechanical

    ;System

    ;FANGJianHui,WANGPeng,andDINGNing

    ;CollegeofPhysicsScienceandTechnology,ChinaUniversityofPetroleum,Dongying257061,China

    ;(ReceivedNovember17,2006)

    ;AbstractTwonewtypesofconservedquantitiesdirectJvdeducedbyMeisymmetryofholonomicmechanicalsystem

    ;arestudied.TedefinitionandcriterionofMeisymmetryforholonomicsystemaregiven.Acoordinationfunction

    ;introduced.theconditionsunderwhichtheMe/symmetrycalldirectJyleadtothetwotypesofconservedquantitiesand

    ;theformsofthetwotypesofconservedquantitiesareobtained.Anillustrativeexampleisgiven.Theresultindicates

    ;thatthecoordinationfunctioncanbeselectedproperlyaccordingtothedemandofthegaugefunction,therebythegauge

    ;functioncanbefoundoutmoreeazily.Furthermore,Mncethechoiceofthecoordinationfunctionhasmultiformity,much

    ;moreconservedquantityofMeisymmetryforholonomicmechanicalsystemcanbeobtained.

    ;PACSnumbers:03.20.+i,11.3O.-j,45.05.+x

    ;Keywords:holonomicmechanicalsystem,Meisymmetry,newconservedquantity

    ;1Introduction

    ;Thereareintimaterelationsbetweensymmetries

    ;andconservedquantities.Themodernmethodsto

    ;findconservedquantitiesbyusingsymmetriesare

    ;mainly:Noethersymmetr1’r’IlJLiesymmetry,[2JandMei

    ;symmetr4.Inrecentyears,moreandmoreattention

    ;hasbeenpaidtotheresearchonthemainthreesym.

    ;metriesandconservedquantitiesofmechanicalsystems. ;Manyachievementshavebeenobtainedinthispopularre. ;searchfield.5lConservedquantitiescanbeinducedby

    ;themainthreesymmetriesbothdirectlyandindirectly. ;Therearemainlythreekindsofconservedquantitiesin. ;ducedbysymmetries:theNoetherconservedquantity,

    ;theHojmanconservedquantity,andtheMeiconserved ;quantity(whichisalsocalledanewtypeofconserved ;quantity1.lNoethersymmetrycaninducetheNoether ;conservedquantitydirectlyandtheHojmanconserved ;quantityaswellastheMeiconservedquantityindirectly: ;LiesymmetrycanleadtotheHojmanconservedquan- ;titydirectlyandtheNoetherandMeiconservedquanti

    ;tiesindirectly:MeisymmetrycaninducedMeiconserved ;quantitydirectlyandtheNoetherandHojmanconserved ;quantitiesindirectly.

    ;Inthispaper,twonewtypesofconservedquantities ;whicharedirectlyinducedbyMeisymmetryofholonomic ;mechanicalsystemarestudied.Theconditionsofexis

    ;tenceofthetwonewtypesofconservedquantitiesaswell ;astheformsofthetwonewtypesofconservedquantities ;leddirectlybyMeisymmetry.areproposed.Anillustra- ;tiveexampleisgiven.TheNoetherconservedquantity ;andMeiconservedquantityofholonomicmechanicalsys

    ;temarespecialcaseofthetwonewconservedquantities ;respectiveIygiveninthiSpaper?

    ;2MeiSymmetryoftheSystemandIts

    ;CriterionEquations

;ConsideramechanicalsystemcomposedofNparti

    ;des,anditsconfigurationisdeterminedbygeneralized ;coordinatesqs(s=1,2,…,).Themotionofthesystem ;issubjectedtotheidealholonomicconstraints,thenthe

    ;equationsofmotionforthesystemare ;daL

    ;dt04

    ;a

    ;OqQs,(1)

    ;whereL=L(t,q,)istheLagrangianofthesystem ;Equation(1)canbewrittensuccinctlyas ;where

    ;E()=Q

    ;da

    ;dt04

    ;a

    ;Oq

    ;Introducetheinfinitesimaltransformations ;tt+e~o(t,q,),(t)qs(t)+E(t,q,),(4)

    ;whereEisaninfinitesimalparameterandandsare ;infinitesimalgenerators.Undertheinfinitesimaltransfor

    ;mations(4),L=L(t,q,)becomesL=L(t,q,),

;QQ(t,q,)becomesQ=Q(t,q,).

    ;DefinitionAfterthedynamicalfunctionsLandQsbe

    ;ingreplacedbythetransformeddynamicalLandQ,if ;theformofequation(2)isinvariant,thatis ;()=Q(5)

    ;theinvarianceiscalledMeisymmetryofholonomicme

    ;chanicalsystem(2).

    ;ExpandingLandQ,wehave

    ;=

    ;L(t,q,)=L(t,q,)+E()()+Oe,(6)

    ;QQ(t,q,)=Q(t,q,)+e2()(Q)+Oe,(7)

    ;where

    ;=+(,

    ;aa..

    ;a...

    ;a

    ;

    ;dtOt++

    ;(8)

    ;

    ;FANGJian-Hui,WANGPeng,andDINGNingVl01.49 ;SubstitutingEqs.(6)and(7)intoEqs.(5),ignoringE.and

    ;thehigher-orderinfinitesimalterms,andbyusingEq.(2), ;weobtain

    ;(()())=()(Q).(10)

    ;Then,wehave

    ;CriteOnForholonomicmechanica1systemf2),ifthe ;infinitesimalgeneratorsand?ssatisfyEqs.(10),the

    ;correspondinginvarianceisMeisymmetryofthesystem. ;Equations(10)arecalledthecriterionequationsofMei ;symmetryforholonomicmechanica1system.

    ;3TWONewConservedQuantitiesofMei

    ;Symmetry

    ;Theconditionsofexistenceofthetwonewconserved ;quantitiesandtheformsofthetwonewconservedquanti

    ;tieswhichareinduceddirectlybyMeisymmetryofholo- ;nomicmechanicalsystemaregiveninthefollowingtheo- ;remS.

    ;Theolr~m1Forholonomicmechanica1systemf2),ifthe ;infinitesimalgenerators?0and?sofMeisymmetryanda

    ;gaugefunctionGM=GM(t,q,q)satisfy

    ;)1)【州驯+(,

    ;()1)(asf)

    ;.

;dGM

    ;.dt

    ;=

    ;0,(ii)

    ;thentheMeisymmetryofthesystemwillleadtothefol-

    ;lowingnewconservedquantity:

    ;1)(+,)+GM

    ;=const.,(12)

    ;where,=f(t,q,)isanarbitraryfunctiontomakeGM ;exist.Wecall{acoordinationfunction. ;P~oo/Takingthederivationof/Mtot,andusing ;Eq.(11),wehave

    ;:

    ;()似??),

    ;+(1)(()

    ;

    ;(.()1)(,)

    ;=

    ;[()1)(Qs)](.

    ;SpecialcaseIf,=,equations(11)and(12)respectivelybecome

    ;()()+()()()+()(Q)(?一吼)+dGM:

    ;.,

;1)(+GMnst.

    ;(13)

    ;(14)

    ;(15)

    ;Equation(14)istheconditionatwhichMeisymmetrycanleadtotheMeiconservedquantity.Equation(15)isjust

    ;theMeiconservedquantityinducedbyMeisymmetryofholonomicmechanicalsystem.Wecallthenewconserved

    ;quantity(12)ageneralizedMeiconservedquantityofholonomicmechanicalsystem.

    ;Theorem2FOrholonomicmechanicalsystem(2),iftheinfinitesimalgenerators?sofMeisymmetryandagauge

    ;functionGN=GN(t,q,)satisfy

    ;+,+6OL+OL()+Qs(已一)+dGN=.,(6)

    ;thenMeisymmetryofthesystemwillleadtothefollowingnewconservedquantity:

    ;=

    ;,+(?一口,)+G?=const.,(17)

    ;where,=f(t,q,

    )isanarbitraryfunctiontomakeGNexist.Wecall,acoordinationfunction. ;P~oo/Takingthederivationof/Ntot,andusingEq.(16),wehave ;=

;,+df+(6

    ;f.

    ;Q

    ;d

    ;OL+OL(一一仉)一苦一,一已差

    ;

    ;OL()_Qs,)

    ;

    ;No.1TwoNewTypesofConservedQuantitiesofMeiSymmetryforHolonomicMechanicMSystem55

    ;,d8LI,8La

    ;qsQ)(一香f)=0

    ;Specialca8eIff=0,equations(16)and(17)respectivelybecome ;+.+

    ;OL

    ;+OL(

    ;IN=L++GM:

    ;Equationf191istheconditiontowhichMeisymmetrycan ;leadtotheNoetherconservedquantity.Equationf201is ;justtheNoetherconservedquantityinducedbyMeisym

    ;metryofholonomicmechanicalsystem.Wecallthenew ;conservedquantityf171ageneralizedNoetherconserved

    ;quantityofholonomicmechanicalsystem. ;4Example

    ;TheLagrangianofamechanicalsystemis ;=

    ;(++)(gg2,

    ;thenon-potentialforcesare

    ;QI=0,Q2:0,Q3=43.(22)

    ;TrytostudyMeisymmetryandthetwonewtypesof ;conservedquantitiesofthesystem. ;Theequationofmotion(1)gives ;1=-1,q2=-1,q3:43

    ;Makingcalculation,wehave

    ;()()f,

    ;dt)+(一香)

    ;(一如)

    ;Choosingtheinfinitesimalgeneratorsas ;=

    ;1,l=1+t,2=3:0

    ;weobtain

    ;()()=一香lt,

    ;thenweget

    ;Es[2()(=()(Q)0.

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