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New Type of Conserved Quantities of Lie Symmetry for Nonholonomic Mechanical Systems in Phase Space

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New Type of Conserved Quantities of Lie Symmetry for Nonholonomic Mechanical Systems in Phase Spaceof,OF,Lie,for,New,Type,new,type,LIE,NEW

    New Type of Conserved Quantities of Lie Symmetry for Nonholonomic Mechanical

    Systems in Phase Space

    Commun.Theor.Phys.(Beijing,China)52(2009)PP.977-980

    ?ChinesePhysicalSocietyandIOPPublishingLtdVo1.52,No.6,December15,2009 NewTypeofConservedQuantitiesofLieSymmetryforNonholonomicMechanical SystemsinPhaseSpace

    PANGTing,FANGJian-Hui,tLINPeng,ZHANGMingJiang,andLUKai

    CollegeofPhysicsScienceandTechnology,ChinaUniversityofPetroleum,Dongying257061,China

    fReceivedJanuary8,2009;RevisedAugust3,2009)

    AbstractThenewtypesofconservedquantities,whicharedirectlyinducedbyLiesymmetryofnonholonomic

    mechanicalsystemsinphasespace,arestudied.Firstly,thecriterionoftheweakLiesymmetryandthestrongLie

    symmetryareWel1.Secondly,theconditionsofexistenceofthenewtypeofconservedquantitiesinducedbytheweak

    LiesymmetryandthestrongLiesymmetrydirectlyareobtained,andtheirformispresented.Finally,anAppell-Hamel

    exampleisdiscussedtofurtherillustratetheapplicationsoftheresults. PACSnumbers:02.20.a,11.30.-j

    Keywords:Liesymmetry,conservedquantity,nonholonomicmechanicalsystem,phasespace

    1Introduction

    Theresearchonsymmetriesandconservedquantities

    ofamechanica1systempossessesimportanttheoretical

    andpracticalsignificance.Themodernmethodstofind conservedquantitiesofamechanicalsystemaremainly Noethersymmetry,Liesymmetry,andMeisymmetry.【一J

    Andtherearemainlythreekindsofconservedquanti

    tiesinducedbythesethreesymmetries:theNoethercon- servedquantity,theHojmanconservedquantity,andthe Meiconservedquantity.Aseriesofimportantachieve

    mentshavebeenobtainedonsymmetriesandconserved quantitiesofmechanicalsystems.[5.]Seekingforother

    newtypeofconservedquantitieshasbecomeanewde- velopmentdirectioninsymmetrytheories.InRefs.[17,

    [XS],and[19],somenewtypesofconservedquantitiesfor Hamiltonsystems,Lagrangesystems,andrelativisticvari

    ablemassmechanicalsysteminphasespacewerestudied respectively.

    Inthispaper,westudyanewtypeofconservedquan- titiesinducedbyLiesymmetryfornonholonomicmechan- icalsystemsinphasespace.Thisnewtypeofconserved quantitiesisagroupofconservedquantities,whichaxe differentfromtheHojmanconservedquantity.Thecon- ditionsofexistenceoftheflewtypeofconservedquan- titiesinducedbytheweakLiesymmetryandthestrong Liesymmetrydirectlyareobtained,andtheirformispre

    sented.

    2EquationsofMotionforSystem

    Consideramechanicalsystemwhoseconfiguration isdeterminedbygeneralizedcoordinatesqs(8=

    1,2.,n).Supposethatthemotionofthesystemissub

    jectedtothefollowinggidealnonholonomicconstraints ofChetaev'stype

(,q,d)=0,=1,2,,g)(1)

    TheChetaev'sconditionthatconstraintsEq.(1)acton

    thevirtualdisplacementsis

    s=1

    :o

    q8

    0,(s=1,2,...,).(2)

    Theequationsofmotionofthesystemcanbewrittenas

    .,n),dtOO 旦丝一一OL:Q+(s:1,2

    s

    

    Oq

    s

    Qs+As,(s1,2,'_',n),

    q=

    s=l

    ,

    whereL=L(t,q,)istheLagrangianofthesystem, Qs=Q8(,q,)arethegeneralizednon-potentialforces, A8arethegeneralizednonholonomicconstraintreacting

    forces,and=,q,)areconstraintmultipliers. ThecanonicalformofEq.(3)inphasespaceis :,:OH++,qs'ps++A'

    (s=1,2.,n),

    whereH=H(t,q,P)istheHamiltonian Q.=Q(,q,p)=Q(,q,(,q,p)),

    A.=A(,q,P)=A(,q,q(t,q,p)).

    ExpandingEq.(5),weget

    =gs(,q,p),=h.(t,q,P),

(8=1,2,,).

    (5)

    (6)

    (7)

    ThenonholonomicconstraintsEq.(1)inphasespacecan beexpressedas

    =

    (t,q,P)=(t,q,q(t,q,p))=0.(8)

    SupportedbytheGraduateStudents'InnovativeFoundationofChinaUniversityofPetroleu

    m(EastChina)underGrantNo?$2009?19

    tCorrespondingauthor,E-mail:fangjh~upc.edu.ca 978PANGTing,FANGJianHui,LINPeng,ZHANGMing-Jiang,andLUKaiVb1.52 3DeftnitionandCriterionofLieSymmetry

    Introducetheinfinitesimaltransformations t=t+(t,q,p),q(t)=q8()+;已(,q,p),

    p)=p@)+;叩,q,p),(9)

    whereisaninfinitesimalparameter,,s,andare

    infinitesimalgenerators.

    Definition1Ifthedifferentialequations(5)andthe constraintsequations(8)remaininvariantunderthein

    finitesimaltransformationsEq.f9),theinvarianceiscalled theweakLiesymmetryofthenonholonomicmechanical systeminphasespace

    Definition2Ifthedifferentialequations(5)andthe constraintsequations(8)remaininvariantunderthein- finitesimaltransformationsEq.(9),andconsideringthe limitationofChetaev'sconditionEq.f21actonthein- finitesimalgenerators,,theinvarianceiscalledthe strongLiesymmetryofthenonholonomicmechanicalsys

    teminphasespace.

Theinvarianceofthedifferentialequations(5)under

    theinfinitesimaltransformationsEq.(9)givesthedeter.

    miningequations

    (一筹)_0j

    ?+OH一国一天)

    oritsequivalentexpression

    (10)

    

    =?(,=?()(1)

    where

    da0a

    

    dtOt+9s+^s,(2)

    )=0

    ,(13)

    =+(一岛o+(s

    ).(14)

    Andtheinvarianceoftheconstraintsequations(8)under

    theinfnitesimaltransf0rmationsEq.(9)givestherestric

    tionequation

    (.[(,q,p):0.(15)

    ConsideringthelimitationofChetaev'scondition Eq.(2)actontheinfinitesimalgenerators0,,wehave

    additionalrestrictionequation .(16)

    Then,accordingtotheabovedefinitions,wecanob

    tainthefollowingcriterions.

    Criterion1Forthenonholonomicmechanicalsystem inphasespaceEqs.(5)and(8),iftheinfinitesimalgener

    ators,,andssatisfythedeterminingequations(10) or(11),andtherestrictionequation(15),theinvariance istheweakLiesymmetryofthesystem.

    Criterion2Forthenonholonomicmechanicalsystem inphasespaceEqs.(5)and(8),iftheinfinitesimalgen

    eratorso}s}andsatisfythedeterminingequations (10)or(11),therestrictionequation(15),andadditional restrictionequation(16),theinvarianceisthestrongLie symmetryofthesystem.

    4NewTypeofConservedQuantitiesDe

    ducedfromLieSymmetryDirectly

    Undercertainconditions,theLiesymmetrycanlead tothenewtypeofconservedquantitiesdirectly. Proposition1Foranonholonomicmechanicalsystem inphasespace,ifthereexistsagaugefunctionGLs= GLs(,q,P)satisfiesthefollowingstructureequation f,Oqk+

    Ohs

    /I\)+(+)(讯一)

    +

    d

    

    G

    :

    L

    

    s

    :0.d

    

    (17)

theweakLiesymmetryleadstoanewtypeofconserved

    quantitydirectly

    /L=矗一g+仉一h8+GL.=const.(18)

    Proposition2Foranonholonomicmechanicalsys

    terninphasespace,ifthereexistsagaugefunction

    GL8=GLs(t,q,P)satisfiesthestructureequation(17),

    thestrongLiesymmetryleadstoanewtypeofconserved

    quantitiesEq.(18)directly.

    rrooy

    dlLs

    :

    孥一一一一.

    +

    鲁一+

    :l9.一一乞.

    一一

    +

    鲁一一?.+.

    一一

    Oh,

    g

    +dGLs.(19)一?.g?.一?.+'

    Usingthestructureequation(17)andthedetermining

    equations(Ii),weobtain

    =堕一一一一一Og8dtdtysdtOtqo0+

    k:T

    

    警一Oh,t.Oh~

    =

    0(20)

    Inaddition,asweknow,undercertainconditionsthe weakLiesymmetryandthestrongLiesymmetryfornon- No6NewTypeofConservedQuantitiesofLieSymmetryforNonholonomicMechanicalSys

    temsinPhaseSpace979

    holonomicmechanicalsystemsinphasespacecanleadto theHojmanconservedquantities.[u]

    Foranonholonomicmechanicalsysteminphasespace, ifthereexistsafunction=(t,q,p)satisfying Oq8++dt.

    s

    .(21)

    theweakLiesymmetrycanleadtotheHojmanconserved quantitydirectly

    ,

    l().1a().1()d

    日—一十—十—?一

    =const.(22)

    Foranonholonomicmechanicalsysteminphasespace, ifthereexistsafunction=(,q,P)satisfyingEq.(21),

    thestrongLiesymmetrycanleadtotheHojmancon- servedquantityEq.(22)directly.

    ContrastoftheconservedquantitiesEqs.(18)and (22),wecanseenewtypeofconservedquantitiesEq.(18) areagroupofconservedquantities,whicharedifferent fromtheHojmanconservedquantityEq.(22). 5AnExample

    TakeanAppell-Hamelexample.Inphasespace,the Hamiltonianoftheproblemis

    H+p;+pD+mgq3,(23)

    theconstraintequationis

,:P+pgp;=0,(24)

    letusstudythenewtypeofconservedquantitiesofLie

    symmetryfornonholonomicmechanicalsystemsinphase

    space.

    Theequationsofmotionforthesystemare

    .1.2.

    ,92q2,93=q3=_,m gl=q1=

    ""

    h1:1:2ffI,h2:2:2,

    h3:3:mg2Ap3

    .

    (25)

    CombiningEq.(24)withEq.(25),wecanobtain

    =

    m2g

    ,(26)

    thenwecanobtain 1=l=Pl,

    g2==P2

    ,3==,

    mmm

    

    ,

    

    1

    ,

    A=,

    

    ,.

    Thedeterminingequations

systemgive

    d1pld1

    mm'…一== (27)

    (11)ofLiesymmetryofthe

    2P22 dt,

    d?3P3d03 dtmdt,m' dt+2

    p3

    

    dt=2p

    3

    +''2

    +mgp2d~o= 2p3z+.2 船班.

    +:0d

    t'2dt' where

    d0.

    pl0.

    P20,

    P30

    

    =一十一一十一一十一一

    .mOql.mOq2.mOq3

    

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