hamiltonian_chaos.pdf

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1、ChaosinHamiltoniansystemsTeemuLaaksoApril19,2013Coursematerial:Chapter7fromOtt1993/2002,ChaosinDynamicalSystems,Cambridgehttp://matriisi.ee.tut./courses/MAT-35006Usefulreading:Goldstein2002,ClassicalMechanics,AddisonWesleyHand&Finch1998,AnalyticalMechanics,Cambri

2、dgeAdvanced:Lichtenberg&Lieberman1992,RegularandChaoticDynamics,SpringerArnold1989,MathematicalMethodsofClassicalMechanics,SpringerContents1Introduction:classicalmechanics22Symplecticstructure43Canonicalchangesofvariables64Hamiltonianmaps75Integrablesystems106Pert

3、urbationsandtheKAMtheorem137Thefateofresonanttori158Transitiontoglobalchaos1711Introduction:classicalmechanicsNewtonianmechanicsConsideraparticleofmassm,subjecttoaforceeldF,inad-dimensionalEuclideanspace(one-bodysystem).Newton'ssecondlaw;mr•=F=)asystemof2drstord

4、erODEsforpositionr2Rdandvelocity_r2Rd.Theseareequationsofmotioninphasespace(thespaceRdRdwhere(r;r_)arecoordinates).ThenumberofdegreesoffreedomofthemechanicalsystemisN=d.Example(harmonicoscillator).Setd=1,F=kr=)r•=kr=m[drawapicture].dr=_rdtd_rk=r:dtmpThesolut

5、ionforinitialvaluesr(0)=r0,_r(0)=0isr(t)=r0costk=m.p[Characteristicequationz2+k=m=0=)z=ik=m=i!=)1;2hr(t)=acos!ht+bsin!ht.]LagrangianmechanicsForanunconstrainedsystemofnbodies:N=dn(2NODEs).Underholo-nomicconstraintsfj(r1;r2;:::;rn;t)=0,j=1;:::;kwecandenegener-al

6、izedcoordinatesqi,i=1;:::;N,whereN=dnk,usingtransformationequationsr1=r1(q1;q2;:::;qN;t)...(1)rn=rn(q1;q2;:::;qN;t):PnDenition.TheLagrangian(function)isL=TV,whereT=i=1mi(_ri
r_i)=2isthekineticenergy,andV=V(r1;r2;:::;rn;t)isthepotentialenergy.2Throughthetransfor

7、mationequations(1),wehaveL=L(q;q;t_).TheHamilton'sprincipleZt2I=0;I=L(q;q;t_)dtt1isavariationalequationforndingapathq(t)fromt1tot2forwhichthelineintegralI(action)isstationary[drawapicture].Solution[e.g.,Goldstein]yieldsthe(Euler-)Lagrangeequationsofmotion:d@L@L

8、=0;i=1;:::;N:dt@q_i@qi[Forallpossiblepaths,thesystemtakestheonerequiringtheleastaction.]Example(harmonicoscillator).L=TV=mr_2=2kr2=2,d@L@Lk=mr;•=kr=)