Goldstein+Classical+Mechanics+Notes

GoldsteinClassicalMechanicsNotesMichaelGoodMay30,20041Chapter1:ElementaryPrinciples1.1MechanicsofaSingleParticleClassicalmechanicsincorporatesspecialrelativity.‘Classical’referstothecon-tradistinctionto‘quantum’mechanics.Velocity:v=drdt.Linearmomentum:p=mv.Force:F=dpdt.Inmostcases,massisconstantandforceissimplified:F=ddt(mv)=mdvdt=ma.Acceleration:a=d2rdt2.Newton’ssecondlawofmotionholdsinareferenceframethatisinertialorGalilean.AngularMomentum:L=r×p.Torque:T=r×F.Torqueisthetimederivativeofangularmomentum:1T=dLdt.Work:W12=Z21F·dr.Inmostcases,massisconstantandworksimplifiesto:W12=mZ21dvdt·vdt=mZ21v·dvdtdt=mZ21v·dvW12=m2(v22−v21)=T2−T1KineticEnergy:T=mv22Theworkisthechangeinkineticenergy.Aforceisconsideredconservativeiftheworkisthesameforanyphysicallypossiblepath.IndependenceofW12ontheparticularpathimpliesthattheworkdonearoundaclosedciruitiszero:IF·dr=0Iffrictionispresent,asystemisnon-conservative.PotentialEnergy:F=−∇V(r).Thecapacitytodoworkthatabodyorsystemhasbyvitureofispositioniscalleditspotentialenergy.Vaboveisthepotentialenergy.Toexpressworkinawaythatisindependentofthepathtaken,achangeinaquantitythatdependsononlytheendpointsisneeded.Thisquantityispotentialenergy.WorkisnowV1−V2.Thechangeis-V.EnergyConservationTheoremforaParticle:Ifforcesactingonaparticleareconservative,thenthetotalenergyoftheparticle,T+V,isconserved.TheConservationTheoremfortheLinearMomentumofaParticlestatesthatlinearmomentum,p,isconservedifthetotalforceF,iszero.TheConservationTheoremfortheAngularMomentumofaParticlestatesthatangularmomentum,L,isconservedifthetotaltorqueT,iszero.21.2MechanicsofManyParticlesNewton’sthirdlawofmotion,equalandoppositeforces,doesnotholdforallforces.Itiscalledtheweaklawofactionandreaction.Centerofmass:R=PmiriPmi=PmiriM.Centerofmassmovesasifthetotalexternalforcewereactingontheentiremassofthesystemconcentratedatthecenterofmass.InternalforcesthatobeyNewton’sthirdlaw,havenoeffectonthemotionofthecenterofmass.F(e)≡Md2Rdt2=XiF(e)i.Motionofcenterofmassisunaffected.Thisishowrocketsworkinspace.Totallinearmomentum:P=Ximidridt=MdRdt.ConservationTheoremfortheLinearMomentumofaSystemofParticles:Ifthetotalexternalforceiszero,thetotallinearmomentumisconserved.Thestronglawofactionandreactionistheconditionthattheinternalforcesbetweentwoparticles,inadditiontobeingequalandopposite,alsoliealongthelinejoiningtheparticles.Thenthetimederivativeofangularmomentumisthetotalexternaltorque:dLdt=N(e).Torqueisalsocalledthemomentoftheexternalforceaboutthegivenpoint.ConservationTheoremforTotalAngularMomentum:Lisconstantintimeiftheappliedtorqueiszero.LinearMomentumConservationrequiresweaklawofactionandreaction.AngularMomentumConservationrequiresstronglawofactionandreaction.TotalAngularMomentum:L=Xiri×pi=R×Mv+Xir0i×p0i.3TotalangularmomentumaboutapointOistheangularmomentumofmo-tionconcentratedatthecenterofmass,plustheangularmomentumofmotionaboutthecenterofmass.Ifthecenterofmassisatrestwrttheoriginthentheangularmomentumisindependentofthepointofreference.TotalWork:W12=T2−T1whereTisthetotalkineticenergyofthesystem:T=12Pimiv2i.Totalkineticenergy:T=12Ximiv2i=12Mv2+12Ximiv02i.Kineticenergy,likeangularmomentum,hastwoparts:theK.E.obtainedifallthemasswereconcentratedatthecenterofmass,plustheK.E.ofmotionaboutthecenterofmass.Totalpotentialenergy:V=XiVi+12Xi,ji6=jVij.IftheexternalandinternalforcesarebothderivablefrompotentialsitispossibletodefineatotalpotentialenergysuchthatthetotalenergyT+Visconserved.Thetermontherightiscalledtheinternalpotentialenergy.Forrigidbodiestheinternalpotentialenergywillbeconstant.Forarigidbodytheinternalforcesdonoworkandtheinternalpotentialenergyremainsconstant.1.3Constraints•holonomicconstraints:thinkrigidbody,thinkf(r1,r2,r3,...,t)=0,thinkaparticleconstrainedtomovealonganycurveoronagivensurface.•nonholonomicconstraints:thinkwallsofagascontainer,thinkparticleplacedonsurfaceofaspherebecauseitwilleventuallyslidedownpartofthewaybutwillfalloff,notmovingalongthecurveofthesphere.1.rheonomousconstraints:timeisanexplicitvariable...example:beadonmovingwire2.scleronomousconstraints:equationsofcontraintareNOTexplicitlyde-pendentontime...example:beadonrigidcurvedwirefixedinspaceDifficultieswithconstraints:41.Equationsofmotionarenotallindependent,becausecoordinatesarenolongerallindependent2.Forcesarenotknownbeforehand,andmustbeobtainedfromsolution.Forholonomicconstraintsintroducegeneralizedcoordinates.Degreesoffreedomarereduced.Useindependentvariables,eliminatedependentcoordi-nates.Thisiscalledatransformation,goingfromonesetofdependentvariablestoanothersetofindependentvariables.Generalizedcoordinatesareworthwhileinproblemsevenwithoutconstraints.Examplesofgeneralizedcoordinates:1.Twoanglesexpressingpositiononthespherethataparticleisconstrainedtomoveon.2.Twoanglesforadoublependulummovinginaplane.3.AmplitudesinaFourierexpansionofrj.4.Quanitieswithwithdimensionsofenergyorangularmomentum.Fornonholonomicconstraintsequationsexpressingtheconstraintcannotbeusedtoeliminatethedependentcoordinates.NonholonomicconstraintsareHARDERTOSOLVE.1.4D’Alembert’sPrincipleandLagr