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1、Math439CourseNotesLagrangianMechanics,Dynamics,andControlAndrewD.LewisJanuary–April2003Thisversion:03/04/2003iiThisversion:03/04/2003PrefaceThesenotesdealprimarilywiththesubjectofLagrangianmechanics.Mattersrelatedtome-chanicsarethedynamicsandcontrolofmechanicalsystems.WhiledynamicsofLagrangiansyst
2、emsisagenerallywell-founded?eld,controlforLagrangiansystemshaslessofahistory.Inconsequence,thecontroltheorywediscusshereisquiteelementary,anddoesnotreallytouchuponsomeofthereallychallengingaspectsofthesubject.However,itishopedthatitwillservetogivea?avourofthesubjectsothatpeoplecanseeiftheareaisone
3、whichthey’dliketopursue.OurpresentationbeginsinChapter1withaverygeneralaxiomatictreatmentofbasicNewtonianmechanics.Inthischapterwewillarriveatsomeconclusionsyoumayalreadyknowaboutfromyourpreviousexperience,butwewillalsoverylikelytouchuponsomethingswhichyouhadnotpreviouslydealtwith,andcertainlythep
4、resentationismoregeneralandabstractthanina?rst-timedynamicscourse.Whilenoneofthematerialinthischapteristechnicallyhard,theabstractionmaybeo?-puttingtosome.Thehope,however,isthatattheendoftheday,thegeneralitywillbringintofocusanddemystifysomebasicfactsaboutthedynamicsofparticlesandrigidbodies.Asfar
5、asweknow,thisisthe?rstthoroughlyGalileantreatmentofrigidbodydynamics,althoughGalileanparticlemechanicsiswell-understood.LagrangianmechanicsisintroducedinChapter2.WheninstigatingatreatmentofLagrangianmechanicsatanotquiteintroductorylevel,onehasadi?cultchoicetomake;doesoneusedi?erentiablemanifoldsor
6、not?Thechoicemadehererunsdownthemiddleoftheusual,“No,itisfartoomuchmachinery,”and,“Yes,theunityofthedi?erentialgeometricapproachisexquisite.”Thebasicconceptsassociatedwithdi?erentialgeometryareintroducedinaratherpragmaticmanner.Theapproachwouldnotbeonerecommendedinacourseonthesubject,buthereserves
7、tomotivatetheneedforusingthegenerality,whileprovidingsomeideaoftheconceptsinvolved.Fortunately,atthislevel,notoverlymanyconceptsareneeded;mainlythenotionofacoordinatechart,thenotionofavector?eld,andthenotionofaon
