Lagrangian Mechanics, Dynamics & Control-Andrew - D. Lewis.pdf

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Math439CourseNotes
LagrangianMechanics,Dynamics,and
Control
AndrewD.Lewis
January–April2003
Thisversion:03/04/2003
ii
Thisversion:03/04/2003
Preface
ThesenotesdealprimarilywiththesubjectofLagrangianmechanics.Mattersrelatedtome-
chanicsarethedynamicsandcontrolofmechanicalsystems.WhiledynamicsofLagrangian
systemsisagenerallywell-foundedfield,controlforLagrangiansystemshaslessofahistory.
Inconsequence,thecontroltheorywediscusshereisquiteelementary,anddoesnotreally
touchuponsomeofthereallychallengingaspectsofthesubject.However,itishopedthat
itwillservetogiveaflavourofthesubjectsothatpeoplecanseeiftheareaisonewhich
they’dliketopursue.
OurpresentationbeginsinChapter 1 withaverygeneralaxiomatictreatmentofbasic
Newtonianmechanics.Inthischapterwewillarriveatsomeconclusionsyoumayalready
knowaboutfromyourpreviousexperience,butwewillalsoverylikelytouchuponsome
thingswhichyouhadnotpreviouslydealtwith,andcertainlythepresentationismore
generalandabstractthaninafirst-timedynamicscourse.Whilenoneofthematerialin
thischapteristechnicallyhard,theabstractionmaybeo-puttingtosome.Thehope,
however,isthatattheendoftheday,thegeneralitywillbringintofocusanddemystify
somebasicfactsaboutthedynamicsofparticlesandrigidbodies.Asfarasweknow,thisis
thefirstthoroughlyGalileantreatmentofrigidbodydynamics,althoughGalileanparticle
mechanicsiswell-understood.
LagrangianmechanicsisintroducedinChapter 2 .Wheninstigatingatreatmentof
Lagrangianmechanicsatanotquiteintroductorylevel,onehasadicultchoicetomake;
doesoneusedierentiablemanifoldsornot?Thechoicemadehererunsdownthemiddle
oftheusual,“No,itisfartoomuchmachinery,”and,“Yes,theunityofthedierential
geometricapproachisexquisite.”Thebasicconceptsassociatedwithdierentialgeometry
areintroducedinaratherpragmaticmanner.Theapproachwouldnotbeonerecommended
inacourseonthesubject,buthereservestomotivatetheneedforusingthegenerality,
whileprovidingsomeideaoftheconceptsinvolved.Fortunately,atthislevel,notoverly
manyconceptsareneeded;mainlythenotionofacoordinatechart,thenotionofavector
field,andthenotionofaone-form.Afterthenecessarydierentialgeometricintroductions
aremade,itisveryeasytotalkaboutbasicmechanics.Indeed,itispossiblethatthe
extratimeneededtounderstandthedierentialgeometryismorethanmadeupforwhen
onegetstolookingatthebasicconceptsofLagrangianmechanics.Alloftheprincipal
playersinLagrangianmechanicsaresimpledierentialgeometricobjects.Specialattention
isgiventothatclassofLagrangiansystemsreferredtoas“simple.”Thesesystemsarethe
onesmostcommonlyencounteredinphysicalapplications,andsoaredeservingofspecial
treatment.What’smore,theypossessanenormousamountofstructure,althoughthisis
barelytoucheduponhere.AlsoinChapter 2 wetalkaboutforcesandconstraints.Totalk
aboutcontrolforLagrangiansystems,wemusthaveathandthenotionofaforce.Wegive
specialattentiontothenotionofadissipativeforce,asthisisoftenthepredominanteect
whichisunmodelledinapurelyLagrangiansystem.Constraintsarealsoprevalentinmany
applicationareas,andsodemandattention.Unfortunately,thehandlingofconstraintsin
theliteratureisoftenexcessivelycomplicated.Wetrytomakethingsassimpleaspossible,
astheideasindeedarenotallthatcomplicated.Whilewedonotintendthesenotesto
beadetaileddescriptionofHamiltonianmechanics,wedobrieflydiscussthelinkbetween
iv
LagrangianHamiltonianmechanicsinSection 2.9 .ThefinaltopicofdiscussioninChapter 2
isthematterofsymmetries.WegiveaNoetheriantreatment.
OnceoneusesthematerialofChapter 2 toobtainequationsofmotion,onewouldliketo
beabletosaysomethingabouthowsolutionstotheequationsbehave.Thisisthesubject
ofChapter 3 .AfterdiscussingthematterofexistenceofsolutionstotheEuler-Lagrange
equations(amatterwhichdeservessomediscussion),wetalkaboutthesimplestpartof
Lagrangiandynamics,dynamicsnearequilibria.ThenotionofalinearLagrangiansystem
andalinearisationofanonlinearsystemarepresented,andthestabilitypropertiesoflinear
Lagrangiansystemsareexplored.Thebehaviourisnongeneric,andsodeservesatreatment
distinctfromthatofgenerallinearsystems.Whenoneunderstandslinearsystems,itis
thenpossibletodiscussstabilityfornonlinearequilibria.Thesubtlerelationshipbetween
thestabilityofthelinearisationandthestabilityofthenonlinearsystemisthetopicof
Section 3.2 .WhileageneraldiscussionthedynamicsofLagrangiansystemswithforcesis
notrealistic,theimportantclassofsystemswithdissipativeforcesadmitsausefuldiscussion;
itisgiveninSection 3.5 .Thedynamicsofarigidbodyissingledoutfordetailedattention
inSection 3.6 .Generalremarksaboutsimplemechanicalsystemswithnopotentialenergy
arealsogiven.Thesesystemsareimportantastheyareextremelystructure,yetalsovery
challenging.Verylittleisreallyknownaboutthedynamicsofsystemswithconstraints.In
Section 3.8 wemakeafewsimpleremarksonsuchsystems.
InChapter 4 wedeliverourabbreviateddiscussionofcontroltheoryinaLagrangian
setting.Aftersomegeneralities,wetalkabout“roboticcontrolsystems,”ageneralisation
ofthekindofsystemonemightfindonashopfloor,doingsimpletasks.Forsystems
ofthistype,intuitivecontrolispossible,sincealldegreesoffreedomareactuated.For
underactuatedsystems,afirststeptowardscontrolistolookatequilibriumpointsand
linearise.InSection 4.4 welookatthespecialcontrolstructureoflinearisedLagrangian
systems,payingspecialattentiontothecontrollabilityofthelinearisation.Forsystems
wherelinearisationsfailtocapturethesalientfeaturesofthecontrolsystem,oneisforced
tolookatnonlinearcontrol.Thisisquitechallenging,andwegiveaterseintroduction,and
pointerstotheliterature,inSection 4.5 .
Pleasepassoncommentsanderrors,nomatterhowtrivial.Thankyou.
AndrewD.Lewis
andrew@mast.queensu.ca
420Jeery
x32395
Thisversion:03/04/2003
TableofContents
1NewtonianmechanicsinGalileanspacetimes 1
1.1Galileanspacetime................................ 1
1.1.1Anespaces............................... 1
1.1.2Timeanddistance............................ 5
1.1.3Observers................................. 9
1.1.4Planarandlinearspacetimes...................... 10
1.2GalileanmappingsandtheGalileantransformationgroup.......... 12
1.2.1Galileanmappings............................ 12
1.2.2TheGalileantransformationgroup................... 13
1.2.3SubgroupsoftheGalileantransformationgroup............ 15
1.2.4Coordinatesystems........................... 17
1.2.5Coordinatesystemsandobservers................... 19
1.3Particlemechanics................................ 21
1.3.1Worldlines................................ 21
1.3.2InterpretationofNewton’sLawsforparticlemotion......... 23
1.4RigidmotionsinGalileanspacetimes...................... 25
1.4.1Isometries................................. 25
1.4.2Rigidmotions.............................. 27
1.4.3Rigidmotionsandrelativemotion................... 30
1.4.4Spatialvelocities............................. 30
1.4.5Bodyvelocities.............................. 33
1.4.6Planarrigidmotions........................... 36
1.5Rigidbodies................................... 37
1.5.1Definitions................................ 37
1.5.2Theinertiatensor............................ 40
1.5.3Eigenvaluesoftheinertiatensor.................... 41
1.5.4Examplesofinertiatensors....................... 45
1.6Dynamicsofrigidbodies............................ 46
1.6.1Spatialmomenta............................. 47
1.6.2Bodymomenta.............................. 49
1.6.3Conservationlaws............................ 50
1.6.4TheEulerequationsinGalileanspacetimes.............. 52
1.6.5SolutionsoftheGalileanEulerequations............... 55
1.7Forcesonrigidbodies.............................. 56
1.8ThestatusoftheNewtonianworldview.................... 57
2Lagrangianmechanics 61
2.1Configurationspacesandcoordinates...................... 61
2.1.1Configurationspaces........................... 62
2.1.2Coordinates................................ 64
2.1.3Functionsandcurves........................... 69
2.2Vectorfields,one-forms,andRiemannianmetrics............... 69

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