1.3. CLASSICALMECHANICSINANUTSHELL 15
1.3 Classical Mechanicsin a Nutshell
AllthemachineryoftheLeastActionPrinciple,theLagrangianFunction,andHamil-
ton’sequations,isoverkillinthecaseofabaseball.Inthatcase,weknewtheequation
ofmotionfromthebeginning. Butformoreinvolveddynamicalsystems,involving,
say,wheels,springs,levers,andpendulums,allcoupledtogetherinsomecomplicated
way,theequationsofmotionareoftenfarfromobvious,andwhatisneededissome
systematicwaytoderivethem.
Foranymechanicalsystem,thegeneralizedcoordinates{qi}areasetofvari-
ablesneededtodescribetheconfigurationofthesystematagiventime.Thesecould
beasetofcartesian coordinatesofanumberofdifferentparticles, ortheangular
displacementofapendulum,orthedisplacementofaspringfromequilibrium,orall
oftheabove. Thedynamicsofthesystem,intermsofthesecoordinates,isgivenbya
LagrangianfunctionL,whichdependsonthegeneralizedcoordinates{qi}andtheir
firsttime-derivatives{q ̇i}.Normally,innon-relativisticmechanics,wefirstspecify
- TheLagrangian
L[{qiq ̇i}]=KineticEnergy−PotentialEnergy (1.35)Onethendefines
- TheAction
S=
∫
dtL[{qi},{q ̇i}] (1.36)FromtheLeastActionPrinciple,followingamethodsimilartotheoneweusedfor
thebaseball(seeProblem4),wederive
- TheEuler-LagrangeEquations
∂L
∂qi−
d
dt∂L
∂q ̇i= 0 (1.37)
Thesearethe2nd-orderequationsofmotion. Togoto1st-orderform,firstdefine
- TheGeneralizedMomenta
pi≡∂L
∂q ̇i(1.38)
whichcanbeinvertedtogivethetime-derivativesq ̇iofthegeneralizedcoordinates
intermsofthegeneralizedcoordinatesandmomenta
q ̇i=q ̇i[{qn,pn}] (1.39)Viewingq ̇asafunctionofpandq,onethendefines