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1.3. CLASSICALMECHANICSINANUTSHELL 17

Example: ThePlanePendulum

Ourpendulumisamassmattheendofaweightlessrigidrodoflengthl,which
pivotsinaplanearoundthepointP.The”generalizedcoordinate”,whichspecifies
thepositionofthependulumatanygiventime,istheangleθ(seeFig. [1.4]).

1.Lagrangian

L=

1

2

ml^2 θ ̇^2 −(V 0 −mglcos(θ)) (1.43)

whereV 0 isthegravitationalpotentialattheheightofpointP,whichthependulum
reachesatθ=π/2. SinceV 0 isarbitrary,wewilljustsetittoV 0 =0.

2.TheAction

S=

∫t 1

t 0

dt

[ 1

2

ml^2 θ ̇^2 +mglcos(θ)

]

(1.44)

3.Euler-LagrangeEquations
Wehave

∂L

∂θ

= −mglsin(θ)

∂L

∂θ ̇

= ml^2 θ ̇ (1.45)

andtherefore
ml^2 θ ̈+mglsin(θ)= 0 (1.46)

istheEuler-Lagrangeformoftheequationsofmotion.

4.TheGeneralizedMomentum

p=

∂L

∂θ ̇

=ml^2 θ ̇ (1.47)

5.TheHamiltonian
Insert
θ ̇= p
ml^2

(1.48)

into

H=pθ ̇−

[

1

2

ml^2 θ ̇^2 +mglcos(θ)

]

(1.49)

toget

H=

1

2

p^2

ml^2

−mglcos(θ) (1.50)