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)