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18 CHAPTER1. THECLASSICALSTATE

6.Hamilton’sequations

θ ̇ = ∂H

∂p

=

p

ml^2

p ̇ = −

∂H

∂θ

=−mglsin(θ) (1.51)

whichareeasilyseentobeequivalenttotheEuler-Lagrangeequations.

Problem- Twopointlikeparticlesmovinginthreedimensionshavemassesm 1 and
m 2 respectively,andinteractviaapotentialV(x% 1 −x% 2 ). FindHamilton’sequations
ofmotionfortheparticles.

Problem - Suppose,instead of arigid rod,the massof the planependulum is
connectedtopointPbyaweightlessspring. Thepotentialenergyofthespringis
1
2 k(l−l^0 )

(^2) ,wherelisthelengthofthespring,andl 0 isitslengthwhennotdisplaced
byanexternalforce.Choosinglandθasthegeneralizedcoordinates,findHamilton’s
equations.
1.4 The Classical State
Predictionisratherimportantinphysics,sincetheonlyreliabletestofascientific
theoryistheability,giventhestateofaffairsatpresent,topredictthefuture.
Statedratherabstractly,theprocessofpredictionworksasfollows: By aslight
disturbanceknownasameasurement,anobjectisassignedamathematicalrepre-
sentationwhichwewillcallitsphysicalstate.Thelawsofmotionaremathematical
rulesbywhich,givenaphysicalstateataparticulartime,onecandeducethephys-
icalstateoftheobjectatsomelatertime. Thelaterphysicalstateistheprediction,
whichcanbecheckedbyasubsequentmeasurementoftheobject(seeFig.[1.5]).
Fromthediscussionsofar,itseasytoseethatwhatismeantinclassicalphysics
bythe”physicalstate”ofasystemissimplyitssetofgeneralizedcoordinatesand
thegeneralizedmomenta{qa,pa}.Thesearesupposedtobeobtained,atsometime
t 0 ,bythemeasurementprocess. Giventhephysicalstateatsometimet,thestate
att+!isobtainedbytherule:
qa(t+!)=qa(t)+!
(
∂H
∂pa
)
t
pa(t+!)=pa(t)−!
(
∂H
∂qa
)
t

(1.52)

Inthisway,thephysicalstateatanylatertimecanbeobtained(inprinciple)toan
arbitrarydegreeofaccuracy,bymakingthetime-step!sufficientlysmall(orelse,if