QMGreensite_merged

10 CHAPTER1. THECLASSICALSTATE

sothatF=macanbeinterpretedasaconditionthatacertainfunctionofxnshould
bestationary.Letusthereforeintroduceaveryimportantexpression,crucialinboth
classicalandquantumphysics, whichisknownas the”action”ofthe trajectory.
Theactionisafunctionwhichdependsonallthepoints{xn}, n= 0 , 1 ,...,N ofthe
trajectory,andinthiscaseitis

S[{xi}]≡

N∑− 1

n=0

[

1

2

m

(xn+1−xn)^2

!

−!V(xn)

]

(1.9)

ThenNewton’sLawF=macanberestatedastheconditionthattheactionfunc-
tionalS[{xi}]isstationarywithrespecttovariationofanyofthexi(exceptforthe
endpointsx 0 andxN,whichareheldfixed).Inotherwords

d

dxk

S[{xi}] =

d

dxk

N∑− 1

n=0

[

1

2

m

(xn+1−xn)^2

!

−!V(xn)

]

=

d

dxk

{

1

2

m

(xk+1−xk)^2

!

+

1

2

m

(xk−xk− 1 )^2

!

−!V(xk)

}

= !{−ma(tk)+F(tk)}

= 0 for k= 1 , 2 ,...,N− 1 (1.10)

ThissetofconditionsisknownasthePrincipleofLeastAction.Itistheprinciple
that theaction S isstationaryat any trajectory {xn} satisfyingthe equationsof
motionF=ma,eq. (1.7),ateverytime{tn}.
Theprocedureforsolvingforthetrajectoryofabaseballbycomputeristopro-
gramthecomputertofindthesetofpoints{xn}whichminimizesthequantity

Q=

∑

k

(

∂S

∂xk

) 2

(1.11)

TheminimumisobtainedatQ=0,whereSisstationary.Thissetofpoints,joined
bystraight-linesegments,givesustheapproximatetrajectoryofthebaseball.

Problem:Doitonacomputerbybothmethods.