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.