1.1. BASEBALL,F=MA,ANDTHEPRINCIPLEOFLEASTACTION 9
andintegralsbysums
∫t 0 +∆t
t 0
dtf(t)→
N∑− 1
n=0
!f(tn) (1.5)
wheref(t)isanyfunctionoftime. Asweknowfromelementarycalculus,theright
handsideof(1.4)and(1.5)equalsthelefthandsideinthelimitthat!→0,which
isalsoknownasthecontinuumlimit.
Wecannowapproximatethelawsofmotion,byreplacingtime-derivativesin(1.2)
bythecorrespondingfinitedifferences,andfind
xn+1 = xn+
(p
n
m
)
!
pn+1 = pn−
(
dV(xn)
dxn
)
! (1.6)
Theseareiterative equations. Givenpositionxandmomentumpattimet= tn,
wecan use(1.6)tofindthepositionandmomentumattimet=tn+1. Thefinite
differenceapproximationofcourseintroducesaslighterror;xn+1andpn+1,computed
fromxnandpnby(1.6)willdifferfromtheirexactvaluesbyanerrorof order!^2.
Thiserrorcanbemadenegligiblebytaking!sufficientlysmall.
Itisthenpossibleusethecomputertofindanapproximationtothetrajectoryin
oneoftwoways: (i)the”hit-or-miss”method;and(ii)themethodofleastaction.
- TheHit-or-MissMethod
Theequationsofmotion(1.2)require asinputbothaninitialposition,inthis
casex 0 =Xin,andaninitialmomentump 0 whichissofarunspecified.Themethod
istomakeaguessfortheinitialmomentump 0 =P 0 ,andthenuse(1.2)tosolvefor
x 1 ,p 1 , x 2 ,p 2 ,andsoon,untilxN,pN. IfxN ≈Xf, thenstop;the set{xn}isthe
(approximate)trajectory. Ifnot,makeadifferentguessp 0 =P 0 ′,andsolveagainfor
{xn,pn}. Bytrialanderror,onecaneventuallyconvergeonaninitialchoiceforp 0
suchthatxN≈Xf. Forthatchoiceofinitialmomentum,thecorrespondingsetof
points{xn},connectedbystraight-linesegments,givestheapproximatetrajectoryof
thebaseball. ThisprocessisillustratedinFig. [1.2].
- TheMethodofLeastAction
Letsreturntothe2nd-orderformofNewton’sLaws,writtenineq. (1.1). Again
using(1.4)toreplacederivativesbyfinitedifferences,theequationF=maateach
timetnbecomes
m
!
{x
n+1−xn
!
−
xn−xn− 1
!
}
=−
dV(xn)
dxn
(1.7)
Theequationshavetobesolvedforn= 1 , 2 ,...,N−1,withx 0 =XinandxN=Xf
keptfixed. Nownoticethateq. (1.7)canbewrittenasatotalderivative
d
dxn
{
1
2
m
(xn+1−xn)^2
!
+
1
2
m
(xn−xn− 1 )^2
!
−!V(xn)
}
= 0 (1.8)