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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 47 "Trajectories in an Attr
active Central Potential" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 
"" 0 "" {TEXT -1 48 "by Jason Janesky, Corinne Manogue & Kerry Browne
" }}{PARA 258 "" 0 "" {TEXT -1 30 "Copyright 1998 Corinne Manogue" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 53 "This wo
rksheet calculates the trajectories of a mass " }{XPPEDIT 18 0 "mu" "6
#%#muG" }{TEXT -1 52 " under the influence of a CENTRAL force numerica
lly." }}{PARA 258 "" 0 "" {TEXT -1 103 "The main example of a central \+
force which we've been studying in class is an attractive inverse squa
re:" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "f(r)=-k/r^2" "6#/-%\"fG6#%\"rG
,$*&%\"kG\"\"\"*$F'\"\"#!\"\"F." }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 34 "restart:with(DEtools):with
(plots):" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }{TEXT 258 18 "Set
up the Problem\n" }{TEXT -1 43 "Choose values for the parameters below
:    " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 
60 "The masses of each of the particles and the reduced mass are" }}
{PARA 258 "> " 0 "" {MPLTEXT 1 0 33 "m1:=2; m2:=2; mu:= m1*m2/(m1+m2);
" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 63 "\nThe angular momentum, tot
al energy, and coupling constant are:" }}}{EXCHG {PARA 258 "> " 0 "" 
{MPLTEXT 1 0 21 "el:=1; E:= -.2; k:=1;" }}}{EXCHG {PARA 258 "" 0 "" 
{TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 50 "The central potential a
nd effective potential are:" }}}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 
0 8 "U:=-k/r;" }}}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 24 "Veff:=U+
el^2/(2*mu*r^2);" }{TEXT -1 0 "" }}}{EXCHG {PARA 258 "> " 0 "" 
{MPLTEXT 1 0 51 "plot([Veff,E],r=0..4,-2..3,\ncolor=[magenta,green]);
" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 77 "\nWhat type of orbit would \+
you guess for these parameters?  What do you think " }{XPPEDIT 18 0 "r
[min]" "6#&%\"rG6#%$minG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "r[max]" 
"6#&%\"rG6#%$maxG" }{TEXT -1 9 " will be?" }}}{EXCHG {PARA 258 "" 0 "
" {TEXT 257 21 "Calculating the Orbit" }{TEXT -1 37 "  \nFirst find th
e force in terms of  " }{XPPEDIT 18 0 "r(t)" "6#-%\"rG6#%\"tG" }{TEXT 
-1 1 ":" }}}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 27 "f:=subs(r=r(t)
,-diff(U,r));" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 37 "Next find the \+
differential equations:" }}}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 
79 "sys:=diff(r(t),t,t)=(1/mu)*f+el^2/(mu^2*r(t)^3),\ndiff(phi(t),t)=e
l/(mu*r(t)^2);" }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 178 "\nThe next s
et of commands tell Maple to solve the system of differential equation
s numerically and plot the result.  The Maple details are unimportant,
 unless you are interested." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "fcns
:=\{r(t),phi(t)\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "rmsol:=solve(
E=U+el^2/(2*mu*r^2), r):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "rm:=min
(rmsol[1],rmsol[2]):" }}{PARA 258 "> " 0 "" {MPLTEXT 1 0 84 "p:=dsolve
(\{sys,r(0)=rm,D(r)(0)=0,phi(0)=0\},\nfcns,type=numeric,output=listpro
cedure):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "p:=dsolve(\{sys,r(0)=rm
,D(r)(0)=0,phi(0)=0\},\nfcns,type=numeric):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 "Pphi:=proc(tt) op(2,p(tt)[2]) end:\nPr:=proc(tt) op(2
,p(tt)[3]) end:" }}{PARA 258 "> " 0 "" {MPLTEXT 1 0 92 "orb:=odeplot(p
,\n[(r(t))*cos(phi(t)),(r(t))*sin(phi(t))],\n-12..12,numpoints = 200,c
olor=red):" }}}{EXCHG {PARA 258 "> " 0 "" {MPLTEXT 1 0 33 "display(orb
,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 35 "\nBonus:  Try some other pote
ntials:" }}{PARA 258 "" 0 "" {TEXT -1 24 "1)  Harmonic Oscillator " }
{XPPEDIT 18 0 "U(r)=(k/2)*r^2" "6#/-%\"UG6#%\"rG*(%\"kG\"\"\"\"\"#!\"
\"F'F+" }}{PARA 258 "" 0 "" {TEXT -1 58 "2)  First order General Relat
ivistic correction to Newton " }{XPPEDIT 18 0 "U(r)=-k/r + delta/r^3" 
"6#/-%\"UG6#%\"rG,&*&%\"kG\"\"\"F'!\"\"F,*&%&deltaGF+*$F'\"\"$F,F+" }
{TEXT -1 7 ",  for " }{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 12 
" very small." }}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }