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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "rotspr.ms   " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Two equal masses a
re connected by a spring, and are rotating about their  CM. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 76 "rotating mass-s
pring system in Lagrangian formalism, patterned after proj.ms" }}
{PARA 0 "" 0 "" {TEXT -1 62 "Maple seems to need a lot of shifting aro
und in the variables." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;with(plo
ts):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "alias(r=r(t),phi=ph
i(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "x1:=r/2*cos(phi):
 y1:=r/2*sin(phi): x2:=-x1:y2:=-y1:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 72 "KE:=simplify(m/2*(diff(x1,t)^2+diff(y1,t)^2+diff(x2,t
)^2+diff(y2,t)^2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#KEG,&*(%\"mG
\"\"\"%\"rG\"\"#-%%diffG6$%$phiG%\"tGF*#F(\"\"%*&F'F(-F,6$F)F/F*F0" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "PE:=k/2*(r-r0)^2;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#PEG,$*&%\"kG\"\"\",&%\"rGF(%#r0G!\"\"\"\"
##F(F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "L:=KE-PE;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG,(*(%\"mG\"\"\"%\"rG\"\"#-%%diff
G6$%$phiG%\"tGF*#F(\"\"%*&F'F(-F,6$F)F/F*F0*&%\"kGF(,&F)F(%#r0G!\"\"F*
#F9F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "valuesin:=\{diff(r
,t)=rdot,diff(phi,t)=phidot,r=ra,phi=phia\};" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%)valuesinG<&/-%%diffG6$%\"rG%\"tG%%rdotG/-F(6$%$phiGF
+%'phidotG/F*%#raG/F0%%phiaG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 49 "Lsvel:=subs(valuesin,L); # L with dummy variables" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%&LsvelG,(*(%\"mG\"\"\"%#raG\"\"#%'phidotGF*#F(
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "pr:=diff(Lsvel,rdot); pphi:=diff(Ls
vel,phidot); # generalized momenta w/ symbols (pj = dLdqjdot)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#prG,$*&%\"mG\"\"\"%%rdotGF(#F(\"\"#
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pphiG,$*(%\"mG\"\"\"%#raG\"\"#%
'phidotGF(#F(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "dLdr:=di
ff(Lsvel,ra);dLdphi:=diff(Lsvel,phia); #  dL/dqj" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%dLdrG,&*(%\"mG\"\"\"%#raGF(%'phidotG\"\"##F(F+*&%\"k
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\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "valuesback:=\{rdot
=diff(r,t),phidot=diff(phi,t),ra=r,phia=phi\};" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%+valuesbackG<&/%%rdotG-%%diffG6$%\"rG%\"tG/%'phidotG-
F)6$%$phiGF,/%#raGF+/%%phiaGF1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 29 "pra:=    subs(valuesback,pr);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%$praG,$*&%\"mG\"\"\"-%%diffG6$%\"rG%\"tGF(#F(\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "pphia:=subs(valuesback,pphi); # gen
eralized p's 'live'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&pphiaG,$*(%
\"mG\"\"\"%\"rG\"\"#-%%diffG6$%$phiG%\"tGF(#F(F*" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 29 "dLdra:=subs(valuesback,dLdr);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%&dLdraG,&*(%\"mG\"\"\"%\"rGF(-%%diffG6$%$phiG%\"
tG\"\"##F(F/*&%\"kGF(,&F)F(%#r0G!\"\"F(F5" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 33 "dLdphia:=subs(valuesback,dLdphi);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%(dLdphiaG\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 51 "eqr:= diff(pra,t)-dLdra = 0; # lagrange's eqn in r." 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eqrG/,(*&%\"mG\"\"\"-%%diffG6$-F+
6$%\"rG%\"tGF0F)#F)\"\"#*(F(F)F/F)-F+6$%$phiGF0F2#!\"\"F2*&%\"kGF),&F/
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{XPPMATH 20 "6#>%&eqphiG/,&**%\"mG\"\"\"%\"rGF)-%%diffG6$%$phiG%\"tGF)
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" 0 "" {MPLTEXT 1 0 49 "eqra:=    subs(r=a(t),phi=b(t),m=1,r0=1,k=2,eq
r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqraG/,*-%%diffG6$-F(6$-%\"a
G6#%\"tGF/F/#\"\"\"\"\"#*&F,F1-F(6$-%\"bGF.F/F2#!\"\"F2F,F2!\"#F1\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "eqphia:=subs(r=a(t),phi
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" {XPPMATH 20 "6#>%'eqphiaG/,&*(-%\"aG6#%\"tG\"\"\"-%%diffG6$-%\"bGF*F
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0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "The square brackets
 are important in the following dsolve command to keep the solutions i
n correct order" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 84 "sol:=dsolve(\{eqra,a(0)=3/2,D(a)(0)=0,eqphia,b(0)=0,D
(b)(0)=10\},[a(t),b(t)],numeric);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>
%$solG:6#%(rkf45_xG6'%\"iG%(rkf45_sG%)outpointG%#r1G%#r2G6#%aoCopyrigh
t~(c)~1993~by~the~University~of~Waterloo.~All~rights~reserved.G6\"C&>8
&-%&evalfG6#9$@$52-%$absG6#,$F3!\"\"-F<6#,&&%,loc_controlG6#\"\"#\"\"
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol(1);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7'/%\"tG\"\"\"/-%\"aG6#F%$\"1jCl;+4n!)!#:/-%%diffG6$F(F
%$!1Z-3z)f[H%F-/-%\"bGF*$\"1T'y<,\"[*e\"F-/-F06$F5F%$\"1\\G)*\\HRdM!#;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "rp:= z -> rhs(sol(z)[2]
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pG:6#%\"zG6\"6$%)operatorG%&arrowGF(-%$rhsG6#&-%$solG6#9$6#\"\"#F(F(" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%phipG:6#%\"zG6\"6$%)operatorG%&ar
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{MPLTEXT 1 0 42 "xp:=z->rhs(sol(z)[2])*cos(rhs(sol(z)[4]));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#xpG:6#%\"zG6\"6$%)operatorG%&arrowGF(*&-%
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "yp:=z->rhs(sol(z)[2])*sin(rh
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)operatorG%&arrowGF(*&-%$rhsG6#&-%$solG6#9$6#\"\"#\"\"\"-%$sinG6#-F.6#
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=plot(phip,0..3,color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
50 "display(\{h,j\},title=` r(blue) and phi(red) vs t`);" }}{PARA 13 "
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