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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 15 "Double Pendulum" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "The double pen
dulum is an interesting example in classical mechanics, as it shows wh
at can happen if one simplifies a problem. Grant Fowles and George Cas
siday in " }{TEXT 257 20 "Analytical Mechanics" }{TEXT -1 381 " (6th e
d., Saunders College Publ. 1999) call it also the problem of two rock \+
climbers dangling on a single rope. Many mechanics texts (and also Dif
ferential Equations with Maple V) discuss only the linearized case whe
re one investigates the normal modes. However, the large-amplitude mot
ion is interesting as it includes chaotic behaviour. The problem is de
alt with in Andre Heck: " }{TEXT 278 21 "Introduction to Maple" }
{TEXT -1 239 " (Springer 1993); he calls the problem the two-link robo
t arm. A perfect classroom demonstration can be made of a more sophist
icated version, namely the double compound pendulum for which one need
s a couple of plates and two ball bearings." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 496 "Fowles and Cassiday almost der
ive the full-blown double pendulum (they could have explained the one \+
missing step easily). We pick the simplified choice of parameters (equ
al mass and equal length), and begin with a graph of the potential ene
rgy. The angles that the two massless bars make with the vertical are \+
called theta and phi respectively. The first term below is the potenti
al energy of the first mass, and the second the potential energy for t
he second mass which is attached to the first." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "Vpot:=(theta,phi)->m*g*l*(1
-cos(theta))+m*g*l*(2-cos(theta)-cos(phi));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%VpotGR6$%&thetaG%$phiG6\"6$%)operatorG%&arrowGF),&**
%\"mG\"\"\"%\"gGF0%\"lGF0,&F0F0-%$cosG6#9$!\"\"F0F0**F/\"\"\"F1F:F2F:,
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1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "co
ntourplot(subs(g=10,m=1,l=1,Vpot(t,f)),t=0..2*Pi,f=0..2*Pi,filled=true
,contours=[5,10,15,20,25,30,35,40,45,50,55]);" }}{PARA 13 "" 1 "" 
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joFdc[m7&Fhc[mFfc[mFeb_lFab_l7&Fhc[mFhc[mFab_lF`b_l7&7$Fc[sF`hoF\\d[mF
ha_lF^b_l7&F\\d[mF\\d[mFha_lFaa_l7&7$Fc[sFj[pF\\d[mFaa_lF[a_l7&F_d[m7$
Fj`pFj[pFj`_lF[a_l7&Fad[mFad[m7$Fc[sF^_pF_d[m7&Fad[mFad[mFcd[m7$Fj`pF^
_p7&Fed[mFed[m7$Fc[sF`^pFcd[m7&Fed[mFed[mFgd[m7$Fj`pF`^p7&7$Fc[sF\\drF
[e[mFid[mFgd[m7&F[e[mF[e[mFid[m7$Fj`pF\\dr7&7$Fj_pFc[sF_e[mF^c[m7$Fj`p
Fc[s7&F_e[mF_e[mF^c[m7$Fj_pFj`p7&Fbe[mFbe[mFbc[mF^c[m7&Fbe[mFbe[mFbc[m
7$Fj_pFj_p7&Fee[mFee[mFfc[mFbc[m7&Ffc[mFee[mFf`_lFb`_l7&Ffc[mFfc[mFb`_
lFeb_l7&Fad[mFad[mFe__lFj`_l7&Fad[mFad[mFe__lFa__l7&Fed[mFad[mFa__lF[_
_l7&Fed[m7$Fj_pF^_pF^]_lF[__l7&F]f[mF]f[mFid[mFed[m7&F]f[mF]f[mFid[mFd
b[m7&Fdb[mFdb[mF]e[mFid[m7&Fdb[mFdb[mF]e[mFfb[m7&Fbe[mF_e[mFi^_lFc^_l7
&Fbe[mFbe[mFc^_lF_^_l7&Fee[mFbe[mF_^_lFi]_l7&Fee[mFee[mFi]_lFe]_l7&Fee
[mFee[mFe]_lFf`_l7&F]f[mF]f[mFh\\_lF^]_l7&F]f[mF]f[mFh\\_lFd\\_l7&Fdb[
mF]f[mFd\\_lF^\\_l7&7$Fc[sFc[sF[g[mF`e[mF`c[m7&F`e[mF`e[mF`c[mF^c[m-F`
[`l6&Fb[`lFc[`lF\\drF\\drFd[`l-%+AXESLABELSG6$%\"tG%\"fG" 1 2 0 1 0 2 
9 1 4 2 1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 88 "#plot3d(subs(g=10,m=1,l=1,Vpot(t,f)),t=0..2*Pi,f=0..2
*Pi,axes=boxed,style=patchcontour);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 107 "To derive the kinetic energy expression one starts with the ex
pression that involves the linear velocities " }{TEXT 260 1 "v" }
{TEXT -1 6 "1 and " }{TEXT 261 1 "v" }{TEXT -1 26 "2 for the two masse
s. T = " }{TEXT 262 1 "m" }{TEXT -1 4 "/2 (" }{TEXT 263 1 "v" }{TEXT 
-1 4 "1^2+" }{TEXT 264 1 "v" }{TEXT -1 78 "2^2). It is important to re
alize that the second mass has a velocity given as " }{TEXT 258 1 "v" 
}{TEXT -1 32 "1 + its own rotation about mass " }{TEXT 259 1 "m" }
{TEXT -1 374 "1. The speeds associated with the two rotations are give
n as length times respective angular velocity. The direction of the ro
tations about the respective support points is given by the theta and \+
phi unit vectors. These have Cartesian components [cos(theta), sin(the
ta)] and [cos(phi), sin(phi)] respectively. Thus, in Cartesian coordin
ates in the plane of swinging we have " }{TEXT 265 1 "v" }{TEXT -1 4 "
1 = " }{TEXT 266 1 "l" }{TEXT -1 12 " diff(theta(" }{TEXT 267 1 "t" }
{TEXT -1 2 ")," }{TEXT 268 1 "t" }{TEXT -1 13 ")*[cos(theta(" }{TEXT 
277 1 "t" }{TEXT -1 13 ")),sin(theta(" }{TEXT 276 1 "t" }{TEXT -1 9 ")
)], and " }{TEXT 270 1 "v" }{TEXT -1 3 "2 =" }{TEXT 269 2 " v" }{TEXT 
-1 3 "1+ " }{TEXT 271 1 "l" }{TEXT -1 9 "diff(phi(" }{TEXT 272 1 "t" }
{TEXT -1 2 ")," }{TEXT 273 1 "t" }{TEXT -1 11 ")*[cos(phi(" }{TEXT 
274 1 "t" }{TEXT -1 10 "),sin(phi(" }{TEXT 275 1 "t" }{TEXT -1 35 "))]
. The kinetic energy reduces to:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 140 "Tkin:=m*l^2*(diff(theta(t),t)^2+1/2*diff(phi(t),t)^2
+diff(theta(t),t)*diff(phi(t),t)*(cos(theta(t))*cos(phi(t))+sin(theta(
t))*sin(phi(t))));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%TkinG*(%\"mG
\"\"\")%\"lG\"\"#\"\"\",(*$)-%%diffG6$-%&thetaG6#%\"tGF5F*F+F'*$)-F06$
-%$phiGF4F5F*F+#F'F**(F/F'F8F',&*&-%$cosG6#F2F'-FA6#F:F'F'*&-%$sinGFBF
'-FGFDF'F'F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "In the linear
ized case the entire bracket with the trig functions is replaced by on
e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "We \+
will now carry out the procedure of Euler and Lagrange. The Lagrange f
unction is defined as:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "L
agr:=Tkin-Vpot(theta(t),phi(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
%LagrG,(*(%\"mG\"\"\")%\"lG\"\"#\"\"\",(*$)-%%diffG6$-%&thetaG6#%\"tGF
6F+F,F(*$)-F16$-%$phiGF5F6F+F,#F(F+*(F0F(F9F(,&*&-%$cosG6#F3F(-FB6#F;F
(F(*&-%$sinGFCF(-FHFEF(F(F(F(F(F(**F'F,%\"gGF(F*F(,&F(F(FA!\"\"F(FM**F
'F,FKF,F*F,,(F+F(FAFMFDFMF(FM" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 245 
"To carry out the variation is not trivial in Maple as we have to diff
erentiate with respect to a function. We can help ourselves with subst
itutions (suggested by A. Heck). An alternative is to download the cal
cvar package from the share library." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 147 "It is important to substitute the deri
vatives first; otherwise the dependent variables in the derivatives ar
e renamed, and then are not recognized." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "L1:=subs(diff(theta(t),t)=thdot,theta(t)=th,diff(phi(
t),t)=phdot,phi(t)=ph,Lagr);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L1G
,(*(%\"mG\"\"\")%\"lG\"\"#\"\"\",(*$)%&thdotGF+F,F(*$)%&phdotGF+F,#F(F
+*(F0F(F3F(,&*&-%$cosG6#%#thGF(-F96#%#phGF(F(*&-%$sinGF:F(-FAF=F(F(F(F
(F(F(**F'F,%\"gGF(F*F(,&F(F(F8!\"\"F(FF**F'F,FDF,F*F,,(F+F(F8FFF<FFF(F
F" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 166 "Now we can do partial deriv
atives with respect to theta, or w.r.t. diff(theta(t),t), etc., but we
 cannot differentiate with respect to time without striking disaster.
" }}{PARA 0 "" 0 "" {TEXT -1 25 "So let us proceed slowly:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "dL_dth:=diff(L1,th);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%'dL_dthG,&*,%\"mG\"\"\")%\"lG\"\"#\"\"\"%&thdo
tGF(%&phdotGF(,&*&-%$sinG6#%#thGF(-%$cosG6#%#phGF(!\"\"*&-F6F3F(-F2F7F
(F(F(F(**F'F,%\"gGF(F*F(F1F,!\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "dL_dthdot:=diff(L1,thdot);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%*dL_dthdotG*(%\"mG\"\"\")%\"lG\"\"#\"\"\",&%&thdotGF*
*&%&phdotGF',&*&-%$cosG6#%#thGF'-F36#%#phGF'F'*&-%$sinGF4F'-F;F7F'F'F'
F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "The latter has to be dif
ferentiated with respect to time. We reverse the substitutions (before
 taking the time derivative on the lhs), and collect the expressions:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "EL1:=diff(subs(thdot=d
iff(theta(t),t),th=theta(t),phdot=diff(phi(t),t),ph=phi(t),dL_dthdot),
t)-subs(thdot=diff(theta(t),t),th=theta(t),phdot=diff(phi(t),t),ph=phi
(t),dL_dth)=0;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$EL1G/,(*(%\"mG\"
\"\")%\"lG\"\"#\"\"\",(-%%diffG6$-%&thetaG6#%\"tG-%\"$G6$F5F,F,*&-F06$
-%$phiGF4F6F),&*&-%$cosG6#F2F)-FA6#F<F)F)*&-%$sinGFBF)-FGFDF)F)F)F)*&-
F06$F<F5F-,**(FFF--F06$F2F5F)FCF-!\"\"*(F@F-FHF-FJF)FP*(F@F-FNF-FHF-F)
*(FFF-FCF-FJF-F)F)F)F)F)*,F(F-F*F-FNF-FJF-,&*&FFF-FCF-FP*&F@F-FHF-F)F)
FP**F(F-%\"gGF)F+F)FFF-F,\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 
"We need to get some order into this..." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 90 "EL1:=collect(EL1/m,[diff(theta(t),t$2),diff(phi(t),t$
2),diff(theta(t),t),diff(phi(t),t)]);" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#>%$EL1G/,**&)%\"lG\"\"#\"\"\"-%%diffG6$-%&thetaG6#%\"tG-%\"$G6$F2F*
\"\"\"F**(F(F+,&*&-%$cosG6#F/F6-F;6#-%$phiGF1F6F6*&-%$sinGF<F6-FCF>F6F
6F6-F-6$F?F3F6F6*(F(F+,&*&F:F+FDF+!\"\"*&FBF+F=F+F6F6)-F-6$F?F2F*F+F6*
(%\"gGF6F)F6FBF+F*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "
EL1:=combine(EL1,trig);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EL1G/,**
&)%\"lG\"\"#\"\"\"-%%diffG6$-%&thetaG6#%\"tG-%\"$G6$F2F*\"\"\"F**(F(F+
-%$cosG6#,&F/F6-%$phiGF1!\"\"F6-F-6$F<F3F6F6*(F(F+-%$sinGF:F6)-F-6$F<F
2F*F+F6*(%\"gGF6F)F6-FC6#F/F6F*\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 160 "This is still a formidable equation, but it is readable at lea
st. It is hard to keep the combined trig functions, and to divide out \+
the length at the same time." }}{PARA 0 "" 0 "" {TEXT -1 51 "We repeat
 the tricks to obtain the second equation." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "dL_dph:=diff(L1,ph);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'dL_dphG,&*,%\"mG\"\"\")%\"lG\"\"#\"\"\"%&thdotGF(%&phdotGF(,&
*&-%$cosG6#%#thGF(-%$sinG6#%#phGF(!\"\"*&-F6F3F(-F2F7F(F(F(F(**F'F,%\"
gGF(F*F(F5F,F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dL_dphdot
:=diff(L1,phdot);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*dL_dphdotG*(%
\"mG\"\"\")%\"lG\"\"#\"\"\",&%&phdotGF'*&%&thdotGF',&*&-%$cosG6#%#thGF
'-F36#%#phGF'F'*&-%$sinGF4F'-F;F7F'F'F'F'F'" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 176 "EL2:=diff(subs(thdot=diff(theta(t),t),th=theta(t)
,phdot=diff(phi(t),t),ph=phi(t),dL_dphdot),t)-subs(thdot=diff(theta(t)
,t),th=theta(t),phdot=diff(phi(t),t),ph=phi(t),dL_dph)=0:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "EL2:=collect(EL2/m,[diff(theta(t),t
$2),diff(phi(t),t$2),diff(theta(t),t),diff(phi(t),t)]):" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 23 "EL2:=combine(EL2,trig);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$EL2G/,**()%\"lG\"\"#\"\"\"-%$cosG6#,&-%&thetaG6#%\"t
G\"\"\"-%$phiGF2!\"\"F4-%%diffG6$F0-%\"$G6$F3F*F4F4*&F(F+-F96$F5F;F4F4
*(F(F+-%$sinGF.F4)-F96$F0F3F*F+F7*(%\"gGF4F)F4-FC6#F5F4F4\"\"!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "The big question is whether Maple
's dsolve[numeric] engine can handle this pair of second-order equatio
ns directly." }}{PARA 0 "" 0 "" {TEXT -1 86 "We pick initial condition
s as one pendulum on top of the other (unstable equilibrium):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eta:=10^(-2);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$etaG#\"\"\"\"$+\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 55 "IC:=theta(0)=Pi,D(theta)(0)=-eta,phi(0)=Pi,D(phi)(0
)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG6&/-%&thetaG6#\"\"!%#PiG
/--%\"DG6#F(F)#!\"\"\"$+\"/-%$phiGF)F+/--F/6#F6F)F*" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 69 "We need to pick values for the constants (e.g.,
 in MKSA or SI units):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "E
L1p:=subs(g=10,l=1,EL1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%EL1pG/,
*-%%diffG6$-%&thetaG6#%\"tG-%\"$G6$F-\"\"#F1*&-%$cosG6#,&F*\"\"\"-%$ph
iGF,!\"\"F7-F(6$F8F.F7F7*&-%$sinGF5F7)-F(6$F8F-F1\"\"\"F7-F?6#F*\"#?\"
\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "EL2p:=subs(g=10,l=1,
EL2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%EL2pG/,**&-%$cosG6#,&-%&th
etaG6#%\"tG\"\"\"-%$phiGF.!\"\"F0-%%diffG6$F,-%\"$G6$F/\"\"#F0F0-F56$F
1F7F0*&-%$sinGF*F0)-F56$F,F/F:\"\"\"F3-F?6#F1\"#5\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "sol:=dsolve(\{EL1p,EL2p,IC\},\{thet
a(t),phi(t)\},numeric,method=lsode,output=listprocedure):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "th:=subs(sol,theta(t)): ph:=subs(so
l,phi(t)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "We have to delay e
valuation of the variables until actual numbers are passed for the tim
e parameter, thus the quotes below:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "plot(['th(t)','ph(t)'],t=0..30,color=[red,blue],numpo
ints=1000);" }}{PARA 13 "" 1 "" {GLPLOT2D 694 299 299 {PLOTDATA 2 "6&-
%'CURVESG6$7din7$\"\"!$\"+aEfTJ!\"*7$$\"+K@$>9$!#6$\"+OsFTJF+7$$\"+![5
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7$$\"+&*=]7:F?$\"+A/'*RJF+7$$\"+bc1)z\"F?$\"+l2fRJF+7$$\"+(Q]P4#F?$\"+
h;<RJF+7$$\"+hza*R#F?$\"+2/pQJF+7$$\"+Q[O/FF?$\"+X<:QJF+7$$\"+ik!z,$F?
$\"+`K_PJF+7$$\"+'4uSH$F?$\"+h[*o8$F+7$$\"+TM(\\g$F?$\"+Xt3OJF+7$$\"+p
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G6$;F(Fhc^l%(DEFAULTG" 1 2 0 1 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Obviously, the pen
dulum goes wild... " }}{PARA 0 "" 0 "" {TEXT -1 82 "1) Note the reluct
ance to get away from the unstable equilibrium at the beginning." }}
{PARA 0 "" 0 "" {TEXT -1 79 "2) Note that the lower pendulum (blue cur
ve) goes into a winding mode at about " }{TEXT 279 1 "t" }{TEXT -1 23 
"=10, and later unwinds." }}{PARA 0 "" 0 "" {TEXT -1 58 "3) The numpoi
nts option was chosen to avoid weird results." }}{PARA 0 "" 0 "" 
{TEXT -1 81 "4) The variables theta(t) and phi(t) keep track of the so
-called winding number. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 108 "Can \+
we animate the solutions? For this we need to reconstruct the Cartesia
n coordinates from the two angles." }}{PARA 0 "" 0 "" {TEXT -1 24 "We \+
pick for the lengths " }{TEXT 280 1 "l" }{TEXT -1 3 "=1." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "x1:=t->sin(th(t)):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "y1:=t->-cos(th(t)):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "x2:=t->x1(t)+sin(ph(t)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "y2:=t->y1(t)-cos(ph(t)):" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Our animations will be made up fr
om primitive straight-line connections." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 121 "PL:=t->plot([[0,0],[x1(t),y1(t)],[x2(t),y2(t)]],styl
e=line,color=magenta,title=cat(\"time= \",convert(evalf(t,3),string)))
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "PLa:=seq(PL(i/10),i=1.
.300):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display(PLa,inseq
uence=true,scaling=constrained,axes=boxed);" }}{PARA 13 "" 1 "" 
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me=~29.9FMFN7&-F(6%7%F+7$$!1+++Ie09**Fen$!1+++Y>C38Fen7$$!1+++yi!R:\"F
3$\"1+++YB%)e&)FenF:FAFE-FJ6#Q*time=~30.FMFN-%(SCALINGG6#%,CONSTRAINED
G-%*AXESSTYLEG6#%$BOXG" 1 2 0 1 0 2 9 1 2 1 1.000000 45.000000 
45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "Observe how there
 are moments when the upper pendulum is almost stationary at the botto
m, while the lower one is spinning at a maximum rate." }}{PARA 0 "" 0 
"" {TEXT -1 241 "Note that the animation can be stopped and continued \+
later, so that one can go back to the upper graph to compare the infor
mation. From Maple 6 onward the title of the graphs is displayed in th
e animation (providing the time for each frame)." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 255 "Exercise1: Change the in
tial conditions by a small amount and compare time histories. Change t
he method of integration (remove method=lsode to use a Runge-Kutta met
hod), and observe whether the results are being reproduced for the sam
e initial conditions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 102 "Exercise2: Extend the problem by allowing different m
asses for the top and bottom pendula (pendulums)." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK 
"44 1 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }