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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 25 "The mathematical pendul
um" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "We
 follow Gerd Baumann's approach in Mathematica to determine the period
 of the mathematical pendulum." }}{PARA 0 "" 0 "" {TEXT -1 165 "First,
 we write down the kinetic and potential energies: the kinetic energy \+
is expressed in terms of the angular velocity and the moment of inerti
a for a point mass " }{TEXT 274 1 "m" }{TEXT -1 23 " suspended at dist
ance " }{TEXT 273 1 "l" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 43 "We call the angular velocity d/dt phi=omega" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "Tkin:=1/2*m*l^2*omega^2;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%%TkinG,$*(%\"mG\"\"\")%\"lG\"\"#F()%&omegaGF+F(#F(F+
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "The potential energy follows \+
from the height of the pendulum bob (point mass " }{TEXT 275 1 "m" }
{TEXT -1 74 "). We choose the potential energy to be zero if the mass \+
is at the bottom." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Upot:=
m*g*l*(1-cos(phi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%UpotG**%\"mG
\"\"\"%\"gGF'%\"lGF',&F'F'-%$cosG6#%$phiG!\"\"F'" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 51 "For plotting we have to pick some parameter values
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "l:=1;" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"lG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 6 "g:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG\"#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m:=1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"mG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
141 "plots[contourplot](Tkin+Upot,phi=-3/2*Pi..3/2*Pi,omega=-10..10,co
ntours=[1,5,9,13,17,21,25,29],axes=boxed,filled=true, coloring=[red,gr
een]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "66-%'CUR
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45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve
 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu
rve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 547 "This phase space diagram is obviously periodic
 with period 2 Pi. For small angular velocities one has a system appro
ximated by a harmonic oscillator, while for larger velocites (or energ
ies, i.e., higher contours) one has solutions that simply revolve (win
ding the pendulum; in the periodic phase space diagrams one leaves e.g
., at the right to re-enter at the same angular velocity at the left. \+
The motion is left to right in the top half (for the open curves), and
 right to the left in the bottom half (look at the sign of the angular
 velocity)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 115 "The regularity of the diagram is very striking. The units of e
nergy (chosen for the contour lines) are in MKSA, if " }{TEXT 259 1 "l
" }{TEXT -1 1 "," }{TEXT 258 1 "g" }{TEXT -1 1 "," }{TEXT 257 1 "m" }
{TEXT -1 19 " are given in MKSA." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 358 "There is a critical energy where the sep
aration occurs between oscillating solutions (closed contours, in the \+
somewhat abstract phase space one obtains elliptic or deformed ellipti
c 'trajectories'; translate those into oscillations using your imagina
tion, by watching the sign of omega, passage through phi=0, i.e., the \+
bottom position of the pendulum, etc.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 229 "The locations phi=+Pi, omega=0, or p
hi=-Pi, omega=0 correspond to the unstable equilibrium (pendulum at th
e top at rest). One can show that the phase-space 'trajectory' become \+
a cosine curves of phi/2 [Marion-Thornton, eq. 4.31]." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 385 "For winding solutions
 we observe how their angular velocity is modulated as the go through \+
the (periodic) potential well. For sufficiently high energies (not sho
wn) the phase space 'trajectories' approach straigh lines, as the pend
ulum bob's kinetic energy is so high that the variation in the potenti
al energy (by being moved up and down in the graviational field) becom
es negligible." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 298 "Time-dependent perturbations of the pendulum by an exter
nal force can lead to a loss of regularity in the phase space diagram \+
(CHAOS). The modulation can come from a simple coupling to a second pe
ndulum, which leads to a time-varying support point (science museum de
monstrations + classroom demo!)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 102 "The determination of the phase-space 'tr
ajectories' was easy. The actual integration of phi(t) is not!" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "The total energy can be expressed
 through the potential energy at the turning point (pendulum bob at re
st), i.e., through the initial angle phi0." }}{PARA 0 "" 0 "" {TEXT 
-1 77 "One introduces the energy parameter (initial phase angle phi0) \+
k=sin(phi0/2);" }}{PARA 0 "" 0 "" {TEXT -1 135 "To recover the general
 expression for the potential energy we unassign the parameters for le
ngth, gravitational acceleration, and mass:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 29 "l:='l': g:='g': m:='m': Upot;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#**%\"mG\"\"\"%\"gGF%%\"lGF%,&F%F%-%$cosG6#%$phiG!\"\"F%
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Upot:=algsubs(cos(phi)=
1-2*sin(phi/2)^2,Upot);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%UpotG,$*
*%\"mG\"\"\"%\"gGF(%\"lGF()-%$sinG6#,$%$phiG#F(\"\"#F2F(F2" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Etot:=subs(phi=phi0,Upot);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%EtotG,$**%\"mG\"\"\"%\"gGF(%\"lGF()
-%$sinG6#,$%%phi0G#F(\"\"#F2F(F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
61 "The total energy for a given trajectory thus depends only on " }
{TEXT 261 1 "k" }{TEXT -1 66 "=sin(phi/2). Therefore, the period is ca
lculated as a function of " }{TEXT 260 1 "k" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 145 "To calculate t
he period for the oscillatory solutions, one uses the energy expressio
n which depends on phi and omega=diff(phi,t), isolates omega." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Tkin+Upot=Etot;" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/,&*(%\"mG\"\"\")%\"lG\"\"#F')%&omegaGF*F'#F'F*
*,F*F'F&F'%\"gGF'F)F')-%$sinG6#,$%$phiGF-F*F'F',$**F&F'F/F'F)F')-F26#,
$%%phi0GF-F*F'F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(%
,omega);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&%\"lG!\"\"*(F%\"\"\"%
\"gGF(,&*$)-%$sinG6#,$%$phiG#F(\"\"#F3F(F&*$)-F.6#,$%%phi0GF2F3F(F(F(F
2F3,$F$!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "DE:=diff(phi
(t),t)=subs(phi=phi(t),%[1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DE
G/-%%diffG6$-%$phiG6#%\"tGF,,$*&%\"lG!\"\"*(F/\"\"\"%\"gGF2,&*$)-%$sin
G6#,$F)#F2\"\"#F<F2F0*$)-F86#,$%%phi0GF;F<F2F2F2F;F<" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 37 "sol:=dsolve(\{DE,phi(0)=phi0\},phi(t));" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG/-%$phiG6#%\"tG-%'RootOfG6#,(
F)\"\"#-%$IntG6$*&%\"lG\"\"\"-%%sqrtG6#,$*&F4F4*(,&-%$cosG6#%#_aG!\"\"
-F=6#%%phi0GF4F4F3F4%\"gGF4F@!\"#F4/F?;\"\"!%#_ZGF@-F06$*(F3F4-F66#F.F
4-F66#*&F4F4*(,&F<F4FAF@F4F3F4FDF4F@F4/F?;FHFCF4" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 266 "Maple 5.0 comes up with a formal solution for phi
(t). It hints at the possibility of finding the solution through ellip
tic integrals. In Maple 6.0 no answer is returned after the program tr
ies to find a solution for a while. Maple 7 goes back to the formal so
lution." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
36 "For the determination of the period " }{TEXT 262 1 "T" }{TEXT -1 
117 " one can proceed differently: the differential equation DE can be
 separated so that an expression for dt is obtained:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "subs(diff(phi(t),t)=dphi/dt,DE);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%%dphiG\"\"\"%#dtG!\"\",$*&%\"lGF(*
(F+F&%\"gGF&,&*$)-%$sinG6#,$-%$phiG6#%\"tG#F&\"\"#F:F&F(*$)-F26#,$%%ph
i0GF9F:F&F&F&F9F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "solve(
%,dt);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(%%dphiG\"\"\"%\"lGF&,$*(
F'F&%\"gGF&,&*$)-%$sinG6#,$-%$phiG6#%\"tG#F&\"\"#F7F&F&*$)-F/6#,$%%phi
0GF6F7F&!\"\"F&F>#F>F7F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 203 "If w
e integrate the right-hand side over phi from 0 to phi0, the left-hand
 side will give us a quarter of the period T (for the full period phi \+
would have to continue through 0 to -phi0 and back to phi0." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 "The integral to
 be carried out falls into a category known as elliptic." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "assume(k>0);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 61 "A required integral can be expressed as an elliptic \+
function:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "Int(1/sqrt((1-
z^2)*(1-k^2*z^2)),z=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6
$*&\"\"\"F'*$-%%sqrtG6#*&,&F'F'*$)%\"zG\"\"#F'!\"\"F',&F'F'*&)%#k|irGF
1F'F/F'F2F'F'F2/F0;\"\"!F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#k|irG!\"\"-%*Ellip
ticFG6$F$*&\"\"\"F*F$F%F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "omega[0]:=sqrt(g/l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&omegaG
6#\"\"!*$-%%sqrtG6#*&%\"gG\"\"\"%\"lG!\"\"F." }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 76 "To help Maple we restrict the inital phase to be positi
ve, and less than Pi." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "as
sume(phi0>0,phi0<Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "T:
=2/omega[0]*Int(1/sqrt(sin(phi0/2)^2-sin(phi/2)^2),phi=0..phi0);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG,$*&*&%\"gG\"\"\"%\"lG!\"\"#F+\"
\"#-%$IntG6$*&F)F)*$-%%sqrtG6#,&*$)-%$sinG6#,$%$phiG#F)F-F-F)F+*$)-F:6
#,$%&phi0|irGF>F-F)F)F)F+/F=;\"\"!FDF)F-" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "T:=value(T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"T
G,$*&*&%\"gG\"\"\"%\"lG!\"\"#F+\"\"#-%*EllipticKG6#-%$sinG6#,$%&phi0|i
rG#F)F-F)\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "For plotting p
urposes let us simplify matters by chosing the pendulum length and the
 gravitational acceleration in MKSA units:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 13 "g:=10; l:=10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"gG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"lG\"#5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "T;" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#,$-%*EllipticKG6#-%$sinG6#,$%&phi0|irG#\"\"\"\"\"#\"\"%" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 67 "We now explore this answer as a function \+
of the initial angle phi0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "plot(T,phi0=0..Pi);" }}{PARA 13 "" 1 "" {GLPLOT2D 696 243 243 
{PLOTDATA 2 "6%-%'CURVESG6$7ao7$$\"\"!F)$\"+3`=$G'!\"*7$$\"+LzxZo!#6$
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$$\"+HeV)y*F6$\"+CV\\\"o'F,7$$\"+*3W'\\5F,$\"+[/IXnF,7$$\"+/9i46F,$\"+
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YN#F,$\"++.c$f*F,7$$\"+O*f2U#F,$\"+'4wP!**F,7$$\"+U!z`[#F,$\"+X%RT-\"!
\")7$$\"+d#HIb#F,$\"+!\\sQ1\"Fgw7$$\"+bW==EF,$\"+E6O26Fgw7$$\"+b]\"[o#
F,$\"+)3R&e6Fgw7$$\"+?R*3v#F,$\"+4f.=7Fgw7$$\"+MNh6GF,$\"+)y#3$G\"Fgw7
$$\"+k]?\")GF,$\"+e&p_P\"Fgw7$$\"+7oK7HF,$\"+%fn^U\"Fgw7$$\"+f&[M%HF,$
\"+C0f#[\"Fgw7$$\"+L3jwHF,$\"+]/,b:Fgw7$$\"+1J\")4IF,$\"+(*f1W;Fgw7$$
\"+THpDIF,$\"+P'*4&p\"Fgw7$$\"+vFdTIF,$\"+?Uu`<Fgw7$$\"+5EXdIF,$\"+,Jj
A=Fgw7$$\"+XCLtIF,$\"+ZT11>Fgw7$$\"+q\\'=3$F,$\"+&fn$f>Fgw7$$\"+'\\(R!
4$F,$\"+m*H4-#Fgw7$$\"+A+$*)4$F,$\"+pBx$4#Fgw7$$\"+ZDY2JF,$\"+xf&H=#Fg
w7$$\"+6)G<6$F,$\"++lLOAFgw7$$\"+u]*f6$F,$\"+N#ozH#Fgw7$$\"+P8E?JF,$\"
+R=(3P#Fgw7$$\"+*fFX7$F,$\"+6z5gCFgw7$$\"+J2mEJF,$\"+c5^8DFgw7$$\"+iQz
GJF,$\"+mI;vDFgw7$$\"+$*p#48$F,$\"+t\\3[EFgw7$$\"+D,1LJF,$\"+%4Ott#Fgw
7$$\"+cK>NJF,$\"+(Q0C&GFgw7$$\"+(QEt8$F,$\"+?))e9IFgw7$$\"+>&f%RJF,$\"
+!eO=H$Fgw7$$\"+]EfTJF,%%FAILG-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESL
ABELSG6$Q&phi0|ir6\"Q!Fhal-%%VIEWG6$;F($\"+aEfTJF,%(DEFAULTG" 1 2 0 1 
10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 108 "In the limit of the intial phase going t
o Pi (unstable equilibrium at the top) we obtain an infinite period:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "limit(T,phi0=Pi,left);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%)infinityG" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "What do we get for sma
ll initial phase angles (we should recover isochronicity in the small-
amplitude limit)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "plot(T,
phi0=0..Pi/16);" }}{PARA 13 "" 1 "" {GLPLOT2D 691 179 179 {PLOTDATA 2 
"6%-%'CURVESG6$7S7$$\"\"!F)$\"+3`=$G'!\"*7$$\"+07')zU!#7$\"++D>$G'F,7$
$\"+bbu.!)F0$\"+k/@$G'F,7$$\"+)Hk\">7!#6$\"+wOC$G'F,7$$\"+]+tS;F;$\"+C
5H$G'F,7$$\"+u@Hg?F;$\"+/?N$G'F,7$$\"+q*z#\\CF;$\"+'*3U$G'F,7$$\"+-V0_
GF;$\"+_Z]$G'F,7$$\"+$4/'oKF;$\"+%)[g$G'F,7$$\"+.!=Qo$F;$\"+g#=PG'F,7$
$\"+Yl\"46%F;$\"+?!\\QG'F,7$$\"+v^5([%F;$\"+kg(RG'F,7$$\"+rWg5\\F;$\"+
'RKTG'F,7$$\"+'pUeL&F;$\"+_NI%G'F,7$$\"+@&Qcu&F;$\"+W>[%G'F,7$$\"+*Qsx
6'F;$\"+%QbYG'F,7$$\"+^bFglF;$\"+#zv[G'F,7$$\"+tP8NpF;$\"+cX2&G'F,7$$
\"+**H=rtF;$\"+w'>`G'F,7$$\"+^/;dxF;$\"+K\"\\bG'F,7$$\"+*))Q1=)F;$\"+o
V\"eG'F,7$$\"+10*Qe)F;$\"+g+3'G'F,7$$\"+k7k/!*F;$\"+K4P'G'F,7$$\"+0=-
\"R*F;$\"+?.l'G'F,7$$\"+;ay2)*F;$\"+o['pG'F,7$$\"+o&oS-\"!#5$\"+%31tG'
F,7$$\"+jEvh5Fds$\"+S^h(G'F,7$$\"+=DX-6Fds$\"+/:'zG'F,7$$\"+^%*\\W6Fds
$\"+_IL)G'F,7$$\"+sTj&=\"Fds$\"+S+r)G'F,7$$\"+HTVD7Fds$\"+/v3*G'F,7$$
\"+'>D'p7Fds$\"+k7_*G'F,7$$\"+NGL48Fds$\"+sT#**G'F,7$$\"+:#H<N\"Fds$\"
+;\"o.H'F,7$$\"+Pm9!R\"Fds$\"+gEy!H'F,7$$\"+Eq9K9Fds$\"+?#\\7H'F,7$$\"
+?hmr9Fds$\"+c4q\"H'F,7$$\"+f\\(H^\"Fds$\"+kj=#H'F,7$$\"++>O`:Fds$\"+#
*Rn#H'F,7$$\"+&GVcf\"Fds$\"+S$)>$H'F,7$$\"+$Gljj\"Fds$\"+Onr$H'F,7$$\"
+3%4!y;Fds$\"+o/E%H'F,7$$\"+*p3$><Fds$\"+wK\"[H'F,7$$\"+e%esv\"Fds$\"+
%=L`H'F,7$$\"+kJv+=Fds$\"+;J%fH'F,7$$\"+[`lR=Fds$\"+s8]'H'F,7$$\"+!>L6
)=Fds$\"+S)4rH'F,7$$\"+xF$3#>Fds$\"+G]q(H'F,7$$\"+0a\\j>Fds$\"+;'e$)H'
F,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q&phi0|ir6\"Q!Fe[l-%
%VIEWG6$;F($\"+4a\\j>Fds%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "taylor(T,phi0=0,5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#++%&phi0|irG,$%#PiG\"\"#\"\"!,$F&#\"\"\"\"\")F',$F&#\"#6\"%O:\"\"%-%
\"OG6#F+\"\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "T1:=algsub
s(sin(phi0/2)=k,T);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T1G,$-%*Elli
pticKG6#%#k|irG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "ser
ies(T1,k=0,8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%#k|irG,$%#PiG\"\"
#\"\"!,$F&#\"\"\"F'F',$F&#\"\"*\"#K\"\"%,$F&#\"#D\"$G\"\"\"'-%\"OG6#F+
\"\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "T1t:=convert(%,pol
ynom);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$T1tG,*%#PiG\"\"#*(#\"\"\"
F'F*F&F*)%#k|irGF'F*F**(#\"\"*\"#KF*F&F*)F,\"\"%F*F**(#\"#D\"$G\"F*F&F
*)F,\"\"'F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "This agrees with
 eq. 2.25 in Baumann (or analytical mechanics texts)." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot(\{Re(T1),T1t\},k=0..1);" }}
{PARA 13 "" 1 "" {GLPLOT2D 691 185 185 {PLOTDATA 2 "6&-%'CURVESG6$7ao7
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45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 50 "An alternative would be to introduce the variable " }
{TEXT 269 1 "k" }{TEXT -1 26 " directly in the integrand" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "T2:=2/omega[0]*int(1/sqrt(k^2-sin(p
hi/2)^2),phi=0..2*arcsin(k));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#T2
G,$-%*EllipticKG6#%#k|irG\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "simplify(T2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%*Elliptic
KG6#%#k|irG\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "This does not
 work in Maple5! However, it does from 6.0 on." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "simplify(T2,symbolic);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$-%*EllipticKG6#%#k|irG\"\"%" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 132 "Maple is reluctant to carry out the cancellation of fu
nction and its inverse. In Maple6 T2 is directly calculated as 4*Ellip
ticK(k)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
75 "A further exploration should look into the calculation of trajecto
ries phi(" }{TEXT 270 1 "t" }{TEXT -1 2 ")!" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 376 "Before we proceed with this st
ep we construct the phase diagram (a plot of the time derivative of th
e dynamic variable [the phase angle phi] against the dynamic variable \+
itself. For this we need the energy expression (we will look at contou
r lines of equal height, i.e. lines of constant energy, as the mechani
cal energy is conserved for a frictionless mathematical pendulum)." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "l:='l': g:='g': m:='m':" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "phi:='phi';" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$phiGF$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 3 "DE;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%diffG6$-%$phiG6#%\"tGF
*,$*&%\"lG!\"\"*(F-\"\"\"%\"gGF0,&*$)-%$sinG6#,$F'#F0\"\"#F:F0F.*$)-F6
6#,$%&phi0|irGF9F:F0F0F0F9F:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 48 "L
et us make use of Gerd Baumann's analytic work:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "omega[0]:=sqrt(g/l);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>&%&omegaG6#\"\"!*$-%%sqrtG6#*&%\"gG\"\"\"%\"lG!\"\"F.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "Etot;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#,$**%\"mG\"\"\"%\"gGF&%\"lGF&)-%$sinG6#,$%&phi0|irG#F
&\"\"#F0F&F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "E_s:=Etot/(
m*l^2*omega[0]^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E_sG,$*$)-%$s
inG6#,$%&phi0|irG#\"\"\"\"\"#F/F.F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 255 "The scaled energy parameter allows to distinguish easily the d
ifferent types of motion: for E_s<2 we have oscillations, for E_s>2 ro
tations, and E_s=2 corresponds to the asymptotic solution where the pe
ndulum bob remains in the unstable equilibrium point." }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "We express the scaled \+
energy in terms of the parameter " }{TEXT 263 1 "k" }{TEXT -1 13 "=sin
(phi0/2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "k0:=7/8;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#k0G#\"\"(\"\")" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 12 "E_s:=2*k0^2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$E_sG#\"#\\\"#K" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "
phi:='phi':" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "We define the diff
erential equation such that positive and negative angular velocities a
re allowed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "DE:=(diff(ph
i(t),t))^2=omega[0]^2*(2*(cos(phi(t))-1+E_s));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#DEG/*$)-%%diffG6$-%$phiG6#%\"tGF.\"\"#\"\"\"*(%\"gGF
0%\"lG!\"\",&-%$cosG6#F+F/#\"#<\"#;F0F0" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 69 "Oscillatory solutions to the equation of motion (E_s<2) a
re given by:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "theta:=(k,t
)->2*arcsin(k*JacobiSN(omega[0]*t,k));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%&thetaGf*6$%\"kG%\"tG6\"6$%)operatorG%&arrowGF),$-%'arcsinG6#*
&9$\"\"\"-%)JacobiSNG6$*&&%&omegaG6#\"\"!F39%F3F2F3\"\"#F)F)F)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "g:=10; l:=1;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"gG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"
lG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "subs(phi(t)=the
ta(k0,t),DE);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)-%%diffG6$,$-%'ar
csinG6#,$-%)JacobiSNG6$*&-%%sqrtG6#\"#5\"\"\"%\"tGF6#\"\"(\"\")F8\"\"#
F7F;F6,&-%$cosG6#F)\"#?#\"#&)F:F6" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*$
)-%)JacobiSNG6$*&-%%sqrtG6#\"#5\"\"\"%\"tGF/#\"\"(\"\")\"\"#F/#!$X#F3#
\"$X#F3F/,&-%$cosG6#,$-%'arcsinG6#,$F'F1F4\"#?#\"#&)F3F/" }}}{EXCHG 
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1 "" {XPPMATH 20 "6#/$\"*[\"H:y!\")$\"+w9H:y!\"*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 83 "Without further work Maple cannot figure out that \+
the expression written for theta(" }{TEXT 271 1 "t" }{TEXT -1 134 ") s
olves the differential equation. Nevertheless, we can verify that with
in numerical accuracy the differential equation is satisfied." }}}
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{TEXT -1 181 "One observes that for moderate energy values the solutio
n still looks like a sine-curve, although the dependence of the period
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Q2k=~.995time=~19.2FCFH7&-F(6%7$F+7$$\"3r*****4/<t$eF1$\"3@+++,hY>\")F
1F4F;F?-FE6#Q2k=~.995time=~19.3FCFH7&-F(6%7$F+7$$\"3F+++9a=+XF1$\"3u**
***f6#>I*)F1F4F;F?-FE6#Q2k=~.995time=~19.4FCFH7&-F(6%7$F+7$$\"3;+++^D#
3[$F1$\"3e******e*RYP*F1F4F;F?-FE6#Q2k=~.995time=~19.5FCFH7&-F(6%7$F+7
$$\"3A+++vS+dFF1$\"3;+++RhV7'*F1F4F;F?-FE6#Q2k=~.995time=~19.6FCFH7&-F
(6%7$F+7$$\"3++++uE[)G#F1$\"3\"******H0@Yt*F1F4F;F?-FE6#Q2k=~.995time=
~19.7FCFH7&-F(6%7$F+7$$\"3))*****>$o%4/#F1$\"3O+++K:^*y*F1F4F;F?-FE6#Q
2k=~.995time=~19.8FCFH7&-F(6%7$F+7$$\"3)*******GZ8%*>F1$\"3m*****fQa\"
*z*F1F4F;F?-FE6#Q2k=~.995time=~19.9FCFH7&-F(6%7$F+7$$\"35+++gz1W@F1$\"
3))*****p_Wuw*F1F4F;F?-FE6#Q1k=~.995time=~20.FCFH-%*AXESSTYLEG6#%$BOXG
-%(SCALINGG6#%,CONSTRAINEDG" 1 2 0 1 10 0 2 9 1 2 1 1.000000 
37.000000 44.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
231 "The slowdown of the pendulum bob as the top position is approache
d is very obvious. Thus the lack of isochronicity, or the dependence o
f the period on the maximum amplitude (energy deposited into the pendu
lum) is made very obvious." }}{PARA 0 "" 0 "" {TEXT -1 298 "In Maple 6
 the title for each frame is actually displayed in the animation. Howe
ver, we found two errors for Maple 6.0 in this worksheet: 1) the conto
urplot is wrong; 2) the numerical evaluation of the JacobiSN functions
 is in error. We can only hope that a service pack will remedy these p
roblems." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 215 "While the demonstration of Maple's handling of elliptic \+
integrals has been impressive, we wish to demonstrate also a general a
pproach that can be applied to any differential equation, namely the n
umerical solution. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "DE:=
k->(diff(phi(t),t))^2=omega[0]^2*(2*(cos(phi(t))-1+2*k^2));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#DEGf*6#%\"kG6\"6$%)operatorG%&arrowGF(/*$
)-%%diffG6$-%$phiG6#%\"tGF5\"\"#\"\"\"*&)&%&omegaG6#\"\"!F6F7,(-%$cosG
6#F2F6F6!\"\"*&\"\"%F7)9$F6F7F7F7F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "omega[0]:=1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&o
megaG6#\"\"!\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "IC:=p
hi(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG/-%$phiG6#\"\"!F)" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "sol:=dsolve(\{DE(0.999),IC
\},phi(t),numeric);" }}{PARA 8 "" 1 "" {TEXT -1 83 "Error, (in DEtools
/convertsys) unable to convert to an explicit first-order system\n" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "Bad news! We need to go back to \+
the original Newton law. Maple 6 is, however, improved in this area. M
aple 7 can't handle it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "
omega:='omega': # it is not good to mix omega(t), and omega[0], which \+
makes omega a table." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "DE:
=diff(phi(t),t)=omega(t),diff(omega(t),t)=-(g/l)*sin(phi(t));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG6$/-%%diffG6$-%$phiG6#%\"tGF--%&
omegaGF,/-F(6$F.F-,$-%$sinG6#F*!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 94 "The initial condition is chosen to be the turning point, where \+
all energy is in the potential." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 23 "phifun:=k->2*arcsin(k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'
phifunGf*6#%\"kG6\"6$%)operatorG%&arrowGF(,$-%'arcsinG6#9$\"\"#F(F(F(
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "phi0:=phifun(0.999);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%phi0G$\"+![U@0$!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "IC:=omega(0)=0,phi(0)=phi0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG6$/-%&omegaG6#\"\"!F*/-%$phiGF)$
\"+![U@0$!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "sol:=dsolv
e(\{DE,IC\},[omega(t),phi(t)],numeric,output=listprocedure);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$solG7%/%\"tGf*6#F'6\"6#%aoCopyright~(c)~1
993~by~the~University~of~Waterloo.~All~rights~reserved.GF*9$F*F*F*/-%&
omegaGF)f*F)6&%$resG%*plistprocG%)outpointG%&ndsolG6#%inCopyright~(c)~
2000~by~Waterloo~Maple~Inc.~All~rights~reserved.GF*C'>8&-%&evalfG6#F->
8%f*6#%(rkf45_xG6.%\"iG%/comp_soln_dataG%(odeprocG%&icvecG%(LR_caseG%
\"YG%$valGF5%#ptG%*stop_procG%+stop_arrayG%&cplexG6#%aoCopyright~(c)~2
000~by~the~University~of~Waterloo.~All~rights~reserved.GE\\s#Q0soln_pr
oceduresF*7%\")#*f^;\")%QOc\"\")W\"Qd\"Q(complexF*%&falseGC2@$-%'membe
rG6$F-7$Q4left_comp_soln_dataF*Q5right_comp_soln_dataF*O\"\"!@$2%'Digi
tsG-%&truncG6#-%'evalhfG6#F`o>F`oFao>8+F<@$4-%%typeG6$Fio.%(numericG@%
-Fgn6$Fio7%Q&startF*Q%leftF*Q&rightF*O$F]oF]oO-.9!F>>F;f*6&%\"NGF'%\"Y
G%#YPGF*6#%A[Y[1]~=~omega(t),~Y[2]~=~phi(t)]GF*C$>&9'6#\"\"\",$-%$sinG
6#&9&6#\"\"#!#5>&FiqFbr&FarFjqF*F*F*>8'=F*6#;F[rFcrE\\[l#F[rFipFcr$\"+
![U@0$!\"*>8-F]o>8.F]o>8)-%$mapG6$%$rhsG-%&parseG6#-%#opG6$\"\"$-%%eva
lG6#F;>8/-F]q6#FX@'/FioFipO7$/F'Fio-%$seqG6$/&Ffs6#8$-F=6#&FirFcu/Fdu;
F[r-%%nopsG6#Ffs1FipFio>8(F[o>F_vFjn>F@-F]q6#F_v@%4-F]p6$7#F@7#.%%list
GC%Z%>F@-%;dsolve/numeric/SC/IVPsolveG6)F;;FipFioFirFbsFdsFft/%'scproc
GFYF*C&@$/Fft%%trueGYF*>FftFhw>F@F_w>FgtFhw>8*&&F@6#Fat6#&F@Fbr>FbvF@C
$>F^xF_x@$53/F_vF[o2F^xFio3/F_vFjn2FioF^xC&@$5/&F@6#\"\")F[r2\"#5FcyC$
>F^xF_x@%FjxY6$QQcannot~evaluate~the~solution~further~right~of~%1F*-&F
=Fdy6#F^xY6$QPcannot~evaluate~the~solution~further~left~of~%1F*F^zZ%>F
@-%<dsolve/numeric/SC/IVPextendG6*F@F;FbwFirFbsFdsFftFcwF*C&@$FgwYF*>F
ftFhw>F@Ffz>FgtFhw>FbvF@>F^xF_x@&FjxC$@$33/%:_Env_smart_dsolve_numeric
GFhw2FipF^x2-F]q6#FgpF^x>Fj[lF^x@$F[y@)/&FbvFdyF[rY6$Qcocannot~evaluat
e~the~solution~further~right~of~%1,~probably~a~singularityF*F^z/F`\\lF
crY6$Qcqcannot~evaluate~the~solution~further~right~of~%1,~maxfun~limit
~exceeded~(see~?dsolve,maxfun~for~details)F*F^z2FgyF`\\lC$@$0%7_Env_ds
olve_nowarnstopGFhw-%(WARNINGG6#-_%,StringToolsG%.FormatMessageG6%Qhoc
annot~evaluate~the~solution~further~right~of~%1,~stop~condition~#%2~vi
olatedF*F^z,&F`\\lF[rFgy!\"\">FioF^xYF\\zF]yC$@$33Ff[l2F^xFip2F^x-F]q6
#Ffp>F`^lF^x@$F^y@)F_\\lY6$Qbocannot~evaluate~the~solution~further~lef
t~of~%1,~probably~a~singularityF*F^zFd\\lY6$Qbqcannot~evaluate~the~sol
ution~further~left~of~%1,~maxfun~limit~exceeded~(see~?dsolve,maxfun~fo
r~details)F*F^zFh\\lC$@$F[]l-F^]l6#-Fa]l6%Qgocannot~evaluate~the~solut
ion~further~left~of~%1,~stop~condition~#%2~violatedF*F^zFf]l>FioF^xYFb
z>F^x-%9dsolve/numeric/SC/IVPvalG6$F@Fio7$F]u-F_u6$/Fbu&F^xFcuFhuF*F*F
*@$4-F]p6$F;F_p@)-Fgn6$F;FdpO-F@F>/F;Q.procedurelistF*O-Fct6#F@0F]q%(u
nknownGOF[qC$>Fir-%(pointtoG6#&-F@6#FSFbrO-.FirF>>Fdu7#-%*traperrorG6#
Fe`l@%-F]p6$&FduFjq%'stringGY-F_tFcu-Fjs6#&FablFbrF*F*F*/-%$phiGF)f*F)
F2F7F*C'>F;F<>F@FA@$F^`l@)Fb`lOFe`lFf`lOFi`lF[alOF[qC$>Fir-Faal6#&Fdal
FaxOFgal>FduFjal@%F_blYFdbl-Fjs6#&FablFaxF*F*F*" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 24 "theta:=subs(sol,phi(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&thetaGf*6#%\"tG6&%$resG%*plistprocG%)outpointG%&ndso
lG6#%inCopyright~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.G
6\"C'>8&-%&evalfG6#9$>8%f*6#%(rkf45_xG6.%\"iG%/comp_soln_dataG%(odepro
cG%&icvecG%(LR_caseG%\"YG%$valGF+%#ptG%*stop_procG%+stop_arrayG%&cplex
G6#%aoCopyright~(c)~2000~by~the~University~of~Waterloo.~All~rights~res
erved.GE\\s#Q0soln_proceduresF/7%\")#*f^;\")%QOc\"\")W\"Qd\"Q(complexF
/%&falseGC2@$-%'memberG6$F67$Q4left_comp_soln_dataF/Q5right_comp_soln_
dataF/O\"\"!@$2%'DigitsG-%&truncG6#-%'evalhfG6#Fhn>FhnFin>8+F3@$4-%%ty
peG6$Fao.%(numericG@%-FU6$Fao7%Q&startF/Q%leftF/Q&rightF/O$FenFenO-.9!
F5>F2f*6&%\"NGF'%\"YG%#YPGF/6#%A[Y[1]~=~omega(t),~Y[2]~=~phi(t)]GF/C$>
&9'6#\"\"\",$-%$sinG6#&9&6#\"\"#!#5>&FaqFjq&FiqFbqF/F/F/>8'=F/6#;FcqF[
rE\\[l#FcqFapF[r$\"+![U@0$!\"*>8-Fen>8.Fen>8)-%$mapG6$%$rhsG-%&parseG6
#-%#opG6$\"\"$-%%evalG6#F2>8/-Fep6#FP@'/FaoFapO7$/F'Fao-%$seqG6$/&F^s6
#8$-F46#&FarF[u/F\\u;Fcq-%%nopsG6#F^s1FapFao>8(FY>FguFX>F8-Fep6#Fgu@%4
-Feo6$7#F87#.%%listGC%Z%>F8-%;dsolve/numeric/SC/IVPsolveG6)F2;FapFaoFa
rFjrF\\sF^t/%'scprocGFQF/C&@$/F^t%%trueGYF/>F^tF`w>F8Fgv>F_tF`w>8*&&F8
6#Fis6#&F8Fjq>FjuF8C$>FfwFgw@$53/FguFY2FfwFao3/FguFX2FaoFfwC&@$5/&F86#
\"\")Fcq2\"#5F[yC$>FfwFgw@%FbxY6$QQcannot~evaluate~the~solution~furthe
r~right~of~%1F/-&F4F\\y6#FfwY6$QPcannot~evaluate~the~solution~further~
left~of~%1F/FfyZ%>F8-%<dsolve/numeric/SC/IVPextendG6*F8F2FjvFarFjrF\\s
F^tF[wF/C&@$F_wYF/>F^tF`w>F8F^z>F_tF`w>FjuF8>FfwFgw@&FbxC$@$33/%:_Env_
smart_dsolve_numericGF`w2FapFfw2-Fep6#F_pFfw>Fb[lFfw@$Fcx@)/&FjuF\\yFc
qY6$Qcocannot~evaluate~the~solution~further~right~of~%1,~probably~a~si
ngularityF/Ffy/Fh[lF[rY6$Qcqcannot~evaluate~the~solution~further~right
~of~%1,~maxfun~limit~exceeded~(see~?dsolve,maxfun~for~details)F/Ffy2F_
yFh[lC$@$0%7_Env_dsolve_nowarnstopGF`w-%(WARNINGG6#-_%,StringToolsG%.F
ormatMessageG6%Qhocannot~evaluate~the~solution~further~right~of~%1,~st
op~condition~#%2~violatedF/Ffy,&Fh[lFcqF_y!\"\">FaoFfwYFdyFexC$@$33F^[
l2FfwFap2Ffw-Fep6#F^p>Fh]lFfw@$Ffx@)Fg[lY6$Qbocannot~evaluate~the~solu
tion~further~left~of~%1,~probably~a~singularityF/FfyF\\\\lY6$Qbqcannot
~evaluate~the~solution~further~left~of~%1,~maxfun~limit~exceeded~(see~
?dsolve,maxfun~for~details)F/FfyF`\\lC$@$Fc\\l-Ff\\l6#-Fi\\l6%Qgocanno
t~evaluate~the~solution~further~left~of~%1,~stop~condition~#%2~violate
dF/FfyF^]l>FaoFfwYFjy>Ffw-%9dsolve/numeric/SC/IVPvalG6$F8Fao7$Fet-Fgt6
$/Fjt&FfwF[uF`uF/F/F/@$4-Feo6$F2Fgo@)-FU6$F2F\\pO-F8F5/F2Q.procedureli
stF/O-F[t6#F80Fep%(unknownGOFcpC$>Far-%(pointtoG6#&-F86#FKFiwO-.FarF5>
F\\u7#-%*traperrorG6#F]`l@%-Feo6$&F\\uFbq%'stringGY-FgsF[u-Fbs6#&FialF
iwF/F/F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "In the plot command \+
we need to put quotes around theta(t), as the expression can only be e
valuated for numerical values of t." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "plot('theta(t)',t=0..2*Pi);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 695 299 299 {PLOTDATA 2 "6%-%'CURVESG6$7jn7$$\"\"!F)$\"+![U@
0$!\"*7$$\"+&eb&p8!#5$\"+OMjVIF,7$$\"+w&)>hDF0$\"+$o87-$F,7$$\"+]dK,RF
0$\"+(=S^(HF,7$$\"+dhL]_F0$\"+]aC)*GF,7$$\"+`\\$Hf'F0$\"+)4,wx#F,7$$\"
+,fpPyF0$\"+)[>$3EF,7$$\"+jPdE\"*F0$\"+#)e0\\BF,7$$\"+4L&f/\"F,$\"+B`
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\"+bO(eV\"F,$!+\\K([**)!#67$$\"+UNj.:F,$!+?4\"R7&F07$$\"+IMRr:F,$!+u%Q
r7*F07$$\"+Y:VR;F,$!+:mky7F,7$$\"+i'puq\"F,$!+!RM2g\"F,7$$\"+%*p.t<F,$
!+9Mln=F,7$$\"+EVgQ=F,$!+FVi$4#F,7$$\"+jrod>F,$!+HE;7CF,7$$\"+v\")G*4#
F,$!+;IzpEF,7$$\"+1GC>AF,$!+ite9GF,7$$\"+f&y(eBF,$!+v0GDHF,7$$\"+V8H#[
#F,$!+2ZQ()HF,7$$\"+VW!yh#F,$!+2WZGIF,7$$\"+h\\%ou#F,$!+WK(z/$F,7$$\"+
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$\"+;%>qF$F,$!+]rCaHF,7$$\"+?&3wR$F,$!+dCQwGF,7$$\"+c![y_$F,$!+03c[FF,
7$$\"+T#)RiOF,$!+PqAZDF,7$$\"+Z$HSz$F,$!+&pE<D#F,7$$\"+!G4x&QF,$!+Md%G
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]um)>\"F,7$$\"+ZGLETF,$!+)p=e\\)F07$$\"+q]')*=%F,$!+f(Req%F07$$\"+z#*p
dUF,$!+%eW*RYF^p7$$\"+([LbK%F,$\"+4Iz)z$F07$$\"+i8+(Q%F,$\"+ClN@vF07$$
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+(p\\it\"F,7$$\"+J55YYF,$\"+#>FZ(>F,7$$\"+\"eJ$4ZF,$\"+r-aw@F,7$$\"+n)
>:%[F,$\"+.vO(\\#F,7$$\"+y!e2(\\F,$\"+bFw3FF,7$$\"+3&eg5&F,$\"+4]laGF,
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\"+R&4P.$F,7$$\"+/ii>gF,$\"+rB-)*HF,7$$\"+$)[mYhF,$\"+0C/SHF,7$$\"+&H&
=$G'F,$\"+>2!4%GF,-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"t
6\"Q!Fe_l-%%VIEWG6$;F($\"+3`=$G'F,%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 26 "Omega:=subs(sol,omega(t));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&OmegaGf*6#%\"tG6&%$resG%*plistprocG%)outpointG%&ndso
lG6#%inCopyright~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.G
6\"C'>8&-%&evalfG6#9$>8%f*6#%(rkf45_xG6.%\"iG%/comp_soln_dataG%(odepro
cG%&icvecG%(LR_caseG%\"YG%$valGF+%#ptG%*stop_procG%+stop_arrayG%&cplex
G6#%aoCopyright~(c)~2000~by~the~University~of~Waterloo.~All~rights~res
erved.GE\\s$Q0soln_proceduresF/7%\")#*f^;\")%QOc\"\")W\"Qd\"Q5right_co
mp_soln_dataF/7-\"\"#\"$p&X*%)anythingG6\"F/[gl'!%\"!!\"d^m\"d^m000000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FACB7A3FE43326E8408F923FE4B478918653AA3FE535CA3ACC17E43FE558E8320188F9
3FE57C062936FA0E3FE59F24206C6B233FE5C24217A1DC383FE640D5459392623FE6BF
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EA2A3C06BA46B13FEA4DC6D26DA6093FEACC9D3C8B92C33FEB4B73A6A97F7D3FEBCA4A
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 152 "It is evident that there are exte
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{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The imaginary values of " }{TEXT 
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