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1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "_cstyle3" -1 211 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold" -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 200 0 "" }}{PARA 18 "" 0 "" {TEXT 212 16 "Planetary Motion" }}{PARA 19 "" 0 "" {TEXT 213 51 "Worksheet by Mi ke May, S.J. \2512006- maymk@slu.edu" }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{PARA 210 "" 0 "" {TEXT 211 220 "One use of parameterized curves is to plot planeta ry motion. That is actually a good example to study since one could m ake a good case for the claim that Newton invented calculus to solve p roblems of planetary motion. " }}{PARA 210 "" 0 "" {TEXT 211 52 "This is a demonstration worksheet without exercises." }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 214 32 "Parameterizing \+ elliptical motion" }{TEXT 214 0 "" }}{PARA 0 "" 0 "" {TEXT 200 197 "Th e first approach to planetary motion in the class is to look at trying to parameterize the elliptical paths of the planets. We want to star t with parameterizing circular motion and modifying it." }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 200 95 "If we want to move around a circle of radius r and period omega the easiest parameteriza tion is" }}{PARA 0 "" 0 "" {TEXT 200 56 "x(t) = a*cos(2*Pi*t/omega), \+ y(t) = a*sin(2*Pi*t/omega)." }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 200 65 "For the earth, r=93 in millions of miles, and \+ omega = 1 in years." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a := \+ 93: omeg := 1:\n" }{MPLTEXT 1 0 61 "x := t -> a*cos(2*Pi*t/omeg): y \+ := t ->a*sin(2*Pi*t/omeg):\n" }{MPLTEXT 1 0 28 "plot([x(t), y(t), t=0. .1] );" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 140 "Next we w ant to take into account that the orbit of the earth is flattened into an ellipse. The eccentricity of the earth's orbit is .02. " }} {PARA 0 "" 0 "" {TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 200 77 "That me ans that the sum is shifted c = e*a = .02*93 million miles off center. " }}{PARA 0 "" 0 "" {TEXT 200 85 "It also means that the short axis is b = sqrt(a^2-c^2) = a*sqrt(1-e^2) million miles." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "e := .02: \n" }{MPLTEXT 1 0 11 "c := a*e;\n" }{MPLTEXT 1 0 19 "b := a*sqrt(1-e^2);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 59 "This leads to a new parameterization of the eart h's motion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "a := 93: ome g := 1:\n" }{MPLTEXT 1 0 47 "e := .02: c := a*e: b := a*sqrt(1-e ^2):\n" }{MPLTEXT 1 0 37 "x := t -> a*cos(2*Pi*t/omeg) + c: \n" } {MPLTEXT 1 0 31 "y := t -> b*sin(2*Pi*t/omeg):\n" }{MPLTEXT 1 0 28 "pl ot([x(t), y(t), t=0..1] );" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 50 "This still looks like a circle to the unaided eye." }} {PARA 0 "" 0 "" {TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 200 73 "If we l ook at the orbit of Mercury, a is 36 million miles and e is .21.\n" } {TEXT 200 177 "We also need to compute the period, which will be (36/9 3)^(3/2). (The period is the size of the orbit, measured in AUs (The \+ size of the earth's orbit) raised to the 3/2 power.)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "a := 36: omeg := evalf((a/93.0)^(3/2)); \n" }{MPLTEXT 1 0 47 "e := .21: c := a*e: b := a*sqrt(1-e^2):\n" }{MPLTEXT 1 0 37 "x := t -> a*cos(2*Pi*t/omeg) + c: \n" }{MPLTEXT 1 0 31 "y := t -> b*sin(2*Pi*t/omeg):\n" }{MPLTEXT 1 0 51 "plot([x(t), y (t), t=0..omeg], scaling=constrained);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 51 "For comparison we plot a circle at the same cent er." }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "xc := t -> a*cos(2*Pi*t/omeg) + c: \n" }{MPLTEXT 1 0 32 "yc := t -> a*sin( 2*Pi*t/omeg):\n" }{MPLTEXT 1 0 82 "plot(\{[xc(t), yc(t), t=0..omeg], [ x(t), y(t), t=0..omeg]\}, scaling=constrained);" }{MPLTEXT 1 0 0 "" }} }{PARA 0 "" 0 "" {TEXT 200 75 "It is still very close to circular in o rbit, even if noticeably off center." }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 200 59 "For Halley's comment the numbers are \+ that a=1680 and e=.97." }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "a := 1680: omeg := evalf((a/93.0)^(3/2));\n" } {MPLTEXT 1 0 47 "e := .97: c := a*e: b := a*sqrt(1-e^2):\n" } {MPLTEXT 1 0 37 "x := t -> a*cos(2*Pi*t/omeg) + c: \n" }{MPLTEXT 1 0 31 "y := t -> b*sin(2*Pi*t/omeg):\n" }{MPLTEXT 1 0 51 "plot([x(t), y(t ), t=0..omeg], scaling=constrained);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 143 "This orbit is clearly an ellipse. It is worth no ting that our model would have the period of Mercury be 88 days and Ha ley's comet be 77 years." }{TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 214 16 "Relative motions" }{TEXT 214 0 "" }}{PARA 210 "" 0 "" {TEXT 200 244 "The next question of interest is to plot relative motion for two planets. In parametric form, we simply take the diffe rence of the two parameterizations. Not that we want to take our time period to be at least the longer of the two periods. " }}{PARA 210 " " 0 "" {TEXT 200 0 "" }}{PARA 210 "" 0 "" {TEXT 200 40 "Consider Mercu ry and the earth together." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "am := 36: omegm := evalf((am/93.0)^(3/2)):\n" }{MPLTEXT 1 0 54 " em := .21: cm := am*em: bm := am*sqrt(1-em^2):\n" }{MPLTEXT 1 0 41 "xm := t -> am*cos(2*Pi*t/omegm) + cm: \n" }{MPLTEXT 1 0 34 "ym := t -> bm*sin(2*Pi*t/omegm):\n" }{MPLTEXT 1 0 46 "ae := 93: omege := \+ evalf((ae/93.0)^(3/2)):\n" }{MPLTEXT 1 0 54 "ee := .02: ce := ae*ee : be := ae*sqrt(1-ee^2):\n" }{MPLTEXT 1 0 41 "xe := t -> ae*cos(2*P i*t/omege) + ce: \n" }{MPLTEXT 1 0 34 "ye := t -> be*sin(2*Pi*t/omege ):\n" }{MPLTEXT 1 0 32 "tmax := max(2*omegm, 2*omege);\n" }{MPLTEXT 1 0 69 "plot([xm(t) - xe(t), ym(t) - ye(t), t=0..tmax], scaling=constrai ned);" }{MPLTEXT 1 0 0 "" }}}{PARA 210 "" 0 "" {TEXT 200 118 "Looking \+ from one planet to the other it appears that the planet is stopping an d starting with some backing up as well." }}{PARA 210 "" 0 "" {TEXT 200 0 "" }}{PARA 210 "" 0 "" {TEXT 200 61 "Consider what happens if we try the earth and Halley's comet." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "am := 1680: omegm := evalf((am/93.0)^(3/2)):\n" } {MPLTEXT 1 0 54 "em := .97: cm := am*em: bm := am*sqrt(1-em^2): \n" }{MPLTEXT 1 0 41 "xm := t -> am*cos(2*Pi*t/omegm) + cm: \n" } {MPLTEXT 1 0 34 "ym := t -> bm*sin(2*Pi*t/omegm):\n" }{MPLTEXT 1 0 46 "ae := 93: omege := evalf((ae/93.0)^(3/2)):\n" }{MPLTEXT 1 0 54 "ee \+ := .02: ce := ae*ee: be := ae*sqrt(1-ee^2):\n" }{MPLTEXT 1 0 41 "xe := t -> ae*cos(2*Pi*t/omege) + ce: \n" }{MPLTEXT 1 0 34 "ye := t \+ -> be*sin(2*Pi*t/omege):\n" }{MPLTEXT 1 0 32 "tmax := max(2*omegm, 2*o mege);\n" }{MPLTEXT 1 0 69 "plot([xm(t) - xe(t), ym(t) - ye(t), t=0..t max], scaling=constrained);" }{MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 200 158 "The motion is obviously wrong. We have neglected that \+ fact that planets move at different speeds on their orbits. They go f aster when they are near the sun." }{TEXT 200 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 214 36 " Finding a path determined by gravity" }{TEXT 214 0 "" }}{PARA 0 "" 0 " " {TEXT 200 220 "If we are to look at planetary motion from calculus, \+ what we know is that the only force that acts on the planets is gravit y and that it is in the direction of the sun with a force inversely pr oportional to the distance." }{TEXT 200 0 "" }}{PARA 0 "" 0 "" {TEXT 200 0 "" }}{PARA 210 "" 0 "" {TEXT 200 48 "In terms of differential eq uations this becomes:" }}{PARA 210 "" 0 "" {TEXT 200 28 "x'' = -x/r^3, y'' = -y/r^3." }}{PARA 210 "" 0 "" {TEXT 200 65 "(Note that x^2 + y^ 2 = r^2 and that sqrt((x'')^2+(y'')^2)=1/r^2.)" }}{PARA 210 "" 0 "" {TEXT 200 0 "" }}{PARA 210 "" 0 "" {TEXT 200 209 "We have only worked \+ with vector fields that are first order differential equations and thi s system is second order. We handle that problem by defining u=x' and v=y' to get a system of 4 first order equations." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "x:='x': y:= 'y': u :='u': v:= 'v': t:= 't':" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ODEa \+ := diff(x(t),t)=u(t):\n" }{MPLTEXT 1 0 28 "ODEb := diff(y(t),t)=v(t): \n" }{MPLTEXT 1 0 51 "ODEc := diff(u(t),t)=-x(t)/(x(t)^2+y(t)^2)^(3/2) :\n" }{MPLTEXT 1 0 51 "ODEd := diff(v(t),t)=-y(t)/(x(t)^2+y(t)^2)^(3/2 ):\n" }{MPLTEXT 1 0 33 "eqns := [ODEa, ODEb, ODEc, ODEd];" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&/-I%diffG%*protectedG6$-I\" xG6\"6#I\"tGF*F,-I\"uGF*F+/-F%6$-I\"yGF*F+F,-I\"vGF*F+/-F%6$F-F,,$*&F( \"\"\",&*$F(\"\"#F;*$F2F>F;#!\"$F>!\"\"/-F%6$F4F,,$*&F2F;F " 0 "" {MPLTEXT 1 0 19 "ICx := x(0) = 1: \n" } {MPLTEXT 1 0 18 "ICy := y(0) = 0:\n" }{MPLTEXT 1 0 18 "ICu := u(0) = 0 :\n" }{MPLTEXT 1 0 18 "ICv := v(0) = 1:\n" }{MPLTEXT 1 0 69 "solcurve \+ := dsolve(\{ODEa, ODEb, ODEc, ODEd, ICx, ICy, ICu, ICv\}, \n" } {MPLTEXT 1 0 40 " [x(t), y(t), u(t), v(t)], numeric);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I(x_rkf45G6\"6)I$resGF%I% dataGF%I%varsGF%I)solnprocGF%I)outpointGF%I&ndsolGF%I\"iGF%6#IinCopyri ght~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.GF%F%C*>I8_Env DSNumericSaveDigitsGF%I'DigitsGF%>F3\"#:@%/I-_EnvInFsolveGF%I%trueG%*p rotectedG>8(-&I&evalfGF:6#F26#9$>F<-F?FA>8%`6$I$GetGF%I$SetGF%b6#I+thi smoduleGF%6#I%DataGF%F%FHF%F%F%F%I*_m6681100GF%FN>8&-_FFFI6#Q(sysvarsF %>8'-FT6#Q/soln_procedureF%@$4-I%typeGF:6$F<.I(numericGF:C$@,-I'member GF:6$FB7+Q&startF%.I&startGF%Q'methodF%.I'methodGF%Q%leftF%Q&rightF%Q) leftdataF%Q*rightdataF%Q+enginedataF%C$>8$-FX6#-I(convertGF:6$FB.I'str ingGF:@&-Fin6$F`p.I&arrayGF:O-I%evalGF:6$F`p\"\"\"0F`pQ)procnameF%OF`p -F`o6$FB7&Q%lastF%.I%lastGF%Q(initialF%.I(initialGF%C$F_p@$-Fin6$F`p.I %listGF:O7#-I$seqGF:6$/&FR6#,&8*FaqFaqFaq&F`pF[s/F]s;\"\"!\"\"%3-Fin6$ FF`p-FX6 #/F[r7$Fas-I#opGF:6#FatYQKinitial~values~must~be~specified~in~a~listF% F_r/F8) -I(pointtoGF:6#&-FT6#Q0soln_proceduresF%6#FasO-.F]vFAZ%C$>F`p-FXF[tFer F%YF%F%F%F%" }}}{PARA 0 "" 0 "" {TEXT 200 74 "Then we can look at wher e the planet is at times Pi/2, Pi, 3Pi/2, and 2Pi." }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 23 "P1 := solcurve(Pi/2);\n" }{MPLTEXT 1 0 21 "P 2 := solcurve(Pi);\n" }{MPLTEXT 1 0 25 "P3 := solcurve(3*Pi/2);\n" } {MPLTEXT 1 0 21 "P4 := solcurve(2*Pi);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "7'/I\"tG6\"$\"0!\\zEjzq:!#9/-I\"xGF%6#F$$!3_y:!=O q\"y[!#C/-I\"yGF%F,$\"2Gr!ynC******!#!R! #B/-I\"uGF%F,$\"3IR27*[*)4z%F5/-I\"vGF%F,$!3H1!yQH,++\"!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "7'/I\"tG6\"$\"0o%Q!)*)Q7Z!#9/-I\"xGF%6#F$$\"3&GJ >^$\\U06!#A/-I\"yGF%F,$!3g*\\&e`e'*****!#=/-I\"uGF%F,$\"3/'=(o))>++5!# !#A/-I\"uGF%F,$!3m&\\lcY\"[3?F5/-I\"vGF%F,$\"31ax'Q\">++5!#<" }}}{PARA 0 "" 0 "" {TEXT 200 67 "We are really interested in the valu es of x and y at those 4 times." }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "P1a := [op(2,P1[2]),op(2,P1[3])];\n" }{MPLTEXT 1 0 35 "P2a := [op(2,P2[2]),op(2,P2[3])];\n" }{MPLTEXT 1 0 35 "P3a := \+ [op(2,P3[2]),op(2,P3[3])];\n" }{MPLTEXT 1 0 33 "P4a := [op(2,P4[2]),op (2,P4[3])];" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$$!3_ y:!=Oq\"y[!#C$\"2Gr!ynC******!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$$! 3WH8mzu(*****!#=$!3=D70%H8>!R!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$$ \"3&GJ>^$\\U06!#A$!3g*\\&e`e'*****!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$$\"3koxfm;'*****!#=$\"3\")4COaF'3)>!#A" }}}{PARA 0 "" 0 "" {TEXT 200 146 "This lets us see that with 6 decimal points of accuracy, the \+ path is a circle with period 2Pi. We can graph the results to see thi s more clearly." }{TEXT 200 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(plots):\n" }{MPLTEXT 1 0 70 "pointplot([seq([op(2,solcurv e(t*.05)[2]),op(2,solcurve(t*.05)[3])], \n" }{MPLTEXT 1 0 35 " \+ t=0..125)], connect=true);" }{MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6#-%'CURVESG6#7jr7$$\"\"\"\"\"!$F*F *7$$\"+rD]()**!#5$\"+:y\"z*\\!#67$$\"+!fT+&**F/$\"+6=M$)**F27$$\"+%o5x ))*F/$\"+%Q\"Q%\\\"F/7$$\"+/cm+)*F/$\"+1Mp')>F/7$$\"+HS7*o*F/$\"+a&RSZ #F/7$$\"+qXO`&*F/$\"+0A?bHF/7$$\"+(oEPR*F/$\"+a\"y*GMF/7$$\"+,&41@*F/$ \"+!G$=%*QF/7$$\"+b.Z/!*F/$\"+/al\\VF/7$$\"+(oA&F/7$$\"+4^N`#)F/$\"+@WUYcF/7$$\"+1k$3'zF/$\"+1R'=0'F /7$$\"+k-U[wF/$\"+[iIS&F/$\"+`xq9%)F/7$$\"+Fqqv\\F/$\"+/(HUn)F/ 7$$\"+1!ef`%F/$\"+?/27*)F/7$$\"+P5([3%F/$\"+VfjF\"*F/7$$\"+INdBOF/$\"+ 4rQ?$*F/7$$\"+Y'>K:$F/$\"+v=%)*[*F/7$$\"+8W)\\n#F/$\"+eqdN'*F/7$$\"+#F/$\"+@%Gsv*F/7$$\"+%)om*p\"F/$\"+4;\\a)*F/7$$\"+tK-07F/$\"+yO7F** F/7$$\"+/KntqF2$\"+fJ%\\(**F/7$$\"+YOVz?F2$\"+b.$y***F/7$$!+h1+?HF2$\" +n\"Gd***F/7$$!+M%R@\"zF2$\"+I=ko**F/7$$!+'4]%)G\"F/$\"+$**Qm\"**F/7$$ !+wcY#y\"F/$\"+9(\\)R)*F/7$$!+$HE?F#F/$\"+#yl%Q(*F/7$$!+Nx!fv#F/$\"+Q2 u7'*F/7$$!+P3!HB$F/$\"+J)))HY*F/7$$!+*o8=q$F/$\"+zSe*G*F/7$$!+hMZhTF/$ \"+*HgH4*F/7$$!+%oJ2h%F/$\"+x)3O())F/7$$!+AfY[]F/$\"+,w2K')F/7$$!+[6et aF/$\"+62(*o$)F/7$$!+m\\,&)eF/$\"+kf%\\3)F/7$$!+X(Rm,V#*F/$\"+M$yl\"QF/7$$!+PZ@A%*F/$\"+s([)\\LF/7$$!+^ ?'yd*F/$\"+AfuuGF/7$$!+!Rp&4(*F/$\"+*HeCR#F/7$$!+^y+<)*F/$\"+42>/>F/7$ $!+,)3***)*F/$\"+)3j6T\"F/7$$!+*)[1e**F/$\"+'o'3Y\"*F27$$!+24L\"***F/$ \"+DJodTF27$$!+eOi****F/$!+PI;6%)!#77$$!+lB#H)**F/$!+:5\"y$eF27$$!+[)o 7%**F/$!+u8*>3\"F/7$$!+,rwu)*F/$!+lu\\x:F/7$$!+YMe$y*F/$!+S.1p?F/7$$!+ \\d%zm*F/$!+VDRf(F/$!+(H%G1lF/7$$!+0TDfsF/$!+/GpxoF/7$$!+G\"Qk!pF/$!+/0\">B (F/7$$!+@-OOlF/$!+390ovF/7$$!+3[%*\\hF/$!+%3w_)yF/7$$!+')o:[dF/$!+3?z# =)F/7$$!+o9+K`F/$!+]Z&)f%)F/7$$!+Q%>D!\\F/$!+/Uo)*F/7$$!+FwT@6F/$!+'G))o$**F/7$$!+G\"ePB'F2$!+Ts^!)**F/7$ $!+GHwP7F2$!+K)*>****F/7$$\"+@rKhPF2$!+\\%*)G***F/7$$\"+#y<5v)F2$!+WDJ'F/7$$\"+nX`h!)F /$!+[I+h)F/$!+)GFE3&F/7$$\"+d;Cb ))F/$!+\")H&ek%F/7$$\"+(frj2*F/$!+NqY(>%F/7$$\"+Q`\"[F*F/$!+W$*eQPF/7$ $\"+]n2]%*F/$!+cjOqKF/7$$\"+&R< " 0 "" {MPLTEXT 1 0 43 "e := 0.8; a := 1; tmax := 2* Pi*a*sqrt(a):\n" }{MPLTEXT 1 0 25 "ICx := x(0) = a*(1-e); \n" } {MPLTEXT 1 0 18 "ICy := y(0) = 0:\n" }{MPLTEXT 1 0 18 "ICu := u(0) = 0 :\n" }{MPLTEXT 1 0 45 "ICv := v(0) = evalf(sqrt((1+e)/(a*(1-e))));\n" }{MPLTEXT 1 0 69 "solcurve := dsolve(\{ODEa, ODEb, ODEc, ODEd, ICx, IC y, ICu, ICv\}, \n" }{MPLTEXT 1 0 42 " [x(t), y(t), u(t), v(t)], nu meric);\n" }{MPLTEXT 1 0 48 "pointplot([seq([op(2,solcurve(t*tmax/100) [2]),\n" }{MPLTEXT 1 0 39 " op(2,solcurve(t*tmax/100)[3])], \n" } {MPLTEXT 1 0 56 " t=0..100)], connect=true, scaling=constraine d);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"\")!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "/-I\"xG6\"6#\"\"!$\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "/-I\"vG 6\"6#\"\"!$\"+++++I!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I(x_rkf45 G6\"6)I$resGF%I%dataGF%I%varsGF%I)solnprocGF%I)outpointGF%I&ndsolGF%I \"iGF%6#IinCopyright~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserv ed.GF%F%C*>I8_EnvDSNumericSaveDigitsGF%I'DigitsGF%>F3\"#:@%/I-_EnvInFs olveGF%I%trueG%*protectedG>8(-&I&evalfGF:6#F26#9$>F<-F?FA>8%`6$I$GetGF %I$SetGF%b6#I+thismoduleGF%6#I%DataGF%F%FHF%F%F%F%I*_m8034984GF%FN>8&- _FFFI6#Q(sysvarsF%>8'-FT6#Q/soln_procedureF%@$4-I%typeGF:6$F<.I(numeri cGF:C$@,-I'memberGF:6$FB7+Q&startF%.I&startGF%Q'methodF%.I'methodGF%Q% leftF%Q&rightF%Q)leftdataF%Q*rightdataF%Q+enginedataF%C$>8$-FX6#-I(con vertGF:6$FB.I'stringGF:@&-Fin6$F`p.I&arrayGF:O-I%evalGF:6$F`p\"\"\"0F` pQ)procnameF%OF`p-F`o6$FB7&Q%lastF%.I%lastGF%Q(initialF%.I(initialGF%C $F_p@$-Fin6$F`p.I%listGF:O7#-I$seqGF:6$/&FR6#,&8*FaqFaqFaq&F`pF[s/F]s; \"\"!\"\"%3-Fin6$FF`p-FX6#/F[r7$Fas-I#opGF:6#FatYQKinitial~values~must~be~spec ified~in~a~listF%F_r/F8)-I(pointtoGF:6#&-FT6#Q0soln_proceduresF%6#FasO-.F]vFA Z%C$>F`p-FXF[tFerF%YF%F%F%F%" }}{PARA 13 "" 1 "" {GLPLOT2D 400 400 400 {PLOTDATA 2 "6$-%'CURVESG6#7aq7$$\"+++++?!#5$\"\"!F,7$$\"+R#eFc\"F*$ \"+q!)zaYDu\\F*7$$!+!f%)GK$F*$\"+DSG.`F*7$ $!+v/@%=%F*$\"+VY,YbF*7$$!+?BF'*\\F*$\"+L)GHs&F*7$$!+DTJjdF*$\"+X$*)z% eF*7$$!+=-@*['F*$\"+>$G6$fF*7$$!+X[OxrF*$\"+@BmzfF*7$$!+L5tIyF*$\"+(zR \"**fF*7$$!+aP'=X)F*$\"+g<(Q*fF*7$$!+Py(H/*F*$\"+?yFnfF*7$$!++3+1'*F*$ \"+(H>@#fF*7$$!+'zhU,\"!\"*$\"+o3mgeF*7$$!+25Vl5Ffp$\"+cJy%y&F*7$$!+' \\QU6\"Ffp$\"+L,2'p&F*7$$!+6-!3;\"Ffp$\"+F`'ef&F*7$$!+<&>_?\"Ffp$\"+p] K&[&F*7$$!+<))eZ7Ffp$\"+BoWl`F*7$$!+Q5*zG\"Ffp$\"+#3.rB&F*7$$!+r4]E8Ff p$\"+$3h55&F*7$$!+Vg=j8Ffp$\"+bj*z&\\F*7$$!+Qt5)R\"Ffp$\"+iJ^3[F*7$$!+ G/KJ9Ffp$\"+NJ:`YF*7$$!+5d(GY\"Ffp$\"+\"4+C\\%F*7$$!+$3>G\\\"Ffp$\"+bR pEVF*7$$!+-D>@:Ffp$\"+kiVcTF*7$$!+BV.[:Ffp$\"+AJ*>)RF*7$$!+^%zLd\"Ffp$ \"+Xup.QF*7$$!+9)fsf\"Ffp$\"+,%e=i$F*7$$!++Yq>;Ffp$\"+UIwOMF*7$$!+!fS2 k\"Ffp$\"+h'y'[KF*7$$!+k@Rg;Ffp$\"+.H&y0$F*7$$!+^9oy;Ffp$\"+$>;X'GF*7$ $!+\\'Gcp\"Ffp$\"+n!)))oEF*7$$!+r?D6w:F#F*7$$!+!R#fQW5F*7$$!+4b,y;Ffp$!+8XrOMF*7$$!+BJF(f\"Ffp$!+<)4=i$F*7$$!+JORt:Ff p$!+$**\\O!QF*7$$!+,&\\![:Ffp$!+)=[>)RF*7$$!+J'37_\"Ffp$!+*G$RcTF*7$$! +\"*f$G\\\"Ffp$!+\"y\\mK%F*7$$!+/N*GY\"Ffp$!+!4cB\\%F*7$$!+_#R8V\"Ffp$ !+%*=6`YF*7$$!+ks7)R\"Ffp$!+:dZ3[F*7$$!+Up?j8Ffp$!+#=fz&\\F*7$$!+GH_E8 Ffp$!+A_-,^F*7$$!+xT,)G\"Ffp$!+T<2P_F*7$$!+_JhZ7Ffp$!+F'=aO&F*7$$!+h]C 07Ffp$!+e\"*H&[&F*7$$!+bq#3;\"Ffp$!+S_%ef&F*7$$!+xnE96Ffp$!+,f0'p&F*7$ $!+q1Yl5Ffp$!+#*=x%y&F*7$$!+#*HH95Ffp$!+&)flgeF*7$$!+f#Hjg*F*$!+M@7AfF *7$$!+RPKV!*F*$!+F)*GnfF*7$$!+\\%GAX)F*$!+%o$*Q*fF*7$$!+2e6JyF*$!+cW<* *fF*7$$!+p9xxrF*$!+(36(zfF*7$$!+^0k*['F*$!+N\\>JfF*7$$!+X-xjdF*$!+0x2[ eF*7$$!+Qnv'*\\F*$!+&)Q/BdF*7$$!+Vgs%=%F*$!+SP;YbF*7$$!+-VVBLF*$!+fxZ. `F*7$$!+Cht3CF*$!+])3X(\\F*7$$!+S:>Q9F*$!+/JjFXF*7$$!+6==kTF5$!+ " 0 "" {MPLTEXT 1 0 49 "ee := 0.02; ae := 1; tmaxe := 2*Pi*ae*sqrt(ae); \n" }{MPLTEXT 1 0 28 "ICxe := x(0) = ae*(1-ee); \n" }{MPLTEXT 1 0 19 " ICye := y(0) = 0:\n" }{MPLTEXT 1 0 19 "ICue := u(0) = 0:\n" }{MPLTEXT 1 0 49 "ICve := v(0) = evalf(sqrt((1+ee)/(ae*(1-ee))));\n" }{MPLTEXT 1 0 74 "solcurvee := dsolve(\{ODEa, ODEb, ODEc, ODEd, ICxe, ICye, ICue , ICve\}, \n" }{MPLTEXT 1 0 42 " [x(t), y(t), u(t), v(t)], numeric );\n" }{MPLTEXT 1 0 53 "em := 0.21; am := 36/93; tmaxm := 2*Pi*am*sqrt (am);\n" }{MPLTEXT 1 0 28 "ICxm := x(0) = am*(1-em); \n" }{MPLTEXT 1 0 19 "ICym := y(0) = 0:\n" }{MPLTEXT 1 0 19 "ICum := u(0) = 0:\n" } {MPLTEXT 1 0 49 "ICvm := v(0) = evalf(sqrt((1+em)/(am*(1-em))));\n" } {MPLTEXT 1 0 74 "solcurvem := dsolve(\{ODEa, ODEb, ODEc, ODEd, ICxm, I Cym, ICum, ICvm\}, \n" }{MPLTEXT 1 0 40 " [x(t), y(t), u(t), v(t)] , numeric);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"\"# !\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I#PiG%*protectedG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "/-I\"xG6\"6#\"\"!$\"#)*!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "/-I\"vG6 \"6#\"\"!$\"+hS??5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I(x_rkf45G 6\"6)I$resGF%I%dataGF%I%varsGF%I)solnprocGF%I)outpointGF%I&ndsolGF%I\" iGF%6#IinCopyright~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved .GF%F%C*>I8_EnvDSNumericSaveDigitsGF%I'DigitsGF%>F3\"#:@%/I-_EnvInFsol veGF%I%trueG%*protectedG>8(-&I&evalfGF:6#F26#9$>F<-F?FA>8%`6$I$GetGF%I $SetGF%b6#I+thismoduleGF%6#I%DataGF%F%FHF%F%F%F%I+_m17795824GF%FN>8&-_ FFFI6#Q(sysvarsF%>8'-FT6#Q/soln_procedureF%@$4-I%typeGF:6$F<.I(numeric GF:C$@,-I'memberGF:6$FB7+Q&startF%.I&startGF%Q'methodF%.I'methodGF%Q%l eftF%Q&rightF%Q)leftdataF%Q*rightdataF%Q+enginedataF%C$>8$-FX6#-I(conv ertGF:6$FB.I'stringGF:@&-Fin6$F`p.I&arrayGF:O-I%evalGF:6$F`p\"\"\"0F`p Q)procnameF%OF`p-F`o6$FB7&Q%lastF%.I%lastGF%Q(initialF%.I(initialGF%C$ F_p@$-Fin6$F`p.I%listGF:O7#-I$seqGF:6$/&FR6#,&8*FaqFaqFaq&F`pF[s/F]s; 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