{VERSION 5 0 "SUN SPARC SOLARIS" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plo t" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "R:=`R`:\nxphug :=[ \+ diff(theta(t),t) = (v(t)^2 - cos(theta(t)))/v(t),\n diff(v(t ),t) = -sin(theta(t)) - R*v(t)^2 ,\n diff(x(t),t) = \+ v(t)*cos(theta(t)),\n diff(y(t),t) = v(t)*sin(theta(t)) ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xphugG7&/-%%diffG6$-%&thetaG6 #%\"tGF-*&,&*$)-%\"vGF,\"\"#\"\"\"F5-%$cosG6#F*!\"\"F5F2F9/-F(6$F2F-,& -%$sinGF8F9*&F4F5F1F5F9/-F(6$-%\"xGF,F-*&F2F5F6F5/-F(6$-%\"yGF,F-*&F2F 5F>F5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "phug := [ xphug[1] , xphug[2] ];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%phugG7$/-%%diffG6$ -%&thetaG6#%\"tGF-*&,&*$)-%\"vGF,\"\"#\"\"\"F5-%$cosG6#F*!\"\"F5F2F9/- F(6$F2F-,&-%$sinGF8F9*&%\"RGF5F1F5F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "phugR:= subs( \{v(t)=v, theta(t)= theta\}, \n \+ map( rhs, phug));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&phugRG7$ *&,&*$)%\"vG\"\"#\"\"\"F,-%$cosG6#%&thetaG!\"\"F,F*F1,&-%$sinGF/F1*&% \"RGF,F)F,F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "J:=jacobian(phugR, [t heta, v]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'matrixG6#7$7$*&-%$sinG6#%&thetaG\"\"\"%\"vG!\" \",&\"\"#F/*&,&*$)F0F3F/F/-%$cosGF-F1F/F0!\"#F17$,$F8F1,$*&%\"RGF/F0F/ F:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "trace(J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%$sinG6#%&thetaG\"\"\"%\"vG!\"\"F)*(\"\"#F )%\"RGF)F*F)F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(J);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,(*(-%$sinG6#%&thetaG\"\"\"%\"RGF +)%\"vG\"\"#F+F/*&-%$cosGF)F+F-F+!\"\"*$)F1F/F+F3F+F.!\"#F3" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "solve( \{phugR[1]=0, phugR[2 ]=0\}, \{theta,v\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%&thetaG-%' arctanG6$,$*&%\"RG\"\"\"-%'RootOfG6#,&!\"\"F,*&)%#_ZG\"\"#F,,&*$)F+F5F ,F,F,F,F,F,F,F1F-/%\"vG-F.6#,&F-F1*$F3F,F," }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 24 "fix:=convert(%,radical);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fixG<$/%&thetaG-%'arctanG6$,$*&%\"RG\"\"\"-%%sqrtG6# *&F.F.,&*$)F-\"\"#F.F.F.F.!\"\"F.F7*$F/F./%\"vG*$)F2#F.\"\"%F." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "trfix:= simplify(subs(fix, t race(J)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&trfixG,$*&%\"RG\"\"\" )*&F(F(,&*$)F'\"\"#F(F(F(F(!\"\"#F(\"\"%F(!\"$" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 36 "detfix:= simplify(subs(fix,det(J)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'detfixG,$*&\"\"\"F'*$-%%sqrtG6#*&F'F',&*$ )%\"RG\"\"#F'F'F'F'!\"\"F'F2F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "solve( (trfix)^2 = 4*detfix, R);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*$-%%sqrtG6#\"\"#\"\"\"F(,$F$!\"#" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+CrU GG!\"*$!+CrUGGF%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(DE tools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "destuff:= xphug , [ theta(t), v(t), x(t), y(t)], t=0..5, \n [[ theta(0)=0, v(0)= 2, \+ x(0)=0, y(0)=2 ]],\n theta=-Pi...Pi, v=0.01..3, x=0..5, y=0..4, ste psize=0.1, linecolor=black;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%(dest uffG6,7&/-%%diffG6$-%&thetaG6#%\"tGF.*&,&*$)-%\"vGF-\"\"#\"\"\"F6-%$co 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"Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Cur ve 22" "__never_display_this_legend_entry" "Curve 24" "Curve 25" "Curv e 26" "Curve 27" "Curve 28" "Curve 29" "Curve 30" "Curve 31" "Curve 32 " "Curve 33" "Curve 34" "Curve 35" "Curve 36" "Curve 37" "Curve 38" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "foo:=DEplot( destuff, scen e=[theta,v]):\nzoom(foo,-1.2..-1.05, 0.6..0.75);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 102 102 {PLOTDATA 2 "6&-%'CURVESG6%7U7$$\"\"!F)$\"\"#F)7 $$\"3n7#zg!oyw5!#=$\"37*Gl9/(oB9!#<7$$\"3^#plZNmT_\"F/$\"3GVa`bzM(4\"F 27$$\"3%*o9-!32C]\"F/$\"3B3?%QU3.())F/7$$\"3<7\"4F0Kg3\"F/$\"3(*QfQ*=' Q@uF/7$$\"3z%Q8'*[f:A$!#>$\"3eSt!)*zv'*R'F/7$$!31)=vPf9#otFE$\"3C$QkZi z=p&F/7$$!3*Q4gzj1>,#F/$\"3ak)4'[$[UB&F/7$$!331L'oe2QR$F/$\"3\">Az%pf$ =)\\F/7$$!3.&Q[:W@Hw%F/$\"3Qa:*=X*Q$*[F/7$$!3znl)***31@gF/$\"3G\"3,2( \\*p#\\F/7$$!3$f_9$\\.Z6rF/$\"3A^E9)3@C/&F/7$$!3*RL$yW]Y. 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