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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "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)) ];\n\nphug := [xphug[1], xphug[2]];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "with(plots): with(DEtools):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %&xphugG7&/-%%diffG6$-%&thetaG6#%\"tGF-*&,&*$)-%\"vGF,\"\"#\"\"\"F5-%$ cosG6#F*!\"\"F5F2F9/-F(6$F2F-,&-%$sinGF8F9*&%\"RGF5F1F5F9/-F(6$-%\"xGF ,F-*&F2F5F6F5/-F(6$-%\"yGF,F-*&F2F5F>F5" }}{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 0 261 "vphug :=[ diff(theta(t),t) \+ = (v(t)^2 - cos(theta(t))),\n diff(v(t),t) = v(t)*(-sin (theta(t)) - R*v(t)^2) ,\n diff(x(t),t) = v(t)^2*cos(the ta(t)),\n diff(y(t),t) = v(t)^2*sin(theta(t)),\n \+ diff(T(t),t) = v(t)];" 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Let's linearize them and see what's what." } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "J:=jacobian( [ v^2-cos(the ta), -v*sin(theta) - 'R'*v^3], [theta, v]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"JG-%'matrixG6#7$7$-%$sinG6#%&thetaG,$%\"vG\"\" #7$,$*&F/\"\"\"-%$cosGF,F4!\"\",&F*F7*(\"\"$F4%\"RGF4)F/F0F4F7" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "evalf(subs(\{v=0, theta=Pi/2 \}, evalm(J)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$$\" \"\"\"\"!$F*F*7$F+$!\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "So \+ its a saddle, with eigenvectors parallel to the coordinate axes." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "evalf(subs(\{v=0, theta=-Pi/ 2\}, evalm(J)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7$7$$! \"\"\"\"!$F*F*7$F+$\"\"\"F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 705 "some:=\n DEplot( vphug, [theta(t),v(t),x(t),y(t),T(t)], t=0..10 ,\n [ seq([theta(0)=0, v(0)=1+i/5, T(0)=0, x(0)=0, y(0)=2], i=0..10)] ,\n theta=-Pi/2..5*Pi/2, v=0..3, T=0..10, x=-1..5, y=0..3,\n linecol or=black, obsrange=true, stepsize=0.1, scene=[theta,v]):\n special:=\n DEplot( vphug, [theta(t),v(t),x(t),y(t),T(t)], t=-10..10,\n [ [the ta(0)=Pi/2, v(0)=0.01,T(0)=0, x(0)=0, y(0)=2],\n [theta(0)=-Pi/2,v( 0)=0.01,T(0)=0, x(0)=0, y(0)=2],\n [theta(0)=3*Pi/2,v(0)=0.01,T(0)= 0, x(0)=0, y(0)=2],\n [theta(0)=5*Pi/2,v(0)=0.01,T(0)=0, x(0)=0, y( 0)=2]],\n theta=-Pi/2..5*Pi/2, v=0..3, T=0..10, x=-1..5, y=0..3,\n l inecolor=red, obsrange=true, stepsize=0.1, scene=[theta,v]):\nplots[di splay](\{some,special\});" }}{PARA 13 "" 1 "" {GLPLOT2D 398 194 194 {PLOTDATA 2 "6&-%'CURVESG6)7ew7$%%FAILGF(F'F'F'F'F'F'F'F'F'F'F'F'F'F'F 'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'F'7$$!3%y#>u4ZH^N !#<$\"3\\e.kF#)GjMF,7$$!3V5.p#HN*zCF,$\"3m<.`H)e6*GF,7$$!3')G2R%ox8m\" F,$\"3'R3;(4')G*p#F,7$$!3`#pVeP&)o))*!#=$\"3g68U7Ld\"e#F,7$$!3(G%\\pg/ *fM%F<$\"3g'**e0]wgU#F,7$$\"31S%Q#)\\D()***!#?$\"3:OAIt10@AF,7$$\"3hmD 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