08-03-11.mw

> with(DEtools):
 

> phug:= [ diff(theta(t),t) = ( v(t)^2 - cos(theta(t))) / v(t),
        diff(v(t),t)     = -sin(theta(t)) - R*v(t)^2 ];
 

>
 

[diff(theta(t), t) = `/`(`*`(`+`(`*`(`^`(v(t), 2)), `-`(cos(theta(t))))), `*`(v(t))), diff(v(t), t) = `+`(`-`(sin(theta(t))), `-`(`*`(R, `*`(`^`(v(t), 2)))))] (1)
 

> R:=0;
DEplot(phug, [ theta(t), v(t) ], t=0..40,
 

>        [[theta(0)=0, v(0)=2.5], [theta(0)=0, v(0)=.8]],
 

>        theta=-Pi/2..3*Pi, v=0..3,
 

>        linecolor=[blue,red], stepsize=0.1);
phase:=%:
 

 

0
Plot_2d
 

>
 

> DEplot(phug, [ theta(t), v(t) ], t=0..10,
 

>       [[theta(0)=0, v(0)=2.5], [theta(0)=0, v(0)=.8]],
 

>        theta=-Pi/2..3*Pi, v=0..3,
      scene=[t,theta],
 

>        linecolor=[blue,red], stepsize=0.1);
thetaplot:=%:
 

>
 

Plot_2d
 

> DEplot(phug, [ theta(t), v(t) ], t=0..10,
 

>    [[theta(0)=0, v(0)=2.5], [theta(0)=0, v(0)=.8]],
 

>        theta=-Pi/2..3*Pi, v=0..3,
      scene=[t,v],
 

>        linecolor=[blue,red], stepsize=0.1);
vplot:=%:
 

>
 

>
 

Plot_2d
 

> with(plots):
 

> display( array ([ phase, thetaplot, vplot ]));
 

Plot_2d Plot_2d Plot_2d

 

> R:='R':
xphug:= [ diff(theta(t),t) = ( v(t)^2 - cos(theta(t))) / v(t),
        diff(v(t),t)     = -sin(theta(t)) - R*v(t)^2 ,
        diff(x(t),t)     = v(t)*cos(theta(t)),
        diff(y(t),t)     = v(t)*sin(theta(t))];
 

[diff(theta(t), t) = `/`(`*`(`+`(`*`(`^`(v(t), 2)), `-`(cos(theta(t))))), `*`(v(t))), diff(v(t), t) = `+`(`-`(sin(theta(t))), `-`(`*`(R, `*`(`^`(v(t), 2))))), diff(x(t), t) = `*`(v(t), `*`(cos(theta(t...
[diff(theta(t), t) = `/`(`*`(`+`(`*`(`^`(v(t), 2)), `-`(cos(theta(t))))), `*`(v(t))), diff(v(t), t) = `+`(`-`(sin(theta(t))), `-`(`*`(R, `*`(`^`(v(t), 2))))), diff(x(t), t) = `*`(v(t), `*`(cos(theta(t...
(2)
 

> R:=0;
display( array([
     DEplot(xphug, [ theta(t), v(t), x(t), y(t) ], t=0..10,
 

>          [[theta(0)=0, v(0)=2.5, x(0)=0, y(0)=4 ],
          [theta(0)=0, v(0)=.8,  x(0)=0, y(0)=1 ]],
 

>         theta=-Pi/2..3*Pi, v=0..3, x=-1..6, y=0..7,
        scene=[theta,v],
 

>         linecolor=[blue,red], stepsize=0.1, title="v,theta phase"),
 

>       DEplot(xphug, [ theta(t), v(t), x(t), y(t) ], t=0..10,
 

>          [[theta(0)=0, v(0)=2.5, x(0)=0, y(0)=4 ],
          [theta(0)=0, v(0)=.8,  x(0)=0, y(0)=1 ]],
 

>         theta=-Pi/2..3*Pi, v=0..3, x=-1..6, y=0..7,
        scene=[x,y], title="path of glider",
 

>         linecolor=[blue,red], stepsize=0.1)
      ]));
 

>
 

 

0

Plot_2d Plot_2d

 

>
 

> R:=0.1;
display( array([
     DEplot(xphug, [ theta(t), v(t), x(t), y(t) ], t=0..10,
 

>          [[theta(0)=0, v(0)=2.5, x(0)=0, y(0)=4 ],
          [theta(0)=0, v(0)=.8,  x(0)=0, y(0)=1 ]],
 

>         theta=-Pi/2..3*Pi, v=0..3, x=-1..6, y=0..7,
        scene=[theta,v],
 

>         linecolor=[blue,red], stepsize=0.1, title="v,theta phase"),
 

>       DEplot(xphug, [ theta(t), v(t), x(t), y(t) ], t=0..10,
 

>          [[theta(0)=0, v(0)=2.5, x(0)=0, y(0)=4 ],
          [theta(0)=0, v(0)=.8,  x(0)=0, y(0)=1 ]],
 

>         theta=-Pi/2..3*Pi, v=0..3, x=-1..6, y=0..7,
        scene=[x,y], title="path of glider",
 

>         linecolor=[blue,red], stepsize=0.1)
      ]));
 

>
 

 

.1

Plot_2d Plot_2d

 

> R:=0.8;
display( array([
     DEplot(xphug, [ theta(t), v(t), x(t), y(t) ], t=0..10,
 

>          [[theta(0)=0, v(0)=2.5, x(0)=0, y(0)=4 ],
          [theta(0)=0, v(0)=.8,  x(0)=0, y(0)=1 ]],
 

>         theta=-Pi/2..3*Pi, v=0..3, x=-1..6, y=0..7,
        scene=[theta,v],
 

>         linecolor=[blue,red], stepsize=0.1, title="v,theta phase"),
 

>       DEplot(xphug, [ theta(t), v(t), x(t), y(t) ], t=0..10,
 

>          [[theta(0)=0, v(0)=2.5, x(0)=0, y(0)=4 ],
          [theta(0)=0, v(0)=.8,  x(0)=0, y(0)=1 ]],
 

>         theta=-Pi/2..3*Pi, v=0..3, x=-1..6, y=0..7,
        scene=[x,y], title="path of glider",
 

>         linecolor=[blue,red], stepsize=0.1)
      ]));
 

>
 

 

.8

Plot_2d Plot_2d

 

> R:=3;
display( array([
     DEplot(xphug, [ theta(t), v(t), x(t), y(t) ], t=0..10,
 

>          [[theta(0)=0, v(0)=2.5, x(0)=0, y(0)=4 ],
          [theta(0)=0, v(0)=.8,  x(0)=0, y(0)=1 ]],
 

>         theta=-Pi/2..3*Pi, v=0..3, x=-1..6, y=0..7,
        scene=[theta,v],
 

>         linecolor=[blue,red], stepsize=0.1, title="v,theta phase"),
 

>       DEplot(xphug, [ theta(t), v(t), x(t), y(t) ], t=0..10,
 

>          [[theta(0)=0, v(0)=2.5, x(0)=0, y(0)=4 ],
          [theta(0)=0, v(0)=.8,  x(0)=0, y(0)=1 ]],
 

>         theta=-Pi/2..3*Pi, v=0..3, x=-1..6, y=0..7,
        scene=[x,y], title="path of glider",
 

>         linecolor=[blue,red], stepsize=0.1)
      ]));
 

>
 

>
 

 

3

Plot_2d Plot_2d

 

> Plot_2d
 

> guts:= [ theta(t), v(t), x(t), y(t) ], t=0..10,
 

>          [[theta(0)=0, v(0)=2.5, x(0)=0, y(0)=4 ],
          [theta(0)=0, v(0)=.8,  x(0)=0, y(0)=1 ],
          [theta(0)=0, v(0)=1,  x(0)=0, y(0)=3 ]],
 

>         theta=-Pi/2..3*Pi, v=0..3, x=-1..6, y=0..7,
 

>         linecolor=[blue,red, black], stepsize=0.1:
 

>
 

> R:=0;
display( array([
     DEplot(xphug, guts,
        scene=[theta,v],title="v,theta phase"),
     DEplot(xphug, guts,
        scene=[x,y],title="glider path")]));
 

>      
 

>
 

 

0

Plot_2d Plot_2d

 

> guts:= [ theta(t), v(t), x(t), y(t) ], t=0..10,
 

>          [seq( [theta(0)=0, v(0)=1+i/10, x(0)=0, y(0)=i+.1 ], i=0..10)],
 

>         theta=-Pi/2..3*Pi, v=0..3, x=-1..6, y=0..13,
 

>         linecolor=[seq(COLOR(HUE,i/10),i=0..10)], stepsize=0.1:
 

>
 

> R:=0;
display( array([
     DEplot(xphug, guts,
        scene=[theta,v],title="v,theta phase"),
     DEplot(xphug, guts,
        scene=[x,y],title="glider path")]));
 

 

0

Plot_2d Plot_2d

 

>