08-03-04.html

> with(plots);

(1)

> fieldplot( [ ( v^2 - cos(theta))/v, -sin(theta) ], theta=-Pi..Pi, v=0..3, arrows=slim);

Plot_2d

> 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 ];

>

(2)

> R:=0;

(3)

> DEplot( phug, [ theta(t), v(t) ], t=0..1, theta=-Pi..Pi, v=0..3);

Plot_2d

> DEplot( phug, [ theta(t), v(t) ], t=0..3,
[[theta(0)=0, v(0)=0.5], [theta(0)=0, v(0)=2]],
theta=-Pi..Pi, v=0..3);

Plot_2d

> DEplot( phug, [ theta(t), v(t) ], t=0..6,
[[theta(0)=0, v(0)=0.5], [theta(0)=0, v(0)=2]],
theta=-Pi/2..2*Pi, v=0..3);

Plot_2d

> DEplot( phug, [ theta(t), v(t) ], t=0..6,
       [seq([theta(0)=0, v(0)=0.5+2*i/10], i=0..10)],
       theta=-Pi/2..2*Pi, v=0..3,
       linecolor=black);

Plot_2d

> DEplot( phug, [ theta(t), v(t) ], t=0..6,

> [seq([theta(0)=0, v(0)=0.5+2*i/10], i=0..10)],

> theta=-Pi/2..2*Pi, v=0..3,

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

>

>

Plot_2d

> DEplot( phug, [ theta(t), v(t) ], t=0..6,

> [seq([theta(0)=0, v(0)=0.5+2*i/10], i=0..10)],

> theta=-Pi/2..2*Pi, v=0..3,

> linecolor=[seq(COLOR(HUE, i/10), i=0..10)], stepsize=0.01);

>

>

Plot_2d

> R:=0.1;

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

> [seq([theta(0)=0, v(0)=0.5+2*i/10], i=0..10)],

> theta=-Pi/2..3*Pi, v=0..3,

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

Plot_2d

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

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

> theta=-Pi/2..3*Pi, v=0..3,

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

>

Plot_2d

> R:=0.25;

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

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

> theta=-Pi/2..3*Pi, v=0..3,

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

>

>

Plot_2d

>