08-03-04.mw

 > with(plots);

 (1)

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

 > 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);

 > 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);

 > 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);

 > 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);

 > 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)]);

 >

 >

 > 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);

 >

 >

 > 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);

 > 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);

 >

 > 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);

 >

 >

 >