08-03-13.mw

 > R:='R': with(DEtools): with(plots): 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))];

 (1)

 > 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")]));

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

 > [seq( [theta(0)=0, v(0)=1.7+i/100, 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", view=[1..3,0..1]),      DEplot(xphug, guts,         scene=[x,y],title="glider path")]));

 > guts:= [ theta(t), v(t), x(t), y(t) ], t=0..20,

 > [seq( [theta(0)=0, v(0)=1+2*i/10, x(0)=0, y(0)=i+.1 ], i=0..12)],

 > theta=-Pi/2..3*Pi, v=0..3.5, x=-1..20, y=0..15,

 > linecolor=[seq(COLOR(HUE,i/12),i=0..12)], stepsize=0.1:

 >

 > R:=0.1; display( array([      DEplot(xphug, guts,         scene=[theta,v],title="v,theta phase"),      DEplot(xphug, guts,         scene=[x,y],title="glider path")]));

 >

 > R:=3;

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

 > [seq( [theta(0)=0, v(0)=1+2*i, x(0)=0, y(0)=i+.1 ], i=0..12)],

 > theta=-Pi/2..3*Pi, v=0..35, x=-1..6, y=0..16,

 > linecolor=[seq(COLOR(HUE,i/12),i=0..12)], stepsize=0.1:

 >

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

 >

 > R:='R': solve( {(v^2-cos(theta))/v = 0,  -sin(theta) - R*v^2 = 0}, {theta,v});

 (2)

 (3)

 > plot( [ seq( exp(a*t), a=-5..5)], t=-2..2, y=0..6);

 > DEplot( [ diff(x(t),t) = 4*x(t), diff(y(t),t)= -2*y(t)],  [x(t), y(t)], t=-5..5,        [seq( [x(0)=i/10, y(0)=1], i=-10..10 ), seq( [x(0)=i/10, y(0)=-1], i=-10..10 )], x=-2..2, y=-2..2, linecolor=black);

 >