08-03-27.mw

 > jack := VectorCalculus[Jacobian](    [  (v^2 - cos(theta))/v, -sin(theta) - R*v^2 ],  [theta, v]);

 (1)

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

 (2)

 > eval(jack, fix);

 (3)

 > simplify(eval(jack, fix));

 (4)

 > eval(%,R=0);

 (5)

 > jill := unapply( simplify(eval(jack, fix)), R):

 > jill(0); jill(R); jill(0.2);

 (6)

 > with(LinearAlgebra):

 > Eigenvalues(jill(0.2));

 (7)

 > [Trace(jill(0.2)), Determinant(jill(0.2)) ];

 (8)

for what R is tr^2 = 4*det?

 > discr := Trace(jill(R))^2 - 4*Determinant(jill(R));

 (9)

 > solve(discr=0, R);

 (10)

 > evalf(eval(discr,R=2*sqrt(2)+.1));

 (11)

 > evalf(eval(discr,R=2*sqrt(2)-.1));

 (12)

 > evalf(eval(discr,R=100));

 (13)

 > evalf(jill(100));

 (14)

 > Determinant(jill(100));

 (15)

 > Determinant(jill(R));

 (16)

 > 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))];

 (17)

 > with(DETools):with(plots):

 > R:=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)       ]));

 >

 > eval(fix);

 (18)

 > 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)       ]));

 >

 > Eigenvectors(jill(3.0)); evalf(fix);

 (19)

 >