Computer Algebra Recipes

178 CHAPTER 4. NONLINEAR ODE MODELS
x(0) = 0:5, y(0) = 0, z(0) = 0, _x(0) = 0, _y(0) = ¡2, and _z(0) = ¡0:2.
> m:=1: q:=1:
> ic:=(x(0)=0.5,y(0)=0,z(0)=0,D(x)(0)=0,D(y)(0)=-2,D(z)(0)=0.2):
Maple is not able to produce an explicit closed-form analytic solution for the
nonlinear ODE system, sys, so a numerical solution is generated, the output
being given as a \listprocedure."
> sol:=dsolve(fsys,icg,fx(t),y(t),z(t)g,numeric,
output=listprocedure):
Evaluating x(t), y(t), and z(t) withsol will allow Vectoria to calculate the
charge's position at an arbitrary time t.
> X:=eval(x(t),sol): Y:=eval(y(t),sol): Z:=eval(z(t),sol):
For example, the charge's x-coordinate at t = 2 time units is now evaluated.
> X(2);
¡0:0727500262205896948
Vectoria will now animate the motion of the charge in the magnetic bottle. She
takes the total time to be T =24:0time units and will have N =200time
steps in her animation. The time step size T=N is then calculated.
> T:=24.0: N:=200: step:=T/N;
step := 0:1200000000
The spacecurve command is used to plot (but not display) the entire trajectory
overthetimeintervalt =0toT , the trajectory being colored with the zhue
option. To obtain a smooth curve, the minimum number of plotting points is
taken to be 1000.
> sc:=spacecurve([X(t),Y(t),Z(t)],t=0..T,shading=zhue,
numpoints=1000):
Using a do loop running from n =0toN, the charge is plotted on each time step
in 3 dimensions as a size-18 red circle superimposed on the entire trajectory.
> for n from 0 to N do
> t:=step*n;
> pp[n]:=pointplot3d([X(t),Y(t),Z(t)],symbol=circle,
symbolsize=18,color=red);
> gr[n]:=display(fsc,pp[n]g);
> end do:
The motion of the charge in the magnetic bottle is animated with the display
command, using the insequence=true option. To see the animation execute
the worksheet, click on the plot, and on the start arrow in the tool bar.
> display(seq(gr[n],n=0..N),insequence=true,axes=frame,
labels=["x","y","z"]);
Vectoria is quite pleased that her piecewise model mimics magnetic bottle behavior.
But now she has to forsake this bottle for another one, as she rushes
o® to join her girlfriends and share a bottle of Asti Spumante.