Creating a movie of an animated trajectory

Question

I would like to make a movie of an animated trajectory but the computation seems to be kind of slow on my machine. Consider the harmonic oscillator

harmonicsol = {x, x'} /. NDSolve[{x''[t] + 0.05 x'[t] + 0.5 x[t] == 0, x[0] == 1, x'[0] == -1},{x, x'}, {t, 0, 100}][[1]];

the animation doesn't seem to be very laggy

Animate[Graphics[{Line[{harmonicsol[[1]][#], harmonicsol[[2]][#]} & /@
  Range[0, t, 0.01]]}, PlotRange -> 2], {t, 0, 100}, AnimationRate -> 1]

but when I compute a table of graphics (to export them later on)

Table[Graphics[{Line[{harmonicsol[[1]][#], harmonicsol[[2]][#]} & /@ Range[0, t, 0.01]]}, PlotRange -> 2], {t, 0, 100, 0.1}]

it takes quite some time. Are there any ways to speed that up or better ways to make a movie out of this?

Answer 1

ParametricPlot could simplify, e.g.

eqn = {y[t] == x'[t], y'[t] == -0.05  y[t] - 0.5 x[t], x[0] == 1, 
   y[0] == -1};
sol = NDSolve[eqn, {x, y}, {t, 0, 100}][[1]];
p[r_] := ParametricPlot[{x[t], y[t]} /. sol, {t, 0, r}, 
    PlotRange -> {{-2, 2}, {-2, 2}}, 
    Epilog -> {Red, PointSize[0.02], 
      Point[{(x[u] /. sol) /. u -> r, 0}]}] /. Line[u__] :> Arrow[u];
tab = Table[p[j], {j, 0.1, 100, 1}];
Export["e:/mse/mseho.gif", tab, 
 "DisplayDurations" -> Table[0.1, Length@tab], 
 "AnimationRepetitions" -> Infinity]