Beispiel Lissajous Figuren
Ausarbeitung Helmut Hörner 8850092

Lösen der Differentialgleichungen

xFkt=DSolve[{x''[t]==-kx^2 x[t],x[0]==1,x'[0]==0}, x, t] //Flatten
yFkt=DSolve[{y''[t]==-ky^2 y[t],y[0]==0,y'[0]==4},y,t]//Flatten

Bei rationalem Verhältnis zwischen kx/ky (z.B. 1/3) ist die Kurve geschlossen:

Manipulate[ParametricPlot[{x[t] /. xFkt /. kx->KX,y[t] /. yFkt /. ky->KY},{t,0,1000},PlotPoints->500],{KX,1},{KY,3},ControlType->InputField]

Bei irrationalem Verhältnis zwischen kx/ky (z.B. 1/π) ist die Kurve nicht geschlossen:

Manipulate[ParametricPlot[{x[t] /. xFkt /. kx->KX,y[t] /. yFkt /. ky->KY},{t,0,1000},PlotPoints->500],{KX,1},{KY,π},ControlType->InputField]

Animation mit kx=9, ky=7 (geschlossene Kurve)

Animate[Show[ParametricPlot[{x[tt] /. xFkt /. kx->9,y[tt] /. yFkt /. ky->7},{tt,0,Evaluate[t]},PlotPoints->500,PlotRange->{{-1.2,1.2},{-1,1}}],Graphics[{ Disk[{x[t] /. xFkt /. kx->9,y[t] /. yFkt /. ky->7},0.05]},PlotRange->{{-1.2,1.2},{-1,1}}]],{t,0,10*2π},AnimationRate->.005]

Animation mit kx=9, ky=π (nicht geschlossene Kurve)

Animate[Show[ParametricPlot[{x[tt] /. xFkt /. kx->9,y[tt] /. yFkt /. ky->π},{tt,0,Evaluate[t]},PlotPoints->500,PlotRange->{{-1.2,1.2},{-1.5,1.5}}],Graphics[{ Disk[{x[t] /. xFkt /. kx->9,y[t] /. yFkt /. ky->π},0.05]},PlotRange->{{-1.2,1.2},{-1.5,1.5}}]],{t,0,10*2π},AnimationRate->.005]

Bewegungsgleichungen mit Dämpfung

xFktD=FullSimplify[DSolve[{x''[t]+kx^2 x[t]+g1 x'[t]==0 ,x[0]==1,x'[0]==0}, x, t]] //Flatten 
yFktD=FullSimplify[DSolve[{y''[t]+ky^2 y[t]+g2 y'[t]==0,y[0]==0,y'[0]==4},y,t]]//Flatten

Plot mit Dämpfung.

Manipulate[ParametricPlot[{x[t] /. xFktD /. {kx->KX, g1->G1},y[t] /. yFktD /. {ky->KY, g2->G2}},{t,0,2π*10},PlotPoints->500],{KX,1},{KY,3},{G1,0.01},{G2,0.03},ControlType->InputField]

Animation mit Dämpfung. kx=1, ky=3, g1=0.01, g2=0.03

Animate[Show[ParametricPlot[{x[tt] /. xFktD /.{kx->1, g1->0.01},y[tt] /. yFktD /. {ky->3, g2->0.03}},{tt,0,Evaluate[t]},PlotPoints->500,PlotRange->{{-1.2,1.2},{-1.5,1.5}}],Graphics[{ Disk[{x[t] /. xFktD /. {kx->1, g1->0.01},y[t] /. yFktD /.  {ky->3, g2->0.03}},0.05]},PlotRange->{{-1.2,1.2},{-1.5,1.5}}]],{t,0,10*2π},AnimationRate->.01]