I have a differential equation for dr/dt and d[Theta]/dt in terms of r and theta, and I am trying to plot the trajectory in the x-y plane for a given initial condition (r,theta).


There is an example on this page for a set of differential equations already in cartesian coordinates, but I don't know what to do for polar.

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    – bbgodfrey
    Commented May 11, 2015 at 23:55
  • 3
    $\begingroup$ Please provide your equations in the question, so readers know what problem you are trying to solve. For instance, I would think you should use ParametricPlot, not StreamPlot. $\endgroup$
    – bbgodfrey
    Commented May 11, 2015 at 23:57
  • $\begingroup$ duplicate for field: StreamPlot in Polar Coordinates, answer for single trajectory: graph in the Polar Plane. TransformedField may be of use. $\endgroup$
    – Kuba
    Commented May 12, 2015 at 6:44

1 Answer 1




Then, after you solve the differential equation, use ParametricPlot:



polarToCartesian[{r_, theta_}] := r*{Cos[theta], Sin[theta]}

s = ParametricNDSolve[
      {r'[t] == r[t] (1 - (r[t])^2) (4 - (r[t])^2), 
       theta'[t] == 2 - (r[t])^2,
       r[0] == r0, theta[0] == theta0},
       {r, theta}, {t, 0, 10}, {r0, theta0}];

solution[r0_, theta0_, t_] := 
 Evaluate[{r[r0, theta0][t], theta[r0, theta0][t]} /. s]

 ParametricPlot[polarToCartesian[solution[r0, theta0, t]], {t, 0, 10},
   PlotRange -> {-2.5, 2.5}]
   ,{r0, 0.01, 2}, {theta0, 0, 2 Pi}]

enter image description here

  • $\begingroup$ could you please try plotting my example? It seems to keep running and won't display anything for me, I think it's having trouble trying to solve the equations. My differential equations are r'[t] == r[t] (1 - (r[t])^2) (4 - (r[t])^2), [Theta]'[t] == 2 - (r[t])^2 $\endgroup$
    – Kenny L
    Commented May 12, 2015 at 0:09
  • 1
    $\begingroup$ @KennyL You should add this important information to your question. $\endgroup$
    – Michael E2
    Commented May 12, 2015 at 0:31
  • $\begingroup$ @KennyL : Done! $\endgroup$
    – Ivan
    Commented May 12, 2015 at 1:34

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