How to plot a Phase-Portrait of three coupled differential equations?

Question

I have the following problem: I would like to plot a StreamPlot (phase portrait) of three coupled, second-order, non-linear differential equations. These are:

m*l^2*θ1''[t] + m*x''[t]*l*Cos[θ1[t]] + 
  c1*θ1[t] + m*g*l*Sin[θ1[t]] == 0

m*l^2*θ2''[t] + m*x''[t]*l*Cos[θ2[t]] + 
 c1*θ2[t] + m*g*l*Sin[θ2[t]]==0

(M + 2 m)*x''[t] + c2*x'[t] + 
 k*x[t] + (m*l*θ1''[t]*Cos[θ1[t]] - 
   m*l*θ1'[t]^2*Sin[θ1[t]]) + (m*l*θ2''[t]*
    Cos[θ2[t]] - m*l*θ2'[t]^2*Sin[θ2[t]])==0

where theta1(t), theta2(t) and x(t) are generalized coordinates and the rest are constants. Has anyone a hint how to plot a StreamPlot of e.g. theta1'(t) vs. theta1(t)?

Thank you very much. Best, Jonas.

Answer 1

Not sure what you want but here's a start: Can solve these numerically as IVPs if given values to all constants and starting conditions. Here's a set up with random values:

   (* set all constants to some vals *)
m = 1;
l = 1;
c1 = 1;
c2 = 1;
g = 1
M = 1
k = 1;
(* create an array of equations *)
theEqns = {m*l^2*\[Theta]1''[t] + m*x''[t]*l*Cos[\[Theta]1[t]] + 
    c1*\[Theta]1[t] + m*g*l*Sin[\[Theta]1[t]] == 0,
  m*l^2*\[Theta]2''[t] + m*x''[t]*l*Cos[\[Theta]2[t]] + 
    c1*\[Theta]2[t] + m*g*l*Sin[\[Theta]2[t]] == 0,
  (M + 2 m)*x''[t] + c2*x'[t] + 
    k*x[t] + (m*l*\[Theta]1''[t]*Cos[\[Theta]1[t]] - 
      m*l*\[Theta]1'[t]^2*Sin[\[Theta]1[t]]) + (m*l*\[Theta]2''[t]*
       Cos[\[Theta]2[t]] - m*l*\[Theta]2'[t]^2*Sin[\[Theta]2[t]]) == 0}
(* define some initial conditions for the IVP *)
theInit = {\[Theta]1[0] == 0, \[Theta]1'[0] == 0.1, \[Theta]2[0] == 
   0.5, \[Theta]2'[0] == 0.17, x[0] == 0, x'[0] == 0.2}

Now use NDSolveValue to solve the system for 0<=t<=25:

{theta1, theta2, theX} = 
 NDSolveValue[
  Join[theEqns, theInit], {\[Theta]1, \[Theta]2, x}, {t, 0, 25}]

Now take the (numeric) derivative of theta1 and plot it vs. theta1[t]:

theta1D[t_] = D[theta1[t], t];
ParametricPlot[{theta1D[t], theta1[t]}, {t, 0, 25}]