Parametric plot of double pendulum

Question

I have a double pendulum system with the following equations

$(m_1+m_2)a_1\ddot{θ_1}+m_2a_2\ddot{θ_2}cos(θ_1-θ_2)+m_2a_2\dot{θ_2}^2sin(θ_1-θ_2)+(m_1+m_2)gsin{θ_1}=0$

$m_2a_2\ddot{θ_2}+m_2a_1\ddot{θ_1}cos(θ_1-θ_2)-m_2a_1\dot{θ_2}^2sin(θ_1-θ_2)+m_2gsin{θ_2}=0$

where $a_1=a_2=m_1=m_2=1$ and $θ_1=θ_2=π/2$, and $\dot{θ_1}=\dot{θ_2}=0$. And I'm trying to use ParametricPlot to show the motion of the 2nd mass only. I have tried doing the following but it's not working,

With[{g = 9.8}, {θ1, θ2} = NDSolveValue[{2*θ1''[t] + θ2''[t]*Cos[θ1[t] - θ2[t]] + θ2'[t]^(2)*Sin[θ1[t] - θ2[t]] + 2*9.8*Sin[θ1[t]] == 0, θ2''[t] + θ1''[t]*Cos[θ1[t] - θ2[t]] - θ1'[t]^(2)*Sin[θ1[t] - θ2[t]] + g*Sin[θ2[t]] == 0, θ1[0] == Pi/2, θ2[0] == Pi/2, θ1'[0] == 0, θ2'[0] == 0}, {θ1[t], θ2[t]}, {t, 0, 100}]]

but when I tried the ParametricPlot nothing showed up. How could I edit this so as to get the plot for only the second mass?

Answer 1

First, let's define the solution variables to be different than the equation variables (harmless in this case, but could result in problems down the line):

With[{g = 9.8}, {θ1Sol, θ2Sol} = 
  NDSolveValue[{2*θ1''[t] + θ2''[t]*
       Cos[θ1[t] - θ2[t]] + θ2'[t]^(2)*
       Sin[θ1[t] - θ2[t]] + 2*9.8*Sin[θ1[t]] == 0, 
       θ2''[t] + θ1''[t]*
       Cos[θ1[t] - θ2[t]] - θ1'[t]^(2)*
       Sin[θ1[t] - θ2[t]] + g*Sin[θ2[t]] == 0, 
       θ1[0] == Pi/2, θ2[0] == Pi/2, θ1'[0] == 0, 
       θ2'[0] == 0}, {θ1, θ2}, {t, 0, 100}]]

Then, I believe you want to be adding θ1Sol + θ2Sol, and converting to cartesian coordinates to plot the location of the second mass:

With[{θ0 = π/2},
 ParametricPlot[{Cos[θ1Sol[t] - θ0], Sin[θ1Sol[t] - θ0]} 
 + {Cos[θ2Sol[t] - θ0], Sin[θ2Sol[t] - θ0]}, {t, 0, 100}]]