I'm staring down a system of equations that look like (taken from http://scienceworld.wolfram.com/physics/DoublePendulum.html)

$ \begin{align} \ddot \theta_1 &= - \frac{m_2 l_2 \ddot{\theta_2}}{l_1(m_1 + m_2)}\cos(\theta_1-\theta_2) + \frac{m_2 l_2 \dot{\theta_2}^2 \sin (\theta_1 - \theta_2)}{l_1 (m_1 + m_2)} - \frac{g}{l} \sin (\theta_1)\\ \ddot \theta_2 &= -\frac{l_1}{l_2} \cos(\theta_1 - \theta_2) + \frac{l_1}{l_2}\ddot{\theta_1}\sin (\theta_1 - \theta_2) - g\sin \theta_2 \end{align} $

where I'd like to solve for $\theta_1$ and $\theta_2$. I'm looking to use MATLAB's ODE45 to do it, and I'm running into problems because the two second derivatives depend on each other. That is, I can't express $\ddot \theta_2$ without using $\ddot \theta_1$. And I'm not sure how to do that. Essentially the problem is that I have to begin by defining one before the other, and by that time I need the other to be defined as well. How do I get around this?

Thank you for your time.

  • $\begingroup$ It's a non linear system of differential equations. If you're looking to solve it for either theta, you're not in luck. There may be things you can determine about such a system though. What are you actually looking to do? $\endgroup$ – Kaynex May 2 '17 at 17:25
  • $\begingroup$ I'd like a graph of the angles over time as part of studying the transition between periodic and chaotic motion. I figured it's possible to solve for both of them numerically; the website linked in the OP says so. I can't think of a way to do it analityically but I figured MATLAB could figure it out. Is it my method that's unsuitable? $\endgroup$ – user93114 May 2 '17 at 17:52

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