Chaotic Pendulum Simulation
Solutions approximated with the Runge-Kutta method
\[
\begin{align}
\ddot{\theta_1} &= \frac{-g(2m_1 + m2)\sin(\theta_1) - m_2g\sin(\theta_1-2\theta_2)
-2\sin(\theta_1-\theta_2)\cdot m_2(l_2\dot{\theta_{2}^2} + l_1 \dot{\theta_1^2}(\cos(\theta_1-\theta_2)))}
{l_1(2m_1+m_2 - m_2\cos(2(\theta_1-\theta_2)))}
\\
\ddot{\theta_2} &= \frac{2\sin(\theta_1-\theta_2)(l_1(m_1+m_2)\dot{\theta_1^2})
+g(m_1+m_2)\cos(\theta_1) + l_2m_2\dot{\theta_2^2}\cos(\theta_1-\theta_2)}
{l_2(2m_1+m_2 - m_2\cos(2(\theta_1-\theta_2)))}\\
\end{align}
\]
Where \(l_1, m_1, \theta_1\) are the length, mass, and angle from vertical of the top pendulum (respectively).
\(l_2, m_2, \theta_2\) are the length, mass, and angle from vertical of the bottom pendulum (respectively).
\(g = 9.81\) is the force of gravity.