I've heard double pendulums are notoriously chaotic in their motion and that no closed-form solution to their ODEs of motion exist.
Setting, by the theorem of kinetic energy $$\dot T= \Pi$$ I end up with:
$$4l_1\dot \theta_1 \ddot \theta_1(\frac {1}{3}m_1+m_2)+\frac{4}{3}l_2^2\dot\theta_2 \ddot \theta_2 m_2+2m_2l_1l_2 \{[\ddot \theta_1 \dot \theta_2+\ddot \theta_2 \dot \theta_1]sin(\theta_1-\theta_2)+\dot \theta_1 \dot \theta_2[\dot \theta_1 - \dot \theta_2]cos(\theta_1-\theta_2)\}=g[(\frac{m_1}{2}+m_2)l_1cos(\theta_1)\dot \theta_1-\frac{m_2l_2}{2}sin(\theta_2)\dot \theta_2]$$ Which is gnarly to say the least. I have only taken an introductory Analytical Mechanics course but I have a feeling this is one of those "no-closed-form-exists" ODEs.
To summarise how I obtained the above formula, $$T=\frac{1}{2}(m_1v_G^2+I_G \dot \theta_1^2+m_2v_H^2+I_H\dot \theta_2^2)$$ And $$\Pi=\frac{dW}{dt}=-m_1g \dot Y_G-m_2g \dot Y_H=g[(\frac{m_1}{2}+m_2)l_1cos(\theta_1)\dot \theta_1-\frac{m_2l_2}{2}sin(\theta_2)\dot \theta_2]$$
(This is assumed to be a 2D analysis of rigid bodies with perfect restraints).
Is there a way to further simplify the equation of motion for this general case with unspecified parameters?