From the Lagrangian I've got the following equations of motion for the double pendulum in 2D. (The masses are different but the lengths of the two pendula are equal.) Let $m_2$ be the lowest-hanging mass.
$$(m_1+m_2)\ddot{\theta_1}+2m_2\ddot\theta_2\cos(\theta_2-\theta_1)=\\ -2m_2\dot\theta_1\dot\theta_2\sin(\theta_1-\theta_2)-(m_1+m_2)g/l\sin(\theta_1)$$
and
$$m_2\ddot{\theta_1}+2m_2\ddot\theta_2\cos(\theta_2-\theta_1)=\\ 2m_2\dot\theta_1\dot\theta_2\sin(\theta_1-\theta_2)-m_2g/l\sin(\theta_1)$$
In the small angle approximation these become, respectively
$$(m_1+m_2)\ddot{\theta_1}+2m_2\ddot\theta_2= -2m_2\dot\theta_1\dot\theta_2(\theta_1-\theta_2)-\theta_1(m_1+m_2)g/l$$
and
$$m_2\ddot{\theta_1}+2m_2\ddot\theta_2= 2m_2\dot\theta_1\dot\theta_2(\theta_1-\theta_2)-\theta_1m_2g/l$$.
Most sources don't have the terms of order $\dot\theta$. This is because they apply the small angle approximation to the Lagrangian before taking the derivatives, thereby ignoring terms of order $\theta.$ What justification do we have for getting rid of these terms?
Best Answer
I think the issue here is that you need to keep a consistent level of approximation in your "small angle approximation." By small angles, we typically mean $\theta_1$ and $\theta_2$ are both of order $\epsilon$, where $\epsilon \ll 1$. Then the question is - to what order in $\epsilon$ do you want to write down the equations of motion?
When you neglect the term $\frac{3}{2} \dot\theta_1 \dot \theta_2 (\theta_1 - \theta_2)^2$ in the Lagrangian, you are saying that terms of size $\epsilon^4$ are small compared to terms like $\dot \theta_1^2$, which is of size $\epsilon^2$. In the equation of motion, you get terms that are $\dot \theta_1 \dot \theta_2 (\theta_1 - \theta_2)$, which are of size $\epsilon^3$, compared to $\theta_1$, which is size $\epsilon$.
So neglecting the additional term in the Lagrangian gets you the same equation of motion as keeping the whole Lagrangian, and then dropping terms that are of size $\epsilon^3$.
This kind of argument is a little handwavy, and (in principle) could blow up if the time derivatives of $\theta_{1,2}$ were large - at some point, you might want to check out some books on perturbation theory in a more formal sense.