Lagrangian mechanics can be used to describe the double pendulum (see here, for example). In this development are the conjugate momenta $p_{\theta_i}$ exactly the angular momenta $m_i l_i \frac{d \theta_i}{dt}$?
In other words, do the following sequences hold:
You are right. If $q$ is a generalized coordinate then $\dot{q}$ is the generalized velocity and hence the generalized momentum is \begin{equation} p = \frac{\partial L}{\partial\dot{q}} \end{equation} Therefore, your sequence looks correct. Further, equations (20) and (21) of the article you have referenced also tell that the $p_{\theta_i}$ are indeed angular momenta. (You may want to check the dimensions.)
Normally, when we speak of angular momentum as a vector, we refer to a certain origin. In this case, we do not have to do it. The conjugate variable $p$ in this case, "happens to be" angular momentum.
