# Is angular momentum the conjugate momentum of an angle?

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:

• Position -> Velocity -> Momentum
• Angle -> Angular velocity -> Angular momentum
• Generalized coordinate -> Generalized velocity -> Conjugate momentum
• I don't know what you mean when you ask "does the following sequences hold". – ACuriousMind Apr 25 '15 at 16:34
• The question seems pretty clear to me. (The answer is just "yes.") – Nathaniel Apr 26 '15 at 12:52
• I'm not convinced - the naming similarity and arguments based on dimensions aren't really compelling. I'll need to think about a bit more. – pdmclean Apr 28 '15 at 23:14
• @ACuriousMind Can you answer the question? – pdmclean Apr 30 '15 at 12:36
• ACuriousMind shows more of his broadmindedness. – pdmclean Apr 30 '15 at 12:40

You are right. If $q$ is a generalized coordinate then $\dot{q}$ is the generalized velocity and hence the generalized momentum is $$p = \frac{\partial L}{\partial\dot{q}}$$ 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.