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
  • $\begingroup$ I don't know what you mean when you ask "does the following sequences hold". $\endgroup$ – ACuriousMind Apr 25 '15 at 16:34
  • $\begingroup$ The question seems pretty clear to me. (The answer is just "yes.") $\endgroup$ – Nathaniel Apr 26 '15 at 12:52
  • $\begingroup$ 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. $\endgroup$ – pdmclean Apr 28 '15 at 23:14
  • $\begingroup$ @ACuriousMind Can you answer the question? $\endgroup$ – pdmclean Apr 30 '15 at 12:36
  • $\begingroup$ ACuriousMind shows more of his broadmindedness. $\endgroup$ – 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 \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.


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