I am a physics undergraduate student currently taking a classical mechanics course, and I am not able to understand what conjugate/canonical momentum is (physically). It is sometimes equal to the linear momentum we know but not all the time. What exactly is potential momentum? It would be great if someone could explain it in simple words.
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I am a physics undergraduate student currently taking a classical mechanics course, and I am not able to understand what conjugate/canonical momentum is (physically).
The terms "canonical momentum," "conjugate momentum," and "generalized momentum" all refer to the same thing, namely: $$ p_i = \frac{\partial L}{\partial \dot q_i}\;, $$ where $L$ is the Lagrangian and $q_i$ is a generalized coordinate (which is linearly independent of the other generalized coordinates) and $\dot q_i$ is its time derivative.
What exactly is potential momentum?
The mechanical momentum is $m \dot q_i$, which is not always the same as the canonical momenutum $\frac{\partial L}{\partial \dot q_i}$.
When they are not the same, the difference can be referred to as the "potential momentum," but I do not typically find such a term very useful...
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$\begingroup$ So, is conjugate momentum essentially just a generalized and more fundamental way of expressing momentum? $\endgroup$ Commented Dec 1, 2023 at 10:49
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$\begingroup$ Yes, you could call it an "extended" or "generalized" form of momentum, but I would not say it is more "fundamental." $\endgroup$– hftCommented Dec 1, 2023 at 10:59
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$\begingroup$ (Just because that latter term is kind of normative and it's not totally clear what it should really mean.) $\endgroup$– hftCommented Dec 1, 2023 at 11:00