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I know that generalised coordinates and their conjugate momentum may or may not have the same dimensions as to that of length and linear momentum, but in one book I saw it was mentioned that their product must always have the dimensions of angular momentum.

Is it true?

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    $\begingroup$ If you like this question, you may also enjoy reading this & this posts. $\endgroup$ – Qmechanic Sep 1 '17 at 8:20
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The Lagrangian $L$ has dimensions of energy, and $$ p_i=\frac{\partial L}{\partial \dot{q}^i}, $$ so (because $\dot{q}$ has dimension of $[q]/T$) $$ \frac{[q]}{T}\cdot[p]=E, \\ [q]\cdot[p]=E\cdot T. $$

And $E\cdot T$ is precisely the dimension of action/angular momentum.

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