I'm just curious about why many physical identities build relationship with the same units as angular momentum like the action, Lagrangian$\cdot$time, Hamiltonian$\cdot$time, phase space area etc?
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Well, OK, this is a resolutely vague question, but there is something special, actually, about angular momentum dimensions.
In quantum mechanics, the fundamental constant, $\hbar$ has dimensions of angular momentum (and is very small in terms of angular momenta, actions, or phase-space areas of our macroscopic world experience). Classical mechanics results as the "small-$\hbar$ limit" of quantum mechanics, when the action (Lagrangian-times-time) of a specific problem is much larger than $\hbar$, as first observed by Dirac and Wentzel about 80 years ago, and exploited by Feynman in developing path-integral quantization. Before that, Bohr had noticed area rules in phase space that led to an early version of quantum mechanics.
So you might say that nature, mysteriously, by dint of quantum mechanics, has chosen a fundamental constant with units of angular momentum.
Cosmas Zachos
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The Lagrangian and Hamiltonian both have units of energy. You can get a long way in classical mechanics by only thinking about energy. In field theories the relevant unit becomes energy density.
The action must have units of energy times time, as it is the time integral of the Lagrangian.
I don't think there's any profound significance to the fact that angular momentum has the same units as the action, any more than to the fact that torque has units of energy, or that pressure has units of energy density. (Actually of those three the pressure one is probably the most interesting.)
rob
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