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I've been studying Liouville's theorem lately. The statistical mechanics textbook by Kardar proves the theorem by showing that $dq\cdot dp$ is unchanged for each coordinate for an infinitessimal rectangle of length and width $dq$ and $dp$.

My thought is:

  • this proof shows that the $2n$-volume of a set in an $2n$ dimensional phase space is unchanged over time.

  • by taking a subset with fixed energy level, and recognizing that it stays within the subspace of that energy level and projecting onto that subspace, the $2n-1$ volume of that subset also stays the same.

  • however, the area of a set as projected on arbitrary coordinates $q_i,p_i$ won't necessarily stay constant, because even though this projection is preserved by a first order approximation, it is not once we take interaction with other variables into account. Hence, we can have that the projection of the phase space onto some variables $q_i,p_i$ increases or decreases over time.

Is this correct?

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The statements mentioned by OP seem to mostly due to that Hamiltonian time-evolution preserves the symplectic 2-form $\omega$, and that the canonical volume-form in phase space is $\omega^{\wedge n}$.

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  • $\begingroup$ unfortunately I still don't know what a 2-form is, or a volume-form or what $\omega ^{\land n}$ means, so I'm not sure whether your answer is actually "yes" or "no" :P. $\endgroup$ – user56834 Jul 12 '20 at 5:47

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