# Sanity check on meaning of Liouville's theorem

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?

## 1 Answer

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 $$\omega^{\wedge n}$$.

• 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. – user56834 Jul 12 '20 at 5:47