User:Eh9/Counterexample

About this page
This page is a work-in-progress on a disproof of a statement I believe false in the article on Liouville's theorem (Hamiltonian). Here is the statement in question:"Another illustration is to consider the trajectory of a cloud of points through phase space. It is straightforward to show that as the cloud stretches in one coordinate – $p_i$ say – it shrinks in the corresponding $q^i $ direction so that the product $\Delta p_i \Delta q^i $ remains constant."

I should first say that this statement has something of the "not even wrong" flavor about it, since the meanings of $$\Delta p_i$$ and $$\Delta q^i $$ aren't defined. The sense of the statement, however badly worded, seems to be that Liouville's theorem applies not only on the whole phase space, but also separately on each 2-dimensional phase subspace as defined by the canonical coordinates.