In the Hamilton-Jacobi equation $-\frac{\partial S}{\partial t} = H$, there is also a Hamiltonian governing time evolution.
That is a recurring theme just like: $\frac{df}{dt}={f,H}$ or the Schrodinger equation, that's all ok.
But what kind of a function (mathematical object?) is the $S$ actually. Hamilton's principal function, a generating function ok, but
Does the $-$ sign in the HJ equation indicate something similar that is going on with the Liouville equation.
There is the thing that:
$\frac{d\rho}{dt}={\rho,H}$ and not $\frac{df}{dt}={f,H}$ because states evolve with $-$ in referenc…