In classical mechanics the liouville’s theorem states that the intgeral element \begin{align} \partial p_0 \partial q_0 = \partial p_t \partial q_t \end{align} where \(p_0, q_0\) and \(p_t, q_t\)are the generalized coordinates at time 0 and t, respectively.

To prove this, we notice the Hamiltonian equation \begin{align} \dot p_0 = -\frac{\partial H}{\partial q_0}, \dot q_0 = \frac{\partial H}{\partial p_0} \end{align} and \begin{align} \dot p_t = -\frac{\partial H}{\partial q_t}, \dot q_t = \frac{\partial H}{\partial p_t} \end{align}

Since \(q_t=q_t(p_0,q_0)\) is a function of \(p_0\), and \(q_0\), we can write \begin{align} \dot q_t = \frac {\partial q_t}{\partial p_0} \cdot \dot p_0 + \frac{\partial q_t}{\partial q_0} \cdot \dot q_0 = \frac{\partial q_t}{\partial p_0} \cdot (-\frac{\partial H}{\partial q_0}) + \frac{\partial q_t}{\partial q_0} \cdot \frac{\partial H}{\partial p_0} = -\frac{\partial q_t}{\partial p_0} \cdot (\frac{\partial p_t}{\partial q_0} \cdot \frac{\partial H}{\partial p_t} + \frac{\partial q_t}{\partial q_0} \cdot \frac{\partial H}{\partial q_t})+ \frac{\partial q_t}{\partial q_0} \cdot (\frac{\partial p_t}{\partial p_0} \cdot \frac{\partial H}{\partial p_t} + \frac{\partial q_t}{\partial p_0} \cdot \frac{\partial H}{\partial q_t}) \
\dot q_t = (-\frac{\partial q_t}{\partial p_0} \cdot \frac{\partial p_t}{\partial q_0} + \frac{\partial q_t}{\partial q_0} \cdot \frac{\partial p_t}{\partial p_0}) \cdot \frac{\partial H}{\partial p_t} + (-\frac{\partial q_t}{\partial p_0} \cdot \frac{\partial q_t}{\partial q_0} + \frac{\partial q_t}{\partial q_0} \cdot \frac{\partial q_t}{\partial p_0}) \cdot \frac{\partial H}{\partial q_t} = (-\frac{\partial q_t}{\partial p_0} \cdot \frac{\partial p_t}{\partial q_0} + \frac{\partial q_t}{\partial q_0} \cdot \frac{\partial p_t}{\partial p_0}) \cdot \frac{\partial H}{\partial p_t} \end{align}

Combing this equation with the Hamiltonian equation for \(\dot q_t\), we obtained \begin{align} \frac{\partial q_t}{\partial q_0} \cdot \frac{\partial p_t}{\partial p_0} - \frac{\partial q_t}{\partial p_0} \cdot \frac{\partial p_t}{\partial q_0} = 1 \end{align}

That is \begin{align} det (J(q_t, p_t, q_0, p_0)) = det(\frac{\partial (q_t, p_t)}{\partial (q_0, p_0)}) = \begin{vmatrix} \frac{\partial q_t}{\partial q_0} & \frac{\partial q_t}{\partial p_0}\\ \frac{\partial p_t}{\partial q_0} & \frac{\partial p_t}{\partial p_0} \end{vmatrix} = 1 \end{align} Where J is the Jacobian matrix. So, \begin{align} \partial p_t \partial q_t = det\lvert J \rvert \cdot\partial p_0 \partial q_0 = \partial p_0 \partial q_0 \end{align}

Q.E.D