11.5: Jacobian for Time Evolution

As we’ve established, time development is equivalent to a canonical coordinate transformation,

\begin{equation}
\left(p_{t}, q_{t}\right) \rightarrow\left(p_{t+\tau}, q_{t+\tau}\right) \equiv(P, Q)
\end{equation}

Since we already know that the number of points inside a closed volume is constant in time, Liouville’s theorem is proved if we can show that the volume enclosed by the closed surface is constant, that is, with \(\begin{equation}
V^{\prime}
\end{equation}\)

denoting the volume V evolves to become, we must prove

\begin{equation}
\int_{V^{\prime}} d Q_{1} \ldots d Q_{s} d P_{1} \ldots d P_{s}=\int_{V} d q_{1} \ldots d q_{s} d p_{1} \ldots d p_{s} ?
\end{equation}

If you’re familiar with Jacobians, you know that (by definition)

\begin{equation}
\int d Q_{1} \ldots d Q_{s} d P_{1} \ldots d P_{s}=\int D d q_{1} \ldots d q_{s} d p_{1} \ldots d p_{s}
\end{equation}

where the Jacobian

\begin{equation}
D=\dfrac{\partial\left(Q_{1}, \ldots, Q_{s}, P_{1}, \ldots, P_{s}\right)}{\partial\left(q_{1}, \ldots, q_{s}, p_{1}, \ldots, p_{s}\right)}
\end{equation}

Liouville’s theorem is therefore proved if we can establish that D=1. If you’re not familiar with Jacobians, or need reminding, read the next section!