10.5: Poisson Brackets under Canonical Transformations

First, note that if Hamilton’s equations have the standard canonical form

\begin{equation}
\dot{q}_{i}=\left[H, q_{i}\right]=\dfrac{\partial H}{\partial p_{i}}, \quad \dot{p}_{i}=\left[H, p_{i}\right]=-\dfrac{\partial H}{\partial q_{i}}
\end{equation}

with respect to a pair of variables \(\begin{equation}
p, q
\end{equation}\)

then those variables are said to be canonically conjugate.

The Poisson bracket is invariant under a canonical transformation, meaning

\begin{equation}
[f, g]_{p, q}=[f, g]_{P, Q}
\end{equation}

Let's begin by establishing that

\begin{equation}
\left[Q_{i}, Q_{k}\right]_{p, q}=0, \quad\left[P_{i}, P_{k}\right]_{p, q}=0, \quad\left[P_{i}, Q_{k}\right]_{p, q}=\delta_{i k}
\end{equation}

We'll show the method by taking just one pair of variables p,q, and a generating function \(\begin{equation}
F(q, Q)
\end{equation}\)

Then

\begin{equation}
[P, Q]_{p, q}=\left(\dfrac{\partial P}{\partial p}\right)_{q}\left(\dfrac{\partial Q}{\partial q}\right)_{p}-\left(\dfrac{\partial P}{\partial q}\right)_{p}\left(\dfrac{\partial Q}{\partial p}\right)_{q}
\end{equation}

With the generating function \(\begin{equation}
F(q, Q), \text { we have } p(q, Q)=(\partial F / \partial q)_{Q}, P(q, Q)=-(\partial F / \partial Q)_{q}, \text { so }
\end{equation}\)

\begin{equation}
\left(\dfrac{\partial P}{\partial p}\right)_{q}=\left(\dfrac{\partial P}{\partial Q}\right)_{q}\left(\dfrac{\partial Q}{\partial p}\right)_{q}
\end{equation}

and

\begin{equation}
\left(\dfrac{\partial P}{\partial q}\right)_{p}=\left(\dfrac{\partial P}{\partial q}\right)_{Q}+\left(\dfrac{\partial P}{\partial Q}\right)_{q}\left(\dfrac{\partial Q}{\partial q}\right)_{p}
\end{equation}

Putting these results into the Poisson bracket,

\begin{equation}
[Q, P]=-\left(\dfrac{\partial Q}{\partial p}\right)_{q}\left(\dfrac{\partial P}{\partial q}\right)_{Q}=\left(\dfrac{\partial Q}{\partial p}\right)_{q} \dfrac{\partial^{2} F}{\partial q \partial Q}=\left(\dfrac{\partial Q}{\partial p}\right)_{q}\left(\dfrac{\partial p}{\partial Q}\right)_{q}=1
\end{equation}

These basic results can then be used to prove the general Poisson bracket is independent of the parametrization of phase space, details in Goldstein and elsewhere.

Landau, on the other hand, offers a one-line proof of the invariance of the Poisson bracket of two dynamical functions \(\begin{equation}
f\left(p_{i}, q_{i}\right), g\left(p_{i}, q_{i}\right)
\end{equation}\) under a canonical transformation: imagine a fictitious system having g as its Hamiltonian. Then \(\begin{equation}
[f, g]_{p, q} \text { is just } d f / d t
\end{equation}\) and cannot depend on the coordinate system used, so must equal \(\begin{equation}
[f, g]_{P, Q}
\end{equation}\).