7.1: The Poisson Bracket

A function \(\begin{equation}
f(p, q, t)
\end{equation}\) of the phase space coordinates of the system and time has total time derivative

\begin{equation}
\frac{d f}{d t}=\frac{\partial f}{\partial t}+\sum_{i}\left(\frac{\partial f}{\partial q_{i}} \dot{q}_{i}+\frac{\partial f}{\partial p_{i}} \dot{p}_{i}\right)
\end{equation}

This is often written as

\begin{equation}
\frac{d f}{d t}=\frac{\partial f}{\partial t}+[H, f]
\end{equation}

where

\[[H, f]=\sum_{i}\left(\frac{\partial H}{\partial p_{i}} \frac{\partial f}{\partial q_{i}}-\frac{\partial H}{\partial q_{i}} \frac{\partial f}{\partial p_{i}}\right) \label{PoissonBracket}\]

is called the Poisson bracket.

Caution

Equation \ref{PoissonBracket} is Landau's definition for the Poisson bracket. It differs in sign from Goldstein, Wikipedia and others.

If, for a phase space function \(\begin{equation}
f\left(p_{i}, q_{i}\right)
\end{equation}\) (that is, no explicit time dependence) \(\begin{equation}
[H, f]=0, \text { then } f\left(p_{i}, q_{i}\right)
\end{equation}\) is a constant of the motion, also called an integral of the motion.

In fact, the Poisson bracket can be defined for any two functions defined in phase space:

\begin{equation}
[f, g]=\sum_{i}\left(\frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial q_{i}}-\frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}}\right)
\end{equation}

It’s straightforward to check the following properties of the Poisson bracket:

\begin{equation}
\begin{aligned}
&[f, g]=-[g, f]\\
&[f, c]=0 \text { for } c \text { a constant }\\
&\left[f_{1}+f_{2}, g\right]=\left[f_{1}, g\right]+\left[f_{2}, g\right]\\
&\left[f_{1} f_{2}, g\right]=f_{1}\left[f_{2}, g\right]+\left[f_{1}, g\right] f_{2}\\
&\frac{\partial}{\partial t}[f, g]=\left[\frac{\partial f}{\partial t}, g\right]+\left[f, \frac{\partial g}{\partial t}\right]
\end{aligned}
\end{equation}

The Poisson brackets of the basic variables are easily found to be:

\begin{equation}
\left[q_{i}, q_{k}\right]=0, \quad\left[p_{i}, p_{k}\right]=0, \quad\left[p_{i}, q_{k}\right]=\delta_{i k}
\end{equation}

Now, using

\begin{equation}
\left[f_{1} f_{2}, g\right]=f_{1}\left[f_{2}, g\right]+\left[f_{1}, g\right] f_{2}
\end{equation}

and the basic variable P.B.’s, we find

\begin{equation}
\left[p, q^{2}\right]=2 q,\left[p, q^{3}\right]=3 q^{2}
\end{equation}

and, in fact, the bracket of p with any reasonably smooth function of q is:

\begin{equation}
[p, f(q)]=d f / d q
\end{equation}