7.1: The Poisson Bracket
- Page ID
- 29568
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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}
\dfrac{d f}{d t}=\dfrac{\partial f}{\partial t}+\sum_{i}\left(\dfrac{\partial f}{\partial q_{i}} \dot{q}_{i}+\dfrac{\partial f}{\partial p_{i}} \dot{p}_{i}\right)
\end{equation}
This is often written as
\begin{equation}
\dfrac{d f}{d t}=\dfrac{\partial f}{\partial t}+[H, f]
\end{equation}
where
\[[H, f]=\sum_{i}\left(\dfrac{\partial H}{\partial p_{i}} \dfrac{\partial f}{\partial q_{i}}-\dfrac{\partial H}{\partial q_{i}} \dfrac{\partial f}{\partial p_{i}}\right) \label{PoissonBracket}\]
is called the Poisson bracket.
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(\dfrac{\partial f}{\partial p_{i}} \dfrac{\partial g}{\partial q_{i}}-\dfrac{\partial f}{\partial q_{i}} \dfrac{\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}\\
&\dfrac{\partial}{\partial t}[f, g]=\left[\dfrac{\partial f}{\partial t}, g\right]+\left[f, \dfrac{\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}