7.5: Example- Angular Momentum Components

A moving particle has angular momentum about the origin \(\begin{equation}
\vec{L}=\vec{r} \times \vec{p}, \text { so }
\end{equation}\)

\begin{equation}
L_{1}=r_{2} p_{3}-r_{3} p_{2}, \quad L_{2}=r_{3} p_{1}-r_{1} p_{3}
\end{equation}

Using the Poisson brackets found above,

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

we have

\begin{equation}
\begin{array}
{\left[L_{1}, L_{2}\right]=\left[r_{2} p_{3}-r_{3} p_{2}, r_{3} p_{1}-r_{1} p_{3}\right]} \\
=\left[r_{2} p_{3}, r_{3} p_{1}\right]+\left[r_{3} p_{2}, r_{1} p_{3}\right] \\
=r_{2} p_{1}-p_{2} r_{1} \\
=-L_{3}
\end{array}
\end{equation}

(Note: we remind the reader that we are following Landau's convention, in which the Poisson brackets have the opposite sign to the more common use, for example in Goldstein and Wikipedia.)

We conclude that if two components of angular momentum are conserved, so is the third.