Poisson brackets of angular momentum

Question

So I'm trying to simplify this Poisson bracket of angular momentum vectors: {$L_1,L_2$}

Where $L=r \times p$

I know that $L_1=r_2p_3-r_3p_2$ and $L_2=r_3p_1-r_1p_3$ (I can easily derive this from cross product of the position $r$ and momentum $p$ vectors). But then I get a Poisson bracket$$\{r_2p_3-r_3p_2, r_3p_1-r_1p_3\}$$ and I don't know how to proceed to break it up from there. Do I distribute or do I use $\{fg,h\}=\{f,h\}g+f\{g,h\}$ property? Any help's appreciated.

Answer 1

You'll need to distribute to get four different Poisson brackets, and then use the expansion property you cited to simplify them. Luckily, most of these brackets will vanish; this is obvious if you already know the pairwise brackets $\{r_i,r_j\}$, $\{p_i,p_j\}$, and $\{r_i,p_j\}$.