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.
Best Answer
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 $\{r_i,p_j\}$.