# Poisson brackets and angular momentum

by Rhi
Tags: angular, brackets, momentum, poisson
 P: 10 1. The problem statement, all variables and given/known data Let f(q, p), g(q, p) and h(q, p) be three functions in phase space. Let Lk = εlmkqlpm be the kth component of the angular momentum. (i) Define the Poisson bracket [f, g]. (ii) Show [fg, h] = f[g, h] + [f, h]g. (iii) Find [qj , Lk], expressing your answer in terms of the permutation symbol. (iv) Show [Lj , Lk] = qjpk−qkpj . Show also that the RHS satisfies qjpk−qkpj = εijkLi. Deduce [Li, |L|2] = 0. [Hint: the identity εijkεklm = δilδjm − δimδjl may be useful in (iv)] 2. Relevant equations n/a 3. The attempt at a solution i) [f,g]=$\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}$ ii) easy to show from the definition in i) iii) after a bit of working, I get εlmkql iv) my working is quite long, but I get [Lj,Lk]=qjpk-qkpj=εijkLi as required. The bit I'm having trouble with is the very last bit of the question, to deduce [Li, |L|2] = 0. Since it's only a small part of the question, it seems as though this part should be fairly simple so maybe I'm overlooking something, but I don't get 0. This is my working: [Li, |L|2]=[Li, LjLj]=Lj[Li, Lj]+[Li, Lj]Lj=2Lj[Li, Lj] I'm not entirely sure where to go from here so any help (or pointing out of any glaring errors) would be great.
 P: 117 So here it goes. Leave the last equality out and when you get $L_j[L_i,L_j]+[L_i,L_j]L_j = L_j\epsilon_{kij} L_k + \epsilon_{kij} L_kL_j = \epsilon_{kij}L_jL_k - \epsilon_{jik} L_kL_j = \epsilon_{kij}L_jL_k - \epsilon_{kij} L_jL_k = 0$ Where in the equality before last I just relabel j to k and vice versa in the second summand.