# Clarifications about Poisson brackets and Levi-Civita symbol

I need some clarifications about Poisson brackets.

I know canonical brackets and the properties of Poisson Brackets and I also know something about Levi-Civita symbol (definition and basic properties), but I have some doubts.

I don't know how I could apply Poisson brackets properties if I have a summation, for example if in the case of a system of N particles I have to solve $[L_i, x_{\alpha j}]$. I know that a generical component of total angular momentum is given by $L_a=\sum_{\alpha=1} ^N l_{\alpha a}$ and also know that the components of algular momentum af a particle is given by $l_a= \epsilon_{aij} x_i p_j$. Now, if I have to calculate $[L_i, x_{\alpha j}]$, I have these doubts:

1) how can I decide the indices of Levi-Civita symbol that I'm going to use for solving the problem?

2) how can I use the property of linearity of Poisson brackets in this case?

and an other (general) question:

3) If I have a Levi-Civita symbol that multiplyes a sum of two terms and each term is mulplied for a Kronecker delta, I have to follow these steps:

a) multiply the Levi-Civita symbol for each term

b) impose the condition thanks to each Kronecker delta isn't equal to zero

c) eventually, substitute these conditions in the two Levi-Civita symbol, but I have to substitute in each Levi-Civita symbol the condition that I found for that Kronecker delta that at the step a) was multipling just that one Levi-Civita symbol

Is it correct this way to go on?

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2.) Linearity implies that if you enter a sum as the argument of a Poisson brackets, you get a sum of Poisson brackets, with each of them having a single term your original sum as the argument, i.e. $$[\Sigma^N_{\alpha=1}B_\alpha,A]=[B_1,A]+[B_2,A]+[B_3,A]+\ldots+[B_N,A]$$ 3.) If you mean something like $$\epsilon_{ijk}(A_{jl}\delta_{lk}+B_{kl}\delta_{jl})=\epsilon_{ijk}A_{jl}\delta_{lk}+\epsilon_{ijk}B_{kl}\delta_{jl}=\epsilon_{ijk}A_{jk}+\epsilon_{ijk}B_{kj},$$ the answer is yes.