(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that:

(i){p_{i},p_{j}}=0

(ii){q_{i},q_{j}}=0

(iii){q_{i},p_{j}}=δ_{ij}

2. Relevant equations

{x,Y} = [(∂x/∂q * ∂Y/∂p) - (∂Y/∂q * ∂x/∂p)]

3. The attempt at a solution

Right I know first of all that q_{ij}and p_{ij}are generalized coordinates so that we dont need to worry about what units etc the answers are in.

I know that if the Poisson Bracket is equal to zero then the point you have used it on is a conserved quantity.

I think (i) and (ii) are ok but stuck on what to do on (iii). I have a feeling it has something to do with the Levi Civita Tensor as that is the last place I came across Kronecker Delta.

(i/ii){q_{i},q_{j}} = [(∂q_{i}/∂q)*(∂q_{j}/∂p) - (∂q_{j}/∂q)*(∂q_{i}/∂p)]

The ∂q_{i}/∂p and the ∂q_{j}/∂p on either side are just '0' I think as it is a partial derivative of a q component with respect to p and it does not have any p component so QED 0 - 0 = 0

And that would be basically the same solution for parts (i) and (ii). Now...

(iii) {q_{i},p_{j}} = [(∂q_{i}/∂q) * (∂p_{j}/∂p) - (∂p_{j}/∂q)*(∂q_{i}/∂p)]

Last part is equal to zero for same reasons as part (i) and (ii) leaving me with

(∂q_{i}/∂q)*(∂p_{j}/∂p)

Now I dont really have a clue what to do like I said before I think it is something to do with Levi.... any help is appreciated.

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# Homework Help: Prove the basic rule's of Poisson's Brackets?

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