# Help with Poisson Brackets

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1. Nov 29, 2014

### mc0210

1. The problem statement, all variables and given/known data
Consider the motion of a particle with charge e in a homogenous magnetic field B_i. The Hamiltonian for this problem is $$H = \frac{1}{2m} \sum_{i=1}^{i=3} \left[ p_i - \frac{e}{2}\epsilon _{ijk}B_j x_k\right]^2.$$ By calculating the Poisson brackets, show that the transformation $$p_i \rightarrow p_i - \frac{e}{2}\epsilon _{ijk}B_j x_k$$ is not canonical.

2. Relevant Equations

3. Attempt at Solution
I feel decently comfortable using Poisson brackets, however I am not sure how to perform this calculation because of the summation from i=1 to i=3. My attempt was to add each component, so start with x_i and p_i and perform the poisson bracket and get 1, then repeat for x_j and p_j, and for x_k and p_k. Since each yields one, I got a total of 3. To be canonical it must equal 1. However, I really don't think this method is correct. Thanks for any help!

2. Dec 4, 2014