# Poisson brackets little problem

1. Aug 6, 2011

### fluidistic

1. The problem statement, all variables and given/known data
For a particle, calculate Poisson brackets formed by:
1)The Cartesian components of the linear momentum $\vec p$ and the angular momentum [/itex]\vec M =\vec r \times \vec p[/itex].
2)The Cartesian components of the angular momentum.

2. Relevant equations

$[u,p]_{q,p}= \sum _k \left ( \frac{\partial q }{\partial q_k } \frac{\partial v }{\partial p _k} -\frac{\partial q }{\partial p_k } \frac{\partial v }{\partial q _k} \right )$.

3. The attempt at a solution
2)Nothing still, waiting to complete 1).
1)I calculated the Cartesian components of M and p.
I don't understand what I have to calculate. $[\vec p, \vec M]$ I'm guessing but with what subscript?
Thanks for any help.

Edit: Hmm I think the subscript is always q,p. But p and M are vectors, so have I to calculate directional derivative?

Last edited: Aug 6, 2011
2. Aug 6, 2011

### sgd37

$$\vec{p} = ( p_x , p_y , p_z ) = p_i$$ and $$\vec{r} = ( x , y , z ) = x_i$$

$$M_i = \epsilon_{ijk} x_j p_k$$ where the einstein summation convention is used

$$[p_i,M_j]= \sum _l \left ( \frac{\partial p_i }{\partial q_l } \frac{\partial M_j }{\partial p _l} -\frac{\partial p_i }{\partial p_l } \frac{\partial M_j }{\partial q _l} \right )$$

hope this helps