Confirming the N-Particle PB Problem

[tendor]

Hi,
I have a small problem about PB. I think I know the answer, but I want to make sure it's correct.

For example let's have cartesian n dimensional problem with N particles, then PB becomes
$$\left\{F(\vec x^1,\cdots,\vec x^N,\vec p^1,\cdots,\vec p^N),G(\vec x^1,\cdots,\vec x^N,\vec p^1,\cdots,\vec p^N)\right\} =\sum\limits_{a=1}^{N}\sum\limits_{i=1}^{n}\frac{\partial F}{\partial x^a_i}\frac{\partial G}{\partial p^a_i}-\frac{\partial F}{\partial p^a_i}\frac{\partial G}{\partial x^a_i}$$

Is there somebody who could confirm me my supposition or send me some link about this (because I haven't found anything, just one particle situations which are not very helpful).

Thanks T.

 

[BigJDubs]

I.A!Yes, your supposition is correct. The equation you have provided is an expression of the Poisson Bracket for an n-dimensional problem with N particles. Here is a good resource which explains this expression in more detail: https://en.wikipedia.org/wiki/Poisson_bracket#Examples.