Poisson Bracket - Constrained system

[vnikoofard]

Hi friends

I am trying to drive constraints of a Lagrangian density by Dirac Hamiltonian method. But I encountered a problem with calculating one type of Poisson Bracket:
{$\varphi,\partial_x\pi$}=?
where $\pi$ is conjugate momentum of $\varphi$. I do not know for this type Poisson Bracket I can use part-by-part integration or not. I mean
{$\varphi,\varphi\partial_x\pi$}= -$\varphi$

Cheeeers!
Vahid

 

[tom.stoer]

$$\{\varphi(x),\pi(y)\} = \delta(x-y)$$
$$\{\varphi(x),\partial_y\pi(y)\} = \partial_y\{\varphi(x),\pi(y)\} = \partial_y \delta(x-y)$$

 

[vnikoofard]

Thanks very much for response.
I wonder myself maybe appear a minus sign in the second line. Are you sure? Maybe I am confusing this situation with part by part integration!

 

[tom.stoer]

The minus sign appears in partial integration or when the derivative is acting on x instead of y:

$$\partial_y \delta(x-y) = \delta(x-y)\partial_y$$
$$\partial_y \delta(x-y) = -\partial_x \delta(x-y) = -\delta^\prime(x-y)$$