Poisson Bracket - Constrained system

  • Thread starter vnikoofard
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  • #1
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Main Question or Discussion Point

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:
{[itex]\varphi,\partial_x\pi[/itex]}=?
where [itex]\pi[/itex] is conjugate momentum of [itex]\varphi[/itex]. I do not know for this type Poisson Bracket I can use part-by-part integration or not. I mean
{[itex]\varphi,\varphi\partial_x\pi[/itex]}= -[itex]\varphi[/itex]

Cheeeers!
Vahid
 

Answers and Replies

  • #2
tom.stoer
Science Advisor
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[tex]\{\varphi(x),\pi(y)\} = \delta(x-y)[/tex]
[tex]\{\varphi(x),\partial_y\pi(y)\} = \partial_y\{\varphi(x),\pi(y)\} = \partial_y \delta(x-y)[/tex]
 
  • #3
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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!
 
Last edited:
  • #4
tom.stoer
Science Advisor
5,766
159
The minus sign appears in partial integration or when the derivative is acting on x instead of y:

[tex]\partial_y \delta(x-y) = \delta(x-y)\partial_y[/tex]
[tex]\partial_y \delta(x-y) = -\partial_x \delta(x-y) = -\delta^\prime(x-y)[/tex]
 

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