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Poisson Bracket - Constrained system

  1. Aug 16, 2012 #1
    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
     
  2. jcsd
  3. Aug 16, 2012 #2

    tom.stoer

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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]
     
  4. Aug 16, 2012 #3
    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: Aug 16, 2012
  5. Aug 17, 2012 #4

    tom.stoer

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    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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