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Poisson brackets and EM Hamiltonian

  • Thread starter hellfire
  • Start date
hellfire
Science Advisor
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Consider the following general Hamiltonian for the electromagnetic field:

[tex]H = \int dx^3 \frac{1}{2} E_i E_i + \frac{1}{4}F_{ij}F_{ij} + E_i \partial_i A_0 + \lambda E_0[/tex]

where [itex]\lambda[/itex] is a free parameter and [itex]E_0[/itex] is the canonical momentum associated to [itex]A_0[/itex], which defines a constraint ([itex]E_0 = 0[/itex] on the constraint surface). [itex]E_i[/itex] is the canonical momentum associated to [itex]A_i[/itex].

I am not able to work out the Poisson brackets to get the following results:

[tex]\dot E_i = \{E_i, H \} = -\partial_{j} F_{ij}[/tex]
[tex]\dot E_0 = \{E_0, H \} = -\partial_{i} E_i[/tex]

Help would be appreciated.
 

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