# Poisson brackets and EM Hamiltonian

1. Sep 11, 2005

### hellfire

Consider the following general Hamiltonian for the electromagnetic field:

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

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

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

$$\dot E_i = \{E_i, H \} = -\partial_{j} F_{ij}$$
$$\dot E_0 = \{E_0, H \} = -\partial_{i} E_i$$

Help would be appreciated.