Poisson brackets and EM Hamiltonian

[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.

 

[member 553083]

The Poisson brackets for the above Hamiltonian are given by:\{E_i(x), E_j(y)\} = 0\{E_i(x), F_{jk}(y)\} = \delta_{ij}\partial_{x_k}\delta(x-y) - \delta_{ik}\partial_{x_j}\delta(x-y)\{E_0(x), A_0(y)\} = \delta(x-y)Using these Poisson brackets, we can calculate the following:\dot E_i = \{E_i, H \} = -\partial_{j} F_{ij}\dot E_0 = \{E_0, H \} = -\partial_{i} E_i