How to Derive the Poisson Brackets for Continuous Systems

Part of my research involves studying the classical mechanics of Electromagnetic fields. I have followed the procedure for determining the Lagrangian and Hamiltonian formulations for continuous systems in this link (I believe this is an old version of Classical Mechanics by Goldstein but I am not sure.): https://courses.physics.ucsd.edu/2016/Fall/physics200a/Hamiltonian%20Formulations%20for%20Continuua-RFS.pdf

Within this context I need to determine the arbitrary Poisson brackets of two functionals:

$F=int d^3xmathcal{F}(eta_i(x,t),pi_i(x,t),t)$, $G=int d^3xmathcal{G}(eta_i(x,t),pi_i(x,t),t)$.

Where $eta_i(x,t)$ are the canonical field coordinates and $pi(x,t)$ are the components of the canonical momentum density. I understand from elementary classical mechanics that the Poisson bracket for a system with finite degrees of freedom (canonical coordinates, $q_i$ and canonical momentum components $p_i$) is given by

$[u,v] = frac{partial u}{partial q_i}frac{partial v}{partial p_i} – frac{partial v}{partial q_i}frac{partial u}{partial p_i}$ (Implicit summation)

however, I do not know how to define the Poisson bracket with respect to the field components $eta_i(x,t)$ and $pi_i(x,t)$. I understand how you can show that:

$[G,H] = iiint left(frac{delta G}{deltaeta_i}frac{partial H}{partial pi_i} – frac{delta H}{delta eta_i} frac{delta G}{deltapi_i}right)d^3x$ (H is Hamiltonian, not an arbitrary functional)

by using

$frac{dG}{dt} = [G,H] + frac{partial G}{partial t}$

and the Hamiltonian equations (see link) but there is no description of the general poisson bracket between two abitrary functionals. Most of the references I have read simply state it is given by:

$[F,G] = iiint left(frac{delta F}{deltaeta_i}frac{partial G}{partial pi_i} – frac{delta G}{delta eta_i} frac{delta F}{deltapi_i}right)d^3x$

either without justification or simply extending the formula for $[G,H]$ but replacing $H$ by an arbitrary functional. Whilst I find it reasonable to expect this formula to be valid for the general Poisson bracket I would argue that simply replacing the Hamiltonian with a different functional and stating that this is the Poisson bracket is a non-sequitur as the case for the Hamiltonian is a specific one. I could be wrong about this but I don’t think so. I have thought about other methods such as using the Jacobi identity

$[H,[F,G]] + [F,[G,H]] + [G,[H,F]] = 0$

To isolate the bracket, $[F,G]$, but I just end up with a mess of integrals. I have worked on this problem for a while now and I can’t really see the wood for the trees anymore. I would appreciate any help that can be given for this problem. Thank you.