# 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:

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

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:

$$[G,H] = \iiint \left(\frac{\delta G}{\delta\eta_i}\frac{\partial H}{\partial \pi_i} - \frac{\delta H}{\delta \eta_i} \frac{\delta G}{\delta\pi_i}\right)d^3x$$ (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:

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

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.

• Some thoughts: 1) you don't derive the form of the PBs you simply define it and check it satisfies some axioms. 2) If you know $[f,H]$ then you can presumably extend this to $[f,G]$ where $G$ generates a $\lambda$ flow rather than a $t$ flow which probably justifies the approach taken in your reference. Nov 26, 2020 at 14:51
• Thanks for your comment! The main reason I need the Poisson bracket is to use in quantum correspondence principle and find quantum commutators For the field quantities. Would these axioms be in agreement with these conditions. Also I briefly considered option 2 but wasn't sure how to go about it I will think about it a bit more though now you suggest it. Nov 26, 2020 at 16:00