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I'm going back over some of my old notes for a current project, and I'm trying to figure out if I made an error or if I once knew something that I've now forgotten.

Consider a local field theory containing a set of fields $\phi^{(a)}$, for which the Lagrangian density is $$ \mathcal{L}(\phi^{(a)}, \dot{\phi}^{(a)}, \partial_i \phi^{(a)} ). $$ Here, $\partial_i$ stands for spatial derivatives only, i.e., we've already done a decomposition into a preferred foliation of spacetime (if necessary.) We can define a set of conjugate field momenta via the usual relation: $$ \pi^{(a)} \equiv \frac{\delta \mathcal{L}}{\delta \dot{\phi}^{(a)} }. $$ The Hamiltonian density will then be $$ \mathcal{H} = \sum_a \pi^{(a)}\dot{\phi}^{(a)} - \mathcal{L}. $$ We can then define a Poisson bracket for field quantities, of the form $$ \{ f, g \} \equiv \sum_a \left[ \frac{ \delta f}{\delta \phi^{(a)}} \frac{ \delta g}{\delta \pi^{(a)}} - \frac{ \delta g}{\delta \phi^{(a)}} \frac{ \delta f}{\delta \pi^{(a)}}\right], $$ where $f$ and $g$ are in principle any two quantities that depend on the fields and the momenta.

Here are my questions:

  1. Is it always the case under these definitions that $$ \{ \partial_i f, g \} \overset{?}{=} \partial_i \{f, g\} $$ for any two quantities $f$ and $g$?
  2. Is it always the case under these definitions that $$ \{ \partial_i f, \mathcal{H} \} \overset{?}{=} \partial_i \{f, \mathcal{H} \} $$ for any quantity $f$?
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  1. Due to OP's mentioned context, we interpret $f$ & $g$ as functions (as opposed to functionals). In particular OP seems to be considering 'same-spacetime' functional derivative (FD), cf. my Phys.SE answer here.

  2. Leibniz rule $$ \{d_x f(x), g(x)\} +\{f(x), d_xg(x)\}~\stackrel{?}{=}~d_x \{f(x),g(x)\} $$ is not always satisfied.

  3. Counterexample: In a hopefully understandable abuse of notation, let $$f(x)~:=~f(\phi(x))\qquad \text{and} \qquad g(x)~:=~g(\pi(x)).$$ Then $$ \{d_x f(x), g(x)\} ~=~ 0~=~\{f(x), d_xg(x)\} $$ while $$ d_x\{f(x),g(x)\} ~=~ d_x\left(f^{\prime}(\phi(x)) g^{\prime}(\pi(x))\right). $$

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