I am studying the effect of infinitesimal transformations on a hamiltonian field theory, $\mathscr{H}(\mathscr{Q}_n,\mathscr{P}_n,\partial_i\mathscr{Q}_n,x)$, with scalar fields and momenta $\mathscr{Q}_n$,$\mathscr{P}_n$. I write the transformation generated by $\mathscr{G}$ in terms of the local Poisson Bracket $$ \delta\mathscr{Q}_n = \{\mathscr{Q}_n,\mathscr{G}\}\delta\epsilon $$ $$ \delta\mathscr{P}_n = \{\mathscr{P}_n,\mathscr{G}\}\delta\epsilon\tag{1} $$ with \begin{equation} \begin{array}{rcl} \{\mathscr{A},\mathscr{B}\} & = & \left[\dfrac{\partial \mathscr{A}}{\partial \mathscr{Q}_n} - \partial_{i}\dfrac{\partial\mathscr{A}}{\partial(\partial_{i}\mathscr{Q}_n)}\right]\left[\dfrac{\partial \mathscr{B}}{\partial \mathscr{P}_n} - \partial_{j}\dfrac{\partial\mathscr{B}}{\partial(\partial_{j}\mathscr{P}_n)}\right] \\ & & \\ & - & \left[\dfrac{\partial \mathscr{A}}{\partial \mathscr{P}_n} - \partial_{i}\dfrac{\partial\mathscr{A}}{\partial (\partial_{i}\mathscr{P}_n)}\right]\left[\dfrac{\partial \mathscr{B}}{\partial \mathscr{Q}_n} - \partial_{j}\dfrac{\partial\mathscr{B}}{\partial (\partial_{j}\mathscr{Q}_n)}\right]\\ & & \\ & = & - \{\mathscr{B},\mathscr{A}\} \end{array}\tag{2} \end{equation} and $$ \delta\mathscr{H} = \{\mathscr{H},\mathscr{G}\}\delta\epsilon + \partial_i\left\{\dfrac{\partial\mathscr{H}}{\partial (\partial_i\mathscr{Q}_n)}\delta\mathscr{Q}_n\right\} + \partial_{\epsilon}[\mathscr{H}]\delta\epsilon.\tag{3} $$
where $\partial_{\epsilon}[\mathscr{H}]$ adds the explicit variation of $\mathscr{H}$ (e.g.for a time translation, it would be the explicit derivative with respect to $t$). The symmetry condition requires $$ \delta S = 0\tag{4} $$ so we implement the variations on the fields, discarding flux terms \begin{equation} \begin{array}{rcl} \delta\mathrm{S} & = & \displaystyle\int \delta\left\{\mathscr{P}_n\dot{\mathscr{Q}}_n - \mathscr{H}\right\}d{^4x}\\ & & \\ & = & \displaystyle \int \left\{\delta\mathscr{P}_n\dot{\mathscr{Q}}_n - \dot{\mathscr{P}}_n\delta\mathscr{Q}_n - \delta\mathscr{H}\right\}d{^4x}\\ & & \\ & = & \displaystyle \int \left\{\{\mathscr{P}_n,\mathscr{G}\}\dot{\mathscr{Q}}_n\delta\epsilon - \dot{\mathscr{P}}_n\{\mathscr{Q}_n,\mathscr{G}\}\delta\epsilon - \delta\mathscr{H}\right\}d{^4x}\\ & & \\ & = & \displaystyle \int \left\{-\left[\dfrac{\partial\mathscr{G}}{\partial\mathscr{Q}_n}-\partial_i\dfrac{\partial\mathscr{G}}{\partial(\partial_i\mathscr{Q}_n)}\right]\dot{\mathscr{Q}}_n\delta\epsilon\right.\\ & & \\ & - & \left. \left[\dfrac{\partial\mathscr{G}}{\partial\mathscr{P}_n}-\partial_i\dfrac{\partial\mathscr{G}}{\partial(\partial_i\mathscr{P}_n)}\right]\dot{\mathscr{P}}_n \delta\epsilon- \delta\mathscr{H}\right\} d{^4x}\\ & & \\ & = & \displaystyle \int \left\{-\dot{\mathscr{G}}\delta\epsilon + \partial_t[\mathscr{G}]\delta\epsilon - \delta\mathscr{H}\right\} d{^4x}. \end{array}\tag{5} \end{equation}
And I understand that this leads to conserved quantities when $\delta\mathscr{H}$ and $\partial_{t}[\mathscr{G}]$ stand for a total derivative. But I keep thinking that the symmetry condition $\delta S = 0$ could be satisfied just as well if $$ -\dot{\mathscr{G}} + \partial_t[\mathscr{G}] - \{\mathscr{H},\mathscr{G}\} - \partial_{\epsilon}[\mathscr{H}]\tag{6} $$ amounted to a total derivative. Then it does not lead to immediate conservation law, and has the on-shell consequence that $$ \partial_{\epsilon}[\mathscr{H}]\tag{7} $$ equals to a total derivative.
Maybe I got my definitions wrong, and symmetries are only called so when $\mathscr{H}$ also varies with a total derivative. Or perhaps conservation laws are only supposed to appear when $$ \mathscr{H} = \mathscr{H}(\mathscr{Q}_n,\mathscr{P}_n,\partial_i\mathscr{Q}_n)\tag{8} $$ and $$ \partial_t[\mathscr{G}] = 0\tag{9} $$ or some other hypothesis. But this severely limits the hamiltonians I can work with, and leads me to some trouble finding the conservation law for Lorentz symmetry, as the generator depends explicitly on time.
Also, I like to understand symmetries as movements that can be naturally carried by the system, and not associate them to conservation laws a priori. However, I really want to derive the center of energy theorem from the lorentz symmetry.
