If this quantity exists, then the Leibnitz rule will work on the Poisson bracket.
If this quantity exists, then the Leibnitz rule will work on the Poisson bracket. Noether's theorem can be extended to this quantity by stating that the group of transformations infinitesimally generated by a constant of the motion will leave it invariant. The Poisson bracket of this quantity with a function will yield that function's time derivative, so if the Poisson bracket is zero, that function is conserved. Obtained by performing a Legendre transform on the Lagrangian, FTP, what is this quantity which in quantum mechanics is the observable representing the total energy of a system?
If this quantity exists, then the Leibnitz rule will work on the Poisson bracket.
{Hamiltonian}