We revisit Poisson brackets and introduce them as an axiomatic away to describe classical mechanics.

We then show that these axioms are equivalent to the previous definition (in Hamiltonian terms).

Symmetries were originally defined as transformations in coordinate $q$-space that don't change the Lagrangian (and hence the action). We now consider symmetries in phase space - that is, transformations in the phase-space coordinates, $q, p$-space, that don't change the energy, or Hamiltonian.