# If Poisson Bracket of Momentum and Position is non-zero, why no Uncertainty Principle?

In Hamiltonian classical mechanics, we have that the Poisson bracket of position and momentum satisfies $$\{q_i, p_j\} = \delta_{ij}$$

But this implies that momentum and position 'generate' changes in each other. I.e. as we move along the flow of one, the other changes. Since this holds in classical mechanics as well as in quantum mechanics, why is there no uncertainty principle in classical mechanics? Why is it that non-commutativity under the Lie bracket operator gives rise to fundamental uncertainty in one case and not in the other?