What is the physical interpretation of the Poisson bracket

Question

Apologies if this is a really basic question, but what is the physical interpretation of the Poisson bracket in classical mechanics? In particular, how should one interpret the relation between the canonical phase space coordinates, $$\lbrace q^{i}, p_{j} \rbrace_{PB}~=~\delta^{i}_{j} $$ I understand that there is a 1-to-1 correspondence between these and the commutation relations in quantum mechanics in the classical limit, but in classical mechanics all observables, such as position and momentum commute, so I'm confused as to how to interpret the above relation?

Answer 1

In a rather general approach you can consider the Poisson bracket $\{g,f\}$ as expressing the rate of change of $g$ as a consequence of a flow induced by $f$. As mentioned by AngusTheMan in the comments, you get the time variation of $g$ if $f=H$ (assuming that quantities are not explicitly time-dependent). Here $g$ and $f$ are any (smooth) functions on the phase space, i.e. observables. When $g=q$ and $f=p$, since the momenta are the generators of translations, the flow generated by $f$ can be interpreted as translations, so that the canonical bracket $$\{q,p\} = 1$$ implies a variation of $\delta q = \{q,p\}\epsilon = \epsilon$. Generalising this to many dimensions you get $$\delta q_i = \{q_i,p_j\}\epsilon = \delta_{ij}\epsilon,$$ which is expressing the fact that $p_j$ generates the translations along the $j$-th coordinate (indeed $q_i$ changes by $\epsilon > 0$ only if $j=i$).