For phase-space functions $A(q,p)$ and $B(q,p)$, the Poisson bracket is $${A,B}=\sum_i\left(\frac{\partial A}{\partial q_i}\frac{\partial B}{\partial p_i}-\frac{\partial A}{\partial p_i}\frac{\partial B}{\partial q_i}\right).$$ It is antisymmetric, bilinear, obeys a product rule and satisfies the Jacobi identity.

The canonical coordinates obey $${q_i,q_j}=0,\qquad {p_i,p_j}=0,\qquad {q_i,p_j}=\delta_{ij}.$$ Hamilton's equations can therefore be written compactly as $$\dot q_i={q_i,H},\qquad \dot p_i={p_i,H}.$$ More generally, for any observable $A(q,p,t)$, $$\frac{dA}{dt}={A,H}+\frac{\partial A}{\partial t}.$$ Thus $A$ is conserved when it has no explicit time dependence and ${A,H}=0$.

For a one-dimensional oscillator with $H=p^2/(2m)+m\omega^2q^2/2$, $${q,H}=\frac{\partial H}{\partial p}=\frac{p}{m},$$ recovering $\dot q=p/m$.

Poisson brackets turn conservation into an algebraic relation and describe how observables generate transformations in phase space. Their structure also foreshadows quantum mechanics, where classical Poisson brackets are replaced, in an appropriate correspondence, by operator commutators divided by $i\hbar$.