= Solution
Hamilton's equations and the multivariable <chain rule> give
$$
\frac{df}{dt}
=\sum_{i=1}^3\left(
\frac{\partial f}{\partial q_i}\dot q_i
+\frac{\partial f}{\partial p_i}\dot p_i
\right)
=\sum_{i=1}^3\left(
\frac{\partial f}{\partial q_i}\frac{\partial H}{\partial p_i}
-\frac{\partial f}{\partial p_i}\frac{\partial H}{\partial q_i}
\right)
=\{f,H\}.
$$
Thus the <Hamiltonian conservation law> gives
$$
\boxed{f\text{ is conserved}\quad\Longleftrightarrow\quad\{f,H\}=0.}
$$
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