Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S7 / 2 / 2 / Solution

Hamilton's equations are ${\dot{q}}_i=H_{p_i}$ and ${\dot{p}}_i=-H_{q_i}$. Using the elementary brackets from the previous part givesA constant of motion is a function that stays unchanged along each physical trajectory. For a function $C(\mathbf{q},\mathbf{p})$ with no explicit time dependence, the chain rule and Hamilton's equations giveThus $\boxed{C\text{ is conserved}⟺\{C,H\}=0}$ on the accessible phase-space domain. If a function explicitly depends on time, the criterion is instead ${\partial }_{t}C+\{C,H\}=0$; the simpler statement uses the source's time-independent $C$ assumption.