Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 9C / i / Solution

The Poisson bracket in canonical coordinates isUsing Hamilton's equations ${\dot{q}}_i=H_{p_i}$ and ${\dot{p}}_i=-H_{q_i}$ in the chain rule yieldsThe partial time derivative holds phase-space coordinates fixed; the total derivative follows the actual trajectory.