Past exam of the classical mechanics course of the University of Oxford / 2023 / 3 / a / ii

In canonical coordinates on phase space, the Poisson bracket isIt treats $t$ as a parameter. Along a solution of Hamilton's equations, the chain rule givesTherefore the Hamiltonian conservation law isThe total derivative follows the trajectory; the explicit partial derivative holds $(\mathbf{q},\mathbf{p})$ fixed.