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

Since $H$ is a conserved quantity, the Hamiltonian energy balance gives $H_t=0$. Since $f$ is a conserved quantity, $f_t+\{f,H\}=0$. Take an explicit partial derivative with respect to time, holding the phase space coordinates fixed:The last Poisson bracket vanishes because $H_t=0$. ThusThis is the time derivative of a conserved quantity for an autonomous Hamiltonian. The conservation hypotheses are understood to hold for every trajectory, as identities on phase space.