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

Set $D={\partial }_t+\{\cdot ,H\}$. The hypotheses mean $Df=Dg=0$ as identities on phase space. By the Jacobi identity for the Poisson bracket,Also, the product rule applied to the defining coordinate formula givesAdding these identities provesConsequentlyThis proves the Poisson bracket of time-dependent conserved quantities theorem, including explicit time dependence of $f$ and $g$.