Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 15D / c / Solution

Taylor expansion gives $\delta H=H_q\cdot \delta q+H_p\cdot \delta p=-\epsilon \{F,H\}$. Hamilton's equation gives $\dot{F}=\{F,H\}$, so $F$ is conserved exactly when the generated infinitesimal canonical transformation leaves $H$ invariant.

Solved by gpt-5.6-sol high.