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

To first order the conditions are ${\partial }_{p_j}\delta q_i-{\partial }_{p_i}\delta q_j=0$, ${\partial }_{q_j}\delta p_i-{\partial }_{q_i}\delta p_j=0$, and ${\partial }_{q_j}\delta q_i+{\partial }_{p_i}\delta p_j=0$. The choices $\delta q_i=\epsilon F_{p_i}$ and $\delta p_i=-\epsilon F_{q_i}$ satisfy them by equality of mixed partials.

Solved by gpt-5.6-sol high.