Past exam of the theoretical physics I course of the University of Cambridge / 2014 / Paper 1 / 3 / a

For a regular Lagrangian define $p_i=\partial L/\partial {\dot{q}}_i$ and $H={\sum }_i{p}_i{\dot{q}}_i-L$. Its differential isThe velocity differentials cancel. Using $\partial L/\partial q_i={\dot{p}}_i$ and comparing independent coefficients givesThe last equality concerns explicit partial derivatives in their respective natural variables; it does not assert that total time derivatives are always negatives.