Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 1 / 15C / a / Solution

Vary $q_a$ and $p_a$ independently, fixing the endpoint positions. Integration by parts givesThe boundary term vanishes, so arbitrary interior variations implyIn the Lagrangian principle only $q$ is varied independently, and $\delta \dot{q}=d(\delta q)/dt$; here positions and momenta are independent paths in phase space. The extremality condition is stationarity of the action, not necessarily a minimum. Along Hamiltonian motion,so an explicitly time-independent Hamiltonian is conserved.