Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 9C / Solution

For variations $q_i+\epsilon {\eta }_i$ with ${\eta }_i(t_1)={\eta }_i(t_2)=0$, differentiation of the action and integration by parts giveThe fundamental lemma of the calculus of variations therefore yields the Euler-Lagrange equations $d{L}_{{\dot{q}}_i}/dt=L_{q_i}$. An ignorable coordinate has $L_{q_j}=0$, so its conjugate momentum $p_j=L_{{\dot{q}}_j}$ is constant.

For the sphere, $\phi$ is ignorable andThe Hamiltonian is the mechanical energy. Along solutions, $\dot{H}=-\partial L/\partial t=0$, because the Lagrangian has no explicit time dependence.