= Solution
For variations $q_i+\varepsilon\eta_i$ with $\eta_i(t_1)=\eta_i(t_2)=0$, differentiation of the <action> and <integration by parts> give
$$
\delta S=\int_{t_1}^{t_2}\left(L_{q_i}-\frac d{dt}L_{\dot q_i}\right)\eta_i\,dt.
$$
The <fundamental lemma of the calculus of variations> therefore yields the <Euler-Lagrange equations> $dL_{\dot q_i}/dt=L_{q_i}$. An <ignorable coordinate> has $L_{q_j}=0$, so its <conjugate momentum> $p_j=L_{\dot q_j}$ is constant.
For the sphere, $\varphi$ is <ignorable> and
$$
\boxed{p_\varphi=ma^2\sin^2\theta\,\dot\varphi,\qquad
H=\frac{ma^2}{2}(\dot\theta^2+\sin^2\theta\,\dot\varphi^2)+V(\theta).}
$$
The <Hamiltonian> is the <mechanical energy>. Along solutions, $\dot H=-\partial L/\partial t=0$, because the <Lagrangian> has no explicit time dependence.
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