= Solution
For independent <generalized coordinates> $q_i$, their <generalized velocities> $\dot q_i$ and a <Lagrangian> $L(q,\dot q,t)$, the <Euler-Lagrange equations> are
$$
\boxed{\frac{d}{dt}\left(\frac{\partial L}{\partial\dot q_i}\right)
-\frac{\partial L}{\partial q_i}=0.}
$$
Here the partial <derivatives> treat $q_i,\dot q_i,t$ as independent arguments, whereas $d/dt$ follows the motion. For a conservative mechanical system, choose coordinates satisfying the <mechanical constraints>, express the <kinetic energy> $T$ and <potential energy> $V$ in those coordinates, set $L=T-V$, and evaluate these equations to obtain the <equations of motion>. If nonconservative <generalized forces> $Q_i$ are present, they appear on the right-hand side.
A <cyclic coordinate> is absent explicitly from $L$: $\partial L/\partial q_i=0$, even though $L$ may depend on $\dot q_i$. Its <canonical momentum> is consequently conserved,
$$
p_i=\frac{\partial L}{\partial\dot q_i},\qquad\boxed{\dot p_i=0.}
$$
This does not require $\dot q_i=0$ or even that $\dot q_i$ be constant; the relation between <canonical momentum> and <generalized velocity> depends on the <Lagrangian>.
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