= Solution
<Generalized coordinates> incorporate <holonomic constraints> without solving separately for every constraint force. For conservative mechanics use $L=T-V$. The <principle of stationary action> makes the <action> $\int L\,dt$ stationary for variations vanishing at the endpoints. <Integration by parts> gives
$$
\boxed{\frac{d}{dt}\frac{\partial L}{\partial\dot q_i}-\frac{\partial L}{\partial q_i}=0.}
$$
The <canonical momentum> is $p_i=\partial L/\partial\dot q_i$. A <cyclic coordinate> gives conserved $p_i$; absent explicit time dependence, $\sum_i p_i\dot q_i-L$ is conserved. These are symmetry–conservation relationships, not assumptions that all momenta are constant. Nonconservative generalized forces enter the right side.
For example, a pendulum has $L=m\ell^2\dot\theta^2/2+mg\ell\cos\theta$, giving $m\ell^2\ddot\theta+mg\ell\sin\theta=0$. The method extends directly to coupled coordinates and velocity-dependent interactions when the correct <Lagrangian> is used; $T-V$ with a position-only potential is not the most general form.
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