= Solution
In <Lagrangian> mechanics, <generalized coordinates> $q_i$ describe the independent configurations allowed by the <constraints>. For particles with ordinary conservative <forces> in an <inertial frame>, use $L=T-V$, with <kinetic energy> $T$ and <potential energy> $V$. <Holonomic constraints> can often be built into the coordinates, avoiding explicit <constraint> <forces>.
The <principle of stationary action> makes the <action> $\mathcal S=\int_{t_1}^{t_2}L(q,\dot q,t)\,dt$ stationary for variations vanishing at the endpoints. Integrating the <velocity> variation by parts gives
$$
\delta\mathcal S=\int_{t_1}^{t_2}\sum_i
\left(\frac{\partial L}{\partial q_i}-\frac{d}{dt}\frac{\partial L}{\partial\dot q_i}\right)\delta q_i\,dt.
$$
Since each independent variation is arbitrary, the <Euler-Lagrange equations> are
$$
\boxed{\frac{d}{dt}\frac{\partial L}{\partial\dot q_i}-\frac{\partial L}{\partial q_i}=0.}
$$
If nonconservative <generalized forces> are present, their components appear on the right. The formalism is invariant under suitable coordinate changes and adding a total time <derivative> to $L$ leaves the fixed-endpoint equations unchanged.
The <generalized momentum> is $p_i=\partial L/\partial\dot q_i$. If $q_i$ is cyclic, $\partial L/\partial q_i=0$ and $p_i$ is conserved. If $L$ has no explicit time dependence, the conserved <energy> function is
$$
E_L=\sum_i\dot q_i\frac{\partial L}{\partial\dot q_i}-L.
$$
For a <kinetic energy> quadratic in <velocities> and a velocity-independent <potential energy>, this equals $T+V$. This connects <symmetry> to <conservation laws>.
For example, a <simple pendulum> of <mass> $m$ and length $\ell$ has
$$
L=\frac12m\ell^2\dot\theta^2-mg\ell(1-\cos\theta),
\qquad \ddot\theta+\frac{g}{\ell}\sin\theta=0.
$$
Its <tension> does not appear because the fixed-length <constraint> is already imposed. Expanding for small $\theta$ gives <simple harmonic motion> with $\omega=\sqrt{g/\ell}$. More generally, expansion of $T$ and $V$ to quadratic order about a <stable equilibrium> leads directly to the normal-mode <eigenvalue> problem.
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