In this post we will give a brief derivation of the Euler-Lagrange equation. Say we want to minimize the following integral:
\[S = \int\limits_{t_i}^{t_f}\dd{t}\mathcal{L}(q, \dot{q}, t)\]This is the integral of a known function $\mathcal{L}$, which in turn depends on an unknown function $q(t)$ its derivative $\dot{q}(t)$, and the time coordinate $t$. In order to minimize $S$, we will apply the functional derivative $\delta$, which behaves similiar to the differential operator $\mathrm{d}$ from ordinary calculus. First, let us examine how $\var{S}$ can be expressed:
\[\begin{align} \delta S &= \delta \qty(\int\limits_{t_i}^{t_f}\dd{t}\mathcal{L}(q, \dot{q})) \\ &= \int\limits_{t_i}^{t_f}\dd{t}\var{\mathcal{L}(q, \dot{q})} \\ &= \int\limits_{t_i}^{t_f}\dd{t}\qty[ \pdv{\mathcal{L}}{q}\delta q + \pdv{\mathcal{L}}{\dot{q}} \delta\dot{q}] \\ &= \int\limits_{t_i}^{t_f}\dd{t}\qty[ \pdv{\mathcal{L}}{q}\delta q + \dv{t}\qty(\pdv{\mathcal{L}}{\dot{q}}\delta q) -\dv{t}\qty(\pdv{\mathcal{L}}{\dot{q}})\delta q] \\ &= \eval{\pdv{\mathcal{L}}{\dot{q}}\delta q}_{t_i}^{t_f} +\int\limits_{t_i}^{t_f}\dd{t}\qty[ \pdv{\mathcal{L}}{q}\delta q -\dv{t}\qty(\pdv{\mathcal{L}}{\dot{q}})\delta q]\\ \end{align}\]If we assume that the function $\delta q(t)$ vanishes at the endpoints, then it follows:
\[\delta S = \int\limits_{t_i}^{t_f}\dd{t}\qty[ \pdv{\mathcal{L}}{q}\delta q -\dv{t}\qty(\pdv{\mathcal{L}}{\dot{q}})\delta q]\\\]And if we enforce $\delta S = 0$, then we conclude:
\[\boxed{\pdv{\mathcal{L}}{q} - \dv{t}\qty(\pdv{\mathcal{L}}{\dot{q}}) = 0}\]