The core idea of this problem is to find a function that is a stationary point for a certain process, expresed as an integral functional, this means that we need to set the functional derivative of the integral to zero.
$$ 0 = \delta S[q|\phi] = \frac{\partial}{\partial \varepsilon} \int_a^b F\circ \Gamma(q+\varepsilon \phi) dt = \int_a^b \frac{\partial}{\partial \varepsilon} F(q+\varepsilon \phi, \dot q+\varepsilon \dot\phi)dt = $$ $$ \int_a^b \frac{\partial F}{\partial q}\phi + \frac{\partial F}{\partial \dot q}\dot\phi dt = \left[\frac{\partial F}{\partial \dot q}\phi\right]^b_a + \int_a^b \left(\frac{\partial F}{\partial q} - \frac{d}{dt}\frac{\partial F}{\partial \dot q}\right)\phi dt $$
We haven't made made any assumption of the form of $q$ and the space $\Omega$ where it resides. On the standard variational problem $\Omega$ is the space of the functions that $q(a)=q_a$ and $q(b)=q_b$, i.e, fixed boundaries, and are sufficiently smooth. Also, we asumed a standart lift: $\Gamma(q) = (q, \dot q)$.
Saying that, now the fundamental lemma of variational calculus enters. The variation, $\phi$, is arbitrary but with $\phi(a)=\phi(b)=0$. The later assertion, makes the term outside the integral zero. On the other hand, by the fundamental lemma, the interior parentheisi is zero, giving:
$$ \frac{\partial F}{\partial q} - \frac{d}{dt}\frac{\partial F}{\partial \dot q} = 0 $$
that is, the Euler-Lagrange equations.
Some variations worth studying
1. Change $\Omega$: the functions do not need to be continous at all points, this gives collisions on a variational setting.
2. Variable end points, i.e. $a$ and $b$ vary with $\varepsilon$
3. Change the interdependencies of the lift $\Gamma$. This will lead to Vakonomic Mechanics.