The following Wikipedia page uses $x_\varepsilon (t) = x(t) + \varepsilon (t)$ in the proof.
https://en.wikipedia.org/wiki/Hamilton%27s_principle#Mathematical_formulation
But in my mechanics book (by David Morin), the author uses something similar to $x_\varepsilon (t) = x(t) + \varepsilon \eta (t), \ \varepsilon \in \mathbb{R}$.
Since the proof using the concept of $\delta \mathcal S$ is written on the Wikipedia page I won't go through that.
Consider the following partial derivative of the action functional (whichever epsilon it is).
$$\frac{\partial \mathcal S}{\partial \varepsilon} = \int_{t_1}^{t_2} \frac{\partial \mathscr L}{\partial \varepsilon} \ dt = \int_{t_1}^{t_2} \frac{\partial \mathscr L}{\partial x_\varepsilon}\frac{\partial x_\varepsilon}{\partial \varepsilon} + \frac{\partial \mathscr L}{\partial x'_\varepsilon}\frac{\partial x'_\varepsilon}{\partial \varepsilon} \ dt $$
Note that I used the notation $x'$ to indicate the time derivative.
If I put $x_\varepsilon (t) = x(t) + \varepsilon \eta (t)$ here then it works and I get the Euler-Lagrange equation.
But if I substitute $x_\varepsilon (t) = x(t) + \varepsilon (t)$ into the integral, I obtain
$$ \frac{\partial x_\varepsilon}{\partial \varepsilon} = 1 $$ and $$ \frac{\partial x'_\varepsilon}{\partial \varepsilon} = \frac{\partial \varepsilon '}{\partial \varepsilon} = 0$$
which lead to an absurd result.
Why?