I study the functional $F_\epsilon$ (see picture) in $\phi$ ($c_1$ and $c_2$ are constants).
From its expression I try to write the Euler-Lagrange equation associated to ϕ. Only for that the author introduces an artificial variable t which represents the time (see second image) in order to solve the equation by a gradient descent.
But I do not understand where the t comes from in view of the Euler-Lagrange equation nor the method. In fact at first I do not know this technique (and I do not understand it) of introducing an artificial time parameter $t$. Thank you in advance for your help.