Given a functional of the form
$$ J[y] = \int_{a}^b F(x,y,y')dx $$
the EL equation is given by
$$ \frac{dJ}{dy} = \left( \frac{\partial}{\partial y} - \frac{d}{dx}\frac{\partial}{\partial y'} \right) F $$
In applications however if often see the following rule for gradient descent
$$ \frac{\partial}{\partial t} y = -\left( \frac{\partial}{\partial y} - \frac{d}{dx}\frac{\partial}{\partial y'} \right) F $$
The questions isn't much about the formula per se, but how do we formalize the gradient descent in such case? I can't manage to find a proof or a derivation of the rule.