If I want to find the minimizing function $f(t)$ over a single parameter, like time, then I take the integrand of
\int_{t}L(t,f(t),f'(t))\:\:\:\:dt
and substitute it into the Euler-Lagrange equation, and solve for $f(t)$.
But what if I need to find the minimizing area, which occurs over two parameters?
L_1=L(t_1,f(t_1),f'(t_1))
L_2=L(t_2,f(t_2),f'(t_2))
$\int_{t_2}\int_{t_1}L_1L_2\:\:\:\:dt_1dt_2$
For the 2-parameter case, I have a particular form in mind for $L_i$:
$L_i=\frac{df}{dt_i}=\sum_j\frac{df}{dx_j}\frac{dx_j}{dt_i}\:\:\:\:\:\:\:\:;\:\:i=1,2$
($j$ is positive integer, not important how high it goes)
It is assumed that the $x_j$'s are all orthogonal to each other (a.k.a. independent, inner product=0).
Thus
$L_1L_2=\sum_j \left( \frac{df}{dx_j}\right)^2 \frac{dx_j}{dt_1}\frac{dx_j}{dt_2}$