@Semiclassical Could I also ask you something else?
According to my notes, the following theorem holds:
If $y$ is a local extremum for the functional $J(y)= \int_a^b L(x,y,y') dx$ with $y \in C^2([a,b]), \ y(a)=y_0, \ y(b)=y_1$ then the extremum $y$ satisfies the ordinary differential equation of second order $L_y(x,y,y')- \frac{d}{dx}L_{y'}(x,y,y')=0$ (Euler's equation).
I want to prove that Euler's equation of the problem $J(y)=\int L(t,y,y') dt$ can be written in the form $L_t- \frac{d}{dt}(L-y' L_{y'})=0$.
Let $V=C^1([a,b])$. If $J$ is a continuous functional for the norm $\|\cdot\|_\infty$ then it is continuous for the norm $\|\cdot\|_1$. But the converse is false.
In my book there is the hint that we can use the functional of arc length, in order to show that the converse does not hold.
The fun...
