I have just started studying Calculus of Variations, and need some help about deriving the Euler-Lagrange equation.
In the book I'm reading, the writer starts by imposing the following inner product on the infinite dimensional function space: $$\langle f;g\rangle=\int_a^bf(x).g(x)dx $$Then he states what he called "the basic directional derivative formula" $$\langle\nabla J[u];v\rangle=\frac{d}{d\varepsilon}J[u+\varepsilon v]$$ at $\varepsilon=0$ and where $J$ is the objective functional.
The minimizing function of $J$ is a one satisfying $\langle \nabla J[u];v\rangle=0$
The writer then works for a formula of $\langle\nabla J[u];v\rangle$ and finally reaches :$$\langle\nabla J[u];v\rangle=\int_a^bv.[\dfrac{\partial L}{\partial u}-\dfrac{d}{dx}(\dfrac{\partial L}{\partial u'})]dx +\dfrac{\partial L}{\partial u'}(b).v(b)-\dfrac{\partial L}{\partial u'}(a)v(a)$$Then states that, for suspect functions (those verifing boundary conditions), $v(a)=v(b)=0$, and finally deduces Euler-Lagrange equation for the minmizing function :$$\dfrac{\partial L}{\partial u}-\dfrac{d}{dx}(\dfrac{\partial L}{\partial u'})=0$$
I have two questions now:
Shouldn't the norm of $v$ be equal to 1 in order to satisfy the formula of directional derivative?
Shouldn't we prove that the region of suspect functions is connected so that we can talk about a directional derivative at a function in the direction of another?
Please help.
1) In principle you are right, and in the definition of $\nabla J$ there should be a division by $\| v \|$ in order to mimick the usual formula from calculus on $\Bbb R ^n$. Nevertheless, keep in mind that you are interested in the solutions of $\nabla J = 0$, and these do not care about the equation being divided by non-zero numbers.
There is a deeper reason, though, for $\| v \|$ not being present in the denominator: in a normed space, you may speak about the Fréchet derivative. In a space without norm, though, what can you do? The concept to use is the Gâteaux derivative, which is exactly the one used in defining $\nabla J$. It happens that you have a norm on your space, but the definition given works on spaces without norm, too; it wouldn't have worked if there had been that $\| v \|$ in the denominator (what would you have replaced it with in the absence of a norm?).
2) I don't see why you require connectedness, but if you really want it then note that the set of functions that vanish on the boundary is even better - it is pathwise connected: for every $f, g$ in it, the segment $tf + (1-t)g$ vanishes on the boundary too. Anyway, you don't need this (think about derivability in $\Bbb R ^n$: where do you use connectedness? nowhere).