I'm reading the proof of the proposition that states that every minimizing curve is a geodesics when it is given an unit speed parametrization.
In the proof appears the following quantity : $$ \delta J= \epsilon \sum_{\alpha }\int_{t_0}^t\big[\frac{\partial f}{\partial \dot{x}^\alpha}(\phi(t), \phi(t)')-\int_{t_0}^t\frac{\partial f}{\partial \dot{x}^\alpha}(\phi(t), \phi(t)') dr\big] \dot{\psi}^\alpha(t) dt.$$
(with $f:(a,b)\rightarrow X$ continuous, $\phi:[t_0,t_1]\rightarrow U\subset X$, with $X$ Riemannian manifold of dim $n$, of class $C^2$ s.t. $\phi(t_0)=x_0$ and $\phi(t_1)=x_1$)
Then it is said that :
$\delta J=0$ $\forall \psi$ with $\psi: [t_0,t_1]\rightarrow \mathbb{R}^n$ of class $C^2$ s.t. $\psi(t_0)=\psi(t_1)=0$
is equivalent to
$$\big[\frac{\partial f}{\partial \dot{x}^\alpha}(\phi(t), \phi(t)')-\int_{t_0}^t\frac{\partial f}{\partial \dot{x}^\alpha}(\phi(t), \phi(t)') dr\big]=cost$$ for every $\alpha=1,...,n$.
Why can I say this? Is it a theorem in calculus of variation?
I know the following lemma: if $f\in L^1_{loc}$ $$\int_a^b f(x) \phi(x)' dx=0 $$ for every $\phi\in C^1_0(I)$ than $f(x)=cost$ a.e.
Is the assertion mentioned above a consequence of a generalization of this lemma? (The thing that confuses me is that there is a summation over $\alpha$ and this index appears also in $\dot{\psi}^\alpha$).
Thanks for the help!
Yes, precisely, it is a generalization of your lemma. Note that your lemma implies that if: $$ \sum_i \int_a^b f_i(x)\, \phi_i'(x)\,dx =0\,, $$
then all the $f_i$ are constant almost everywhere.