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.

  • $\begingroup$ thanks for your answer...but why have I this implication? Is there because $\sum_i \int_a^b f_i(x)\, \phi_i'(x)\,dx =0$ implies $\int_a^b f_i(x) \, \phi_i'(x)\,dx$ for every $i$? But why should all these integrals be positive? Thank you very much! $\endgroup$ – Gggl. Sep 2 '15 at 8:26
  • 1
    $\begingroup$ The key is: the integral must be zero for every such $\phi_i$. So the only possibility is that the $f_i$ have all zero weak derivatives, i.e. they are constant a.e.. $\endgroup$ – geodude Sep 2 '15 at 9:13

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