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I am now reading the book Calculus of Variations written by Jost and I have a problem in the proof of Noether's theorem:

Theorem 1.5.1. Let $F\in C^2([a, b]\times \mathbb R^d \times \mathbb R^d, \mathbb R)$ and a one-parameter family of maps $$h_s:\mathbb R^d\rightarrow \mathbb R^d$$ be of class $C^2\bigl((-\varepsilon_0, \varepsilon_0)\times \mathbb R^d, \mathbb R\bigr)$ for some $\varepsilon_0>0$ with $$h_0(z)=z \quad \forall z\in \mathbb R^d$$ satisfying $$\int_a^b F\Big(t,h_s\bigl(u(t)\bigr),\frac {d}{dt}h_s\bigl(u(t)\bigr)\Big) dt=\int_a^bF\Big(t,u(t),\dot{u}(t)\Big)dt$$ for all $s\in(-\varepsilon_0, \varepsilon_0)$ and all $u\in C^2([a, b], \mathbb R^d)$.

Then, for any solution $u(t)$ of the Euler-Lagrange equation for $$I(u)=\int_a^bF\Big(t,u(t),\dot{u}(t)\Big)dt,$$ $$F_p\bigl(t,u(t),\dot{u}(t)\bigr) \frac{d}{ds}h_s\bigl(u(t)\bigr)\Bigl|_{s=0}$$ is a constant $\forall t\in [a, b]$.

The proof begins with saying the invariance of the integral gives $\forall t_0\in [a, b]$, $$\frac {d}{ds} \int_a^{t_0} F\Big(t,h_s\bigl(u(t)\bigr),\frac {d}{dt}h_s\bigl(u(t)\bigr)\Big) dt\Bigl|_{s=0}=0.$$However, this is where I find my difficulties. I can understand it when $t_0=b$, but I cannot see the reason why otherwise. I do think it is possible since we require the integral to be unchanged $\forall u\in C^2$. I have tried changing the variable $t$ so that the case where $t_0\in [a,b]$ is arbitrary is reduced to $t_0=b$, but I have yet to make any progress.

So is there any hint that anyone can give me? It would be of great help if there are any and thanks in advance!

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