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In the derivation of energy conservation, there is the transformation $q(t)\rightarrow q'(t)=q(t+\epsilon)$, whose end points are kind of fuzzy. The original path $q(t)$ is only defined from $t_1$ to $t_2$. If this transformation rule is imposed, $q'(t_2-\epsilon)=q(t_2)$ to $q'(t_2)=q(t_2+\epsilon)$ is not defined in the original path. Then how could the Lagrangian be integrated?

On P.98 of Jakob Schwichtenberg's book Physics from Symmetry, he stated that $\delta q(t_1)=\delta q(t_2)=0$ whereas Kleinert stated in his Particles and Quantum Fields $\delta q_s(t_a)$ and $\delta q_s(t_b)$ are not necessarily $0$. Who's correct?

This question is different from the endpoint questions since it is already clear that $q(t_2+\epsilon)\neq q(t_2)$.

Ladmon Draxngfüskiii
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    Possible duplicates: https://physics.stackexchange.com/q/464730/2451 , https://physics.stackexchange.com/q/299111/2451 , https://physics.stackexchange.com/q/189690/2451 and links therein. – Qmechanic Sep 11 '19 at 10:04
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    @Qmechanic My main concern here is not about whether the end point conditions still hold in noether's theorem, but how can the endpoints be defined when t is varied. – Ladmon Draxngfüskiii Sep 11 '19 at 10:53
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    It's not a duplicate. – 7th808s Sep 11 '19 at 12:02

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