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 $q(t_2+\epsilon)\neq q(t_2)$.