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- Conflicts arise on boundary when proving energy conservation using Noether's theorem. Different statement appear in Physics from Symmetry and Kleinert's Particles and Quantum Fields.

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)##.

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)##.

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