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  • $\begingroup$ Thanks, that's a very nice observation. Two quick questions: 1) If there is a term of order $\sqrt{\epsilon}$, does that not imply that the variational derivative $\delta S/\delta x$ is ill defined (infinite)? 2) Am I allowed to use $E=const$ in carrying out the variation ($E=const$ only for stationary paths)? $\endgroup$ Commented Sep 6, 2020 at 19:30
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    $\begingroup$ 1) It is certainly not a stationary path. 2) $E=const$ is true for cycloid segments, but not for generic paths. $\endgroup$ Commented Sep 6, 2020 at 19:46