The total change in the action S now comprises changes brought by every interval in play. Parts, where variation itself vanishes, i.e outside [ t 0 , t 1 ] bring no Δ Δ S . The middle part does not change the action either, because its transformation φ φ is a symmetry and thus preserves the Lagrangian L and the action S = ∫ ∫ L . The only remaining parts are the "buffering" pieces. In these regions both the coordinate q and velocity q ˙ ˙ change, but q ˙ ˙ changes by δ δ q / τ τ , and the change δ δ q in the coordinate is negligible by comparison since the time span τ τ of the buffering is small (taken to the limit of 0), so δ δ q / τ τ ≫ ≫ δ δ q . So the regions contribute mostly through their "slanting" q ˙ ˙ → → q ˙ ˙ ± ± δ δ q / τ τ .