Here's a screenshot of derivation of Euler-Lagrange from feynman lecture https://www.feynmanlectures.caltech.edu/II_19.html
My doubt is in the last paragraph. I get that $\eta = 0$ at both ends, but we also have velocity $d\underline{x} / dt$ which is a function of $t$ and which may not be the same at the two points $t1$ and $t2$ to make the integrated term = 0. Can anybody kindly explain? I saw the same thing in Landau-Lifshitz as well.
If $\eta(t_i) = \eta(t_f) = 0$, $$ m \dot{x}(t_f) \eta(t_f) - m \dot{x}(t_i) \eta(t_i) = 0 \cdot (m \dot{x}(t_f) - m \dot{x}(t_i)) = 0 $$ no matter what are the values of $\dot{x}$ at the endpoints, as long as they are finite.
I think you are puzzled because you are confusing with the situation where $\eta(t_i) = \eta(t_f) = \eta_c \neq 0$, in which case the full derivative term would contribute $ \eta_c \cdot (m \dot{x}(t_f) - m \dot{x}(t_i))$ and would vanish only if $\dot{x}(t_i) = \dot{x}(t_f)$. But here $\eta_c = 0$
