[QUE/CM-02013]

Let $C$ and $C^\prime$ be two neighbouring paths in configuration space $$ C: q_k=q_k(t) \qquad C^\prime : q_k = q_k(t) + \delta q_k(t) \qquad t_1 < t < t_2 $$ which do not have the same end points. Show that the difference in action along the two paths receives contribution only from the end points if and only if Euler Lagrange equations of motion are obeyed. In other words, show that the action integral depends on the differences $\Delta q_1 $ and $\Delta q_2$ at the end points $t_1$ and $t_2$ only and not on values of $\delta q(t)$ for other values of $t$ between $t_1$ and $t_2$