[QUE/CM-03003]

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.( 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 $t_2$