# Physical meaning of boundary conditions in local variation of action

In the procedure to obtain Euler Lagrange equation of motion, as a consequence of the local variation of the action I obtain the term (see picture). To eliminate it, I must assume the asymptotic radial behavior for which $u$ goes to zero. I was wondering what physical meaning it might have?

And the ultimate origin of this weirdness is the required vanishing of these boundary terms, $\left[f(x, \dot x, \dots) ~\delta x(t)\right]_{t_0}^{t_1} = 0.$ It forces $\delta x(t_0) = \delta x(t_1) = 0$ which means that the path is not being perturbed at its boundary, but only in its main body: so we must in the Lagrangian approach treat the starting and ending point as fixed and then cook up the appropriate trajectories in the middle.