When we derive Euler-Lagrange equations in classical mechanics following the Lagrange approach we introduce Boundary conditions at the starting and ending points of the path in the configuration space. Usually,though not necessarily, one admits that q(t_initial)=q(t_final)=0. But the standard problem in classical mechanics is rather to assume Initial (not Boundary) conditions, in practice it is assumed as known the initial position q(t_initial) and initial velocity q'(t_initial). Is there any way to modify the Lagrangian such as the optimization of the Action would yield along with the Euler-Lagrange equations the above mentioned Initial conditions?

When we derive Euler-Lagrange equations in classical mechanics following the Lagrangian approach we introduce Boundary conditions at the starting- and end-points of the path in the configuration space. Usually,though not necessarily, one requires that $q(t_{initial})=q(t_{final})=0$. But the standard problem in classical mechanics is to instead assume Initial (not Boundary) conditions, in practice it is assumed that the initial position $q(t_{initial})$ and initial velocity $q'(t_{initial})$ are known.

Is there any way to modify the Lagrangian such that the extremisation of the Action would yield the Euler-Lagrange equations together with the above mentioned Initial conditions?