This Euler-lagrange equation must be satisfied in 3D and even in the 4D of relativity. Even if the number of parameters increase it must be satisfied over lines and surfaces of classical string theory. So what makes this Action integral so necessary to physics? Is there some underlying geometry that necessitate the Action Integral?

Can the various functions of physical situations be considered as a type of boundary conditions of the Euler-lagrange differential equation? For example, the momentum is the derivative of the Lagrangian which is a function derived from the particular physical situation, but it is also a specified first order diff eq to the second order diff eq of the Euler-lagrange eq. Isn't this a boundary value problem for a second order diff eq?