What conditions do I need on a functional such that Euler Lagrange is consistent with boundary condition?

Question

Suppose we have a functional integral between some end points and in order to find the function which optimizes it, I apply the Euler-Lagrange to the integrand. What I receive is a differential equation which I can put in the initial or final condition into. The doubt I have is, what conditions are required on the functional equation such that the curve I get in the end is consistent with both boundary condition?

Answer 1

It seems OP is putting the cart before the horse. One cannot derive$^1$ the Euler-Lagrange (EL) equations without assuming appropriate boundary conditions (BCs) in the first place.

For a first-order Lagrangian, there are the following possible BCs:

1. Essential/Dirichlet BCs,

2. Natural BCs,

3. Combinations thereof,

cf. e.g. my Math.SE answer here.

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$^1$ Of course one can always write down the EL equations, but without appropriate BCs, there is no guarantee that the EL equations are relevant for the variational problem at hand.