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

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?

• The derivation of the Euler-Lagrange equation assumes Dirichlet boundary conditions.
– Ian
May 29, 2022 at 22:31
• what does that mean in English o_o May 29, 2022 at 22:33
• Ok I see, that regularity thing I said is there is assumed May 29, 2022 at 23:26
• It means boundary conditions of the form $y(a)=y_a,y(b)=y_b$. (I'm a little surprised you know anything about calculus of variations but have never heard of Dirichlet boundary conditions.)
– Ian
May 29, 2022 at 23:53
• My knowledge is basically from a physics context. Maybe I did hear of it before, but it didn't really click stick to me. @Ian May 29, 2022 at 23:54

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.