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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?

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    $\begingroup$ The derivation of the Euler-Lagrange equation assumes Dirichlet boundary conditions. $\endgroup$
    – Ian
    Commented May 29, 2022 at 22:31
  • $\begingroup$ what does that mean in English o_o $\endgroup$ Commented May 29, 2022 at 22:33
  • $\begingroup$ Ok I see, that regularity thing I said is there is assumed $\endgroup$ Commented May 29, 2022 at 23:26
  • $\begingroup$ 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.) $\endgroup$
    – Ian
    Commented May 29, 2022 at 23:53
  • $\begingroup$ 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 $\endgroup$ Commented May 29, 2022 at 23:54

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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.

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