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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Qmechanic
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tryst with freedom
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1The derivation of the Euler-Lagrange equation assumes Dirichlet boundary conditions. – Ian May 29 '22 at 22:31
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what does that mean in English o_o – tryst with freedom May 29 '22 at 22:33
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Ok I see, that regularity thing I said is there is assumed – tryst with freedom May 29 '22 at 23:26
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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 '22 at 23:53
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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 – tryst with freedom May 29 '22 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:
Essential/Dirichlet BCs,
Natural BCs,
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.
Qmechanic
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