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on the second term. The second term on the second line vanishes because at and by definition. Also, as previously mentioned the left side of the equation is zero so that

According to the fundamental lemma of calculus of variations, the fact that this equation holds for any choice of implies that the part of the integrand in parentheses is zero, i.e.

which is called the Euler-Lagrange equation. The left hand side of this equation is called the functional derivative

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