Du Bois Reymond's Theorem
The discussion thus far has assumed that extremal functions possess two continuous derivatives, although the existence of the integral A requires only first derivatives of trial functions. The condition that the first variation vanish at an extremal may be regarded as a weak form of the Euler-Lagrange equation. The theorem of du Bois Reymond asserts that this weak form implies the strong form. If L has continuous first and second derivatives with respect to all of its arguments, and if
then has two continuous derivatives, and it satisfies the Euler-Lagrange equation.
Read more about this topic: Calculus Of Variations
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