Consider the following minimization problem. Given the *Lagrange function* which is sufficiently regular (e.g. ∈ *C*^{2}) and the function , find function such that

*Definition* 1. A function is called a candidate for a solution of (A.1) if it is defined on Ω, obeys the boundary condition, and is piece-wise differentiable.

A necessary condition for a candidate , to be a solution of (A.1) is that the following system of differential equations applies:

This fundamental theorem was proven by Leonhard Euler in 1744. With this proof, he created ...

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