Weierstrass's Necessary Condition

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Theorem

Let $\mathbf y: \R \to \R^n$ be an $n$-dimensional vector-valued function such that $\map {\mathbf y} a = A$ and $\map {\mathbf y} b = B$.

Let $J$ be a functional such that:

$\ds J \sqbrk {\mathbf y} = \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$

Let $\mathbf w$ be an $n$-dimensional vector such that $\mathbf w \in \R^n$.

Let $\gamma$ be a strong minimum of $J$.



Then along $\gamma$ and for every $\mathbf w$:

$\map E {x, \mathbf y, \mathbf y', \mathbf w} \ge 0$



where $E$ stands for the Weierstrass E-Function.


Proof


Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.


Sources