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 $J$.

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

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

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

Proof

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Source of Name

This entry was named for Karl Theodor Wilhelm Weierstrass.

Sources

• 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 6.34$: The Weierstrass E-Function. Sufficient Conditions for a Strong Extremum