Weierstrass's Necessary Condition

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:


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

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.