Weierstrass's Necessary Condition

Theorem
Let $ \mathbf y $ be an $ n $-dimensional vector such that $ \mathbf y \left ( { a } \right ) = A $ and $ \mathbf y \left ( { b } \right ) = B $.

Let $ J $ be a functional such that:


 * $ \displaystyle J \left [ { \mathbf y } \right ] = \int_a^b F \left ( { x, \mathbf y, \mathbf y' } \right ) \mathrm d x $

Let $ \mathbf w $ be a finite vector.

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

Then:
 * $E \left ( { x, \mathbf y, \mathbf y', \mathbf w } \right ) \ge 0 $

along $ \gamma $ and for every $ \mathbf w $, and $ E $ stands for Weierstrass E-Function.