Legendre's Condition

Theorem
Let $ y = y \left ( { x } \right )$ be a real function, such that:


 * $ y \left ( { a } \right )= A, \quad y \left ( { b } \right ) = B$

Let $ J \left [ { y } \right ]$ be a functional, such that:


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

Then a necessary condition for $ J \left [ { y } \right ]$ to have a minimum for $ y = \hat { y }$ is


 * $ F_{ y' y' } \big \vert_{ y = \hat { y } } \ge 0 \quad \forall x \in \left [ { { a } \,. \,. \, { b } } \right ] $.