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$

where


 * $ F \in C^2 \left [ { a \,. \,. \, b } \right ]$

all its variables, and $ C $ stands for differentiability class.

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 ] $.

Lemma 1
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$

where


 * $ F \in C^2 \left [ { a \,. \,. \, b } \right ] $

all its variables.

Then


 * $ \displaystyle \delta^2 J \left [ { y; h } \right ] = \int_a^b \left ( { P \left ( { x, y \left ( { x } \right ) } \right ) h'^2+Q \left ( { x, y \left ( { x } \right ) } \right ) h^2 } \right ) \mathrm d x$

where


 * $ \displaystyle P \left ( { x, y \left ( { x } \right ) } \right ) = \frac{ 1 }{ 2 } F_{ y' y' }, \quad Q \left ( { x, y \left ( { x } \right ) } \right ) = \frac{ 1 }{ 2 } \left ( { F_{ y y } - \frac{ \mathrm d }{ \mathrm d x } F_{ y y'} } \right )$

Lemma 2
Let $ h$ be a real function such that:


 * $ h \in C^1 \left ( { a, b } \right ), \quad h \left ( { a } \right )= 0, \quad h \left ( { b } \right ) = 0 $

Let


 * $ \displaystyle \delta^2 J \left [ { y; h } \right ] = \int_a^b \left ( { P \left ( { x, y \left ( { x } \right ) } \right ) h'^2+Q \left ( { x, y \left ( { x } \right ) } \right ) h^2 } \right ) \mathrm d x $

where $ P \in C^0 \left [ { a \,. \,. \, b } \right ] $.

Then a necessary condition for


 * $ \delta^2 J \left [ { y; h } \right ] \ge 0$

is


 * $ P \left ( { x, y \left ( { x } \right ) } \right ) \ge 0 \quad \forall x \in \left [ { a \,. \,. \, b } \right ] $