Legendre's Condition

Theorem
Let $y =\map y x$ be a real function, such that:


 * $\map y a = A,\quad \map y b = B$

Let $J \sqbrk y$ be a functional, such that:


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

where


 * $F \in C^2 \closedint a b$

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

Then a necessary condition for $J \sqbrk y$ to have a minimum at $y = \hat y$ is:


 * $\bigintlimits {F_{y'y'} } {y \mathop = \hat y} {} \ge 0 \quad \forall x \in \closedint a b$

Lemma 1
Let $y = \map y x$ be a real function, such that:


 * $\map y a = A, \quad \map y b = B$

Let $J \sqbrk y$ be a functional, such that:


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

where:


 * $F \in C^2 \closedint a b$

all its variables.

Then:


 * $\displaystyle \delta^2 J \sqbrk {y; h} = \int_a^b \paren {\map P {x, \map y x } h'^2 + \map Q {x, \map y x} h^2} \rd x$

where:


 * $\map P {x, \map y x} = \dfrac 1 2 F_{y' y'}, \quad \map Q {x, \map y x} = \dfrac 1 2 \paren {F_{yy} - \dfrac \d {\d x} F_{yy'} }$

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


 * $h \in C^1 \openint a b, \quad \map h a = 0, \quad \map h b = 0$

Let:


 * $\displaystyle \delta^2 J \sqbrk {y; h} = \int_a^b \paren {\map P {x, \map y x} h'^2 + \map Q {x, \map y x} h^2} \rd x$

where $P \in C^0 \closedint a b$.

Then a necessary condition for:


 * $\delta^2 J \sqbrk {y; h} \ge 0$

is:


 * $\map P {x, \map y x} \ge 0 \quad \forall x \in \closedint a b$