Definition:Jacobi's Equation of Functional

Definition
Let:


 * $(1): \quad \displaystyle \int_a^b \map F {x, y, y'} \rd x$

be a functional such that:
 * $\map y a = A$
 * $\map y b = B$

Let:


 * $(2): \quad \displaystyle \int_a^b \paren {P h'^2 + Q h^2} \rd x$

be a quadratic functional such that:
 * $P = \dfrac 1 2 F_{y'y'}$
 * $Q = \dfrac 1 2 \paren {F_{yy} - \dfrac \d {\d x} F_{yy'} }$

Then Euler's equation of functional $(2)$:


 * $-\map {\dfrac \d {\d x} } {P h'} + Q h = 0$

is called Jacobi's Equation of functional $(1)$.