Definition:Jacobi's Equation of Functional

Definition
Let


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

be a functional, where $ y \left ( { a } \right ) = A$ and $ y \left ( { b } \right ) = B$.

Let


 * $ \displaystyle \int_a^b \left ( { P h'^2 + Q h^2 } \right ) \mathrm d x $

be a quadratic functional, where $ \displaystyle P = \frac{ 1 }{ 2 } F_{ y' y' }$ and $ \displaystyle Q = \frac{ 1 }{ 2 } \left ( { F_{ y y } - \frac{ \mathrm d }{ \mathrm d x } F_{ y y' } } \right ) $

Then the Euler's equation of the latter functional:


 * $ \displaystyle - \frac{ \mathrm d }{ \mathrm d x} \left ( { Ph' } \right ) + Q h = 0$

is called Jacobi's Equation of the former functional.