Definition:Jacobi's Equation of Functional/Dependent on N Functions

Definition
Let


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

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

Let


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

be a quadratic functional, where $ \displaystyle P_{ i j } = \frac{ 1 }{ 2 } F_{ y_i' y_j' }$ and $ \displaystyle Q_{ i j } = \frac{ 1 }{ 2 } \left ( { F_{ y_i y_j } - \frac{ \mathrm d }{ \mathrm d x } F_{ y_i y_j' } } \right ) $

Then the Euler's equation of the latter functional:


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

is called Jacobi's Equation of the former functional.