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

Definition
Let:


 * $\ds \int_a^b \map F {x, \mathbf y, \mathbf y'} \rd x$

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

Let:


 * $\ds \int_a^b \paren {\mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$

be a quadratic functional, where:
 * $P_{ij} = \dfrac 1 2 F_{y_i'y_j'}$
 * $Q_{ij} = \dfrac 1 2 \paren {F_{y_i y_j} - \dfrac \d {\d x} F_{y_i y_j'} }$

Then the Euler's equation of the latter functional:


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

is called Jacobi's Equation of the former functional.