Definition:Conjugate Point (Calculus of Variations)/Dependent on N Functions

Definition
Let $K$ be a functional such that:


 * $\displaystyle K \left [ { h } \right ] = \int_a^b \left ( { \mathbf h' \mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \right) \rd x$

Consider Euler's equation related to the functional $K$:


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

where $\mathbf P$ and $\mathbf Q$ are symmetric matrices.

Let the set of solutions to this equation be


 * $\left \{ \mathbf h^{ \left ( { i } \right ) } = \left ( { \langle h_{ij} \rangle } \right ):  i,j \in \N_{ \le N }  \right \}$

Suppose


 * $\exists j : \forall k \ne j : \left ( { \mathbf h^{ \left ( { j } \right ) } \left ( { a } \right ) = 0 } \right ) \land \left ( { h_{ j j }' \left ( { a } \right ) = 1, h'_{ j k} = 0 } \right)$

Let the determinant, built from $h_{ i j }$, be such that:


 * $\left \vert h_{ i j } \left ( { \tilde a } \right ) \right \vert = 0$

Here $i$ denotes rows, and $j$ denotes columns.

Then $\tilde a$ is said to be conjugate to point $a$ the functional $K$.