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

Definition
Let $K$ be a functional such that:


 * $\displaystyle K\sqbrk h=\int_a^b \paren {\mathbf h'\mathbf P\mathbf h'+\mathbf h\mathbf Q\mathbf h}\rd x$

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


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

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

Let the set of solutions to this equation be


 * $\lbrace {\mathbf h}^{\paren i}=\paren {\sequence {h_{ij} } }:i,j\in\N_{\le N}\rbrace$

Suppose


 * $\exists j:\forall k\ne j:\paren {\map {\mathbf h^{\paren j} } a=0}\land \paren {\map {h_{jj}'} a=1,h'_{jk}=0}$

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


 * $\size {h_{ij} }\paren{\tilde a}=0$

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

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