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 ) \mathrm d x $

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


 * $ \displaystyle - \frac{ \mathrm d }{ \mathrm 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 $.