Definition:Conjugate Point (Calculus of Variations)/Dependent on N Functions
Jump to navigation
Jump to search
Definition
Let $K$ be a functional such that:
- $\ds K \sqbrk h = \int_a^b \paren {\mathbf h'\mathbf P \mathbf h' + \mathbf h \mathbf Q \mathbf h} \rd x$
Consider Euler's equation related to the functional $K$:
- $-\map {\dfrac \d {\d x} } {\mathbf P \mathbf h'} + \mathbf Q \mathbf h = 0$
where $\mathbf P$ and $\mathbf Q$ are symmetric matrices.
Let the general solution to this equation be:
- $\set {\mathbf h^{\paren i} = \paren {\sequence {h_{ij} } }: i,j \in \N_{\le N} }$
Let:
- $\exists j: \forall k \ne j: \paren {\map {\mathbf h^{\paren j} } a = 0} \land \paren {\map {h_{j j}'} a = 1, h'_{j k} = 0}$
Let the determinant, built from $h_{ij}$, be such that:
- $\size {h_{i j} } \paren {\tilde a} = 0$
Here $i$ denotes rows, and $j$ denotes columns.
Then $\tilde a$ is said to be conjugate to point $a$ with respect to the functional $K$.
Sources
1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 5.29$: Generalization to n Unknown Functions