# Jacobi's Necessary Condition/Dependent on N Functions

## Theorem

Let $J$ be a functional, such that:

- $J\sqbrk{\mathbf y}=\displaystyle\int_a^b \map F {x,\mathbf y,\mathbf y'}\rd x$

where $\mathbf y=\paren{\sequence {y_i}_{1\le i\le N} }$ is an N-dimensional real vector.

Let $\map {\mathbf y} x$ correspond to the minimum of $J$.

Let the $N\times N$ matrix $\mathbf P=F_{y_i'y_j'}$ be positive definite along $\map {\mathbf y} x$.

Then the open interval $\openint a b$ contains no points conjugate to $a$.

## Proof

By Necessary Condition for Twice Differentiable N Function dependent Functional to have Minimum, $J$ is minimised by $y=\map {\mathbf{\hat y} } x$ if

- $\displaystyle\delta^2 J\sqbrk{\mathbf{\hat y};\mathbf h}\ge 0$

for all admissable real functions $\mathbf h$.

By lemma 1 of Legendre's Condition,

- $\displaystyle\delta^2 J\sqbrk{\mathbf y;\mathbf h}=\int_a^b\paren{\mathbf h'\mathbf P\mathbf h'+\mathbf h\mathbf Q\mathbf h}\rd x$

where

- $\mathbf P=F_{y_i'y_j'}$

By Nonnegative Quadratic N function dependent Functional implies no Interior Conjugate Points, $\openint a b$ does not contain any conjugate points with respect to $J$.

$\blacksquare$

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 5.29$: Generalization to n Unknown Functions