Jacobi's Necessary Condition/Dependent on N Functions

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Theorem

Let $J$ be a functional, such that:

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

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:

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

for all admissable real functions $\mathbf h$.

By lemma 1 of Legendre's Condition:

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

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