Jacobi's Necessary Condition

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Theorem

Let $J$ be a functional, such that:

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

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

Let

$F_{y'y'}>0$

along $\map y x$.


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

Proof

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

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

for all admissable real functions $h$.

By lemma 1 of Legendre's Condition,

$\displaystyle\delta^2 J\sqbrk{y;h}=\int_a^b\paren{Ph'^2+Qh^2}\rd x$

where

$P=F_{y'y'}$

By Nonnegative Quadratic 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): $\S 5.27$: Jacobi's Necessary Condition. More on Conjugate Points