Jacobi's Necessary Condition

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Theorem

Let $J$ be a functional, such that:

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

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:

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

for all admissable real functions $h$.

By lemma 1 of Legendre's Condition,

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

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 $J$.

$\blacksquare$


Source of Name

This entry was named for Carl Gustav Jacob Jacobi.


Sources