Jacobi's Necessary Condition

From ProofWiki
Jump to navigation Jump to search


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 $J$.



along $\map y x$.

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


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,

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


$P = F_{y' y'}$

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


Source of Name

This entry was named for Carl Gustav Jacob Jacobi.