Jacobi's Necessary Condition

Theorem
Let $ J $ be a functional, such that:


 * $ J \left [ { y } \right ] = \displaystyle \int_a^b F \left ( { x, y, y' } \right ) \mathrm d x$

Let $ y \left ( { x } \right ) $ correspond to the minimum of $J$.

Let


 * $ F_{ y'y' } > 0$

along $ y \left ( { x } \right ) $.

Then the open interval $ \left ( { a \,. \,. \, b } \right )$ contains no points conjugate to $ a $.

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


 * $ \displaystyle \delta^2 J \left [ { \hat y; h } \right ] \ge 0$

for all admissable real functions $ h $.

By lemma 1 of Legendre's Condition,


 * $ \displaystyle \delta^2 J \left [ { y; h } \right ] = \int_a^b \left ( { P h'^2+Qh^2 } \right ) \mathrm d x$

where


 * $ P = F_{ y' y' }$

By Nonnegative Quadratic Functional implies no Interior Conjugate Points, $ \left ( { a \,. \,. \, b } \right )$ does not contain any conjugate points $ J $.