Jacobi's Equation is Variational Equation of Euler's Equation

Theorem
The Variational equation of Euler's equation is Jacobi's equation.

Proof
Let Euler's equation be


 * $ \displaystyle F_y \left ( { x, y, y' } \right ) - \frac{ \mathrm d }{ \mathrm d x } F_{ y' } \left ( { x, y, y' } \right ) = 0 $

Let $ \hat y \left ( { x } \right )$ and $ \hat y \left ( { x } \right ) = y \left ( { x } \right ) + h \left ( { x } \right ) $ be solutions of Euler's equation.

By Taylor's theorem: