# Definition:Jacobi's Equation of Functional

## Definition

Let

- $\displaystyle\int_a^b\map F {x,y,y'}\rd x$

be a functional, where $\map y a=A$ and $\map y b=B$.

Let

- $\displaystyle\int_a^b\paren{Ph'^2+Qh^2}\rd x$

be a quadratic functional, where $\displaystyle P=\frac 1 2 F_{y'y'}$ and $\displaystyle Q=\frac 1 2 \paren{F_{yy}-\frac \d {d x} F_{yy'} }$

Then the Euler's equation of the latter functional:

- $\displaystyle -\frac \d {\d x}\paren{Ph'}+Qh=0$

is called **Jacobi's Equation** of the former functional.

## Source of Name

This entry was named for Carl Gustav Jacob Jacobi.

## 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