# Definition:Jacobi's Equation of Functional

## Definition

Let:

$(1): \quad \ds \int_a^b \map F {x, y, y'} \rd x$

be a functional such that:

$\map y a = A$
$\map y b = B$

Let:

$(2): \quad \displaystyle \int_a^b \paren {P h'^2 + Q h^2} \rd x$

be a quadratic functional such that:

$P = \dfrac 1 2 F_{y'y'}$
$Q = \dfrac 1 2 \paren {F_{yy} - \dfrac \d {\d x} F_{yy'} }$

Then Euler's equation of functional $(2)$:

$-\map {\dfrac \d {\d x} } {P h'} + Q h = 0$

is called Jacobi's Equation of functional $(1)$.

## 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