\begin{definition}[Definition:Jacobi's Equation of Functional]
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:
:$(2): \quad \ds \int_a^b \paren {P h'^2 + Q h^2} \rd x$
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 $(1)$.
{{NamedforDef|Carl Gustav Jacob Jacobi|cat = Jacobi}}
\end{definition}