Definition:Jacobi's Equation of Functional

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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