Euler's Equation/Independent of y'

Theorem

Let $y$ be a mapping.

Let $J$ a functional be such that

$\displaystyle J \sqbrk y = \int_a^b \map F {x,y} \rd x$

Then the corresponding Euler's Equation can be reduced to:

$F_y = 0$

Furthermore, this is an algebraic equation.

Proof

Assume that:

$\displaystyle J \sqbrk y = \int_a^b \map F {x,y} \rd x$

Then Euler's Equation for $J$ is:

$F_y = 0$

Since $F$ is independent of $y'$, the equation is algebraic.

$\blacksquare$