Conditions for Functional to be Extremum of Two-variable Functional over Canonical Variable p

Theorem
Let $y=\map y x$ and $\map F {x,y,y'}$ be real functions.

Let $\displaystyle\frac{\partial^2 F}{\partial {y'}^2}\ne 0$.

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

Let $J\sqbrk{y,p}=\int_a^b\paren{-\map H {x,y,p}+py'}\rd x$, where $H$ is the Hamiltonian of $J\sqbrk y$.

Then $\displaystyle J\sqbrk y=J\sqbrk{y,p}\big\vert_{\frac{\delta J\sqbrk{y,p} }{\delta p}=0}$

Proof
Euler's equation for $J\sqbrk{y,p}$:

Substitute this result back into the functional $J\sqbrk {y,p}$: