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 $\dfrac {\partial^2 F} {\partial {y'}^2} \ne 0$.

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

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

Then $\displaystyle J \sqbrk y = \bigvalueat {J \sqbrk {y, p} } {\frac {\delta J \sqbrk{y, p} } {\delta p} \mathop = 0}$

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

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