# 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 $J\sqbrk{y,p}$:

\(\displaystyle \frac{\delta J\sqbrk{y,p} }{\delta p}\) | \(=\) | \(\displaystyle \frac{\partial}{\partial p}\paren{-\map H {x,y,p}+py'}\) | $\quad$ Depends only on $p$ and not its derivatives | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle -\frac{\partial H}{\partial p}+y'\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 0\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(\implies\) | \(\displaystyle y'=\frac{\partial H}{\partial p}\) | $\quad$ | $\quad$ |

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

\(\displaystyle J\sqbrk{y,p}\big\vert_{\frac{\delta J\sqbrk{y,p} }{\delta p}=0}\) | \(=\) | \(\displaystyle \int_a^b\paren{-H+p\frac{\partial H}{\partial p} }\rd x\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_a^b\paren{\map F {x,y,y'}-py'+p\frac{\partial\paren{-\map F {x,y,y'}+py'} }{\partial p} } \rd x\) | $\quad$ Definition of Hamiltonian | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_a^b\paren{\map F {x,y,y'}-py'+py'}\rd x\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \int_a^b \map F {x,y,y'}\rd x\) | $\quad$ | $\quad$ |

$\blacksquare$

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 4.18$: The Legendre Tranformation