Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation

Theorem
Let $\Phi=\map {\Phi} {x,\langle y_i\rangle_{1\le i\le n},\langle p_i\rangle_{1\le i\le n} }$ be a real function.

Let $H$ be Hamiltonian.

Then a necessary and sufficient condition for $\Phi$ to be the first integral of Euler's Equations is


 * $\dfrac {\partial\Phi} {\partial x}+\sqbrk{\Phi,H}=0$

Proof
For $\Phi$ to be the first integral:
 * $\dfrac {\d\Phi} {\d x}=0$

Hence the result.