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

Theorem
Let $\Phi = \Phi \left({x, \langle y_i \rangle_{1 \le i \le n}, \langle p_i \rangle_{1 \le i \le n} } \right)$ 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


 * $\dfrac {\partial \Phi} {\partial x} + \left[{\Phi, H}\right] = 0$

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

Hence the result.