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

Theorem
Let $\Phi = \map {\Phi} {x, \family {y_i}_{1 \mathop \le i \mathop \le n}, \family {p_i}_{1 \mathop \le i \mathop \le n} }$ 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.