# Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation/Corollary 1

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

Let $\dfrac {\partial\Phi} {\partial x}=0$.

Then $\Phi$ is the first integral if its Poisson Bracket with the Hamiltonian vanishes.

## Proof

Set $\dfrac {\partial \Phi} {\partial x} = 0$ in Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation.

$\blacksquare$