# Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation/Corollary 2

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

Let $\Phi = H$.

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

Then $H$ is the first integral of Euler's Equations.

## Proof

The statment is proven from Conditions for Function to be First Integral of Euler's Equations for Vanishing Variation

by setting $\Phi = H$ and $\dfrac {\partial H} {\partial x} = 0$, and noticing that $\sqbrk{H, H} = 0$.

$\blacksquare$