# 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

Consider the 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.

$\blacksquare$

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 4.17$: First Integrals of the Euler Equations