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
Consider the 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.
Context needed: what is $H$, what is $\Phi$?
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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 $\sqbrk{H, H} = 0$.
... from where?
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$\blacksquare$
Sources
- 1963: I.M. Gelfand and S.V. Fomin: Calculus of Variations ... (previous) ... (next): $\S 4.17$: First Integrals of the Euler Equations