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

Jump to navigation
Jump to search

This article needs to be linked to other articles.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

## 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

- $\dfrac {\partial \Phi} {\partial x} + \sqbrk{\Phi, H} = 0$

### Corollary 1

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

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

### Corollary 2

Let $\Phi = H$.

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

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

## Proof

\(\ds \dfrac {\d \Phi} {\d x}\) | \(=\) | \(\ds \frac {\partial\Phi} {\partial x} + \sum_{i \mathop = 1}^n \frac {\partial \Phi} {\partial y_i} \frac {\partial y_i} {\partial x} + \sum_{i \mathop = 1}^n \frac {\partial \Phi} {\partial p_i} \frac{\partial p_i} {\partial x}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac {\partial \Phi} {\partial x} + \sum_{i \mathop = 1}^n \frac {\partial \Phi} {\partial y_i} \frac {\partial H} {\partial p_i} - \sum_{i \mathop = 1}^n \frac {\partial \Phi} {\partial p_i} \frac{\partial H} {\partial y_i}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac {\partial \Phi} {\partial x} + \sqbrk{\Phi, H}\) |

For $\Phi$ to be the first integral:

- $\dfrac {\d \Phi} {\d x} = 0$

Hence the result.

$\blacksquare$

This needs considerable tedious hard slog to complete it.In particular: get proof for the other conditionTo discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1963: I.M. Gelfand and S.V. Fomin:
*Calculus of Variations*... (previous) ... (next): $\S 4.16$: The Canonical Form of the Euler's Equations