Euler-Lagrange Integral

Given that the Euler-Lagrange equation (not the usual physics form) is

$\frac { d }{ dx } \left( \frac { \partial F }{ \partial y' } \right) -\frac { \partial F }{\partial y } =0$

show that if $F$ does not explicitly depend on $x$ , then the Euler equation can be integrated as

$F-y' \frac{\partial F}{\partial y'} = C.$

Solution

Part 1.

We must first show that $\frac { d }{ dx } \left( F-y' \frac { \partial F }{ \partial y' } \right) -\frac { \partial F }{\partial x} =0.$

We observe that $\frac{dF}{dx} = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \frac{dy}{dx} + \frac{\partial F}{\partial y'} \frac{dy'}{dx} = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} y'+ \frac{\partial F}{\partial y'} y''.$

Since $\frac{d}{dx} \left(y' \frac{\partial F}{\partial y'} \right)=y' \frac{d}{dx} \left(\frac{\partial F}{\partial y'} \right)+\frac{\partial F}{\partial y'} y''.$

By subtraction $\frac{d}{dx} \left(F-y' \frac{\partial F}{\partial y'} \right) = \frac{\partial F}{\partial x} + y' \left[\frac{\partial F}{\partial y} - \frac{d}{dx}\left( \frac{\partial F}{\partial y'} \right) \right].$

Since $\frac { d }{ dx } \left( \frac { \partial F }{ \partial y' } \right) -\frac { \partial F }{\partial y } =0$ , we find that $\frac { d }{ dx } \left( F-y' \frac { \partial F }{ \partial y' } \right) -\frac { \partial F }{\partial x} =0.$

Part 2.

Since $F$ does not explicitly depend on $x$ , then $\frac { \partial F }{\partial x} =0$ .

By integration $F-y' \frac{\partial F}{\partial y'} = C$ for some arbitrary constant $C$ .

Check out my other notes at Proof, Disproof, and Derivation

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
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`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
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Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$