Euler- Lagrange equation proof

[member 731016]

Homework Statement

Please see below

Relevant Equations

Please see below

For this problem,

The solution is,

However, I have a question about the solution. Does someone please know why they write out $\frac{dF}{dx} = \frac{\partial F}{\partial y}y' + \frac{\partial F}{\partial y'}y''$ since we already know that $\frac{dF}{dx} = 0$?

Thanks!

 

[PhDeezNutz]

I believe you are confusing total derivatives with partial derivatives

$\frac{dF}{dx}$ and $\frac{\partial F}{\partial x}$ are not the same thing.

 

[Orodruin]

To expand in the above:

In general, without the condition $\partial F/\partial x = 0$, we would have
$$
\frac{dF}{dx} =
\frac{\partial F}{\partial x} +
\frac{\partial F}{\partial y} y’ +
\frac{\partial F}{\partial y’} y’’
$$
by virtue of the chain rule for derivatives. Apply the condition to obtain what is in the proof.