What is the Proof of the Euler Lagrange Equation?

[ercagpince]

[SOLVED] Euler Lagrange Equation

Hi there ,
I am missing a crucial point on the proof of Euler Lagrange equation , here is my question :
$$\frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{df}{dy^{'}}\right)=0$$ (Euler-Lagrange equation)

If the function "f" doesn't depend on x explicitly but implicitly and if y satisfies the Euler-Lagrange equation then ;
$$\frac{\partial f}{\partial x}=0$$

Why is that so ? While ,supposedly , f is dependent to 3 variables : x,y,y' how van that statement be true ?

 

[Rainbow Child]

If the function $$f$$ doesn't depend on x explicitly then

$$\frac{\partial f}{\partial x}=0$$

and this has nothing to do with the

...and if y satisfies the Euler-Lagrange equation...

I don't understand what is the question

 

[ercagpince]

Still , the function f does depend on x through y and y' . That is why I asked basically .

 

[Rainbow Child]

That's why we use partial differention. Let

$$f\left(x,y(x),y'(x)\right)=x^2\,e^{y(x)}\,\ln{y'(x)}+\sin\left(x\,y(x)\right)$$

then $\frac{\partial\,f}{\partial\,x}$ means

$$\frac{\partial\,f}{\partial\,x}=2\,x\,e^{y(x)}\,\ln{y'(x)}+y(x)\,\cos\left(x\,y(x)\right)$$

i.e. you treat $$x,y(x),y'(x)$$ as independent variables.

 

[ercagpince]

Thanks for helping out!