What Are the Implications of the y,y'-Domain in Euler-Lagrange Equations?

[JanEnClaesen]

In the Euler-Lagrange equations y and y' are independent variables, while for a given curve y(x), they are related by the differential equation y=f(x)y'. If you draw arbitrary curves on the y,y'-plane, it is immediately clear that most curves do not correspond to a curve y(x), is it fruitful to consider this domain further?
Why are y and y' independent variables? It looks like the y,y'-domain has a different name, perhaps the frequency domain?

 

[D H]

It looks like the y,y'-domain has a different name, perhaps the frequency domain?

It's called phase space. Frequency domain is a different beast. Apply a Fourier transform to $y(t)$ you'll get $\hat y(\omega)$. (Note well: nomenclature varies!)

 

[JanEnClaesen]

Frequency domain occurred to me because the solution of y=f(x)y' is y=exp(int(f(x))), the frequency is also an argument of an exponential function. Why are y and y' independent variables?