In some lecture slides for a class I am taking, this is stated:
I don't see how this is true. The E-L equation merely tells us that at extrema (i.e. the minimum, which is an attained infimum):
$$\frac{\partial L}{\partial y} = \frac{d}{dx}\frac{\partial L}{\partial y'}$$
At most I would get that:
$$\int \frac{\partial L}{\partial y} dx = \int\frac{d}{dx}\frac{\partial L}{\partial y'} dx$$
Which is:
$$\int \frac{\partial L}{\partial y} dx = \frac{\partial L}{\partial y'} + C$$
Where are $v$ and $f$ coming from? How is the expression being put in terms of a single integral? Why doesn't $\bar{y}$ show up in this expression ?
I am very lost. This is the only slide related to the topic, the class has no book and due to covid, things are online, but my internet isn't great and I already suffer from attention problems, so I have not been able to follow the material very well.
Take your Euler Lagrange equation and multiply by $v(x)$. Now integrate both sides and use integration by parts on the right side.