# Help with derivation of euler-lagrange equations

1. Nov 21, 2009

### teeeeee

Hi,

I am trying to follow a derivation of the euler lagrange equations in one of my textbooks. It says that

$$\int ( f\frac{dL}{dx} + f'\frac{dL}{dx'}) dt$$

=

$$f\frac{dL}{dx'} + \int f ( \frac{dL}{dx} - \frac{d}{dt}(\frac{dL}{dx'}) ) dt$$

where f is an arbitrary function and L is the Lagrangian.

I'm not sure how to perform this step. I think it has something to do with integration by parts but can't work it out. Any help would be appreciated.
Thanks
teeeeee

2. Nov 21, 2009

### CompuChip

The integral over f (dL/dx) can be ignored, it simply sits in both expressions. So your question is, how do you get from

$$\int \frac{df}{dt} \frac{dL}{dx'}$$
to
$$f \frac{dL}{dx'} - \int f \frac{d}{dt} \frac{dL}{dx'}$$

right?
Because that is just partial integration in its purest form:
$$\int f' g = f g - \int f g'$$
where g = dL/dx'.