Verifying that the Euler-Lagrange equation uses generalized coordinates

[Saketh]

This is a question that I'm asking myself for my own understanding, not a homework question.

I realize that in most derivations of the Euler-Lagrange equations the coordinate system is assumed to be general. However, just to make sure, I want to apply the "brute force" method (as Shankar calls it) to verify that the Euler-Lagrange equations indeed use generalized coordinates.

So, here's the problem. Given the Euler-Lagrange equations in a Cartesian coordinate system $x_1, x_2, ... x_n$, show, by change of variables, that the equations retain the same form under a coordinate transformation.

$$\frac{d}{dt} \left (\frac{\partial L}{\partial \dot{x_i}} \right ) = \frac{\partial L}{\partial x_i}$$

I have no idea how to go about transforming coordinates. I created a coordinate system $q_i$ which could be written in terms of the $x_i$, but I wasn't sure how to use brute force methods to verify that the E-L equations use generalized coordinates. If someone could show me how to do it, I would appreciate it greatly.

Thanks for helping me understand this!

 

[robphy]

http://www.uio.no/studier/emner/matnat/fys/FYS3120/v05/undervisningsmateriale/Symmetry.pdf may be useful.