Verifying that the Euler-Lagrange equation uses generalized coordinates

  • Thread starter Saketh
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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 [itex]x_1, x_2, ... x_n[/itex], show, by change of variables, that the equations retain the same form under a coordinate transformation.

[tex]
\frac{d}{dt} \left (\frac{\partial L}{\partial \dot{x_i}} \right ) = \frac{\partial L}{\partial x_i}
[/tex]

I have no idea how to go about transforming coordinates. I created a coordinate system [itex]q_i[/itex] which could be written in terms of the [itex]x_i[/itex], 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!
 
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robphy
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http://www.uio.no/studier/emner/matnat/fys/FYS3120/v05/undervisningsmateriale/Symmetry.pdf [Broken] may be useful.
 
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