- #1
Saketh
- 258
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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 x_1, x_2, ... x_n, show, by change of variables, that the equations retain the same form under a coordinate transformation.
<br /> \frac{d}{dt} \left (\frac{\partial L}{\partial \dot{x_i}} \right ) = \frac{\partial L}{\partial x_i}<br />
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!
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.
<br /> \frac{d}{dt} \left (\frac{\partial L}{\partial \dot{x_i}} \right ) = \frac{\partial L}{\partial x_i}<br />
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!
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