Rewrite a Lagrange function to Euler-Lagrange equation in polar coordinate

Question

If we have a Lagrange function in the form $L(p, q) = \frac{p^2}{2} + q^2$, how could it be re-written as a form of Euler-Lagrange equation in polar coordinates ?

Answer 1

Not so clear: $q$, and $p$ are your generalized coordinates and momenta. You first have to define your Lagrangian as a function of $x$ or $x$ and $y$ or whatever, and then you perform the coordinate changing.

For example, the simplest Lagrangian is given by

$$\mathcal{L} = \frac{m}{2} v^2 = \frac{m}{2}(\dot{x}^2 + \dot{y}^2) $$

which would be your kinetic energy term. Assuming the potential is zero. Performing a coordinate change, you get

$$\mathcal{L} = \frac{m}{2}(\dot{R}^2 + R\dot{\theta}^2)$$

Final Remark

Having $p$ and $q$ means nothing. Thou have to find what $p$ and $q$ means, with respect upon the problem you're looking at.