I'm new to Lagrangian mechanics and I wanted to draw an x-y plot for motion of a small bead on a wire which is the radius of a wheel rolling on the ground.
I started by writing Lagrange equations in spherical coordinates
Using the Lagrange method I got
Now I don't know if this can be solved or not. But let's simplify it and say the angular velocity of the wheel is constant so
$$\theta''=0$$ $$2r'\theta'=-g*cos(\theta)$$ $$r''=r\theta'^2-g*sin(\theta)$$
How can I get the equations of motion from a point of view outside the wheel? (Like someone who is standing on the street where this wheel is rolling on)
Are my equations of motion right or I should include the kinetic energy of the wheel in the Lagrangian too?