I want to find the equations of motion for a ball travelling down a cycloid under gravity so I can model the brachistochrone problem on matlab. I've looked online everywhere and can only find the equations for x and y with respect to theta (of the cycloid). I'm guessing then that I'd have to find an equation for the angle with respect to the time and then work with that to find x(t) but I have no idea how to do it.
Equations for cycloid:
I derived this equation (with the help of Ray Vickson) where you can just sub in f(x) but obviously its a little more difficult when you've got parametric equations for a cycloid.
The Attempt at a Solution
Well using those cycloid equations, I can find
dx/d0 = a - cos0 (0 is supposed to be theta, im not sure how you generate equations)
and dy/d0 = sin0
then from that we can find dy/dx = sin0/(a - cos0)
I know i've got to bring the time into it somehow but I have no idea how. I've thought about using the chain rule but I don't know how to apply that. To be fair I'm not even entirely sure what I'm trying to find, whether it be d0/dt or what.
Any tips you have are helpful, thanks!
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