Here is the equation of Norton's dome used to demonstrate an indeterministic system within Newtonian formalism:
h(r) = $-\frac{2}{3g}r^{\frac{3}{2}}$
where r is defined as "radial distance coordinate".
I struggle to understand this equation- I presume r really just stands for arc length but in this case how can it serve as an independent variable ? Is it possible to introduce standard Cartesian coordinates (so h becomes y) and rewrite this equation in x-y terms ?
Another (possibly related) question concerns the application of Newton' s equation of motion. The author states that
$sin(\theta) = \frac{dh}{dr}$
where $\theta$ is the angle between the tangent to the surface in the radial direction in horizontal.
How come this is the case ? How to bind this derivative with the geometry of our picture ?
If anything, I am referring to this paper (pages 2-4):
http://www.pitt.edu/~jdnorton/papers/DomePSA2006.pdf
Any help will be much appreciated !!!