I was recently looking into generalisations of the brachistochrone problem: for example, in [this article][1] the authors study the brachistochrone with Amontons-Coulomb friction where a bead slides along a wire from one fixed point to another under the influence of gravity and friction.

In [this one][2] the author considers the brachistochrone for a sphere which rolls without friction.  It seems like in the literature no-one has studied the brachistochrone/tautochrone curves for a homogeneous sphere with slippage between sphere and surface ie. a sphere which slides and rolls with friction (I am happy to be corrected if I am wrong on this).  Is there any reason why this solution would be impossible to obtain or something to do with the effect of slip being too difficult to analyze?

I note that in the first article it is possible to express the solution in terms of elementary functions but perhaps that is no longer possible for the rolling sphere problem.


  [1]: https://ui.adsabs.harvard.edu/abs/1975AmJPh..43..902A/abstract
  [2]: https://aapt.scitation.org/doi/abs/10.1119/1.1990827