The work provides a definition of a generalised form of the Brachistochrone problem for arbitrary mechanical systems. The problem then is solved for mechanical systems including Coulomb friction and/or rotational motion. By studying the solution curves, an interesting relation between the solution curve for systems with Coulomb friction and the cycloid has been shown. To estimate influences of simplifications made in solving the problem for systems with Coulomb friction, a simulation has been implemented. Finally, an experimental analysis is conducted, and it is shown that the investigated mathematical models describe a real-world mechanical system more accurate than the mechanical system investigated by Bernoulli in his work on the problem.