Geometric phase transition in the problem of brachistochrone

It is found analytical solution of problem I. Bernoulli on the brahistohrone with frictional forces (viscous, proportional to speed motion and dry). It is shown that its solution is represented only in the form quadratures. With the aid of the numerical calculation are given different figures of optimum trajectories under dissipation conditions. It is proven that in the absence frictional forces any motion along the curvilinear chute under the action only of gravitational force is always reduced to the task about the brahistohrone. It is found the point “geometric phase transition”. This point corresponds to the disruption of brahistohrones from one class of trajectories to qualitatively another class. It is shown that with steady state the trajectory takes the form of the usual parabola, motion along which is periodically like the pendulum.