By using the Euler-Lagrange Equation how do you solve the following functional?
Brachistochrone functional taking friction into consideration: $$\large t=\int_a^b \sqrt{\frac{({y'}^2+1)}{2g(y-\mu x)}}~dx\tag{28}$$ where $g$ is a constant ($9.81\mathrm{m/s^2}$) and $\mu$ is the coefficient of friction (another constant).
This problem has been already solved and documented, unfortunately not freely accessible Brachistochrone with Coulomb friction
N. Ashby, W. E. Brittin, W. F. Love, and W. Wyss, American Journal of Physics, Volume 43, Issue 10, 1975
The problem of finding the curve of most rapid descent of a bead sliding along a wire, from one fixed point to another, under the simultaneous influence of gravity and friction, is solved. A remarkable feature of the problem is that the solution can be expressed in closed form, in terms of elementary functions.