Find the curve joining two points along which a particle falling from rest accelerated by gravity travels in the least time. From conservation of Energy,
The function to be varied is then
and the Euler-Lagrange Differential Equation gives
If Kinetic Friction is included, terms corresponding to the normal component of weight and the normal component of the acceleration (present because of path curvature) must be included. Including both terms requires a constrained variational technique (Ashby et al. 1975), but including the normal component of weight only gives an elementary solution. The tangent and normal vectors are
Gravity and friction are then
and the components along the curve are
so Newton's First Law gives
Using the Euler-Lagrange Differential Equation gives
This can be reduced to
the solution is
Ashby, N.; Brittin, W.E.; Love, W.F.; and Wyss, W. ``Brachistochrone with Coulomb Friction.'' Amer. J.
Phys. 43, 902-905, 1975.
Haws, L. and Kiser, T. ``Exploring the Brachistochrone Problem.'' Amer. Math. Monthly 102,
Haws, L. and Kiser, T. ``Exploring the Brachistochrone Problem.'' Amer. Math. Monthly 102, 328-336, 1995.
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