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## Brachistochrone Problem

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,  so  The function to be varied is then and the Euler-Lagrange Differential Equation gives    where Let  Now let then 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 But   so Using the Euler-Lagrange Differential Equation gives This can be reduced to Now letting the solution is References

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, 328-336, 1995.