Find the shape of the Curve down which a bead sliding from rest and Accelerated by
gravity will slip (without friction ) from one point to another in the least
time. This was one of the earliest problems posed in the Calculus of Variations. The solution, a segment of a
Cycloid, was found by Leibniz, L'Hospital, Newton, and the two
Bernoullis.
The time to travel from a point to another point is given by the Integral

(1) 
The Velocity at any point is given by a simple application of
energy conservation equating kinetic energy to
gravitational potential energy,

(2) 
so

(3) 
Plugging this into (1) then gives

(4) 
The function to be varied is thus

(5) 
To proceed, one would normally have to apply the fullblown EulerLagrange Differential Equation

(6) 
However, the function is particularly nice since does not appear explicitly. Therefore,
, and we can immediately use the Beltrami Identity

(7) 
Computing

(8) 
subtracting
from , and simplifying then gives

(9) 
Squaring both sides and rearranging slightly results in

(10) 
where the square of the old constant has been expressed in terms of a new (Positive) constant . This equation is
solved by the parametric equations
which arelo and beholdthe equations of a Cycloid.
If kinetic friction is included, the problem can also be solved analytically, although the
solution is significantly messier. In that case, 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 Second Law gives

(19) 
But

(20) 

(21) 

(22) 
so

(23) 
Using the EulerLagrange Differential Equation gives

(24) 
This can be reduced to

(25) 
Now letting

(26) 
the solution is
See also Cycloid, Tautochrone Problem
References
Ashby, N.; Brittin, W. E.; Love, W. F.; and Wyss, W. ``Brachistochrone with Coulomb Friction.''
Amer. J. Phys. 43, 902905, 1975.
Haws, L. and Kiser, T. ``Exploring the Brachistochrone Problem.'' Amer. Math. Monthly 102,
328336, 1995.
Wagon, S. Mathematica in Action. New York: W. H. Freeman, pp. 6066 and 385389, 1991.
© 19969 Eric W. Weisstein
19990526